From 87bfe58c4025e760d349c591d848fb21d6be19bb Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Fri, 16 Jan 2026 15:47:03 +0100 Subject: [PATCH 01/25] writing scheme PC96 in doc --- docs/web/CMakeLists.txt | 1 + docs/web/PonteCastaneda1992.md | 6 +- docs/web/PonteCastaneda1996.md | 323 ++++++++++++++++++++++ docs/web/bibliography.bib | 28 ++ docs/web/downloads/PonteCastaneda1992.zip | Bin 3163 -> 3106 bytes docs/web/tfel-material.md | 3 +- 6 files changed, 357 insertions(+), 4 deletions(-) create mode 100644 docs/web/PonteCastaneda1996.md diff --git a/docs/web/CMakeLists.txt b/docs/web/CMakeLists.txt index 4587e2aac..386657091 100644 --- a/docs/web/CMakeLists.txt +++ b/docs/web/CMakeLists.txt @@ -95,6 +95,7 @@ pandoc_html(Taylor "--toc" "--toc-depth=2") pandoc_html(Sachs "--toc" "--toc-depth=1") pandoc_html(BetaRule "--toc" "--toc-depth=2") pandoc_html(PonteCastaneda1992 "--toc" "--toc-depth=2") +pandoc_html(PonteCastaneda1996 "--toc" "--toc-depth=2") pandoc_html(Norton-web) pandoc_html(Norton-full) pandoc_html(coverity-scan) diff --git a/docs/web/PonteCastaneda1992.md b/docs/web/PonteCastaneda1992.md index 2e21af653..ef17ff70f 100644 --- a/docs/web/PonteCastaneda1992.md +++ b/docs/web/PonteCastaneda1992.md @@ -126,13 +126,13 @@ where \(W_0^{\mathrm{eff}}\) is the effective energy of a linear comparison comp The macroscopic stress $\overline{\tsigma}$ is also shown to be (see [@ponte_castaneda_nonlinear_1998], eq. (4.41)): \begin{aligned} - \tsigma=\deriv{\overline{w}}{\overline{\tepsilon}}=\deriv{W_0^{\mathrm{eff}}}{\overline{\tepsilon}}=\tenseurq C_0^{\mathrm{eff}}\dbldot\overline{\tepsilon} + \overline{\tsigma}=\deriv{\overline{w}}{\overline{\tepsilon}}=\deriv{W_0^{\mathrm{eff}}}{\overline{\tepsilon}}=\tenseurq C_0^{\mathrm{eff}}\dbldot\overline{\tepsilon} \end{aligned} where $\tenseurq C_0^{\mathrm{eff}}$ is the effective elasticity of the linear comparison composite. In our case, we use Hashin-Shtrikman bounds. The tangent operator is given by \begin{aligned} - \dfrac{\mathrm{d}\tsigma}{\mathrm{d}\overline{\tepsilon}}=\tenseurq C_0^{\mathrm{eff}}+\dfrac{\mathrm{d}\tenseurq C_0^{\mathrm{eff}}}{\mathrm{d}\overline{\tepsilon}} + \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}=\tenseurq C_0^{\mathrm{eff}}+\dfrac{\mathrm{d}\tenseurq C_0^{\mathrm{eff}}}{\mathrm{d}\overline{\tepsilon}} \end{aligned} Here, the derivative of $\tenseurq C_0^{\mathrm{eff}}$ w.r.t. $\overline{\tepsilon}$ can be computed by derivating the Hashin-Shtrikman moduli w.r.t. the secant moduli $\mu_0^r$ and the derivatives of these moduli w.r.t. $\overline{\tepsilon}$. However, it is tedious and in the implementation, we see that the convergence of the Newton Raphson algorithm is good if we only retain the first term \(\tenseurq C_0^{\mathrm{eff}}\) in the tangent operator. @@ -171,7 +171,7 @@ means that the beginning of the `mfront` file reads: @Behaviour PC_VB_92; @Author Martin Antoine; @Date 12 / 12 / 25; -@Description{"Ponte Castaneda second-order estimates for homogenization of non-linear elasticity (one potential), based on second-moments computation"}; +@Description{"Ponte Castaneda variational bounds for homogenization of non-linear elasticity (one potential), based on second-moments computation"}; @UseQt true; @Algorithm NewtonRaphson_NumericalJacobian; @PerturbationValueForNumericalJacobianComputation 1e-10; diff --git a/docs/web/PonteCastaneda1996.md b/docs/web/PonteCastaneda1996.md new file mode 100644 index 000000000..4151c00a7 --- /dev/null +++ b/docs/web/PonteCastaneda1996.md @@ -0,0 +1,323 @@ +--- +title: Ponte-Castaneda second-order estimates for homogenization of non-linear behaviours +author: Antoine Martin +date: 15/01/2026 +lang: en-EN +link-citations: true +colorlinks: true +geometry: + - margin=2cm +papersize: a4 +figPrefixTemplate: "$$i$$" +tblPrefixTemplate: "$$i$$" +secPrefixTemplate: "$$i$$" +eqnPrefixTemplate: "($$i$$)" +--- + +\newcommand{\tenseur}[1]{\underline{#1}} +\newcommand{\tenseurq}[1]{\underline{\mathbf{#1}}} +\newcommand{\tenseurt}[1]{\underline{\underline{\underline{#1}}}} +\newcommand{\tns}[1]{\underset{\tilde{}}{\mathbf{#1}}} +\newcommand{\transpose}[1]{#1^{\mathop{T}}} +\newcommand{\Ka}{\mathcal K} +\newcommand{\tsigma}{\tenseur{\sigma}} +\newcommand{\dbldot}{:} +\newcommand{\tepsilon}{\tenseur{\varepsilon}} +\newcommand{\te}{\tenseurq{e}} +\newcommand{\sigmaeq}{\sigma_{\mathrm{eq}}} +\newcommand{\epsiloneq}{\varepsilon_{\mathrm{eq}}} + +\newcommand{\tepsilonto}{\underline{\epsilon}^{\mathrm{to}}} +\newcommand{\tepsilonel}{\underline{\epsilon}^{\mathrm{el}}} + +\newcommand{\trace}[1]{\mathrm{tr}\paren{#1}} +\newcommand{\Frac}[2]{{\displaystyle \frac{\displaystyle #1}{\displaystyle #2}}} +\newcommand{\deriv}[2]{{\displaystyle \frac{\displaystyle \partial #1}{\displaystyle \partial #2}}} +\newcommand{\derivdeux}[2]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2^2}}} +\newcommand{\dtot}{{\mathrm{d}}} +\newcommand{\paren}[1]{\left(#1\right)} +\newcommand{\nom}[1]{\textsc{#1}} +\renewcommand{\div}{\mathrm{div}} + + +We present here an implementation of Ponte-Castaneda second-order estimates [@castaneda_exact_1996]. +It deals with the homogenization of a composite whose phases are governed by one potential. +It can be used in non-linear elasticity, but an example was also treated in [@bornert_second-order_2001] +for the case of a polycrystalline material. Here the idea is to show that many implementations +are possible in `MFront`. If the user wants to improve the accuracy of the homogenized response, +he may want to use morphological tensors, for example. Note that it will also increase the cost of +the integration. + +This tutorial first recalls the methodology from [@castaneda_exact_1996], and then presents +different kinds of possible implementations. A particular implementation is given in MFront. + +# Ponte-Castaneda second-order estimates + +## The non-linear homogenization problem + +The variational approach used by Ponte-Castaneda is based on one potential $\phi$ which is of the form: +\begin{aligned} + \phi(\tepsilon)= \sum_{r=1}^{N}\chi_r\,\phi_r (\tepsilon) + \end{aligned} +where $N$ is the number of phases and $\chi_r$ is characteristic function of phase $r$. The local +potential $\phi_r$ is a non-linear function of the strain $\tepsilon$, but it can +be also a function of the strain rate $\dot{\tepsilon}$. + +For isotropic materials, the potential is of the form: + \begin{aligned} + \phi_r(\tepsilon)=\Frac92 k_r\, \varepsilon_m^2+f_r (\epsiloneq^2) + \end{aligned} + where +\begin{aligned} + \epsiloneq= \sqrt{\Frac23\tepsilon^d\dbldot\tepsilon^d}\qquad\text{where}\quad \tepsilon^d = \tepsilon - \Frac13 \trace{\tepsilon}\tenseur{1} \qquad\text{and}\quad \varepsilon_m=\Frac13 \trace{\tepsilon} + \end{aligned} + +For polycrystalline materials, the potential is of the form: + +... + + +In the variational approach, the homogenization problem is equivalent to a minimization problem: +\begin{aligned} + \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tepsilon\in\mathcal K(\overline{\tepsilon})}{\textrm{min}} \langle \phi(\tepsilon)\rangle + \end{aligned} + where the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$ is introduced depending on the boundary conditions used, and it makes intervene the loading $\overline{\tepsilon}$ (which will be our total strain $\texttt{eto}$). The $\langle.\rangle$ denotes the average on the representative volume element. Besides, $\Phi^{\mathrm{eff}}$ is the effective energy. + +## The second-order estimates + +Ponte-Castaneda's idea is to make a second-order Taylor development around a given "reference" strain $\tepsilon^r$ in phase $r$: +\begin{aligned} + \phi_r(\tepsilon)\approx\phi_r(\tepsilon^r)+\tsigma_r(\tepsilon^r)\dbldot\left(\tepsilon-\tepsilon^r\right)+\dfrac12\left(\tepsilon-\tepsilon^r\right)\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\tepsilon-\tepsilon^r\right) + \end{aligned} + where + \begin{aligned} + \tsigma_r(\tepsilon^r)=\deriv{\phi_r}{\tepsilon}\left(\tepsilon^r\right)\qquad\text{and}\qquad\tenseurq L_r\left(\tepsilon^r\right)=\derivdeux{\phi_r}{\tepsilon}\left(\tepsilon^r\right) + \end{aligned} + give a "tangent" estimation of the stress $\tsigma_r$ around $\tepsilon^r$: + \begin{aligned} + \tsigma_r(\tepsilon)\approx\tsigma_r(\tepsilon^r)+\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\tepsilon-\tepsilon^r\right) + \end{aligned} + and can be viewed as the behaviour of a so-called "thermoelastic comparison composite". + + Hence, by noting $\tepsilon=\overline{\tepsilon}+\tepsilon'$, where $\tepsilon'$ is the fluctuating part of $\tepsilon$, we obtain + \begin{aligned} + \phi_r(\tepsilon)\approx v_r\left(\overline{\tepsilon}\right)+\tenseur\tau^r\dbldot\tepsilon'+\dfrac12\tepsilon'\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\tepsilon' + \end{aligned} + with + \begin{aligned} + &v_r\left(\overline{\tepsilon}\right)=\phi_r(\tepsilon^r)+\tsigma_r(\tepsilon^r)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right)+\dfrac12\left(\overline{\tepsilon}-\tepsilon^r\right)\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right)\\ + &\tenseur\tau^r=\tsigma_r(\tepsilon^r)+\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right) + \end{aligned} + + After that, we notice that for each $\tepsilon\in\Ka\left(\overline{\tepsilon}\right)$ corresponds a unique $\tepsilon'$ in $\Ka\left(\tenseur 0\right)$, so that the effective energy $\overline{\phi}$ is estimated by + \begin{aligned} + \overline{\phi}\left(\overline{\tepsilon}\right)\approx\sum_r c_r\,v_r\left(\overline{\tepsilon}\right)+\underset{\tepsilon'\in\mathcal K(0)}{\textrm{min}}\sum_r c_r\, \langle \tenseur\tau^r\dbldot\tepsilon'+\dfrac12\tepsilon'\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\tepsilon'\rangle_r + \end{aligned} + where $c_r$ is the volume fraction of the phase $r$, and $\langle.\rangle_r$ denotes the average on phase $r$. Of course, the last term in previous expression corresponds to the effective energy of a linear thermoelastic composite. This effective energy is also shown to be (see Eq. (5.32) of [@ponte_castaneda_nonlinear_1998]): + \begin{aligned} + \overline{\phi}\left(\overline{\tepsilon}\right)\approx\sum_r c_r\,\left[\phi_r\left(\tepsilon^r\right)+\dfrac12 \tsigma_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right)\right] + \end{aligned} + +The last question is the choice of the reference strain $\tepsilon^r$. A discussion in [@castaneda_exact_1996] leads to the simple expression: +\begin{aligned} +\tepsilon^r = \overline{\tepsilon}+\langle\tepsilon'\rangle_r +\end{aligned} + where $\tepsilon'$ here stands for the solution of the minimization problem above, that is, the strain field solution of the homogenization of the thermoelastic composite. This solution can be estimated by a mean-field homogenization scheme (or another more accurate method). + +## Macroscopic stress and tangent operator + +The macroscopic stress $\overline{\tsigma}$ can be obtained by derivation of the estimated effective energy. It gives (see Eq. (5.30) of [@ponte_castaneda_nonlinear_1998]): + \begin{aligned} + \overline{\tsigma}\approx\sum_r c_r\,\left[\tsigma_r\left(\tepsilon^r\right)+\dfrac12 \langle\left(\tepsilon-\tepsilon^r\right)\dbldot\tenseurt N_r\dbldot\left(\tepsilon-\tepsilon^r\right)\rangle_r\dbldot\deriv{\tepsilon^r}{\overline{\tepsilon}}\right] + \end{aligned} +where +\begin{aligned} +\tenseurt N_r = \deriv{\tenseurq L_r}{\tepsilon}\left(\tepsilon^r\right) +\end{aligned} + +The tangent operator is given by +\begin{aligned} + \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}=\tenseurq C_0^{\mathrm{eff}}+\dfrac{\mathrm{d}\tenseurq C_0^{\mathrm{eff}}}{\mathrm{d}\overline{\tepsilon}} + \end{aligned} + Here, the derivative of $\tenseurq C_0^{\mathrm{eff}}$ w.r.t. $\overline{\tepsilon}$ can be computed by derivating the Hashin-Shtrikman + moduli w.r.t. the secant moduli $\mu_0^r$ and the derivatives of these moduli w.r.t. $\overline{\tepsilon}$. However, it is tedious and in the implementation, we see that the convergence of the Newton Raphson algorithm is good if we only retain the first term \(\tenseurq C_0^{\mathrm{eff}}\) in the tangent operator. + + +## Summary + +The resolution consists in +\[ + \text{Find}\quad\tepsilon^r=\overline{\tepsilon}+\langle\tepsilon'\rangle_r +\] +where \(\tepsilon'\) is the strain field solution of a homogenization problem of the form: +\begin{aligned} +&\div\left[ \tenseurq L_r\left(\tepsilon^r\right)\dbldot\tepsilon'+\tenseur\tau^r\right]=0,\qquad\tepsilon'\in\Ka\left(\tenseur 0\right)\\ +&\tenseurq L_r\left(\tepsilon^r\right)=\derivdeux{\phi_r}{\tepsilon}\left(\tepsilon^r\right)\qquad\text{and}\qquad\tenseur\tau^r=\deriv{\phi_r}{\tepsilon}\left(\tepsilon^r\right)+\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right) +\end{aligned} + + +# Possible implementations + + + +# Implementation in MFront + +In the application of the implementation, we take the example of the following behaviour: + \begin{aligned} + w_r(\tepsilon)=\dfrac{9}{2}k_r\,\varepsilon_{m}^2+\dfrac{\sigma_r^0}{n+1}\,\varepsilon_{eq}^{n+1} + \end{aligned} + where $n$ is between $0$ and $1$, which means that $f_r(e)=\dfrac{\sigma_r^0}{n+1}\,e^{\frac{n+1}2}$ (note that the function + $f_r$ is effectively concave relatively to $e$, because $0\lt n\leq 1$). + + Moreover, we make the choice of purely elastic inclusions. Hence, only the matrix is non-linear and has a + secant modulus. + +The unknowns of our non-linear problem are the $e^r$. The secant moduli $\mu_0^r$ can be computed by derivation of the function $f_r$ which is given by the behaviour. In the precise case detailed above, we have +\begin{aligned} +\deriv{f_r}{e}(e)=\dfrac{\sigma_r^0}{2}\,e^{\frac{n-1}2}\qquad\text{and}\quad\mu_0^r=\dfrac{\sigma_r^0}{3}\,\left(e^r\right)^{\frac{n-1}2} +\end{aligned} +Then, the effective energy of the linear comparison composite can be computed via a mean-field scheme. In our implementation, we proposed to use the Hashin-Shtrikman bound. The $e^r$ must then cancel the following residue: +\begin{aligned} +r_{e^r}=e^r - \Frac2{3\,c_r}\deriv{W_0^{\mathrm{eff}}}{\mu_0^r} +\end{aligned} +which means that we only need the derivative of the Hashin-Shtrikman bound. This will be given by the `tfel::material::homogenization` functionalities. + +## Details of implementation + +All the files are available [here](./downloads/PonteCastaneda1992.zip) + +For the jacobian, we adopt the `Numerical Jacobian`, which +means that the beginning of the `mfront` file reads: + +~~~~ {#Begin .cpp .numberLines} +@DSL ImplicitII; +@Behaviour PC_VB_92; +@Author Martin Antoine; +@Date 12 / 12 / 25; +@Description{"Ponte Castaneda variational bounds for homogenization of non-linear elasticity (one potential), based on second-moments computation"}; +@UseQt true; +@Algorithm NewtonRaphson_NumericalJacobian; +@PerturbationValueForNumericalJacobianComputation 1e-10; +@Epsilon 1e-12; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Moreover, the following libraries will be needed for computation +of the stress and secant modulus: + +~~~~ {#Begin .cpp .numberLines} +@TFELLibraries {"Material"}; +@Includes{ +#include "TFEL/Material/IsotropicModuli.hxx" +#include "TFEL/Material/HomogenizationSecondMoments.hxx" +#include "TFEL/Material/LinearHomogenizationBounds.hxx" +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We define two state variables which are the averages (on each phase) of the squares +of the equivalent strains, and correspond to the $e^r$: + +~~~~ {#Begin .cpp .numberLines} +@StateVariable real e_0; +e_0.setEntryName("MatrixSquaredEquivalentStrain"); +@StateVariable real e_r; +e_r.setEntryName("InclusionSquaredEquivalentStrain"); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The secant moduli are computed by derivation of function $f_r$, which is detailed above. Hence + +~~~~ {#Integrator .cpp .numberLines} +@Integrator { +//secant modulus///////////////////////////////// +const auto po = std::pow(max(e_0+de_0,real(1e-10)),0.5*(n_-1)); +const auto mu0 = sig0/3.*po; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The second moment is given by the function `computeMeanSquaredEquivalentStrain` +(see [documentation](tfel-material.html#second-moments-of-the-strains)): + +~~~~ {#Integrator .cpp .numberLines} +//second moments///////////////////////////////// +const auto em2 = tfel::math::trace(eto+deto)/3.; +const auto ed = tfel::math::deviator(eto+deto); +const auto eeq2 = 2./3.*(ed|ed); +using namespace tfel::material; +const auto kg0 = KGModuli(k_m,mu0); +const auto kgr = KGModuli(kar,mur); +using namespace tfel::material::homogenization::elasticity; +const auto eeq2_ = computeMeanSquaredEquivalentStrain(kg0,fr,kgr,em2,eeq2); +const auto eeq20=std::get<0>(eeq2_); +const auto eeq2r=std::get<1>(eeq2_); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Now we can write the residues: + +~~~~ {#Integrator .cpp .numberLines} +//residues///////////////////////////////////// +fe_0 = e_0+de_0 - eeq20; +fe_r = e_r+de_r - eeq2r; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +There is no jacobian because we use the `NumericalJacobian`. The macroscopic stress +is given by the Hashin-Shtrikman lower bound (indeed, the lower bound is better suited +for soft matrix and rigid inclusions): + +~~~~ {#Integrator .cpp .numberLines} +//sigma////////////////////////////////////// +vector tab_f={1-fr,fr}; +vector tab_k = {k_m,kar}; +vector tab_mu = {mu0,mur}; +const auto HSB=computeIsotropicHashinShtrikmanBounds<3,stress>(tab_f,tab_k,tab_mu); +const auto LB=std::get<0>(HSB); +const auto kHS=std::get<0>(LB); +const auto muHS=std::get<1>(LB); +Chom=3*kHS*J+2*muHS*K; +sig = Chom*(eto+deto); +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +(Note that `Chom` is the effective elastic tensor, defined here as +a local variable). +Afterwards, we can give the tangent operator: + +~~~~ {#tangent .cpp .numberLines} +@TangentOperator{ +Dt=Chom; +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +# Results + +We then use `MTest` to simulate a strain-imposed test. + +MTest file is the following: + +~~~~ {#PC_VB_92 .mtest .numberLines} +@ModellingHypothesis 'Tridimensional'; +@Behaviour 'src/libBehaviour.so' 'PC_VB_92'; +@ExternalStateVariable 'Temperature' {0 : 1000,1:1000}; + +@MaterialProperty 'sig0' 1.e9; +@MaterialProperty 'n_' 0.05; +@MaterialProperty 'k_m' 1.e9; +@MaterialProperty 'kar' 1.e15; +@MaterialProperty 'mur' 1.e15; +@MaterialProperty 'fr' 0.25; + +@ImposedStrain 'EXX' {0 : 0, 100 : 0.8660254}; +@ImposedStrain 'EYY' {0 : 0, 100 : -0.8660254}; +@ImposedStrain 'EZZ' {0 : 0, 100 : 0}; +@ImposedStrain 'EXY' {0 : 0, 100 : 0}; +@ImposedStrain 'EXZ' {0 : 0, 100 : 0}; +@ImposedStrain 'EYZ' {0 : 0, 100 : 0}; +@Times {0.,10 in 100}; +~~~~~~~~~~~~~~ + +The macroscopic stress $\sigma_{xx}$ is represented below as a function +of the macroscopic strain $\varepsilon_{xx}$: + +![Macroscopic stress as function of the macroscopic strain](./img/PC_VB_92.png) + +We also plotted the behaviour of the matrix when there is no inclusions (it gives +an idea of the reinforcement given by the inclusions). + + diff --git a/docs/web/bibliography.bib b/docs/web/bibliography.bib index 0de2b3ec8..127c9a607 100644 --- a/docs/web/bibliography.bib +++ b/docs/web/bibliography.bib @@ -12593,3 +12593,31 @@ @book{ponte_castaneda_nonlinear_1998 author = {Ponte Castaneda, Pedro and Suquet, Pierre}, year = {1998}, } + +@article{castaneda_exact_1996, + title = {Exact second-order estimates for the effective mechanical properties of nonlinear composite materials}, + volume = {44}, + copyright = {https://www.elsevier.com/tdm/userlicense/1.0/}, + issn = {00225096}, + url = {https://linkinghub.elsevier.com/retrieve/pii/0022509696000154}, + doi = {10.1016/0022-5096(96)00015-4}, + language = {en}, + number = {6}, + urldate = {2026-01-15}, + journal = {Journal of the Mechanics and Physics of Solids}, + author = {Castañeda, P.Ponte}, + month = jun, + year = {1996}, + pages = {827--862}, +} + +@article{bornert_second-order_2001, + title = {Second-order estimates for the e'ective behaviour of viscoplastic polycrystalline materials}, + volume = {49}, + abstract = {This paper is concerned with the application of the “second-order” nonlinear homogenisation procedure (Ponte Casta˜neda, J.Mech.Phys.Solids 44 (6) (1996) 827) to generate estimates of the self-consistent type for the e'ective behaviour of fcc and hcp viscoplastic polycrystals. The method has the distinctive property that it leads to estimates that are exact to second-order in the heterogeneity contrast, and which are expected to be more accurate, particularly when compared to rigorous bounds, than those resulting from earlier homogenisation schemes such as the Hill “incremental” method or its “total” formulation (Hutchinson) for pure power-law viscous materials.Special attention is paid to large grain anisotropy leading to correspondingly large heterogeneity contrast, and to highly nonlinear behaviour.Comparisons are also carried out with estimates derived from other more recent homogenisation schemes such as the “tangent” and “a7ne” methods.The results, illustrated for zirconium- and ice-type polycrystals, show that the second-order procedure o'ers the potential for signi9cantly improved results, at least relative to the Hill incremental formulation. ? 2001 Elsevier Science Ltd.All rights reserved.}, + language = {en}, + journal = {Journal of the Mechanics and Physics of Solids}, + author = {Bornert, M and Masson, R and Castaneda, P Ponte and Zaoui, A}, + year = {2001}, + pages = {2737--2764}, +} diff --git a/docs/web/downloads/PonteCastaneda1992.zip b/docs/web/downloads/PonteCastaneda1992.zip index 28720ef73a0757559ba85ffd0e05a7e9c8b1d80b..793151450e4db91fa24c7c3b369344ddebd9b830 100644 GIT binary patch delta 1262 zcmcaDu}ES6;55UlkFmQw>>o^Jw6QQmH)>7mr_ORrA_hye`n=Tht;U zp%=fU`wp}zU(@fuSp1vUXN7B@Q~k-(&rY6BGKI}CPDgHvUT=yP_P-e>GMb?MPyb5dQLx6Ejhq}VDQ z{R@@Zc^+59ndK)g)H{$idEFGT87XgU|2U}Hf3z|5Vte1svxTFw>Xh?Vkr<(GOeXbW zRWEI7pJp=VPuP*m^re+EEXq~a zUs}G-Up3#k^UVQq6?T)-qYK!QX9#M0-I-_0TwkFRcz@4x$qL`!D{o5uZ(Z5=%z^E3 z8&8&uq(E~0n$rr><}X%kd~y1)$-2&CrE`ycZQim*K}yzCseluey+l0BCC-?DIok@S2f9)ib=7;dzfu}`I zEf5IcezBspUbQtUMt`4pjC0AKy#nDjtexCfVmyMj9G|{z-z&`>fm@}%?^ez&c0Kg} zNA!Y@WX@wX%f3uWbKmme@o$Cnnqz6deV70CO*ebv!?or{=HfKr9>uSJdy*ghW^-Th z!oo(w%Ixg)efHfpmuh}Vyt%UIoSgIA$ScP@9V0UPro7vA@x63C$4-}NH{a^|{NFgA zZS5MzW85J+;W>wR-1g01EgyGmz3rUrby7=s{7cv0Ziey{YX_rS^Y$QS~CgFv@hz zKkaGRhhDiq{#Ez&nrtUP0qe>Zoyi|eBN8F3+fwB zSNy-v6yw<8CBI8H>BMwLn@IlM81STu2g*mTMJc2(`{bMM+%1j>r<3f{FotlDnvmu9Cc$KIp_GM-uf zRdp}7L>}X}MRR1fbl$P6waSVP_@#mI>NFd7wAH^f9lW z&ZSxP%o_tIF>cOfQ)1DyXBNRxj;Js|fI`EP#y~`=q6sclxEK_`vWyH83=E0?=cTsZ zVMM1VcXMem&trj=M^=-|xaFAD*g$+opd8QSCqRB32Z%2-`3r}NSb#Sx8^}~vAe3if LV0a5G&lng0dKNg? delta 1228 zcmV;-1T*`h7~2?-R<{hlR*X+laC4y2US!J00&&? zlWPVUf0b6tQsX!j-TNz4nT?Zno=h*sAru7YCJ-2A!m#OuONr%Jg)Ie1reV_jdmqWB zgTrek3ss_f&VApj>>tghWRkBkzTjpu8G8F;meM;e3q>vuqw8_>#{icHg-N9%=TsRk z$bm4D3x@k6Y8dGcNVhx&pY0vf3&mH4OR;WTe@X$wA=QQo7E_|xLW;O8Rm>D&+VC72 zG)bU5mAPCp!T+U}N@PNW6zvR#D4-csSRWsVF9jnjX_zpaX3c=isb(<&rBa^D95PK7 zGG7(OX=^;fv41rC%Mhaq>-j;pl!}`)Cl~Cyk>YQW9)Z-P&1`JB1joR_J2pEz|XuhYk} z5OKwOL?BP#OSpsVJP>UZ)0$|$MBw)9e`scog;5U|G-rOpUNxG-SD1>?jEHFHiD-)> zqSqOft(zzxdN{UNxOtyO6=+m#)2MovmKUJm*rMU)ov;#cl5I-j7@)|QW$5qa$phLs zWtwYprFhIyBaNKX%=TXTplzF#1t3=h^Y^XSAt7mfvr1-YId+YHqVtTEMp=Xsf8f|M z3l9~)pWPNzvH1A5;CD1bGt7*lT(}rs)upVi?GRk6S|8t0=j~4=9RHk zVcrih@WYoNC;XjE-WHv{EjqJiCxV-r91}GenS^Cwh;@^Op*8~}aqR9Cfemvn3@5P< zpHDx6om$k{N2oK4dUUQBwp7$0f84d#?P5VuVMuPLx6s{>J|0F^8$t^svGB+Utr&;l zN`CiqdhescpJE)&sP9%zvl;X{pIg3&+WjV6c*f6*o>hP;?Cx}0D~Uuof3OrtwJ`se zKP(?W-}5}6@C)WMBjJ^iO_-@UfE8WG$Lx;V%xoxiJiE0P1|93K&*C2}e+B?eCzeE@ z3H1u*HVWPF)fC(;ZNz6^%U!u!j`5z4^$;K#%`L2Y6-yPc>O;F>Sbx02aPwZh2r7c5 z@@Zn{jJ`ENKHLuy6~MCqxehGzWrBJm*Ah!+c6)oi)BX~qHlY3ngz_Uu9vAH6!n@)p z=ry?h)OMs@dN8yCHgb?CJdc&M zDvguh!0Ou86LoqWkrx(%a@#UK)-rQ48;`0o{CZYTs7|?)D=;YxRJMtu{NE)d17+_K-dY^ecc}l%t&%$gR^)gO&xgFwWc~i{PxV^?Zb| zcJM!g3Us6S6Hk#Hv9)((M&-VJ^nLyE(DXt3MK+k0F&+t zPavRVDO`22_MaLA008p|001HYAOHXW00000005+c*#VR83Lgw+a&K;QRa6ZC2VCfr qYX%vUp9>6=JqsQIW0PD9Z34RplU)lPljsXD3L63d04fRq000015H*hg diff --git a/docs/web/tfel-material.md b/docs/web/tfel-material.md index cacb087cf..f896acbc4 100644 --- a/docs/web/tfel-material.md +++ b/docs/web/tfel-material.md @@ -746,7 +746,8 @@ The number of phases is arbitrary, and the dimension is 2 or 3. ## Second moments of the strains -Some functions allow to compute the second moments of the strains +Some functions (defined in the header HomogenizationSecondMoments.hxx) +allow to compute the second moments of the strains if considering a Hashin-Shtrikman type microstructure. More precisely, we consider isotropic spherical inclusions embedded in an isotropic matrix, and we compute the following moments: From d0787718beea51ef2d2219f67de64a0d86da972f Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Tue, 27 Jan 2026 14:07:37 +0100 Subject: [PATCH 02/25] continued doc on PC96 and add images in doc --- docs/web/PonteCastaneda1992.md | 43 +- docs/web/PonteCastaneda1996.md | 629 +++++++++++++++------ docs/web/bibliography.bib | 37 +- docs/web/img/ParticulateMicrostructure.png | Bin 0 -> 174947 bytes docs/web/img/distributions.png | Bin 0 -> 1177594 bytes docs/web/img/distributions_PCW.pdf | Bin 0 -> 161825 bytes docs/web/img/distributions_PCW.png | Bin 0 -> 380237 bytes docs/web/img/ellipsoide_C0.png | Bin 0 -> 113832 bytes docs/web/img/polycrystal.png | Bin 0 -> 42985 bytes docs/web/tfel-material.md | 43 +- docs/web/tfel-python.md | 26 +- 11 files changed, 593 insertions(+), 185 deletions(-) create mode 100644 docs/web/img/ParticulateMicrostructure.png create mode 100644 docs/web/img/distributions.png create mode 100644 docs/web/img/distributions_PCW.pdf create mode 100644 docs/web/img/distributions_PCW.png create mode 100644 docs/web/img/ellipsoide_C0.png create mode 100644 docs/web/img/polycrystal.png diff --git a/docs/web/PonteCastaneda1992.md b/docs/web/PonteCastaneda1992.md index ef17ff70f..b0cc0617a 100644 --- a/docs/web/PonteCastaneda1992.md +++ b/docs/web/PonteCastaneda1992.md @@ -111,17 +111,6 @@ which can be shown to be equivalent to \langle \epsiloneq^2\rangle_r= \Frac{2}{3c_r}\deriv{W_0^{\mathrm{eff}}}{\mu_0^r} \end{aligned} -## Summary - -The resolution hence consists in -\[ - \text{Find}\,e^r=\langle \epsiloneq^2\rangle_r\,\text{such that}\quad\langle \epsiloneq^2\rangle_r= \Frac{2}{3c_r}\deriv{W_0^{\mathrm{eff}}}{\mu_0^r} -\] -where \(W_0^{\mathrm{eff}}\) is the effective energy of a linear comparison composite whose elastic moduli are \(k_r\) annd \(\mu_0^r\) such that: -\[ -\mu_0^r= \Frac23\deriv{f_r}{e}\left(\langle \epsiloneq^2\rangle_r\right). -\] - ## Macroscopic stress and tangent operator The macroscopic stress $\overline{\tsigma}$ is also shown to be (see [@ponte_castaneda_nonlinear_1998], eq. (4.41)): @@ -136,9 +125,41 @@ The tangent operator is given by \end{aligned} Here, the derivative of $\tenseurq C_0^{\mathrm{eff}}$ w.r.t. $\overline{\tepsilon}$ can be computed by derivating the Hashin-Shtrikman moduli w.r.t. the secant moduli $\mu_0^r$ and the derivatives of these moduli w.r.t. $\overline{\tepsilon}$. However, it is tedious and in the implementation, we see that the convergence of the Newton Raphson algorithm is good if we only retain the first term \(\tenseurq C_0^{\mathrm{eff}}\) in the tangent operator. + + +## Summary and possible implementations + +The resolution hence consists in +\[ + \text{Find}\,e^r=\langle \epsiloneq^2\rangle_r\,\text{such that}\quad\langle \epsiloneq^2\rangle_r= \Frac{2}{3c_r}\deriv{W_0^{\mathrm{eff}}}{\mu_0^r} +\] +where \(W_0^{\mathrm{eff}}\) is the effective energy of a linear comparison composite whose elastic moduli are \(k_r\) annd \(\mu_0^r\) such that: +\[ +\mu_0^r= \Frac23\deriv{f_r}{e}\left(\langle \epsiloneq^2\rangle_r\right). +\] + +### Possible implementations + +The iterative resolution of the non-linear equation can be summarized as +\[ +\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\mu_0^r= \Frac23\deriv{f_r}{e}\left(\langle \epsiloneq^2\rangle_r\right)}\quad\rightarrow\quad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{W_0^{\mathrm{eff}}\left(\mu_0^r\right)= \dfrac12\overline{\tepsilon}\dbldot\tenseurq C_0^{\mathrm{eff}}\dbldot\overline{\tepsilon}}\quad\rightarrow\quad\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\texttt{tfel::material::homogenization}}}{\langle \epsiloneq^2\rangle_r= \Frac{2}{3c_r}\deriv{W_0^{\mathrm{eff}}}{\mu_0^r}} +\] + +And the macroscopic stress must also be computed: +\[ +\overline{\tsigma}=\tenseurq C_0^{\mathrm{eff}}\dbldot\overline{\tepsilon} +\] + +The first step of the resolution consists in computing the secant modulus $\mu_0^r$ and it can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the function $f_r$ in an external file (in our example, the function is to simple to do that). + +The second step consists in computing the effective energy $W_0^{\mathrm{eff}}$. This will be done via `tfel::material::homogenization`. This `namespace` provides classical mean-field schemes (we use a Hashin-Shtrikman bound in our example below). In another [tutorial](./PonteCastaneda1996.html) we show that we can take into account more specifically the morphology of the composite by providing interaction tensors. + +The third step consists in the computation of the second-moments. This is also done via `tfel::material::homogenization`. Here again, there are two strategies: analytical computation when possible, and finite difference or automatic differentiation when the analytical derivation is too tedious or even impossible. In our example below, the computation is analytic. # Implementation in MFront +## Example used for the implementation + In the application of the implementation, we take the example of the following behaviour: \begin{aligned} w_r(\tepsilon)=\dfrac{9}{2}k_r\,\varepsilon_{m}^2+\dfrac{\sigma_r^0}{n+1}\,\varepsilon_{eq}^{n+1} diff --git a/docs/web/PonteCastaneda1996.md b/docs/web/PonteCastaneda1996.md index 4151c00a7..6d23fc5e7 100644 --- a/docs/web/PonteCastaneda1996.md +++ b/docs/web/PonteCastaneda1996.md @@ -1,5 +1,5 @@ --- -title: Ponte-Castaneda second-order estimates for homogenization of non-linear behaviours +title: Ponte-Castaneda second-order estimates for homogenization of a viscoplastic polycrystal author: Antoine Martin date: 15/01/2026 lang: en-EN @@ -20,6 +20,7 @@ eqnPrefixTemplate: "($$i$$)" \newcommand{\tns}[1]{\underset{\tilde{}}{\mathbf{#1}}} \newcommand{\transpose}[1]{#1^{\mathop{T}}} \newcommand{\Ka}{\mathcal K} +\newcommand{\Sa}{\mathcal S} \newcommand{\tsigma}{\tenseur{\sigma}} \newcommand{\dbldot}{:} \newcommand{\tepsilon}{\tenseur{\varepsilon}} @@ -34,290 +35,600 @@ eqnPrefixTemplate: "($$i$$)" \newcommand{\Frac}[2]{{\displaystyle \frac{\displaystyle #1}{\displaystyle #2}}} \newcommand{\deriv}[2]{{\displaystyle \frac{\displaystyle \partial #1}{\displaystyle \partial #2}}} \newcommand{\derivdeux}[2]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2^2}}} +\newcommand{\pderiv}[2]{{\dfrac{\partial #1}{\partial #2}}} +\newcommand{\pderivdeux}[2]{{\dfrac{\partial^2 #1}{\partial #2^2}}} +\newcommand{\derivderiv}[3]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2\,\displaystyle \partial #3}}} \newcommand{\dtot}{{\mathrm{d}}} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\nom}[1]{\textsc{#1}} \renewcommand{\div}{\mathrm{div}} -We present here an implementation of Ponte-Castaneda second-order estimates [@castaneda_exact_1996]. -It deals with the homogenization of a composite whose phases are governed by one potential. -It can be used in non-linear elasticity, but an example was also treated in [@bornert_second-order_2001] -for the case of a polycrystalline material. Here the idea is to show that many implementations -are possible in `MFront`. If the user wants to improve the accuracy of the homogenized response, -he may want to use morphological tensors, for example. Note that it will also increase the cost of -the integration. +We present here an implementation of Ponte-Castaneda second-order estimates [@castaneda_exact_1996] +for the homogenization of a viscoplastic polycrystal, example also treated in [@bornert_second-order_2001]. +The procedure can also be used in non-linear elasticity, providing that the behaviour is governed +by only one potential. -This tutorial first recalls the methodology from [@castaneda_exact_1996], and then presents -different kinds of possible implementations. A particular implementation is given in MFront. +Here, the idea is to show that many implementations of that procedure are possible in `MFront`. +We particularly present the possibility to take into account the morphology of the polycrystal +by using interaction tensors, which are provided before the integration. It results in better +estimates than classical ones. Note that the resulting non-linear system involved +couples the unknowns, and hence increases the cost of the integration. +We also show that we can use a self-consistent scheme, making use of +`tfel::material::homogenization`. -# Ponte-Castaneda second-order estimates +This tutorial first presents the homogenization problem, recalls Ponte-Castaneda's methodology +from [@castaneda_exact_1996], presents different kinds of possible implementations, +and show the details of the `mfront` file. -## The non-linear homogenization problem +# The viscoplastic polycrystal -The variational approach used by Ponte-Castaneda is based on one potential $\phi$ which is of the form: -\begin{aligned} - \phi(\tepsilon)= \sum_{r=1}^{N}\chi_r\,\phi_r (\tepsilon) - \end{aligned} -where $N$ is the number of phases and $\chi_r$ is characteristic function of phase $r$. The local -potential $\phi_r$ is a non-linear function of the strain $\tepsilon$, but it can -be also a function of the strain rate $\dot{\tepsilon}$. +## Behaviour -For isotropic materials, the potential is of the form: - \begin{aligned} - \phi_r(\tepsilon)=\Frac92 k_r\, \varepsilon_m^2+f_r (\epsiloneq^2) +We consider a polycrystalline material, which means that each phase $r$ is associated to a crystal with +corresponding slip systems $\tenseur \mu_k^r$ ($1\leq k\leq K$). The strain rate in each crystal $r$ is given by + \begin{aligned} + \dot{\tepsilon}=\sum_k \dot{\gamma}_k^r\, \tenseur \mu_k^r\qquad\text{with}\quad \tenseur\mu_k^r = \dfrac12\left(\tenseur n_k^r\otimes\tenseur m_k^r + \tenseur m_k^r\otimes\tenseur n_k^r\right) \end{aligned} - where -\begin{aligned} - \epsiloneq= \sqrt{\Frac23\tepsilon^d\dbldot\tepsilon^d}\qquad\text{where}\quad \tepsilon^d = \tepsilon - \Frac13 \trace{\tepsilon}\tenseur{1} \qquad\text{and}\quad \varepsilon_m=\Frac13 \trace{\tepsilon} + where $\dot{\gamma}_k^r$ is the shear strain rate on the $k^{th}$ slip system of crystal $r$, and is given as a + function of the shear stress $\tau_k^r=\tsigma\dbldot\tenseur \mu_k^r$ by means of a potential $\psi_k^r$: + \begin{aligned} + \dot{\gamma}_k^r= \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right) + \end{aligned} + The expressions above show that on phase (or crystal) $r$: + \begin{aligned} + \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\text{with}\quad\psi_r \left(\tsigma\right)=\sum_k \psi_k^r\left(\tau_k^r\right) \end{aligned} + + Now, the behaviour of the polycrystal is governed by a potential $\psi$: + \begin{aligned} + \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\psi(\tsigma)= \sum_{r=1}^{N}\chi_r\,\psi_r (\tsigma) + \end{aligned} +where $N$ is the number of phases (or crystals) and $\chi_r$ is characteristic function of phase $r$. -For polycrystalline materials, the potential is of the form: + In all the sequel, we just note $\tepsilon$ for $\dot{\tepsilon}$. + +## The non-linear homogenization problem -... +We impose a strain rate $\overline{\tepsilon}$ to the polycrystal and look for the solution $\tepsilon,\tsigma$ such that: +\begin{aligned} + &\div\,\tsigma=\tenseur 0\\ + &\tepsilon\in\Ka(\overline{\tepsilon})\\ + &\tepsilon = \deriv{\psi}{\tsigma}\left(\tsigma\right) +\end{aligned} -In the variational approach, the homogenization problem is equivalent to a minimization problem: +where we introduced the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$, depending on the boundary conditions used (in the implementation, we work with periodic boundary conditions). +The effective energy of the composite is then defined as the following average on the +entire Representative Volume Element (RVE): +\begin{aligned} +\overline{\phi}\left(\overline{\tepsilon}\right)= \langle\phi\left(\tepsilon\right)\rangle +\end{aligned} +where $\tepsilon$ is the solution of the above problem, $\langle.\rangle$ denotes the average +on the RVE, and $\phi$ is the primal potential, +defined as the Legendre-Fenchel transform of $\psi$: +\begin{aligned} +\phi\left(\tepsilon\right)=\underset{\tsigma}{\mathrm{max}}\left\{\tsigma\dbldot\tepsilon-\psi\left(\tsigma\right)\right\} +\end{aligned} + +# Ponte-Castaneda second-order estimates + +## The variational approach + + The homogenization problem is equivalent to a maximization problem: + \begin{aligned} + \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tsigma\in\Sa}{\textrm{max}} \left\{ \langle\tsigma\rangle\dbldot\overline{\tepsilon} - \langle\psi(\tsigma)\rangle\right\} + \end{aligned} + where $\Sa$ is the set of divergence-free fields. + + Note that a dual minimization problem can also be solved to find the effective energy \begin{aligned} \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tepsilon\in\mathcal K(\overline{\tepsilon})}{\textrm{min}} \langle \phi(\tepsilon)\rangle \end{aligned} - where the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$ is introduced depending on the boundary conditions used, and it makes intervene the loading $\overline{\tepsilon}$ (which will be our total strain $\texttt{eto}$). The $\langle.\rangle$ denotes the average on the representative volume element. Besides, $\Phi^{\mathrm{eff}}$ is the effective energy. + However, because $\phi$ is known only through the Legendre-Fenchel transform of $\psi$, it will be more + easy to use the maximization problem above. ## The second-order estimates -Ponte-Castaneda's idea is to make a second-order Taylor development around a given "reference" strain $\tepsilon^r$ in phase $r$: +Ponte-Castaneda's idea is to make a second-order Taylor development of each $\psi_r$ around a given "reference" stress $\tsigma^r$ in phase $r$: \begin{aligned} - \phi_r(\tepsilon)\approx\phi_r(\tepsilon^r)+\tsigma_r(\tepsilon^r)\dbldot\left(\tepsilon-\tepsilon^r\right)+\dfrac12\left(\tepsilon-\tepsilon^r\right)\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\tepsilon-\tepsilon^r\right) + \psi_r(\tsigma)\approx\psi_r(\tsigma^r)+\tepsilon_r(\tsigma^r)\dbldot\left(\tsigma-\tsigma^r\right)+\dfrac12\left(\tsigma-\tsigma^r\right)\dbldot\tenseurq M_r\left(\tsigma^r\right)\dbldot\left(\tsigma-\tsigma^r\right) \end{aligned} where \begin{aligned} - \tsigma_r(\tepsilon^r)=\deriv{\phi_r}{\tepsilon}\left(\tepsilon^r\right)\qquad\text{and}\qquad\tenseurq L_r\left(\tepsilon^r\right)=\derivdeux{\phi_r}{\tepsilon}\left(\tepsilon^r\right) + \tepsilon_r(\tsigma^r)=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseurq M_r\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right) \end{aligned} - give a "tangent" estimation of the stress $\tsigma_r$ around $\tepsilon^r$: + give a "tangent" estimation of the strain $\tepsilon_r$ around $\tsigma^r$: \begin{aligned} - \tsigma_r(\tepsilon)\approx\tsigma_r(\tepsilon^r)+\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\tepsilon-\tepsilon^r\right) + \tepsilon_r(\tsigma)\approx\tepsilon_r(\tsigma^r)+\tenseurq M_r\left(\tsigma^r\right)\dbldot\left(\tsigma-\tsigma^r\right) \end{aligned} - and can be viewed as the behaviour of a so-called "thermoelastic comparison composite". - - Hence, by noting $\tepsilon=\overline{\tepsilon}+\tepsilon'$, where $\tepsilon'$ is the fluctuating part of $\tepsilon$, we obtain + and can be viewed as the behaviour of a so-called "thermoelastic comparison composite". We hence obtain \begin{aligned} - \phi_r(\tepsilon)\approx v_r\left(\overline{\tepsilon}\right)+\tenseur\tau^r\dbldot\tepsilon'+\dfrac12\tepsilon'\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\tepsilon' + \psi_r(\tsigma)\approx v_r\left(\tsigma^r\right)+\tenseur e^r\left(\tsigma^r\right)\dbldot\tsigma+\dfrac12\tsigma\dbldot\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma \end{aligned} with \begin{aligned} - &v_r\left(\overline{\tepsilon}\right)=\phi_r(\tepsilon^r)+\tsigma_r(\tepsilon^r)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right)+\dfrac12\left(\overline{\tepsilon}-\tepsilon^r\right)\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right)\\ - &\tenseur\tau^r=\tsigma_r(\tepsilon^r)+\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right) + &v_r\left(\tsigma^r\right)=\psi_r(\tsigma^r)-\tepsilon_r(\tsigma^r)\dbldot\tsigma^r+\dfrac12\tsigma^r\dbldot\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r\\ + &\tenseur e^r\left(\tsigma^r\right)=\tepsilon_r(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r \end{aligned} - After that, we notice that for each $\tepsilon\in\Ka\left(\overline{\tepsilon}\right)$ corresponds a unique $\tepsilon'$ in $\Ka\left(\tenseur 0\right)$, so that the effective energy $\overline{\phi}$ is estimated by + Afterwards, the effective energy is found by the maximization problem: \begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)\approx\sum_r c_r\,v_r\left(\overline{\tepsilon}\right)+\underset{\tepsilon'\in\mathcal K(0)}{\textrm{min}}\sum_r c_r\, \langle \tenseur\tau^r\dbldot\tepsilon'+\dfrac12\tepsilon'\dbldot\tenseurq L_r\left(\tepsilon^r\right)\dbldot\tepsilon'\rangle_r + \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tsigma\in\Sa}{\textrm{max}} \sum_r c_r\,\left[\langle\tsigma\rangle_r\dbldot\left(\overline{\tepsilon}-\tenseur e^r\right)-v_r-\dfrac12\langle\tsigma\dbldot\tenseurq M_r\dbldot\tsigma\rangle_r\right] \end{aligned} - where $c_r$ is the volume fraction of the phase $r$, and $\langle.\rangle_r$ denotes the average on phase $r$. Of course, the last term in previous expression corresponds to the effective energy of a linear thermoelastic composite. This effective energy is also shown to be (see Eq. (5.32) of [@ponte_castaneda_nonlinear_1998]): + where $c_r$ is the volume fraction of the phase $r$, and $\langle.\rangle_r$ denotes the average on phase $r$. + The effective energy is hence given by \begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)\approx\sum_r c_r\,\left[\phi_r\left(\tepsilon^r\right)+\dfrac12 \tsigma_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right)\right] + \overline{\phi}\left(\overline{\tepsilon}\right)\approx P\left(\overline{\tepsilon},\tsigma^r\right) -\sum_r c_r\,v_r\qquad\text{with}\quad P\left(\overline{\tepsilon},\tsigma^r\right)=\underset{\tsigma\in\Sa}{\textrm{max}}\sum_r c_r\, \langle \tenseur d^r\dbldot\tsigma-\dfrac12\tsigma\dbldot\tenseurq M_r\dbldot\tsigma\rangle_r \end{aligned} + where $\tenseur d^r=\overline{\tepsilon}-\tenseur e^r$. Of course, the term $P\left(\overline{\tepsilon},\tsigma^r\right)$ in previous expression corresponds to the effective energy of a linear thermoelastic composite. The maximization leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by + \begin{aligned} + \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) + \end{aligned} + where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). -The last question is the choice of the reference strain $\tepsilon^r$. A discussion in [@castaneda_exact_1996] leads to the simple expression: +The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@castaneda_exact_1996] leads to the simple assumption that these reference stresses are the averages of the stress in the thermoelastic composite: \begin{aligned} -\tepsilon^r = \overline{\tepsilon}+\langle\tepsilon'\rangle_r +\tsigma^r = \langle\tsigma\rangle_r \end{aligned} - where $\tepsilon'$ here stands for the solution of the minimization problem above, that is, the strain field solution of the homogenization of the thermoelastic composite. This solution can be estimated by a mean-field homogenization scheme (or another more accurate method). + where $\tsigma$ here stands for the stress solution of the thermoelastic problem mentioned above. ## Macroscopic stress and tangent operator -The macroscopic stress $\overline{\tsigma}$ can be obtained by derivation of the estimated effective energy. It gives (see Eq. (5.30) of [@ponte_castaneda_nonlinear_1998]): - \begin{aligned} - \overline{\tsigma}\approx\sum_r c_r\,\left[\tsigma_r\left(\tepsilon^r\right)+\dfrac12 \langle\left(\tepsilon-\tepsilon^r\right)\dbldot\tenseurt N_r\dbldot\left(\tepsilon-\tepsilon^r\right)\rangle_r\dbldot\deriv{\tepsilon^r}{\overline{\tepsilon}}\right] - \end{aligned} -where +The above maximization on the set of statically admissible (divergence-free) fields leads to the condition that the field $\tenseur d^r-\tenseurq M_r\dbldot\tsigma$ belongs to $\Ka\left(\tenseur 0\right)$. Hence, by multiplying this field by $\tsigma$, which is divergence-free, and integrating over the entire volume, we obtain (apply Hill's lemma): \begin{aligned} -\tenseurt N_r = \deriv{\tenseurq L_r}{\tepsilon}\left(\tepsilon^r\right) +\sum_r c_r\,\langle\left(\tenseur d^r-\tenseurq M_r\dbldot\tsigma\right)\dbldot\tsigma\rangle_r=0 +\end{aligned} +which leads to +\begin{aligned} +\sum_r c_r\, \langle \tenseur d^r\dbldot\tsigma-\dfrac12\tsigma\dbldot\tenseurq M_r\dbldot\tsigma\rangle_r=\dfrac12\sum_r c_r\, \tenseur d^r\dbldot\langle\tsigma\rangle_r=\dfrac12\sum_r c_r\, \tenseur d^r\dbldot\tsigma^r \end{aligned} -The tangent operator is given by +Hence, using the expressions of $v_r$, $\tenseur e^r$ and $\tenseur d^r$ we obtain (see also expression (B.7) in [@bornert_second-order_2001]) \begin{aligned} - \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}=\tenseurq C_0^{\mathrm{eff}}+\dfrac{\mathrm{d}\tenseurq C_0^{\mathrm{eff}}}{\mathrm{d}\overline{\tepsilon}} + \overline{\phi}\left(\overline{\tepsilon}\right)\approx\sum_r c_r\,\left[\dfrac12\tsigma^r\dbldot\left(\tepsilon_r+\overline{\tepsilon}\right)-\psi_r\left(\tsigma^r\right)\right] + \end{aligned} + The macroscopic stress hence can be obtained by derivation w.r.t. $\overline{\tepsilon}$: + \begin{aligned} +\overline{\tsigma}\approx\dfrac12\sum_r c_r\,\left[\tsigma^r+\left(\overline{\tepsilon}-\tepsilon_r+\tsigma^r\dbldot\tenseurq M_r\right)\dbldot\deriv{\tsigma^r}{\overline{\tepsilon}}\right] \end{aligned} - Here, the derivative of $\tenseurq C_0^{\mathrm{eff}}$ w.r.t. $\overline{\tepsilon}$ can be computed by derivating the Hashin-Shtrikman - moduli w.r.t. the secant moduli $\mu_0^r$ and the derivatives of these moduli w.r.t. $\overline{\tepsilon}$. However, it is tedious and in the implementation, we see that the convergence of the Newton Raphson algorithm is good if we only retain the first term \(\tenseurq C_0^{\mathrm{eff}}\) in the tangent operator. + + The tangent operator is obtained by derivation: + \begin{aligned} + \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}\approx\sum_r c_r\,\left\{\deriv{\tsigma^r}{\overline{\tepsilon}}+\dfrac12\left[\left(\deriv{\tsigma^r}{\overline{\tepsilon}}\right)^T\dbldot\left(\tsigma^r\dbldot\tenseurt O_r\right)\dbldot\deriv{\tsigma^r}{\overline{\tepsilon}}+\left(\overline{\tepsilon}-\tepsilon_r+\tsigma^r\dbldot\tenseurq M_r\right)\dbldot\derivdeux{\tsigma^r}{\overline{\tepsilon}}\right]\right\} + \end{aligned} + where $\tenseurt O_r=\deriv{\tenseurq M_r}{\tsigma^r}$. +In the expressions of the macroscopic stress and tangent operator, we can compute the term $\deriv{\tsigma^r}{\overline{\tepsilon}}$ by means of the jacobian matrix, also done and explained in the documentation of the [Implicit DSL](https://thelfer.github.io/tfel/web/implicit-dsl.html#computation-of-the-consistent-tangent-operator). See below for more details. + +# Summary and possible implementations ## Summary The resolution consists in \[ - \text{Find}\quad\tepsilon^r=\overline{\tepsilon}+\langle\tepsilon'\rangle_r + \text{Find}\quad\tsigma^r\quad\text{such that}\quad\tsigma^r=\langle\tsigma\rangle_r \] -where \(\tepsilon'\) is the strain field solution of a homogenization problem of the form: +where \(\tsigma\) is the stress field solution of a homogenization problem of the form: \begin{aligned} -&\div\left[ \tenseurq L_r\left(\tepsilon^r\right)\dbldot\tepsilon'+\tenseur\tau^r\right]=0,\qquad\tepsilon'\in\Ka\left(\tenseur 0\right)\\ -&\tenseurq L_r\left(\tepsilon^r\right)=\derivdeux{\phi_r}{\tepsilon}\left(\tepsilon^r\right)\qquad\text{and}\qquad\tenseur\tau^r=\deriv{\phi_r}{\tepsilon}\left(\tepsilon^r\right)+\tenseurq L_r\left(\tepsilon^r\right)\dbldot\left(\overline{\tepsilon}-\tepsilon^r\right) +&\div\,\tsigma=\tenseur 0\\ +&\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\overline{\tepsilon}\right)\\ +&\tenseurq M_r=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r \end{aligned} + - -# Possible implementations +## Possible implementations +The iterative resolution of the non-linear system can be summarized as +\[ +\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq M_r= \pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right),\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\tsigma^r = \tenseurq A_r\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} +\] -# Implementation in MFront +The first step of the resolution consists in computing the moduli $\tenseurq M_r$ and free strains $\tenseur e^r$. It can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the local behaviour in an external file (in our example, the potential is to simple to do that). -In the application of the implementation, we take the example of the following behaviour: - \begin{aligned} - w_r(\tepsilon)=\dfrac{9}{2}k_r\,\varepsilon_{m}^2+\dfrac{\sigma_r^0}{n+1}\,\varepsilon_{eq}^{n+1} - \end{aligned} - where $n$ is between $0$ and $1$, which means that $f_r(e)=\dfrac{\sigma_r^0}{n+1}\,e^{\frac{n+1}2}$ (note that the function - $f_r$ is effectively concave relatively to $e$, because $0\lt n\leq 1$). - - Moreover, we make the choice of purely elastic inclusions. Hence, only the matrix is non-linear and has a - secant modulus. - -The unknowns of our non-linear problem are the $e^r$. The secant moduli $\mu_0^r$ can be computed by derivation of the function $f_r$ which is given by the behaviour. In the precise case detailed above, we have +The second step is the computation of the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$. This can be achieved by means of mean-field schemes (the `namespace` `tfel::material::homogenization` provides some localisation tensors). Another approach is to use morphological tensors. The two approaches are shown below in the next subsection. + +We note that the equation relative to the second step is the non-linear equation to solve and will hence correspond to our residue. +The macroscopic stress and the tangent operator must also be computed with the formulas shown above. + +## Computation of tensors Ar and Brs + +### Computation via morphological tensors + +In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\overline{\tepsilon}\right)$, we have, because, $\div\,\tsigma=0$, +\begin{aligned} +\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\end{aligned} +where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives +\begin{aligned} +\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau +\end{aligned} +and hence (due to the expressions of $\tenseur\tau$ and $\tepsilon$) +\begin{aligned} +\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) +\end{aligned} + +\begin{aligned} +\tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq \Delta_0\left(\tenseur e^r\right) +\end{aligned} +with $\tenseurq\delta\tenseurq M_r=\tenseurq M_r-\tenseurq M_0$, $\tenseurq M_0=\tenseurq L_0^{-1}$, and $\tenseurq \Delta_0=\tenseurq L_0-\tenseurq L_0\dbldot\tenseurq \Gamma_0\tenseurq L_0$. We will not resolve this equation exactly, but we consider its variational formulation and consider an approximation of the stress field, piece-wise uniform. It is exactly equivalent to apply the dual version of Hashin-Shtrikman principle. +\begin{aligned} +\langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s +\end{aligned} +where +\begin{aligned} +\tenseurq \Delta_{rs} = \sum_j\langle\,\tenseurq\Delta_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r +\end{aligned} +is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor makes intervene the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). +Hence, the $\langle\tsigma\rangle_r$ 's are solution of the following linear system: +\begin{aligned} +\sum_{s=1}^N\,\left[\delta_{rs}\,\tenseurq I + \tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\right]\dbldot\langle\tsigma\rangle_s=\tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s +\end{aligned} +where $\delta_{rs}$ is Krönecker symbol and $\tenseurq I$ is the fourth-order identity tensor. +Depending on the number of phases $N$, the solution of this system is more or less costly. The solution directly gives +the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ (compare with the expressions above). + +### Computation via a self-consistent scheme + +We restart here with the equation +\begin{aligned} +\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\end{aligned} +which can be discretized (considering again piece-wise uniforms fields and a variational framework) as +\begin{aligned} +\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s +\end{aligned} +with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r$. +A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ so that +\begin{aligned} +\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\tenseurq P_0\dbldot\langle\tenseur\tau\rangle_r+\tenseurq P_0\dbldot\langle\tenseur\tau\rangle +\end{aligned} +and due to the expression of $\tenseur\tau$: +\begin{aligned} +\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\overline{\tepsilon}\right)=-\left[\langle\tsigma\rangle_r-\langle\tsigma\rangle\right] +\end{aligned} +and noting $\tenseurq M_0^*=\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]^{-1}$, with $\tepsilon=\tenseurq M\dbldot\tsigma+\tenseur e$, and $\langle\tsigma\rangle_r=\tsigma^r$ +\begin{aligned} +\left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\overline{\tepsilon}\right)=-\tenseurq M_0^*\dbldot\left[\tsigma^r-\langle\tsigma\rangle\right] +\end{aligned} +which is +\begin{aligned} +\tsigma^r=\left(\tenseurq M_r+\tenseurq M_0^*\right)^{-1}\dbldot\left[\overline{\tepsilon}-\tenseur e^r+\tenseurq M_0^*\dbldot\langle\tsigma\rangle\right] +\end{aligned} +and by taking the average +\begin{aligned} +\langle\tsigma\rangle=\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\left[\overline{\tepsilon}+\tenseurq M_0^*\dbldot\langle\tsigma\rangle\right]-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\dbldot\tenseur e\rangle +\end{aligned} +and then +\begin{aligned} +\langle\tsigma\rangle=\left[\tenseurq I-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\tenseurq M_0^*\right]^{-1}\dbldot\left[\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\overline{\tepsilon}-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\dbldot\tenseur e\rangle\right] +\end{aligned} +Now, the self-consistent approach consists in choosing $\tenseurq M_0$ such that +\begin{aligned} +\tenseurq M_0=\left\{\left[\tenseurq I-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\tenseurq M_0^*\right]^{-1}\dbldot\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\right\}^{-1} +\end{aligned} +This is also \begin{aligned} -\deriv{f_r}{e}(e)=\dfrac{\sigma_r^0}{2}\,e^{\frac{n-1}2}\qquad\text{and}\quad\mu_0^r=\dfrac{\sigma_r^0}{3}\,\left(e^r\right)^{\frac{n-1}2} +\tenseurq M_0=\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle^{-1}-\tenseurq M_0^* \end{aligned} -Then, the effective energy of the linear comparison composite can be computed via a mean-field scheme. In our implementation, we proposed to use the Hashin-Shtrikman bound. The $e^r$ must then cancel the following residue: +This equation can be solved by an iterative procedure, and this is exactly what does the function `computeSelfConsistent` +from the library `tfel::material::homogenization`. +Afterwards, noting $\tenseurq M_0=\tenseurq M_{\mathrm{SC}}$, we have \begin{aligned} -r_{e^r}=e^r - \Frac2{3\,c_r}\deriv{W_0^{\mathrm{eff}}}{\mu_0^r} +\langle\tsigma\rangle=\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\overline{\tepsilon}-\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\left[\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\rangle\right]^{-1}\dbldot\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\tenseur e\rangle \end{aligned} -which means that we only need the derivative of the Hashin-Shtrikman bound. This will be given by the `tfel::material::homogenization` functionalities. +which is, because $\tenseurq M_{\mathrm{SC}}+\tenseurq M_{\mathrm{SC}}^*=\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\rangle^{-1}$, +\begin{aligned} +\langle\tsigma\rangle=\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\overline{\tepsilon}-\left[\tenseurq I+\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\tenseurq M_{\mathrm{SC}}^*\right]\dbldot\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\tenseur e\rangle +\end{aligned} +and then +\begin{aligned} +&\tenseurq A_r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq M_{\mathrm{SC}}^{-1}\right)\\ +&\tenseurq B_{rs}=-\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\delta_{rs}\tenseurq I +\tenseurq M_{\mathrm{SC}}^*\dbldot{}^t\tenseurq A_s\right) +\end{aligned} +In fact, because $\langle\tepsilon\rangle_r = \tenseurq M_r\dbldot\tsigma^r + \tenseur e^r$, we have also +\begin{aligned} +\langle\tepsilon\rangle_r = \mathcal A_r\dbldot\overline{\tepsilon}+ \tenseur e^r +\sum_s \tenseurq M_r\dbldot\tenseurq B_{rs}\dbldot\tenseur e^s +\end{aligned} +where $\mathcal A_r=\tenseurq M_r\dbldot\tenseurq A_r$ is the strain localisation tensor, +also provided by the function `computeSelfConsistent` (see below for details). + +# Implementation in MFront + +All the files are available in the [`MFrontGallery`](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization) project. We also provide in [this archive](./downloads/PonteCastaneda1996.zip) +a `python` script to generate the morphological tensors. + +ARCHIVE AND MFRONTGALLERY !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + +## Example used for the implementation + + The geometry of our polycrystal is generated with [`merope`](https://github.com/MarcJos/Merope/), + with a Voronoi tesselation. + This geometry is saved as an array with an integer representing the number of the phase (or crystal). + The image below shows an example of such a polycrystal. There are here 10 crystals. The orientations + and volume fractions are given in the file `polycrystal.csv`. The geometry is used to + compute the morphological tensors (see the `python` script). + + ![Geometry of the polycrystal](./img/polycrystal.png){width=50%} + + The potential $\psi_k^r$ will be taken of the form + \begin{aligned} + \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^{k,r}}{n+1}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n+1} + \end{aligned} + where the strain rate $\dot{\gamma}_0$ and the creep exponent $n\geq 1$ will be chosen identical for all $k,r$, and the resolved shear stress $\tau_0^{k,r}$ depends on $k,r$. + +The derivatives of the potential are + \begin{aligned} + &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r\\ + &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r + \end{aligned} + and then defining $\overline{\tau}_k^r=\tsigma^r\dbldot\tenseur \mu_k^r$ where $\tsigma^r$ is the reference stress, we have + \begin{aligned} + &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r\\ + &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r + \end{aligned} ## Details of implementation -All the files are available [here](./downloads/PonteCastaneda1992.zip) +### Headers -For the jacobian, we adopt the `Numerical Jacobian`, which -means that the beginning of the `mfront` file reads: +For the jacobian, we adopt the `Numerical Jacobian`, so that +the beginning of the `mfront` file reads: ~~~~ {#Begin .cpp .numberLines} @DSL ImplicitII; -@Behaviour PC_VB_92; +@Behaviour PC_SOE_96; @Author Martin Antoine; -@Date 12 / 12 / 25; -@Description{"Ponte Castaneda variational bounds for homogenization of non-linear elasticity (one potential), based on second-moments computation"}; -@UseQt true; +@Date 26 / 01 / 26; +@Description{"Ponte Castaneda second-order estimates for homogenization of a viscoplastic polycrystal, based on Green interaction tensors"}; +@UseQt false; @Algorithm NewtonRaphson_NumericalJacobian; @PerturbationValueForNumericalJacobianComputation 1e-10; @Epsilon 1e-12; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Moreover, the following libraries will be needed for computation -of the stress and secant modulus: +Moreover, we need to include a header file which contains our interaction tensors. +This file must be present in a repository `extra-headers/TFEL/Material/` +located at the same place as the `.mfront` file. In our implementation, +we generate a polycrystal and compute the tensors via FFT (see the `python` script). + +We will also need to handle the slip systems, and hence include `ExtendedPolyCrystalsSlidingSystems.hxx`. +This header is supposed to be in the same repository as `tensors.hxx`. ~~~~ {#Begin .cpp .numberLines} @TFELLibraries {"Material"}; @Includes{ -#include "TFEL/Material/IsotropicModuli.hxx" -#include "TFEL/Material/HomogenizationSecondMoments.hxx" -#include "TFEL/Material/LinearHomogenizationBounds.hxx" +#include "../extra-headers/TFEL/Material/tensors.hxx" +#include "../extra-headers/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +### Sliding systems + +The sliding systems are loaded via an external file (the same +procedure as for the tutorial on the [Berveiller-Zaoui](ExplicitBerveillerZaouiPolyCrystals.html) scheme). +This is done as follows : + +~~~~ {#Begin .cpp .numberLines} +@CrystalStructure HCP; +@SlidingSystems{<1, 1, -2, 0>{1, -1, 0, 0}, + <-2, 1, 1, 3>{1, -1, 0, 1}, + <-2, 1, 1, 0>{0, 0, 0, 1}, + <1, 1, -2, 0>{1, -1, 0, 1}}; + +@Members { + using PCSlidingSystems = + ExtendedPolyCrystalsSlidingSystems, real>; } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -We define two state variables which are the averages (on each phase) of the squares -of the equivalent strains, and correspond to the $e^r$: +### Poperties and variables + +We can now define a state variable on each phase, which correspond to +the reference stresses $\tsigma^r$. We also define an auxiliary variable +for $\deriv{\psi}{\tsigma}\left(\tsigma^r\right)$. + +~~~~ {#Begin .cpp .numberLines} +@StateVariable Stensor sigma[Np]; +sigma.setEntryName("PhaseReferenceStress"); +@AuxiliaryStateVariable Stensor dpsi_dsigma[Np]; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Among the different local variables, note the following which +allows to store the morphological tensors: ~~~~ {#Begin .cpp .numberLines} -@StateVariable real e_0; -e_0.setEntryName("MatrixSquaredEquivalentStrain"); -@StateVariable real e_r; -e_r.setEntryName("InclusionSquaredEquivalentStrain"); +@LocalVariable std::array,Np>,Np> DELTA; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The secant moduli are computed by derivation of function $f_r$, which is detailed above. Hence +Note also that `Np` is the number of phases. This number is a constant parameter, +otherwise, the definition of the `LocalVariable DELTA` above would not be possible +like that. However, we want that this parameter can change. Hence, we define it +before like that: + +~~~~ {#Begin .cpp .numberLines} +@IntegerConstant Np = @Np@; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +and the parameter will be set at the compilation of the behaviour: + +~~~~ {#bash .sh} +mfront --obuild --interface=generic --@Np@=10 PC_SOE_96.mfront +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +### Tangent modulus and free-strain + +There is no difficulty to compute the tangent modulus and free-strain: ~~~~ {#Integrator .cpp .numberLines} @Integrator { -//secant modulus///////////////////////////////// -const auto po = std::pow(max(e_0+de_0,real(1e-10)),0.5*(n_-1)); -const auto mu0 = sig0/3.*po; + using namespace tfel::math; + Stensor4 M[Np]; + Stensor e[Np]; + for (int r=0;r(k_m,mu0); -const auto kgr = KGModuli(kar,mur); -using namespace tfel::material::homogenization::elasticity; -const auto eeq2_ = computeMeanSquaredEquivalentStrain(kg0,fr,kgr,em2,eeq2); -const auto eeq20=std::get<0>(eeq2_); -const auto eeq2r=std::get<1>(eeq2_); + tmatrix<6*Np,6*Np,real> MAT; + tmatrix<6*Np,6*Np,real> G; + tmatrix<6*Np,6,real> E; + for (int r=0;r(E,6*r,0)=L0; + for (int s=0;s(MAT,6*r,6*s) +=DELTA[r][s]*dM; + map_derivative(G,6*r,6*s) -=DELTA[r][s]; + } + } + map_derivative(MAT,6*r,6*r) +=I; + } + + TinyMatrixInvert<6*Np,real>::exe(MAT); + tmatrix<6*Np,6,real> A = MAT*E; + tmatrix<6*Np,6*Np,real> B = MAT*G; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Now we can write the residues: +### Residues and jacobian + +The residues are ~~~~ {#Integrator .cpp .numberLines} -//residues///////////////////////////////////// -fe_0 = e_0+de_0 - eeq20; -fe_r = e_r+de_r - eeq2r; + for (int r=0;r(A,6*r,0); + fsigma[r] = sigma[r]+dsigma[r]-Ar*(eto+deto); + for (int s=0;s(B,6*r,6*s); + fsigma[r]-=Brs*e[s]; + } + } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -There is no jacobian because we use the `NumericalJacobian`. The macroscopic stress -is given by the Hashin-Shtrikman lower bound (indeed, the lower bound is better suited -for soft matrix and rigid inclusions): +There is no jacobian because we use the `NumericalJacobian`. Here, +an analytical jacobian would have been tedious (see the expressions +of $\tenseurq A_r$ and $\tenseurq B_{rs}$ !). + +### Macroscopic stress + +The computation of the macroscopic stress necessitates the computation of $\deriv{\tsigma^r}{\overline{\tepsilon}}$. +To do that, we use a technique widely used in `MFront`. +The idea is to write the non-linear system to solve as: +\begin{aligned} +\tsigma^r - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)\equiv \tenseur R_r(\overline{\tepsilon},\tsigma^1,...,\tsigma^N)=0 +\end{aligned} +and by derivation of $\tenseur R_r$ w.r.t. $\overline{\tepsilon}$: +\begin{aligned} +-\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\overline{\tepsilon}}=0 +\end{aligned} +where $\tenseurq J_{rs}=\deriv{\tenseur R_r}{\tsigma^s}$ stands for the sub-block $(r,s)$ of the Jacobian. Hence +\begin{aligned} +\deriv{\tsigma^r}{\overline{\tepsilon}}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) +\end{aligned} +where $\mathbf {iJ}$ is the inverse of the Jacobian. + +The implementation is ~~~~ {#Integrator .cpp .numberLines} -//sigma////////////////////////////////////// -vector tab_f={1-fr,fr}; -vector tab_k = {k_m,kar}; -vector tab_mu = {mu0,mur}; -const auto HSB=computeIsotropicHashinShtrikmanBounds<3,stress>(tab_f,tab_k,tab_mu); -const auto LB=std::get<0>(HSB); -const auto kHS=std::get<0>(LB); -const auto muHS=std::get<1>(LB); -Chom=3*kHS*J+2*muHS*K; -sig = Chom*(eto+deto); + tmatrix<6*Np,6*Np,real> iJ = 0.000001*ID+jacobian; + TinyMatrixInvert<6*Np,real>::exe(iJ); + dsigma_deto=iJ*A; + sig=Stensor::zero(); + for (int r=0;r(dsigma_deto,6*r,0); + const auto term = M[r]*(sigma[r]+dsigma[r])-dpsi_dsigma[r]+eto+deto; + const auto term2 = term*dsigma_deto_r+sigma[r]+dsigma[r]; + sig += 0.5*frac[r]*term2; + } } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -(Note that `Chom` is the effective elastic tensor, defined here as -a local variable). -Afterwards, we can give the tangent operator: +Note the use of the matrix `ID` HERE ############# + +### Tangent operator + +As for the tangent operator, due to its complexity, we will only retain the first term in the implementation: +\begin{aligned} + \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}\approx\sum_r c_r\,\deriv{\tsigma^r}{\overline{\tepsilon}} + \end{aligned} +which only requires the computation of $\deriv{\tsigma^r}{\overline{\tepsilon}}$ that +was already computed. Hence, we have ~~~~ {#tangent .cpp .numberLines} @TangentOperator{ -Dt=Chom; + for (int r=0;r(dsigma_deto,6*r,0); + Dt+=frac[r]*dsigma_deto_r; + } } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Results +## Results + +For the tests, we use `mtest`. However, note that the strain we impose +is in fact a strain rate. The idea for us is to keep this strain rate +constant and to make vary the parameters $n^r$ with time. -We then use `MTest` to simulate a strain-imposed test. +Indeed, in Ponte-Castaneda's works (but also in Bornert's works and other works), +when all exponents $n_k^r$ are equal to a unique $n$, +the effective stress is shown to be of the form -MTest file is the following: +\begin{aligned} +\overline{\tsigma}_{11}=\tilde{\tau}_0\left(\dfrac{\dot{\tepsilon}_{11}}{\dot{\gamma_0}}\right)^{1/n} +\end{aligned} -~~~~ {#PC_VB_92 .mtest .numberLines} +VERIFY THE COORDINATES !!!!!!!!!!!!!!!!!!! + +which means that for a constant $\dot{\tepsilon}$, where we introduced +an effective resolved shear stress $\tilde{\tau}_0$. Afterwards, these +authors generally plot this $\tilde{\tau}_0$ as a function of the +parameter $n$. Hence, the idea here is to reproduce this +curve by varying $n$ and keeping $\dot{\tepsilon}$ constant. + +The `.mtest` file is the following: + +~~~~ {#PC_SOE_96 .mtest .numberLines} @ModellingHypothesis 'Tridimensional'; -@Behaviour 'src/libBehaviour.so' 'PC_VB_92'; +@Behaviour 'src/libBehaviour.so' 'PC_SOE_96'; @ExternalStateVariable 'Temperature' {0 : 1000,1:1000}; -@MaterialProperty 'sig0' 1.e9; -@MaterialProperty 'n_' 0.05; -@MaterialProperty 'k_m' 1.e9; -@MaterialProperty 'kar' 1.e15; -@MaterialProperty 'mur' 1.e15; -@MaterialProperty 'fr' 0.25; - -@ImposedStrain 'EXX' {0 : 0, 100 : 0.8660254}; -@ImposedStrain 'EYY' {0 : 0, 100 : -0.8660254}; -@ImposedStrain 'EZZ' {0 : 0, 100 : 0}; -@ImposedStrain 'EXY' {0 : 0, 100 : 0}; -@ImposedStrain 'EXZ' {0 : 0, 100 : 0}; -@ImposedStrain 'EYZ' {0 : 0, 100 : 0}; -@Times {0.,10 in 100}; +@MaterialProperty 'n1' "exp(2*t*2.3026)"; +@MaterialProperty 'n2' "exp(2*t*2.3026)"; +@MaterialProperty 'tau1' 20.; +@MaterialProperty 'tau2' 30.; +@ImposedStrain 'EXX' {0 : -1, 1 : -1}; +@ImposedStrain 'EYY' {0 : 0, 1 : 0}; +@ImposedStrain 'EZZ' {0 : 0, 1 : 0}; +@ImposedStrain 'EXY' {0 : 0, 1 : 0}; +@ImposedStrain 'EXZ' {0 : 0, 1 : 0}; +@ImposedStrain 'EYZ' {0 : 0, 1 : 0}; +@Times {0.,1 in 100}; ~~~~~~~~~~~~~~ -The macroscopic stress $\sigma_{xx}$ is represented below as a function -of the macroscopic strain $\varepsilon_{xx}$: -![Macroscopic stress as function of the macroscopic strain](./img/PC_VB_92.png) +![Effective resolved shear stress as a function of the parameter $n$](./img/PC_VB_92.png) + +FIGURE !!!!!!!!!!!!!!!!!!!!!!!!! -We also plotted the behaviour of the matrix when there is no inclusions (it gives -an idea of the reinforcement given by the inclusions). +In this figure we plotted the behaviour of a single crystal when it is alone +(to show the heterogeneity), and the two implementations of the second-order estimates +(SOE), with morphological tensors (SOE_T) and with self-consistent scheme (SOE_SC). diff --git a/docs/web/bibliography.bib b/docs/web/bibliography.bib index 127c9a607..60d307d0d 100644 --- a/docs/web/bibliography.bib +++ b/docs/web/bibliography.bib @@ -12612,7 +12612,7 @@ @article{castaneda_exact_1996 } @article{bornert_second-order_2001, - title = {Second-order estimates for the e'ective behaviour of viscoplastic polycrystalline materials}, + title = {Second-order estimates for the effective behaviour of viscoplastic polycrystalline materials}, volume = {49}, abstract = {This paper is concerned with the application of the “second-order” nonlinear homogenisation procedure (Ponte Casta˜neda, J.Mech.Phys.Solids 44 (6) (1996) 827) to generate estimates of the self-consistent type for the e'ective behaviour of fcc and hcp viscoplastic polycrystals. The method has the distinctive property that it leads to estimates that are exact to second-order in the heterogeneity contrast, and which are expected to be more accurate, particularly when compared to rigorous bounds, than those resulting from earlier homogenisation schemes such as the Hill “incremental” method or its “total” formulation (Hutchinson) for pure power-law viscous materials.Special attention is paid to large grain anisotropy leading to correspondingly large heterogeneity contrast, and to highly nonlinear behaviour.Comparisons are also carried out with estimates derived from other more recent homogenisation schemes such as the “tangent” and “a7ne” methods.The results, illustrated for zirconium- and ice-type polycrystals, show that the second-order procedure o'ers the potential for signi9cantly improved results, at least relative to the Hill incremental formulation. ? 2001 Elsevier Science Ltd.All rights reserved.}, language = {en}, @@ -12621,3 +12621,38 @@ @article{bornert_second-order_2001 year = {2001}, pages = {2737--2764}, } + +@article{castaneda_effect_1995, + title = {The effect of spatial distribution on the effective behavior of composite materials and cracked media}, + volume = {43}, + copyright = {https://www.elsevier.com/tdm/userlicense/1.0/}, + issn = {00225096}, + url = {https://linkinghub.elsevier.com/retrieve/pii/002250969500058Q}, + doi = {10.1016/0022-5096(95)00058-Q}, + abstract = {Estimates of the Hashin-Shtrikman type are developed for the overall moduli of composites consisting of a matrix containing one or more populations of inclusions, when the spatial correlations of inclusion locations take particular “ellipsoidal” forms. Inclusion shapes can be selected independently of the shapes adopted for the spatial correlations. The formulae that result are completely explicit and easy to use. They are likely to be useful, in particular, for composites that have undergone a prior macroscopically uniform large deformation. To the extent that the statistics that are assumed may not be realized exactly, the formulae provide approximations. Since, however, they are derived as variational approximations for composites with some explicit statistics that are realizable, they are free from some of the drawbacks of competitor approximations such as that of Mori and Tanaka (1973 Acta Metall. 21,571-574), which can generate tensors of effective moduli which fail to satisfy a necessary symmetry requirement. The new formulae are also the only ones known that take explicit account, at least approximately, of inclusion shape and spatial distribution independently.}, + language = {en}, + number = {12}, + urldate = {2026-01-20}, + journal = {Journal of the Mechanics and Physics of Solids}, + author = {Castañeda, P}, + month = dec, + year = {1995}, + pages = {1919--1951}, +} + +@article{willis_bounds_1977, + title = {Bounds and self-consistent estimates for the overall properties of anisotropic composites}, + volume = {25}, + copyright = {https://www.elsevier.com/tdm/userlicense/1.0/}, + issn = {00225096}, + url = {https://linkinghub.elsevier.com/retrieve/pii/0022509677900229}, + doi = {10.1016/0022-5096(77)90022-9}, + language = {en}, + number = {3}, + urldate = {2026-01-20}, + journal = {Journal of the Mechanics and Physics of Solids}, + author = {Willis, J.R.}, + month = jun, + year = {1977}, + pages = {185--202}, +} diff --git a/docs/web/img/ParticulateMicrostructure.png 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zjYww;_JU|4v$%T+RiKp>rOQ9$h-5XiAV9NUg@ zE1{gWVWR)n_P|h=XtfOnS^_R$3bqCATJ`|h=s=*)fMo!FJ~#p<91zIv3XLHmh-)BH S@IDD>2ALY(HmKKgj`=^#fWf5z literal 0 HcmV?d00001 diff --git a/docs/web/tfel-material.md b/docs/web/tfel-material.md index f896acbc4..c78b89ae8 100644 --- a/docs/web/tfel-material.md +++ b/docs/web/tfel-material.md @@ -456,14 +456,17 @@ The expressions of Eshelby tensor can be found in [@torquato_2002] for the spheroidal inclusions and in [@eshelby_1957] for the general ellipsoid (three different semi-axes). +![Ellipsoidal inclusion](./img/ellipsoide_C0.png){width=35%} + Now if we consider an ellipsoid whose elasticity is \(\tenseurq C_i\), embedded in an infinite homogeneous medium whose elasticity is \(\tenseurq C_0\), submitted to a external uniform strain field at infinity \(\tenseur E\), the strain field within the ellipsoid is uniform and given by -\(\tenseur \varepsilon = \tenseurq A:\tenseur E\) +\(\tenseur \varepsilon = \tenseurq A_i:\tenseur E,\qquad\text{where}\quad + \tenseurq A_i = \left[\tenseurq I + \tenseurq P_0:\left(\tenseurq C_i -\tenseurq C_0\right)\right]^{-1}\) -where \(\tenseurq A \) is the strain localisation (or concentration) tensor. +where \(\tenseurq A_i \) is the strain localisation (or concentration) tensor. ### Computation in isotropic reference medium @@ -487,7 +490,7 @@ The second one computes the Hill tensor for an axisymmetrical ellipsoid (or sphe The user must provides the normal vector `n_a` for the axis, and `e` for the aspect ratio. The third line computes the Hill tensor of a more general ellipsoid whose semi-axis lengths are `a`,`b`,`c`. The axis `a` is related to direction given by `n_a` and `b` is related to the -direction given by `n_b`, which must be normal to `n_a`. +direction given by `n_b`, which must be normal to `n_a` (see the figure above). An `IsotropicModuli` can also be passed for the elasticity, as follows: @@ -567,6 +570,8 @@ const auto A_C = computePlaneStrainLocalisationTensor(IM0,C_i,n_a,a,b); The header `AnisotropicEshelbyTensor.hxx` introduces the computation of the Eshelby tensors and Hill tensors of general ellipsoids embedded in an anisotropic medium. +The anisotropic elasticity \(\tenseurq C_0\) must be +expressed in the global basis. These tensors can be computed as follows: @@ -615,7 +620,7 @@ The available schemes are: - dilute scheme - Ponte Castaneda and Willis scheme -Each scheme is based on the average of the localisation tensor \(\tenseur A \) +Each scheme is based on the average of the localisation tensor \(\tenseurq A_i \) defined above. This average is computed assuming different distributions of ellipsoids. Hence different cases are considered: @@ -625,6 +630,8 @@ of ellipsoids. Hence different cases are considered: - transverse isotropic distribution of orientations (one axis \(\tenseur n_a\) of the ellipsoid is fixed, the others are uniformly distributed in the transverse plane) +![The three distributions of orientations considered here: oriented, isotropic, and transverse isotropic](./img/distributions.png){width=100%} + Hence we can compute the homogenized stiffness returned by the available schemes. For example, for the distribution of spheres: @@ -657,6 +664,16 @@ be created by the user. It is defined by two vectors \(\tenseur n_a,\tenseur n_b Distribution D = {.n_a = n_a, .a = a, .n_b = n_b, .b = b, .c = c}; ~~~~ +![The three distributions of orientations can be considered with PCW scheme, but a big ellipsoid defines the spatial distribution of inclusions](./img/distributions_PCW.png){width=100%} + +Indeed, in Ponte-Castaneda and Willis scheme, there is a difference between +the ellipsoid which defines the distribution of the inclusions, and the +ellipsoid which defines the shape of the inclusions (in the image above, we represent +the distribution `D` of inclusions by a big ellipsoid, with colored axes). +A bigger axis for `D` means that along this axis, the distribution of inclusions +is more diluted. A short axis means that, on the contrary, the distribution is denser +along this axis. + For the isotropic distribution of ellipsoids, we can do: ~~~~{.cpp} @@ -669,7 +686,7 @@ Here, the two first schemes return `KGModuli` objects, whereas `computeIsotropicPCWScheme` returns a `st2tost2` object. For this latter case, the ellipsoids have indeed a uniform isotropic distribution of orientations, but the user might use a non-isotropic -`Distribution D`. +`Distribution D` (it corresponds to the configuration at the center on the image). And finally, we can consider a transverse isotropic distribution of inclusions: @@ -680,7 +697,7 @@ const auto C_PCW = computeTransverseIsotropicPCWScheme(IM0,f,IMi,n_a,a,b ~~~~ Here, the three above schemes return `st2tost2` objects. -Because the functions are based on the average of the localisation tensor \(\tenseur A \) +Because the functions are based on the average of the localisation tensor \(\tenseurq A_i \) associated with each distribution, a `Base` function is also defined for each scheme, that only takes in argument the average of the localisation tensor `A_av`. We then can compute a homogenized stiffness with a very general averaged localisator: @@ -789,6 +806,8 @@ via 2 template parameters: `ParticulateMicrostructure` with `N` the dimension. For the details, see the file 'MicrostructureDescription.hxx' which introduces the `class`. +![The `ParticulateMicrostructure class` is made of a matrix which embeds different distributions of inclusions](./img/ParticulateMicrostructure.png){width=50%} + A `ParticulateMicrostructure` consists on a matrix, in which are embedded several distributions of inclusions. The class has three (private) attributes: @@ -1020,7 +1039,7 @@ It is `12` by default. A last method of the `ParticulateMicrostructure` object allows to replace the matrix phase: -~~~~{.py} +~~~~{.cpp} micro_1.replaceMatrixPhase(C0); std::cout<< micro_1.get_matrix_elasticity()<< std::endl; std::cout<< micro_1.is_isotropic_matrix()<< std::endl; @@ -1066,16 +1085,16 @@ the computation considers an isotropic matrix when computing the Hill tensors relative to the inclusions, at each iteration of the algorithm. Indeed, the homogenized stiffness may be non isotropic, so that the user can make the choice of isotropizing this homogenized stiffness at each iteration. -Otherwise, he can put `False`, so that a numerical integration (resulting +Otherwise, he can put `false`, so that a numerical integration (resulting in a slower computation) will be performed to compute the Hill tensors. Moreover, an integer parameter can be added after the boolean, that indicates the number of subdivisions in the numerical integration. This value is `12` by default: -~~~~{.py} +~~~~{.cpp} micro_2.addInclusionPhase(distrib_O); -hmSC_iso=computeSelfConsistent<3u,stress>(micro_2,10,True); -hmSC_aniso=computeSelfConsistent<3u,stress>(micro_2,10,False,10); +hmSC_iso=computeSelfConsistent<3u,stress>(micro_2,10,true); +hmSC_aniso=computeSelfConsistent<3u,stress>(micro_2,10,false,10); std::cout<< "SC iso: "<< hmSC_iso.homogenized_stiffness<< std::endl; std::cout<< "SC aniso: "<< hmSC_aniso.homogenized_stiffness<< std::endl; ~~~~ @@ -1092,7 +1111,7 @@ Moreover, if anisotropic, another parameter can be passed to specify the number of subdivisions in the numerical integration (this value is `12` by default): -~~~~{.py} +~~~~{.cpp} micro_1.replaceMatrixPhase(C0); micro_1.removeInclusionPhase(0); micro_1.addInclusionPhase(distrib_O); diff --git a/docs/web/tfel-python.md b/docs/web/tfel-python.md index 9422e85c5..68943627f 100644 --- a/docs/web/tfel-python.md +++ b/docs/web/tfel-python.md @@ -156,6 +156,8 @@ import tfel.material as tmat import tfel.math as tm ~~~~ +![Ellipsoidal inclusion](./img/ellipsoide_C0.png){width=35%} + ### Hill tensors The Hill tensors are available: @@ -247,7 +249,7 @@ Note that if the elasticity of the inclusion is not isotropic, an anisotropic elasticity `C_i` can be provided, assuming that this elasticiy is expressed in the same basis as the one defined by `n_a,n_b` (the local basis of the inclusion): -~~~~{.cpp} +~~~~{.py} A_aniso = hm.computeLocalisationTensor(IM0,C_i,n_a,a,n_b,b,c) ~~~~ @@ -280,7 +282,7 @@ The following schemes are available for biphasic media with - dilute scheme - Ponte Castaneda and Willis scheme -Here are some examples of computation: +Here are some examples of computation for the spherical inclusions: ~~~~{.py} young=1e9 @@ -298,7 +300,16 @@ KGDS_IM=hm.computeSphereDiluteScheme(IM,f,IMi) KGMT_IM=hm.computeSphereMoriTanakaScheme(IM,f,IMi) print(EnuDS.young,EnuDS.nu,KGDS_IM.kappa,KGDS_IM.mu) print(EnuMT.young,EnuMT.nu,KGMT_IM.kappa,KGMT_IM.mu) +~~~~ + +And we can also consider distribution of ellipsoidal inclusions, +with three kind of distributions of orientations. +![The three distributions of orientations considered here: oriented, isotropic, and transverse isotropic](./img/distributions.png){width=100%} + +Hence, here are the examples to compute the homogenized properties: + +~~~~{.py .numberLines} # Ellipsoidal inclusions a=10. b=1. @@ -334,6 +345,15 @@ print(C_O_MT) print(C_O_PCW) ~~~~ +![The three distributions of orientations can be considered with PCW scheme, but a big ellipsoid defines the spatial distribution of inclusions](./img/distributions_PCW.png){width=100%} + +In Ponte-Castaneda and Willis scheme (PCW), there is a difference between +the ellipsoid which defines the distribution of the inclusions, and the +ellipsoid which defines the shape of the inclusions (in the image above, we represent +the distribution `D` of inclusions by a big ellipsoid, with colored axes). +A bigger axis for `D` means that along this axis, the distribution of inclusions +is more diluted. A short axis means that, on the contrary, the distribution is denser +along this axis. The object `D` is defined above at line 12. ### Homogenization bounds @@ -406,6 +426,8 @@ in the namespace `tfel::material::homogenization::elasticity` for the construction and homogenization of general microstructures. The reader may want to consult this documentation [here](tfel-material.html#homogenization-of-general-microstructures). +![The `ParticulateMicrostructure` object is made of a matrix which embeds different distributions of inclusions](./img/ParticulateMicrostructure.png){width=50%} + The `ParticulateMicrostructure` object is defined and can be instantiated in various ways: From 1f101a62e60fbcd69c0d074f3889db65314beb23 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Mon, 2 Feb 2026 00:34:29 +0100 Subject: [PATCH 03/25] update tuto MassonAffineFormulation.md --- docs/web/CMakeLists.txt | 2 +- docs/web/MassonAffineFormulation.md | 633 ++++++++++++++++++ docs/web/PonteCastaneda1996.md | 634 ------------------- docs/web/bibliography.bib | 11 + docs/web/downloads/Morphological_tensors.zip | Bin 0 -> 4801 bytes docs/web/img/Affine_formulation.png | Bin 0 -> 20539 bytes 6 files changed, 645 insertions(+), 635 deletions(-) create mode 100644 docs/web/MassonAffineFormulation.md delete mode 100644 docs/web/PonteCastaneda1996.md create mode 100644 docs/web/downloads/Morphological_tensors.zip create mode 100644 docs/web/img/Affine_formulation.png diff --git a/docs/web/CMakeLists.txt b/docs/web/CMakeLists.txt index ddabe7c72..a558debbf 100644 --- a/docs/web/CMakeLists.txt +++ b/docs/web/CMakeLists.txt @@ -95,7 +95,7 @@ pandoc_html(Taylor "--toc" "--toc-depth=2") pandoc_html(Sachs "--toc" "--toc-depth=1") pandoc_html(BetaRule "--toc" "--toc-depth=2") pandoc_html(PonteCastaneda1992 "--toc" "--toc-depth=2") -pandoc_html(PonteCastaneda1996 "--toc" "--toc-depth=2") +pandoc_html(MassonAffineFormulation "--toc" "--toc-depth=2") pandoc_html(Norton-web) pandoc_html(Norton-full) pandoc_html(coverity-scan) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md new file mode 100644 index 000000000..632cc8819 --- /dev/null +++ b/docs/web/MassonAffineFormulation.md @@ -0,0 +1,633 @@ +--- +title: Affine formulation for homogenization of a viscoplastic polycrystal +author: Antoine Martin +date: 30/01/2026 +lang: en-EN +link-citations: true +colorlinks: true +geometry: + - margin=2cm +papersize: a4 +figPrefixTemplate: "$$i$$" +tblPrefixTemplate: "$$i$$" +secPrefixTemplate: "$$i$$" +eqnPrefixTemplate: "($$i$$)" +--- + +\newcommand{\tenseur}[1]{\underline{#1}} +\newcommand{\tenseurq}[1]{\underline{\mathbf{#1}}} +\newcommand{\tenseurt}[1]{\underline{\underline{\underline{#1}}}} +\newcommand{\tns}[1]{\underset{\tilde{}}{\mathbf{#1}}} +\newcommand{\transpose}[1]{#1^{\mathop{T}}} +\newcommand{\Ka}{\mathcal K} +\newcommand{\Sa}{\mathcal S} +\newcommand{\tsigma}{\tenseur{\sigma}} +\newcommand{\dbldot}{:} +\newcommand{\tepsilon}{\tenseur{\varepsilon}} +\newcommand{\te}{\tenseurq{e}} +\newcommand{\sigmaeq}{\sigma_{\mathrm{eq}}} +\newcommand{\epsiloneq}{\varepsilon_{\mathrm{eq}}} + +\newcommand{\tepsilonto}{\underline{\epsilon}^{\mathrm{to}}} +\newcommand{\tepsilonel}{\underline{\epsilon}^{\mathrm{el}}} + +\newcommand{\trace}[1]{\mathrm{tr}\paren{#1}} +\newcommand{\Frac}[2]{{\displaystyle \frac{\displaystyle #1}{\displaystyle #2}}} +\newcommand{\deriv}[2]{{\displaystyle \frac{\displaystyle \partial #1}{\displaystyle \partial #2}}} +\newcommand{\derivdeux}[2]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2^2}}} +\newcommand{\pderiv}[2]{{\dfrac{\partial #1}{\partial #2}}} +\newcommand{\pderivdeux}[2]{{\dfrac{\partial^2 #1}{\partial #2^2}}} +\newcommand{\derivderiv}[3]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2\,\displaystyle \partial #3}}} +\newcommand{\dtot}{{\mathrm{d}}} +\newcommand{\paren}[1]{\left(#1\right)} +\newcommand{\nom}[1]{\textsc{#1}} +\renewcommand{\div}{\mathrm{div}} + + +We present here an implementation of the affine formulation for homogenization [@masson_affine_2000] +for the homogenization of a viscoplastic polycrystal, example which is treated in [@bornert_second-order_2001] +but with Ponte-Castaneda second-order estimates. + +Here, the idea is to show that many implementations of that procedure are possible in `MFront`. +We show here a first implementation based on the self-consistent scheme, provided by the `namespace` +`tfel::material::homogenization`. A second approach which uses morphological tensors computed by FFT, +on a given geometry of polycrystal, is also detailed. This allows to improve the estimates, but +it necessitates an offline computation. + +This tutorial first presents the homogenization problem, recalls the methodology +of the affine formulation, presents different kinds of possible implementations, +and show the details of the `mfront` file. + +# The viscoplastic polycrystal + +## Behaviour + +We consider a polycrystalline material, which means that each phase $r$ is associated to a crystal with +corresponding slip systems $\tenseur \mu_k^r$ ($1\leq k\leq K$). The strain rate in each crystal $r$ is given by + \begin{aligned} + \dot{\tepsilon}=\sum_k \dot{\gamma}_k^r\, \tenseur \mu_k^r\qquad\text{with}\quad \tenseur\mu_k^r = \dfrac12\left(\tenseur n_k^r\otimes\tenseur m_k^r + \tenseur m_k^r\otimes\tenseur n_k^r\right) + \end{aligned} + where $\dot{\gamma}_k^r$ is the shear strain rate on the $k^{th}$ slip system of crystal $r$, and is given as a + function of the shear stress $\tau_k^r=\tsigma\dbldot\tenseur \mu_k^r$ by means of a potential $\psi_k^r$: + \begin{aligned} + \dot{\gamma}_k^r= \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right) + \end{aligned} + The expressions above show that on phase (or crystal) $r$: + \begin{aligned} + \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\text{with}\quad\psi_r \left(\tsigma\right)=\sum_k \psi_k^r\left(\tau_k^r\right) + \end{aligned} + + Now, the behaviour of the polycrystal is governed by a potential $\psi$: + \begin{aligned} + \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\psi(\tsigma)= \sum_{r=1}^{N}\chi_r\,\psi_r (\tsigma) + \end{aligned} +where $N$ is the number of phases (or crystals) and $\chi_r$ is characteristic function of phase $r$. + + In all the sequel, we just note $\tepsilon$ for $\dot{\tepsilon}$. + +## The non-linear homogenization problem + +We impose a strain rate $\overline{\tepsilon}$ to the polycrystal and look for the solution $\tepsilon,\tsigma$ such that: + +\begin{aligned} + &\div\,\tsigma=\tenseur 0\\ + &\tepsilon\in\Ka(\overline{\tepsilon})\\ + &\tepsilon = \deriv{\psi}{\tsigma}\left(\tsigma\right) +\end{aligned} + +where we introduced the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$, depending on the boundary conditions used (in the implementation, we work with periodic boundary conditions). + +# The affine formulation + +## The affine linearization + +The idea is to linearize the behaviour around a reference stress $\tsigma^r$: + \begin{aligned} + \tepsilon_r(\tsigma)\approx\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma+\tenseur e^r + \end{aligned} + where + \begin{aligned} + \tenseurq M_r\left(\tsigma^r\right)=\deriv{\tepsilon_r}{\tsigma}\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\tenseur e^r = \tepsilon_r(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r + \end{aligned} + This affine behaviour can be viewed as a so-called "thermoelastic comparison composite", and this composite + can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by + \begin{aligned} + \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) + \end{aligned} + where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). + +The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@masson_affine_2000] leads to the simple assumption that these reference stresses are the averages of the stress in the thermoelastic composite: +\begin{aligned} +\tsigma^r = \langle\tsigma\rangle_r +\end{aligned} + where $\tsigma$ here stands for the stress solution of the thermoelastic homogenization problem. + +## Macroscopic stress and tangent operator + + The macroscopic stress hence can be obtained by the classical relation: + \begin{aligned} +\overline{\tsigma}=\sum_rc_r\,\langle\tsigma\rangle_r=\tenseurq C^{\mathrm{eff}}\dbldot\overline{\tepsilon}+\tenseur\tau^{\mathrm{eff}} + \end{aligned} + + where + \begin{aligned} + \tenseurq C^{\mathrm{eff}}=\sum_rc_r\,\tenseurq A_r\qquad\text{and}\qquad\tenseur\tau^{\mathrm{eff}}=\sum_{r,s}c_r\,\tenseurq B_{rs}\dbldot\tenseur e^s + \end{aligned} + + The tangent operator is obtained by derivation: + \begin{aligned} + \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}=\sum_r c_r\,\deriv{\tsigma^r}{\overline{\tepsilon}} + \end{aligned} + +In the expressions of the tangent operator, we can compute the term $\deriv{\tsigma^r}{\overline{\tepsilon}}$ by means of the jacobian matrix, also done and explained in the documentation of the [Implicit DSL](https://thelfer.github.io/tfel/web/implicit-dsl.html#computation-of-the-consistent-tangent-operator). We explain it below. + +# Summary and possible implementations + +## Summary + +The resolution consists in +\[ + \text{Find}\quad\tsigma^r\quad\text{such that}\quad\tsigma^r=\langle\tsigma\rangle_r +\] +where \(\tsigma\) is the stress field solution of a homogenization problem of the form: +\begin{aligned} +&\div\,\tsigma=\tenseur 0\\ +&\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\overline{\tepsilon}\right)\\ +&\tenseurq M_r=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r +\end{aligned} + + +## Possible implementations + +The iterative resolution of the non-linear system can be summarized as + +\[ +\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq M_r= \pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right),\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\tsigma^r = \tenseurq A_r\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} +\] + +The first step of the resolution consists in computing the moduli $\tenseurq M_r$ and free strains $\tenseur e^r$. It can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the local behaviour in an external file (in our example, the potential is to simple to do that). + +The second step is the computation of the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$. This can be achieved by means of mean-field schemes (the `namespace` `tfel::material::homogenization` provides the good tensors). This can also be done via morphological tensors, computed on a given geometry of polycrystal. We will present below the two possibilities. + +We note that the equation relative to the second step is the non-linear equation to solve and will hence correspond to our residue. +The macroscopic stress and the tangent operator must also be computed with the formulas shown above. + +## Computation of tensors Ar and Brs + +### Computation via morphological tensors + +In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\overline{\tepsilon}\right)$, we have, because, $\div\,\tsigma=0$, +\begin{aligned} +\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\end{aligned} +where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives +\begin{aligned} +\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau +\end{aligned} +and hence (due to the expressions of $\tenseur\tau$ and $\tepsilon$) +\begin{aligned} +\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) +\end{aligned} + +\begin{aligned} +\tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq \Delta_0\left(\tenseur e^r\right) +\end{aligned} +with $\tenseurq\delta\tenseurq M_r=\tenseurq M_r-\tenseurq M_0$, $\tenseurq M_0=\tenseurq L_0^{-1}$, and $\tenseurq \Delta_0=\tenseurq L_0-\tenseurq L_0\dbldot\tenseurq \Gamma_0\tenseurq L_0$. We will not resolve this equation exactly, but we will consider the average of the fields: +\begin{aligned} +\langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s +\end{aligned} +where +\begin{aligned} +\tenseurq \Delta_{rs} = \sum_j\langle\,\tenseurq\Delta_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r +\end{aligned} +is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor makes intervene the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). +Hence, the $\langle\tsigma\rangle_r$ 's are solution of a linear system. +Depending on the number of phases $N$, the computation of the $\tenseurq \Delta_{rs}$ is more or less costly, but it is achieved +at the beginning, once and for all. + +We note that the reference medium can be adjusted with a scalar coefficient. Indeed, if the user wants to choose +a reference medium $r_0=10$ times stiffer, it is possible by dividing $\tenseurq M_0$ by $r_0$. It will be also necessary +to change $\tenseurq \Delta_{rs}$ and to multiply it by $r_0$. + +### Computation via a self-consistent scheme + +The `tfel::material::homogenization` will carry out the homogenization +with the self consistent scheme. The library provides the homogenized stiffness +$\tenseurq L_{\mathrm{SC}}$ needed for the computation of the macroscopic stress, +but also the effective polarization $\tenseurq \tau_{\mathrm{SC}}$ which permit +to compute the macroscopic stress: + +\begin{aligned} +\overline{\tsigma} = \tenseurq L_{\mathrm{SC}}\dbldot\overline{\tepsilon}+\tenseur \tau_{\mathrm{SC}} +\end{aligned} + +However, we also need the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ for computing the average of +the stress field $\langle\tsigma\rangle_r$ (i.e. the residue). To that extent, we go back to the equation +\begin{aligned} +\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\end{aligned} +and consider averages of the fields: +\begin{aligned} +\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s +\end{aligned} +with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r$. +A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ (i.e. the grains have a spherical shape) so that +\begin{aligned} +\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\tenseurq P_0\dbldot\left(\langle\tenseur\tau\rangle_r-\langle\tenseur\tau\rangle\right) +\end{aligned} +and due to the expression of $\tenseur\tau$: +\begin{aligned} +\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\overline{\tepsilon}\right)=-\left[\langle\tsigma\rangle_r-\overline{\tsigma}\right] +\end{aligned} +We know that the reference medium, in the self-consistent approach, is identified with the homogenized medium. Hence, we note +$\tenseurq P_0=\tenseurq P_{\mathrm{SC}}$, $\tenseurq L_0=\tenseurq L_{\mathrm{SC}}$, and $\tenseurq M_{\mathrm{SC}}^*=\left[\tenseurq P_{\mathrm{SC}}^{-1}-\tenseurq L_{\mathrm{SC}}\right]^{-1}$, so that, with $\langle\tepsilon\rangle_r=\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r$, and $\langle\tsigma\rangle_r=\tsigma^r$ +\begin{aligned} +\left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\overline{\tepsilon}\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tsigma^r-\overline{\tsigma}\right] +\end{aligned} +which gives +\begin{aligned} +\tsigma^r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left[\overline{\tepsilon}-\tenseur e^r+\tenseurq M_{\mathrm{SC}}^*\dbldot\overline{\tsigma}\right] +\end{aligned} +which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s$ (we use $\overline{\tsigma}=\tenseurq L_{\mathrm{SC}}\dbldot\overline{\tepsilon}+\tenseur \tau_{\mathrm{SC}}$): +\begin{aligned} +\tenseurq A_r&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_{\mathrm{SC}}\right)\\ +\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseur \tau_{\mathrm{SC}}-\tenseur e^r\right) +\end{aligned} +(and this is sufficient to compute the residue). However, we note that the library `tfel::material::homogenization` provides +the tensor $\tenseurq A_{\mathrm{SC}}$ via the variable `mean_strain_localisation_tensors`, which relates +$\langle\tepsilon\rangle_r$ and $\overline{\tepsilon}$. This tensor is also equal to $\tenseurq A_{\mathrm{SC}}=\tenseurq M_r\dbldot\tenseurq A_r$, due to the expression of $\langle\tsigma\rangle_r$. This will be more simple for the implementation. + + +# Implementation in MFront + +All the files are available in the `MFrontGallery` project, [here](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization/), under the name `MassonAffineFormulation`. + +## Example used for the implementation + + The geometry of our polycrystal is generated with [`merope`](https://github.com/MarcJos/Merope/), + with a Voronoi tesselation. + This geometry is saved as an array with an integer representing the number of the phase (or crystal). + The image below shows an example of such a polycrystal. There are here 10 crystals. The orientations + and volume fractions are given in the file `polycrystal.csv`. The geometry is used to + compute the morphological tensors (see the `python` script). + + ![Geometry of the polycrystal](./img/polycrystal.png){width=50%} + + The potential $\psi_k^r$ will be of the form + \begin{aligned} + \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^{k,r}}{n+1}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n+1} + \end{aligned} + where the strain rate $\dot{\gamma}_0$ and the creep exponent $n\geq 1$ will be chosen identical for all $k,r$, and the resolved shear stress $\tau_0^{k,r}$ depends on $k,r$. + +The derivatives of the potential are + \begin{aligned} + &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r\\ + &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r + \end{aligned} + and then defining $\overline{\tau}_k^r=\tsigma^r\dbldot\tenseur \mu_k^r$ where $\tsigma^r$ is the reference stress, we have + \begin{aligned} + &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r\\ + &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r + \end{aligned} + +## Details of implementation + +### Headers + +For the jacobian, we adopt the `Numerical Jacobian`, so that +the beginning of the `mfront` file reads: + +~~~~ {#Begin .cpp .numberLines} +@DSL ImplicitII; +@Behaviour Affine_formulation; +@Author Martin Antoine; +@Date 24 / 01 / 26; +@Description{"Affine formulation for homogenization of a viscoplastic polycrystal, Masson et al. 2001."}; +@UseQt false; +@Algorithm NewtonRaphson_NumericalJacobian; +@PerturbationValueForNumericalJacobianComputation 1e-10; +@Epsilon 1e-14; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Moreover, we need to include header files, those +for the homogenization of the thermoelastic composite: + + - IsotropicEshelbyTensor.hxx + - MicrostructureDescription.hxx + - MicrostructureLinearHomogenization.hxx + +and the header for handling the rotations of the grains with their +slip systems: + + - ExtendedPolyCrystalsSlidingSystems.hxx + +which is provided as an external file + +~~~~ {#Begin .cpp .numberLines} +@TFELLibraries {"Material"}; +@Includes{ +#include "TFEL/Material/IsotropicModuli.hxx" +#include "TFEL/Material/IsotropicEshelbyTensor.hxx" +#include "TFEL/Material/MicrostructureDescription.hxx" +#include "TFEL/Material/MicrostructureLinearHomogenization.hxx" +#include "../extra-headers/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For the approach based on morphological tensors, we do not need all these headers, +but we need to include a file which contains the morphological tensors. +This file is present in a repository `extra-headers/TFEL/Material/` +located at the same place as the `.mfront` file. + +~~~~ {#Begin .cpp .numberLines} +@TFELLibraries {"Material"}; +@Includes{ +#include "../extra-headers/TFEL/Material/tensors.hxx" +#include "../extra-headers/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We will also generate a polycrystal and compute the tensors via FFT (see the `python` script [here](./downloads/Morphological_tensors.zip)). + +### Sliding systems + +The sliding systems will be generated using these lines (the same +procedure as for the tutorial on the [Berveiller-Zaoui](ExplicitBerveillerZaouiPolyCrystals.html) scheme): + +~~~~ {#Begin .cpp .numberLines} +@ModellingHypothesis Tridimensional; +@OrthotropicBehaviour; +@CrystalStructure HCP; +@SlidingSystems{<1, 1, -2, 0>{1, -1, 0, 0}, + <-2, 1, 1, 3>{1, -1, 0, 1}, + <-2, 1, 1, 0>{0, 0, 0, 1}, + <1, 1, -2, 0>{1, -1, 0, 1}}; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +but the rotations of the grains will be performed later, in the `Integrator` +code block: + +~~~~ {#Integrator .cpp .numberLines} +using PolyCrystalsSlidingSystems = + ExtendedPolyCrystalsSlidingSystems, real>; +const auto& gs = + PolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems("polycrystal.csv"); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +### Poperties and variables + +The state variables are the reference stresses $\tsigma^r$ on each phase. + +~~~~ {#Begin .cpp .numberLines} +@StateVariable Stensor sigma[Np]; +sigma.setEntryName("PhaseReferenceStress"); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Note also that `Np` is the number of phases (or crystals). We define it +before like that: + +~~~~ {#Begin .cpp .numberLines} +@IntegerConstant Np = 10; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Note that among several local variables, one of them, for +morphological tensors, is + +~~~~ {#Begin .cpp .numberLines} +@LocalVariable std::array,Np>,Np> DELTA; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This matrix will store the morphological tensors. Indeed, in the +`InitLocalVariables` code block, we can load these coefficients, +which are in fact situated in the header `` + +~~~~ {#Begin .cpp .numberLines} +DELTA=Delta::get_tensor(); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +### Affine modulus and free-strain + +There is no difficulty in computing the tangent modulus and free-strain, +in the `Integrator` code block: + +~~~~ {#Integrator .cpp .numberLines} +using namespace tfel::math; +Stensor4 LSC; +Stensor e[Np]; +Stensor dpsi_dsigma[Np]; + +for (int r=0;r(k0,mu0); +auto micro=ParticulateMicrostructure<3u,stress>(IM0); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Note that here the matrix phase is defined by the elasticity `(k0,mu0)` +but this elasticity will play no role in the sequel. +Then, we will add some `SphereDistribution` objects, +each of them representing a grain with its shape (`Sphere`), volume fraction +and elasticity: + +~~~~ {#Integrator .cpp .numberLines} +const auto sphere=Sphere(); +std::vector polarisations; +const auto zero = Stensor::zero(); +polarisations.resize(Np+1, zero); + +for (int r=0 ; r(sphere,frac[r],L[r]); + micro.addInclusionPhase(distribution); + polarisations[r+1]=-L[r]*e[r]; +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Because the sum of the volume fractions are one, it means that the matrix +phase will have a volume fraction equal to zero. This is not a problem, +but do not forget that this phase still exist in the `ParticulateMicrostructure` +object (see below). + +The self-consistent scheme is performed afterwards: + +~~~~ {#Integrator .cpp .numberLines} +const auto hmSC=computeSelfConsistent<3u,stress>(micro,1e-6,true,0,polarisations); +LSC = hmSC.homogenized_stiffness; +const auto tauSC = hmSC.effective_polarisation; +const auto A_SC=hmSC.mean_strain_localisation_tensors; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Note that we want here a tolerance of `1e-6` and we consider an isotropic +homogenized stiffness in the iterative algorithm with `true`. + +We will need the following tensors: + +~~~~ {#Integrator .cpp .numberLines} +auto KGSC = computeKGModuli(LSC); +const auto PSC = computeSphereHillPolarisationTensor(KGSC); +const auto MSC_star = invert(invert(PSC)- LSC); +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Note here that we isotropize the homogenized stiffness with the function +`computeKGModuli`, on line 1. This is more consistent with the fact that we used `true` +in the `computeSelfConsistent` function. + +### Residues and jacobian + +For the residues, we work on the strains (because `MFront` is more +suited to strain type residues. Hence, we will define the residues as + +\begin{aligned} +f_{\sigma^r}=\tenseurq M_r\dbldot\left(\tsigma^r-\tenseurq A_r\dbldot\overline{\tepsilon}-\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s\right) +\end{aligned} + +~~~~ {#Integrator .cpp .numberLines} +for (int r=0;r(A,6*r,0)=Ar; + auto KGAr = computeKGModuli(Ar); + fsigma[r] = M[r]*(sigma[r]+dsigma[r])-Ar*(eto+deto); + fsigma[r]-= M[r]*(invert(M[r]+MSC_star))*(MSC_star*tauSC - e[r]); +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +There is no jacobian because we use the `NumericalJacobian`. Here, +an analytical jacobian would have been tedious, although not impossible. + +The computation of the macroscopic stress is simple: + +~~~~ {#Integrator .cpp .numberLines} +sig=LSC*(eto+deto)+tauSC; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + + +### Tangent operator + +The computation of the tangent operator necessitates the computation of $\deriv{\tsigma^r}{\overline{\tepsilon}}$. +To do that, we use a technique widely used in `MFront`. +The idea is to write the non-linear system to solve as: +\begin{aligned} +\tenseurq M_r\dbldot\left[\tsigma^r - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)\right]= f_{\sigma^r}(\overline{\tepsilon},\tsigma^1,...,\tsigma^N)=0 +\end{aligned} +and by derivation of $f_{\sigma^r}$ w.r.t. $\overline{\tepsilon}$: +\begin{aligned} +-\tenseurq M_r\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\overline{\tepsilon}}=0 +\end{aligned} +where $\tenseurq J_{rs}=\deriv{f_{\sigma^r}}{\tsigma^s}$ stands for the sub-block $(r,s)$ of the Jacobian. Hence +\begin{aligned} +\deriv{\tsigma^r}{\overline{\tepsilon}}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq M_s\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) +\end{aligned} +where $\mathbf {iJ}$ is the inverse of the Jacobian. The implementation is + +~~~~ {#tangent .cpp .numberLines} +@TangentOperator{ + tmatrix<6*Np,6*Np,real> iJ = 0.001*ID+jacobian; + TinyMatrixInvert<6*Np,real>::exe(iJ); + dsigma_deto=iJ*A; + Dt=Stensor4::zero(); + for (int r=0;r(dsigma_deto,6*r,0); + Dt+=frac[r]*dsigmar_deto; + } +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Note the use of the matrix `ID` here, that was initialized at the beginning +as a local variable: + +~~~~ {#tangent .cpp .numberLines} +@LocalVariable tmatrix<6*Np,6*Np,real> ID; +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +with, in the `InitLocalVariables` code block: + +~~~~ {#tangent .cpp .numberLines} +for (int r=0;r(ID,6*r,6*r) =I; +} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This is because the jacobian is null + +## Results + +For the tests, we use `mtest`. However, note that the strain we impose +is in fact a strain rate. The idea for us is to keep this strain rate +constant and to make vary the parameter $n$ with time. +Besides, we choose two different values for $\tau_0^r$. +5 crystals are such that $\tau_0=10 MPa$, and 5 others +such that $\tau_0=50 MPa$. + +The `.mtest` file is the following: + +~~~~ {#Affine_formulation .mtest .numberLines} +@ModellingHypothesis 'Tridimensional'; +@Behaviour 'src/libBehaviour.so' 'Affine_tensors'; +@ExternalStateVariable 'Temperature' {0 : 1000,1:1000}; + +@MaterialProperty 'nexp' "1+0.9*(exp(2.4*t)-1)"; +@MaterialProperty 'tau1' 10.; +@MaterialProperty 'tau2' 50.; + +@MaterialProperty 'kap' "0.001"; + +@ImposedStrain 'EXX' {0 : 1, 1 : 1}; +@ImposedStrain 'EYY' {0 : -1, 1 : -1}; +@ImposedStrain 'EZZ' {0 : 0, 1 : 0}; +@ImposedStrain 'EXY' {0 : 0, 1 : 0}; +@ImposedStrain 'EXZ' {0 : 0, 1 : 0}; +@ImposedStrain 'EYZ' {0 : 0, 1 : 0}; +@Times {0.,1 in 50}; + +@OutputFilePrecision 14; +~~~~~~~~~~~~~~ + +For the morphological tensor approach, we talked above +about the coefficient $r_0$ that allows to adjust the reference medium. +This coefficient can hence be set as a material property, and we will add +this line in the `mtest` file: + +~~~~ {#Affine_formulation .mtest .numberLines} +@MaterialProperty 'r0' "1+4*t"; +~~~~~~~~~~~~~~ + +Here, we choose to make vary the reference medium because when `t` +increases, `n` increases, and hence the homogenized stiffness increases. + +![Effective resolved shear stress as a function of the parameter $n$](./img/Affine_formulation.png) + + +In this figure we plotted the results when all the $\tau_0^r$ are equal to $10 Mpa$ +(to show the heterogeneity) ("FFT 0"), the implementation of the affine approach +with self-consistent scheme (AFF_SC), and with morphological tensors (AFF_MT). +FFT results are also plotted (FFT). + + diff --git a/docs/web/PonteCastaneda1996.md b/docs/web/PonteCastaneda1996.md deleted file mode 100644 index 6d23fc5e7..000000000 --- a/docs/web/PonteCastaneda1996.md +++ /dev/null @@ -1,634 +0,0 @@ ---- -title: Ponte-Castaneda second-order estimates for homogenization of a viscoplastic polycrystal -author: Antoine Martin -date: 15/01/2026 -lang: en-EN -link-citations: true -colorlinks: true -geometry: - - margin=2cm -papersize: a4 -figPrefixTemplate: "$$i$$" -tblPrefixTemplate: "$$i$$" -secPrefixTemplate: "$$i$$" -eqnPrefixTemplate: "($$i$$)" ---- - -\newcommand{\tenseur}[1]{\underline{#1}} -\newcommand{\tenseurq}[1]{\underline{\mathbf{#1}}} -\newcommand{\tenseurt}[1]{\underline{\underline{\underline{#1}}}} -\newcommand{\tns}[1]{\underset{\tilde{}}{\mathbf{#1}}} -\newcommand{\transpose}[1]{#1^{\mathop{T}}} -\newcommand{\Ka}{\mathcal K} -\newcommand{\Sa}{\mathcal S} -\newcommand{\tsigma}{\tenseur{\sigma}} -\newcommand{\dbldot}{:} -\newcommand{\tepsilon}{\tenseur{\varepsilon}} -\newcommand{\te}{\tenseurq{e}} -\newcommand{\sigmaeq}{\sigma_{\mathrm{eq}}} -\newcommand{\epsiloneq}{\varepsilon_{\mathrm{eq}}} - -\newcommand{\tepsilonto}{\underline{\epsilon}^{\mathrm{to}}} -\newcommand{\tepsilonel}{\underline{\epsilon}^{\mathrm{el}}} - -\newcommand{\trace}[1]{\mathrm{tr}\paren{#1}} -\newcommand{\Frac}[2]{{\displaystyle \frac{\displaystyle #1}{\displaystyle #2}}} -\newcommand{\deriv}[2]{{\displaystyle \frac{\displaystyle \partial #1}{\displaystyle \partial #2}}} -\newcommand{\derivdeux}[2]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2^2}}} -\newcommand{\pderiv}[2]{{\dfrac{\partial #1}{\partial #2}}} -\newcommand{\pderivdeux}[2]{{\dfrac{\partial^2 #1}{\partial #2^2}}} -\newcommand{\derivderiv}[3]{{\displaystyle \frac{\displaystyle \partial^2 #1}{\displaystyle \partial #2\,\displaystyle \partial #3}}} -\newcommand{\dtot}{{\mathrm{d}}} -\newcommand{\paren}[1]{\left(#1\right)} -\newcommand{\nom}[1]{\textsc{#1}} -\renewcommand{\div}{\mathrm{div}} - - -We present here an implementation of Ponte-Castaneda second-order estimates [@castaneda_exact_1996] -for the homogenization of a viscoplastic polycrystal, example also treated in [@bornert_second-order_2001]. -The procedure can also be used in non-linear elasticity, providing that the behaviour is governed -by only one potential. - -Here, the idea is to show that many implementations of that procedure are possible in `MFront`. -We particularly present the possibility to take into account the morphology of the polycrystal -by using interaction tensors, which are provided before the integration. It results in better -estimates than classical ones. Note that the resulting non-linear system involved -couples the unknowns, and hence increases the cost of the integration. -We also show that we can use a self-consistent scheme, making use of -`tfel::material::homogenization`. - -This tutorial first presents the homogenization problem, recalls Ponte-Castaneda's methodology -from [@castaneda_exact_1996], presents different kinds of possible implementations, -and show the details of the `mfront` file. - -# The viscoplastic polycrystal - -## Behaviour - -We consider a polycrystalline material, which means that each phase $r$ is associated to a crystal with -corresponding slip systems $\tenseur \mu_k^r$ ($1\leq k\leq K$). The strain rate in each crystal $r$ is given by - \begin{aligned} - \dot{\tepsilon}=\sum_k \dot{\gamma}_k^r\, \tenseur \mu_k^r\qquad\text{with}\quad \tenseur\mu_k^r = \dfrac12\left(\tenseur n_k^r\otimes\tenseur m_k^r + \tenseur m_k^r\otimes\tenseur n_k^r\right) - \end{aligned} - where $\dot{\gamma}_k^r$ is the shear strain rate on the $k^{th}$ slip system of crystal $r$, and is given as a - function of the shear stress $\tau_k^r=\tsigma\dbldot\tenseur \mu_k^r$ by means of a potential $\psi_k^r$: - \begin{aligned} - \dot{\gamma}_k^r= \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right) - \end{aligned} - The expressions above show that on phase (or crystal) $r$: - \begin{aligned} - \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\text{with}\quad\psi_r \left(\tsigma\right)=\sum_k \psi_k^r\left(\tau_k^r\right) - \end{aligned} - - Now, the behaviour of the polycrystal is governed by a potential $\psi$: - \begin{aligned} - \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\psi(\tsigma)= \sum_{r=1}^{N}\chi_r\,\psi_r (\tsigma) - \end{aligned} -where $N$ is the number of phases (or crystals) and $\chi_r$ is characteristic function of phase $r$. - - In all the sequel, we just note $\tepsilon$ for $\dot{\tepsilon}$. - -## The non-linear homogenization problem - -We impose a strain rate $\overline{\tepsilon}$ to the polycrystal and look for the solution $\tepsilon,\tsigma$ such that: - -\begin{aligned} - &\div\,\tsigma=\tenseur 0\\ - &\tepsilon\in\Ka(\overline{\tepsilon})\\ - &\tepsilon = \deriv{\psi}{\tsigma}\left(\tsigma\right) -\end{aligned} - -where we introduced the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$, depending on the boundary conditions used (in the implementation, we work with periodic boundary conditions). -The effective energy of the composite is then defined as the following average on the -entire Representative Volume Element (RVE): -\begin{aligned} -\overline{\phi}\left(\overline{\tepsilon}\right)= \langle\phi\left(\tepsilon\right)\rangle -\end{aligned} -where $\tepsilon$ is the solution of the above problem, $\langle.\rangle$ denotes the average -on the RVE, and $\phi$ is the primal potential, -defined as the Legendre-Fenchel transform of $\psi$: -\begin{aligned} -\phi\left(\tepsilon\right)=\underset{\tsigma}{\mathrm{max}}\left\{\tsigma\dbldot\tepsilon-\psi\left(\tsigma\right)\right\} -\end{aligned} - -# Ponte-Castaneda second-order estimates - -## The variational approach - - The homogenization problem is equivalent to a maximization problem: - \begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tsigma\in\Sa}{\textrm{max}} \left\{ \langle\tsigma\rangle\dbldot\overline{\tepsilon} - \langle\psi(\tsigma)\rangle\right\} - \end{aligned} - where $\Sa$ is the set of divergence-free fields. - - Note that a dual minimization problem can also be solved to find the effective energy -\begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tepsilon\in\mathcal K(\overline{\tepsilon})}{\textrm{min}} \langle \phi(\tepsilon)\rangle - \end{aligned} - However, because $\phi$ is known only through the Legendre-Fenchel transform of $\psi$, it will be more - easy to use the maximization problem above. - -## The second-order estimates - -Ponte-Castaneda's idea is to make a second-order Taylor development of each $\psi_r$ around a given "reference" stress $\tsigma^r$ in phase $r$: -\begin{aligned} - \psi_r(\tsigma)\approx\psi_r(\tsigma^r)+\tepsilon_r(\tsigma^r)\dbldot\left(\tsigma-\tsigma^r\right)+\dfrac12\left(\tsigma-\tsigma^r\right)\dbldot\tenseurq M_r\left(\tsigma^r\right)\dbldot\left(\tsigma-\tsigma^r\right) - \end{aligned} - where - \begin{aligned} - \tepsilon_r(\tsigma^r)=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseurq M_r\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right) - \end{aligned} - give a "tangent" estimation of the strain $\tepsilon_r$ around $\tsigma^r$: - \begin{aligned} - \tepsilon_r(\tsigma)\approx\tepsilon_r(\tsigma^r)+\tenseurq M_r\left(\tsigma^r\right)\dbldot\left(\tsigma-\tsigma^r\right) - \end{aligned} - and can be viewed as the behaviour of a so-called "thermoelastic comparison composite". We hence obtain - \begin{aligned} - \psi_r(\tsigma)\approx v_r\left(\tsigma^r\right)+\tenseur e^r\left(\tsigma^r\right)\dbldot\tsigma+\dfrac12\tsigma\dbldot\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma - \end{aligned} - with - \begin{aligned} - &v_r\left(\tsigma^r\right)=\psi_r(\tsigma^r)-\tepsilon_r(\tsigma^r)\dbldot\tsigma^r+\dfrac12\tsigma^r\dbldot\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r\\ - &\tenseur e^r\left(\tsigma^r\right)=\tepsilon_r(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r - \end{aligned} - - Afterwards, the effective energy is found by the maximization problem: - \begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)=\underset{\tsigma\in\Sa}{\textrm{max}} \sum_r c_r\,\left[\langle\tsigma\rangle_r\dbldot\left(\overline{\tepsilon}-\tenseur e^r\right)-v_r-\dfrac12\langle\tsigma\dbldot\tenseurq M_r\dbldot\tsigma\rangle_r\right] - \end{aligned} - where $c_r$ is the volume fraction of the phase $r$, and $\langle.\rangle_r$ denotes the average on phase $r$. - The effective energy is hence given by - \begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)\approx P\left(\overline{\tepsilon},\tsigma^r\right) -\sum_r c_r\,v_r\qquad\text{with}\quad P\left(\overline{\tepsilon},\tsigma^r\right)=\underset{\tsigma\in\Sa}{\textrm{max}}\sum_r c_r\, \langle \tenseur d^r\dbldot\tsigma-\dfrac12\tsigma\dbldot\tenseurq M_r\dbldot\tsigma\rangle_r - \end{aligned} - where $\tenseur d^r=\overline{\tepsilon}-\tenseur e^r$. Of course, the term $P\left(\overline{\tepsilon},\tsigma^r\right)$ in previous expression corresponds to the effective energy of a linear thermoelastic composite. The maximization leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by - \begin{aligned} - \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) - \end{aligned} - where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). - -The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@castaneda_exact_1996] leads to the simple assumption that these reference stresses are the averages of the stress in the thermoelastic composite: -\begin{aligned} -\tsigma^r = \langle\tsigma\rangle_r -\end{aligned} - where $\tsigma$ here stands for the stress solution of the thermoelastic problem mentioned above. - -## Macroscopic stress and tangent operator - -The above maximization on the set of statically admissible (divergence-free) fields leads to the condition that the field $\tenseur d^r-\tenseurq M_r\dbldot\tsigma$ belongs to $\Ka\left(\tenseur 0\right)$. Hence, by multiplying this field by $\tsigma$, which is divergence-free, and integrating over the entire volume, we obtain (apply Hill's lemma): -\begin{aligned} -\sum_r c_r\,\langle\left(\tenseur d^r-\tenseurq M_r\dbldot\tsigma\right)\dbldot\tsigma\rangle_r=0 -\end{aligned} -which leads to -\begin{aligned} -\sum_r c_r\, \langle \tenseur d^r\dbldot\tsigma-\dfrac12\tsigma\dbldot\tenseurq M_r\dbldot\tsigma\rangle_r=\dfrac12\sum_r c_r\, \tenseur d^r\dbldot\langle\tsigma\rangle_r=\dfrac12\sum_r c_r\, \tenseur d^r\dbldot\tsigma^r -\end{aligned} - -Hence, using the expressions of $v_r$, $\tenseur e^r$ and $\tenseur d^r$ we obtain (see also expression (B.7) in [@bornert_second-order_2001]) -\begin{aligned} - \overline{\phi}\left(\overline{\tepsilon}\right)\approx\sum_r c_r\,\left[\dfrac12\tsigma^r\dbldot\left(\tepsilon_r+\overline{\tepsilon}\right)-\psi_r\left(\tsigma^r\right)\right] - \end{aligned} - The macroscopic stress hence can be obtained by derivation w.r.t. $\overline{\tepsilon}$: - \begin{aligned} -\overline{\tsigma}\approx\dfrac12\sum_r c_r\,\left[\tsigma^r+\left(\overline{\tepsilon}-\tepsilon_r+\tsigma^r\dbldot\tenseurq M_r\right)\dbldot\deriv{\tsigma^r}{\overline{\tepsilon}}\right] - \end{aligned} - - The tangent operator is obtained by derivation: - \begin{aligned} - \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}\approx\sum_r c_r\,\left\{\deriv{\tsigma^r}{\overline{\tepsilon}}+\dfrac12\left[\left(\deriv{\tsigma^r}{\overline{\tepsilon}}\right)^T\dbldot\left(\tsigma^r\dbldot\tenseurt O_r\right)\dbldot\deriv{\tsigma^r}{\overline{\tepsilon}}+\left(\overline{\tepsilon}-\tepsilon_r+\tsigma^r\dbldot\tenseurq M_r\right)\dbldot\derivdeux{\tsigma^r}{\overline{\tepsilon}}\right]\right\} - \end{aligned} - where $\tenseurt O_r=\deriv{\tenseurq M_r}{\tsigma^r}$. - -In the expressions of the macroscopic stress and tangent operator, we can compute the term $\deriv{\tsigma^r}{\overline{\tepsilon}}$ by means of the jacobian matrix, also done and explained in the documentation of the [Implicit DSL](https://thelfer.github.io/tfel/web/implicit-dsl.html#computation-of-the-consistent-tangent-operator). See below for more details. - -# Summary and possible implementations - -## Summary - -The resolution consists in -\[ - \text{Find}\quad\tsigma^r\quad\text{such that}\quad\tsigma^r=\langle\tsigma\rangle_r -\] -where \(\tsigma\) is the stress field solution of a homogenization problem of the form: -\begin{aligned} -&\div\,\tsigma=\tenseur 0\\ -&\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\overline{\tepsilon}\right)\\ -&\tenseurq M_r=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r -\end{aligned} - - -## Possible implementations - -The iterative resolution of the non-linear system can be summarized as - -\[ -\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq M_r= \pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right),\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\tsigma^r = \tenseurq A_r\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} -\] - -The first step of the resolution consists in computing the moduli $\tenseurq M_r$ and free strains $\tenseur e^r$. It can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the local behaviour in an external file (in our example, the potential is to simple to do that). - -The second step is the computation of the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$. This can be achieved by means of mean-field schemes (the `namespace` `tfel::material::homogenization` provides some localisation tensors). Another approach is to use morphological tensors. The two approaches are shown below in the next subsection. - -We note that the equation relative to the second step is the non-linear equation to solve and will hence correspond to our residue. -The macroscopic stress and the tangent operator must also be computed with the formulas shown above. - -## Computation of tensors Ar and Brs - -### Computation via morphological tensors - -In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\overline{\tepsilon}\right)$, we have, because, $\div\,\tsigma=0$, -\begin{aligned} -\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) -\end{aligned} -where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives -\begin{aligned} -\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau -\end{aligned} -and hence (due to the expressions of $\tenseur\tau$ and $\tepsilon$) -\begin{aligned} -\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) -\end{aligned} - -\begin{aligned} -\tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq \Delta_0\left(\tenseur e^r\right) -\end{aligned} -with $\tenseurq\delta\tenseurq M_r=\tenseurq M_r-\tenseurq M_0$, $\tenseurq M_0=\tenseurq L_0^{-1}$, and $\tenseurq \Delta_0=\tenseurq L_0-\tenseurq L_0\dbldot\tenseurq \Gamma_0\tenseurq L_0$. We will not resolve this equation exactly, but we consider its variational formulation and consider an approximation of the stress field, piece-wise uniform. It is exactly equivalent to apply the dual version of Hashin-Shtrikman principle. -\begin{aligned} -\langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s -\end{aligned} -where -\begin{aligned} -\tenseurq \Delta_{rs} = \sum_j\langle\,\tenseurq\Delta_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r -\end{aligned} -is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor makes intervene the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). -Hence, the $\langle\tsigma\rangle_r$ 's are solution of the following linear system: -\begin{aligned} -\sum_{s=1}^N\,\left[\delta_{rs}\,\tenseurq I + \tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\right]\dbldot\langle\tsigma\rangle_s=\tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s -\end{aligned} -where $\delta_{rs}$ is Krönecker symbol and $\tenseurq I$ is the fourth-order identity tensor. -Depending on the number of phases $N$, the solution of this system is more or less costly. The solution directly gives -the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ (compare with the expressions above). - -### Computation via a self-consistent scheme - -We restart here with the equation -\begin{aligned} -\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) -\end{aligned} -which can be discretized (considering again piece-wise uniforms fields and a variational framework) as -\begin{aligned} -\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s -\end{aligned} -with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r$. -A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ so that -\begin{aligned} -\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\tenseurq P_0\dbldot\langle\tenseur\tau\rangle_r+\tenseurq P_0\dbldot\langle\tenseur\tau\rangle -\end{aligned} -and due to the expression of $\tenseur\tau$: -\begin{aligned} -\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\overline{\tepsilon}\right)=-\left[\langle\tsigma\rangle_r-\langle\tsigma\rangle\right] -\end{aligned} -and noting $\tenseurq M_0^*=\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]^{-1}$, with $\tepsilon=\tenseurq M\dbldot\tsigma+\tenseur e$, and $\langle\tsigma\rangle_r=\tsigma^r$ -\begin{aligned} -\left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\overline{\tepsilon}\right)=-\tenseurq M_0^*\dbldot\left[\tsigma^r-\langle\tsigma\rangle\right] -\end{aligned} -which is -\begin{aligned} -\tsigma^r=\left(\tenseurq M_r+\tenseurq M_0^*\right)^{-1}\dbldot\left[\overline{\tepsilon}-\tenseur e^r+\tenseurq M_0^*\dbldot\langle\tsigma\rangle\right] -\end{aligned} -and by taking the average -\begin{aligned} -\langle\tsigma\rangle=\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\left[\overline{\tepsilon}+\tenseurq M_0^*\dbldot\langle\tsigma\rangle\right]-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\dbldot\tenseur e\rangle -\end{aligned} -and then -\begin{aligned} -\langle\tsigma\rangle=\left[\tenseurq I-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\tenseurq M_0^*\right]^{-1}\dbldot\left[\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\overline{\tepsilon}-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\dbldot\tenseur e\rangle\right] -\end{aligned} -Now, the self-consistent approach consists in choosing $\tenseurq M_0$ such that -\begin{aligned} -\tenseurq M_0=\left\{\left[\tenseurq I-\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\dbldot\tenseurq M_0^*\right]^{-1}\dbldot\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle\right\}^{-1} -\end{aligned} -This is also -\begin{aligned} -\tenseurq M_0=\langle\left(\tenseurq M+\tenseurq M_0^*\right)^{-1}\rangle^{-1}-\tenseurq M_0^* -\end{aligned} -This equation can be solved by an iterative procedure, and this is exactly what does the function `computeSelfConsistent` -from the library `tfel::material::homogenization`. -Afterwards, noting $\tenseurq M_0=\tenseurq M_{\mathrm{SC}}$, we have -\begin{aligned} -\langle\tsigma\rangle=\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\overline{\tepsilon}-\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\left[\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\rangle\right]^{-1}\dbldot\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\tenseur e\rangle -\end{aligned} -which is, because $\tenseurq M_{\mathrm{SC}}+\tenseurq M_{\mathrm{SC}}^*=\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\rangle^{-1}$, -\begin{aligned} -\langle\tsigma\rangle=\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\overline{\tepsilon}-\left[\tenseurq I+\tenseurq M_{\mathrm{SC}}^{-1}\dbldot\tenseurq M_{\mathrm{SC}}^*\right]\dbldot\langle\left(\tenseurq M+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\tenseur e\rangle -\end{aligned} -and then -\begin{aligned} -&\tenseurq A_r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq M_{\mathrm{SC}}^{-1}\right)\\ -&\tenseurq B_{rs}=-\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\delta_{rs}\tenseurq I +\tenseurq M_{\mathrm{SC}}^*\dbldot{}^t\tenseurq A_s\right) -\end{aligned} -In fact, because $\langle\tepsilon\rangle_r = \tenseurq M_r\dbldot\tsigma^r + \tenseur e^r$, we have also -\begin{aligned} -\langle\tepsilon\rangle_r = \mathcal A_r\dbldot\overline{\tepsilon}+ \tenseur e^r +\sum_s \tenseurq M_r\dbldot\tenseurq B_{rs}\dbldot\tenseur e^s -\end{aligned} -where $\mathcal A_r=\tenseurq M_r\dbldot\tenseurq A_r$ is the strain localisation tensor, -also provided by the function `computeSelfConsistent` (see below for details). - -# Implementation in MFront - -All the files are available in the [`MFrontGallery`](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization) project. We also provide in [this archive](./downloads/PonteCastaneda1996.zip) -a `python` script to generate the morphological tensors. - -ARCHIVE AND MFRONTGALLERY !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! - -## Example used for the implementation - - The geometry of our polycrystal is generated with [`merope`](https://github.com/MarcJos/Merope/), - with a Voronoi tesselation. - This geometry is saved as an array with an integer representing the number of the phase (or crystal). - The image below shows an example of such a polycrystal. There are here 10 crystals. The orientations - and volume fractions are given in the file `polycrystal.csv`. The geometry is used to - compute the morphological tensors (see the `python` script). - - ![Geometry of the polycrystal](./img/polycrystal.png){width=50%} - - The potential $\psi_k^r$ will be taken of the form - \begin{aligned} - \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^{k,r}}{n+1}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n+1} - \end{aligned} - where the strain rate $\dot{\gamma}_0$ and the creep exponent $n\geq 1$ will be chosen identical for all $k,r$, and the resolved shear stress $\tau_0^{k,r}$ depends on $k,r$. - -The derivatives of the potential are - \begin{aligned} - &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r\\ - &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r - \end{aligned} - and then defining $\overline{\tau}_k^r=\tsigma^r\dbldot\tenseur \mu_k^r$ where $\tsigma^r$ is the reference stress, we have - \begin{aligned} - &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r\\ - &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r - \end{aligned} - -## Details of implementation - -### Headers - -For the jacobian, we adopt the `Numerical Jacobian`, so that -the beginning of the `mfront` file reads: - -~~~~ {#Begin .cpp .numberLines} -@DSL ImplicitII; -@Behaviour PC_SOE_96; -@Author Martin Antoine; -@Date 26 / 01 / 26; -@Description{"Ponte Castaneda second-order estimates for homogenization of a viscoplastic polycrystal, based on Green interaction tensors"}; -@UseQt false; -@Algorithm NewtonRaphson_NumericalJacobian; -@PerturbationValueForNumericalJacobianComputation 1e-10; -@Epsilon 1e-12; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Moreover, we need to include a header file which contains our interaction tensors. -This file must be present in a repository `extra-headers/TFEL/Material/` -located at the same place as the `.mfront` file. In our implementation, -we generate a polycrystal and compute the tensors via FFT (see the `python` script). - -We will also need to handle the slip systems, and hence include `ExtendedPolyCrystalsSlidingSystems.hxx`. -This header is supposed to be in the same repository as `tensors.hxx`. - -~~~~ {#Begin .cpp .numberLines} -@TFELLibraries {"Material"}; -@Includes{ -#include "../extra-headers/TFEL/Material/tensors.hxx" -#include "../extra-headers/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -### Sliding systems - -The sliding systems are loaded via an external file (the same -procedure as for the tutorial on the [Berveiller-Zaoui](ExplicitBerveillerZaouiPolyCrystals.html) scheme). -This is done as follows : - -~~~~ {#Begin .cpp .numberLines} -@CrystalStructure HCP; -@SlidingSystems{<1, 1, -2, 0>{1, -1, 0, 0}, - <-2, 1, 1, 3>{1, -1, 0, 1}, - <-2, 1, 1, 0>{0, 0, 0, 1}, - <1, 1, -2, 0>{1, -1, 0, 1}}; - -@Members { - using PCSlidingSystems = - ExtendedPolyCrystalsSlidingSystems, real>; -} -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -### Poperties and variables - -We can now define a state variable on each phase, which correspond to -the reference stresses $\tsigma^r$. We also define an auxiliary variable -for $\deriv{\psi}{\tsigma}\left(\tsigma^r\right)$. - -~~~~ {#Begin .cpp .numberLines} -@StateVariable Stensor sigma[Np]; -sigma.setEntryName("PhaseReferenceStress"); -@AuxiliaryStateVariable Stensor dpsi_dsigma[Np]; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Among the different local variables, note the following which -allows to store the morphological tensors: - -~~~~ {#Begin .cpp .numberLines} -@LocalVariable std::array,Np>,Np> DELTA; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Note also that `Np` is the number of phases. This number is a constant parameter, -otherwise, the definition of the `LocalVariable DELTA` above would not be possible -like that. However, we want that this parameter can change. Hence, we define it -before like that: - -~~~~ {#Begin .cpp .numberLines} -@IntegerConstant Np = @Np@; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -and the parameter will be set at the compilation of the behaviour: - -~~~~ {#bash .sh} -mfront --obuild --interface=generic --@Np@=10 PC_SOE_96.mfront -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -### Tangent modulus and free-strain - -There is no difficulty to compute the tangent modulus and free-strain: - -~~~~ {#Integrator .cpp .numberLines} -@Integrator { - using namespace tfel::math; - Stensor4 M[Np]; - Stensor e[Np]; - for (int r=0;r MAT; - tmatrix<6*Np,6*Np,real> G; - tmatrix<6*Np,6,real> E; - for (int r=0;r(E,6*r,0)=L0; - for (int s=0;s(MAT,6*r,6*s) +=DELTA[r][s]*dM; - map_derivative(G,6*r,6*s) -=DELTA[r][s]; - } - } - map_derivative(MAT,6*r,6*r) +=I; - } - - TinyMatrixInvert<6*Np,real>::exe(MAT); - tmatrix<6*Np,6,real> A = MAT*E; - tmatrix<6*Np,6*Np,real> B = MAT*G; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -### Residues and jacobian - -The residues are - -~~~~ {#Integrator .cpp .numberLines} - for (int r=0;r(A,6*r,0); - fsigma[r] = sigma[r]+dsigma[r]-Ar*(eto+deto); - for (int s=0;s(B,6*r,6*s); - fsigma[r]-=Brs*e[s]; - } - } -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -There is no jacobian because we use the `NumericalJacobian`. Here, -an analytical jacobian would have been tedious (see the expressions -of $\tenseurq A_r$ and $\tenseurq B_{rs}$ !). - -### Macroscopic stress - -The computation of the macroscopic stress necessitates the computation of $\deriv{\tsigma^r}{\overline{\tepsilon}}$. -To do that, we use a technique widely used in `MFront`. -The idea is to write the non-linear system to solve as: -\begin{aligned} -\tsigma^r - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)\equiv \tenseur R_r(\overline{\tepsilon},\tsigma^1,...,\tsigma^N)=0 -\end{aligned} -and by derivation of $\tenseur R_r$ w.r.t. $\overline{\tepsilon}$: -\begin{aligned} --\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\overline{\tepsilon}}=0 -\end{aligned} -where $\tenseurq J_{rs}=\deriv{\tenseur R_r}{\tsigma^s}$ stands for the sub-block $(r,s)$ of the Jacobian. Hence -\begin{aligned} -\deriv{\tsigma^r}{\overline{\tepsilon}}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) -\end{aligned} -where $\mathbf {iJ}$ is the inverse of the Jacobian. - -The implementation is - -~~~~ {#Integrator .cpp .numberLines} - tmatrix<6*Np,6*Np,real> iJ = 0.000001*ID+jacobian; - TinyMatrixInvert<6*Np,real>::exe(iJ); - dsigma_deto=iJ*A; - sig=Stensor::zero(); - for (int r=0;r(dsigma_deto,6*r,0); - const auto term = M[r]*(sigma[r]+dsigma[r])-dpsi_dsigma[r]+eto+deto; - const auto term2 = term*dsigma_deto_r+sigma[r]+dsigma[r]; - sig += 0.5*frac[r]*term2; - } -} -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Note the use of the matrix `ID` HERE ############# - -### Tangent operator - -As for the tangent operator, due to its complexity, we will only retain the first term in the implementation: -\begin{aligned} - \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}\approx\sum_r c_r\,\deriv{\tsigma^r}{\overline{\tepsilon}} - \end{aligned} -which only requires the computation of $\deriv{\tsigma^r}{\overline{\tepsilon}}$ that -was already computed. Hence, we have - -~~~~ {#tangent .cpp .numberLines} -@TangentOperator{ - for (int r=0;r(dsigma_deto,6*r,0); - Dt+=frac[r]*dsigma_deto_r; - } -} -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -## Results - -For the tests, we use `mtest`. However, note that the strain we impose -is in fact a strain rate. The idea for us is to keep this strain rate -constant and to make vary the parameters $n^r$ with time. - -Indeed, in Ponte-Castaneda's works (but also in Bornert's works and other works), -when all exponents $n_k^r$ are equal to a unique $n$, -the effective stress is shown to be of the form - -\begin{aligned} -\overline{\tsigma}_{11}=\tilde{\tau}_0\left(\dfrac{\dot{\tepsilon}_{11}}{\dot{\gamma_0}}\right)^{1/n} -\end{aligned} - -VERIFY THE COORDINATES !!!!!!!!!!!!!!!!!!! - -which means that for a constant $\dot{\tepsilon}$, where we introduced -an effective resolved shear stress $\tilde{\tau}_0$. Afterwards, these -authors generally plot this $\tilde{\tau}_0$ as a function of the -parameter $n$. Hence, the idea here is to reproduce this -curve by varying $n$ and keeping $\dot{\tepsilon}$ constant. - -The `.mtest` file is the following: - -~~~~ {#PC_SOE_96 .mtest .numberLines} -@ModellingHypothesis 'Tridimensional'; -@Behaviour 'src/libBehaviour.so' 'PC_SOE_96'; -@ExternalStateVariable 'Temperature' {0 : 1000,1:1000}; - -@MaterialProperty 'n1' "exp(2*t*2.3026)"; -@MaterialProperty 'n2' "exp(2*t*2.3026)"; -@MaterialProperty 'tau1' 20.; -@MaterialProperty 'tau2' 30.; -@ImposedStrain 'EXX' {0 : -1, 1 : -1}; -@ImposedStrain 'EYY' {0 : 0, 1 : 0}; -@ImposedStrain 'EZZ' {0 : 0, 1 : 0}; -@ImposedStrain 'EXY' {0 : 0, 1 : 0}; -@ImposedStrain 'EXZ' {0 : 0, 1 : 0}; -@ImposedStrain 'EYZ' {0 : 0, 1 : 0}; -@Times {0.,1 in 100}; -~~~~~~~~~~~~~~ - - -![Effective resolved shear stress as a function of the parameter $n$](./img/PC_VB_92.png) - -FIGURE !!!!!!!!!!!!!!!!!!!!!!!!! - -In this figure we plotted the behaviour of a single crystal when it is alone -(to show the heterogeneity), and the two implementations of the second-order estimates -(SOE), with morphological tensors (SOE_T) and with self-consistent scheme (SOE_SC). - - diff --git a/docs/web/bibliography.bib b/docs/web/bibliography.bib index c380a6571..70a3de89d 100644 --- a/docs/web/bibliography.bib +++ b/docs/web/bibliography.bib @@ -12639,6 +12639,17 @@ @article{bornert_second-order_2001 pages = {2737--2764}, } +@article{masson_affine_2000, + title = {An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals}, + volume = {48}, + abstract = {Variational approaches for nonlinear elasticity show that Hill's incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too sti and may even violate rigorous bounds. This paper aims at proposing an alternative `aÅne' formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is Ærst described for nonlinear elasticity and specified by making use of Hashin±Shtrikman estimates for the linear comparison composite; the associated aÅne self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castanƒ eda, P., 1996. Exact second-order estimates for the e ective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827±862) and some limitations of the aÅne method are pointed out; explicit comparisons between di erent procedures are performed for isotropic, two-phase materials. Finally, the aÅne formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it o ers an improved alternative to Hill's incremental formulation. 7 2000 Elsevier Science Ltd. All rights reserved.}, + language = {en}, + journal = {Journal of the Mechanics and Physics of Solids}, + author = {Masson, R and Bornert, M and Suquet, P and Zaoui, A}, + year = {2000}, + pages = {1203--1227}, +} + @article{castaneda_effect_1995, title = {The effect of spatial distribution on the effective behavior of composite materials and cracked media}, volume = {43}, diff --git a/docs/web/downloads/Morphological_tensors.zip b/docs/web/downloads/Morphological_tensors.zip new file mode 100644 index 0000000000000000000000000000000000000000..e229115ea40d277047bb8a0d326e874e55c88a08 GIT binary patch literal 4801 zcmbW5cT|(<^2dWg2q0C8NG~Etk)l*Vs+0hURFU2mLJPeJf;6QBL@Cmi8Yvn;AoSjg z(n}TsEJTXZL_)6@_V?qiXYYE>-aGG^ocH}_<~%viXTCE71p^6b0p})5plI^P%m4q- z1K0staCcXGxD(tCW^L(Y?rH1d0eAPft*=K8AXc3Yf;pVI)tP(wk^u+^BWM5sLV7{~ z0QyHv5C8;N2K*VbhX`;M@nHQAD3R49ZpGFzS z>@-cs5ptQ{PMeAhvIfqoN}ei7pl`kA@yt+tWH*da_9I&F7x)xgh)9v zkcv`Nw>w!4ugMrUawn4j!|u_^37UK61xu`XA%^C9VQMf5*Gs8f>kjYcu)hDLn#z-C 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2001 From: MARTIN Antoine Date: Mon, 2 Feb 2026 01:00:18 +0100 Subject: [PATCH 04/25] update tuto MassonAffineFormulation.md --- docs/web/MassonAffineFormulation.md | 27 +++++++++------------------ docs/web/img/Affine_formulation.png | Bin 20539 -> 25680 bytes 2 files changed, 9 insertions(+), 18 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 632cc8819..5d8e70751 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -292,6 +292,9 @@ The derivatives of the potential are ## Details of implementation +We will mainly present the approach based on self-consistent +scheme, we invite the reader to see the behaviour for the other approach. + ### Headers For the jacobian, we adopt the `Numerical Jacobian`, so that @@ -549,7 +552,7 @@ where $\mathbf {iJ}$ is the inverse of the Jacobian. The implementation is ~~~~ {#tangent .cpp .numberLines} @TangentOperator{ - tmatrix<6*Np,6*Np,real> iJ = 0.001*ID+jacobian; + tmatrix<6*Np,6*Np,real> iJ = jacobian; TinyMatrixInvert<6*Np,real>::exe(iJ); dsigma_deto=iJ*A; Dt=Stensor4::zero(); @@ -560,22 +563,8 @@ where $\mathbf {iJ}$ is the inverse of the Jacobian. The implementation is } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Note the use of the matrix `ID` here, that was initialized at the beginning -as a local variable: - -~~~~ {#tangent .cpp .numberLines} -@LocalVariable tmatrix<6*Np,6*Np,real> ID; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -with, in the `InitLocalVariables` code block: - -~~~~ {#tangent .cpp .numberLines} -for (int r=0;r(ID,6*r,6*r) =I; -} -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -This is because the jacobian is null +Note the use on line 7 of the function `tfel::math::map_derivative` +which allows to extract (but also to fill) a block from a `tmatrix`. ## Results @@ -616,11 +605,13 @@ This coefficient can hence be set as a material property, and we will add this line in the `mtest` file: ~~~~ {#Affine_formulation .mtest .numberLines} -@MaterialProperty 'r0' "1+4*t"; +@MaterialProperty 'r0' "1+t"; ~~~~~~~~~~~~~~ Here, we choose to make vary the reference medium because when `t` increases, 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zzlz(?*r-R$3&L(aJhn=6VI202M-rP2AUmVc0uSgF#Fl^s@qeePE zbUGO|tbvWWJ*Btt9v>wkI}GajqLT z-L2lLbp}@6cJER_dQQG_$y-DNFdAZh`Vqiu!*nvNhzL~w_aEbQw_mm9@Ali!F~`ql Nb#VA9dEl1pKLKXm4n_a~ From c5a5321731e23b19fe71bde49bfe313af4d69270 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Mon, 2 Feb 2026 08:32:34 +0100 Subject: [PATCH 05/25] added ExtendedPolycrystalsSlidingSystems --- docs/web/MassonAffineFormulation.md | 6 +- include/CMakeLists.txt | 2 + .../ExtendedPolyCrystalsSlidingSystems.hxx | 70 ++++++++++++++++ .../ExtendedPolyCrystalsSlidingSystems.ixx | 81 +++++++++++++++++++ 4 files changed, 155 insertions(+), 4 deletions(-) create mode 100644 include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx create mode 100644 include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 5d8e70751..78d1c8205 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -323,8 +323,6 @@ and the header for handling the rotations of the grains with their slip systems: - ExtendedPolyCrystalsSlidingSystems.hxx - -which is provided as an external file ~~~~ {#Begin .cpp .numberLines} @TFELLibraries {"Material"}; @@ -333,7 +331,7 @@ which is provided as an external file #include "TFEL/Material/IsotropicEshelbyTensor.hxx" #include "TFEL/Material/MicrostructureDescription.hxx" #include "TFEL/Material/MicrostructureLinearHomogenization.hxx" -#include "../extra-headers/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +#include "TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For the approach based on morphological tensors, we do not need all these headers, @@ -345,7 +343,7 @@ located at the same place as the `.mfront` file. @TFELLibraries {"Material"}; @Includes{ #include "../extra-headers/TFEL/Material/tensors.hxx" -#include "../extra-headers/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +#include "TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We will also generate a polycrystal and compute the tensors via FFT (see the `python` script [here](./downloads/Morphological_tensors.zip)). diff --git a/include/CMakeLists.txt b/include/CMakeLists.txt index 0a0b72fb1..31a4f0c13 100644 --- a/include/CMakeLists.txt +++ b/include/CMakeLists.txt @@ -575,6 +575,8 @@ install_header(TFEL/Material MetallicCFCGenericSlidingSystemsInteractionMatrix.h install_header(TFEL/Material MetallicCFCGenericSlidingSystemsInteractionMatrix.ixx) install_header(TFEL/Material PolyCrystalsSlidingSystems.hxx) install_header(TFEL/Material PolyCrystalsSlidingSystems.ixx) +install_header(TFEL/Material ExtendedPolyCrystalsSlidingSystems.hxx) +install_header(TFEL/Material ExtendedPolyCrystalsSlidingSystems.ixx) install_header(TFEL/Material OrthotropicAxesConvention.hxx) install_header(TFEL/Material OrthotropicAxesConvention.ixx) install_header(TFEL/Material GursonTvergaardNeedleman.hxx) diff --git a/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx b/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx new file mode 100644 index 000000000..10796a6d3 --- /dev/null +++ b/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx @@ -0,0 +1,70 @@ +/*! + * \file include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx + * \brief + * \author Thomas Helfer + * \brief 16 mai 2013 + * \copyright Copyright (C) 2006-2018 CEA/DEN, EDF R&D. All rights + * reserved. + * This project is publicly released under either the GNU GPL Licence + * or the CECILL-A licence. A copy of thoses licences are delivered + * with the sources of TFEL. CEA or EDF may also distribute this + * project under specific licensing conditions. + */ + +#ifndef TFEL_MATERIAL_EXTENDEDPOLYCRISTALSSLIDINGSYSTEMS_HXX +#define TFEL_MATERIAL_EXTENDEDPOLYCRISTALSSLIDINGSYSTEMS_HXX 1 + +#include +#include "TFEL/Math/stensor.hxx" +#include "TFEL/Math/vector.hxx" + +namespace tfel { + + namespace material { + + /*! + * \param[in] Np : number of phases + * \param[in] GS : class describing the sliding system of one phase + * \param[in] NumType : numeric type used + */ + template + struct ExtendedPolyCrystalsSlidingSystems { + //! number of sliding systems of the mono crystal + static constexpr unsigned short Nss = GS::Nss; + //! a simple alias + typedef NumType real; + //! a simple alias + typedef tfel::math::stensor<3u, real> StrainStensor; + //! return the uniq instance of the class + static const ExtendedPolyCrystalsSlidingSystems &getPolyCrystalsSlidingSystems( + const std::string &); + //! tensor of directional senses, sorted by phases + tfel::math::vector> mus; + //! \brief volume fractions per phases + tfel::math::vector volume_fractions; + + private: + /*! + * default constructor (declared private~: it can only be used + * by the getExtendedPolyCrystalsSlidingSystems method) + * \param[in] f : file where the angles are declared + */ + ExtendedPolyCrystalsSlidingSystems(const std::string &); + /*! + * copy constructor (declared private to disable it) + */ + ExtendedPolyCrystalsSlidingSystems(const ExtendedPolyCrystalsSlidingSystems &); + /*! + * assignement operator (declared private to disable it) + */ + ExtendedPolyCrystalsSlidingSystems &operator=( + const ExtendedPolyCrystalsSlidingSystems &); + }; + + } // end of namespace material + +} // end of namespace tfel + +#include "./ExtendedPolyCrystalsSlidingSystems.ixx" + +#endif /* TFEL_MATERIAL_EXTENDEDPOLYCRISTALSSLIDINGSYSTEMS_HXX */ diff --git a/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx b/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx new file mode 100644 index 000000000..b682be8b1 --- /dev/null +++ b/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx @@ -0,0 +1,81 @@ +/*! + * \file include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx + * \brief + * \author Thomas Helfer + * \brief 16 mai 2013 + * \copyright Copyright (C) 2006-2018 CEA/DEN, EDF R&D. All rights + * reserved. + * This project is publicly released under either the GNU GPL Licence + * or the CECILL-A licence. A copy of thoses licences are delivered + * with the sources of TFEL. CEA or EDF may also distribute this + * project under specific licensing conditions. + */ + +#ifndef TFEL_MATERIAL_EXTENDEDPOLYCRYSTALSSLIDINGSYSTEMS_IXX +#define TFEL_MATERIAL_EXTENDEDPOLYCRYSTALSSLIDINGSYSTEMS_IXX 1 + +#include +#include "TFEL/Utilities/CxxTokenizer.hxx" + + +namespace tfel { + + namespace material { + + template + const ExtendedPolyCrystalsSlidingSystems + &ExtendedPolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems( + const std::string &f) { + static ExtendedPolyCrystalsSlidingSystems gs(f); + return gs; + } + + template + ExtendedPolyCrystalsSlidingSystems:: + ExtendedPolyCrystalsSlidingSystems(const std::string &f) + : mus(Np), volume_fractions(Np) { + using namespace std; + using namespace tfel::math; + using namespace tfel::utilities; + // the sliding systems of one phase + const auto &gs = GS::getSlidingSystems(); + const real pi = real(4) * std::atan(real(1)); + // reading from the specified file + CxxTokenizer file(f); + file.stripComments(); + CxxTokenizer::const_iterator p = file.begin(); + const CxxTokenizer::const_iterator pe = file.end(); + for (unsigned short i = 0; i != Np; i++) { + this->mus[i].resize(Nss); + tmatrix<3u, 3u, real> drot; + const real psi = CxxTokenizer::readDouble(p, pe) * pi / 180.0; + const real the = CxxTokenizer::readDouble(p, pe) * pi / 180.0; + const real phi = CxxTokenizer::readDouble(p, pe) * pi / 180.0; + const real cospsi = std::cos(psi); + const real costhe = std::cos(the); + const real cosphi = std::cos(phi); + const real sinpsi = std::sin(psi); + const real sinthe = std::sin(the); + const real sinphi = std::sin(phi); + drot(0, 0) = cosphi * cospsi - sinphi * costhe * sinpsi; + drot(0, 1) = cosphi * sinpsi + sinphi * costhe * cospsi; + drot(0, 2) = sinphi * sinthe; + drot(1, 0) = -sinphi * cospsi - cosphi * costhe * sinpsi; + drot(1, 1) = -sinphi * sinpsi + cosphi * costhe * cospsi; + drot(1, 2) = cosphi * sinthe; + drot(2, 0) = sinthe * sinpsi; + drot(2, 1) = -sinthe * cospsi; + drot(2, 2) = costhe; + for (unsigned short j = 0; j != Nss; j++) { + auto &mu = this->mus[i][j]; + mu = gs.mus[j]; + mu.changeBasis(drot); + } + this->volume_fractions[i] = CxxTokenizer::readDouble(p, pe); + } + } + } // end of namespace material + +} // end of namespace tfel + +#endif /* TFEL_MATERIAL_EXTENDEDPOLYCRYSTALSSLIDINGSYSTEMS_IXX */ From 394b2fa17504095aa6e9936c1a4ccc55d9a1e501 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Mon, 2 Feb 2026 11:37:26 +0100 Subject: [PATCH 06/25] minor modif --- docs/web/MassonAffineFormulation.md | 65 ++++++++++++++++++++--------- 1 file changed, 45 insertions(+), 20 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 78d1c8205..b19253970 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -44,7 +44,7 @@ eqnPrefixTemplate: "($$i$$)" \renewcommand{\div}{\mathrm{div}} -We present here an implementation of the affine formulation for homogenization [@masson_affine_2000] +We present here an implementation of the affine formulation [@masson_affine_2000] for the homogenization of a viscoplastic polycrystal, example which is treated in [@bornert_second-order_2001] but with Ponte-Castaneda second-order estimates. @@ -266,13 +266,19 @@ All the files are available in the `MFrontGallery` project, [here](https://githu The geometry of our polycrystal is generated with [`merope`](https://github.com/MarcJos/Merope/), with a Voronoi tesselation. - This geometry is saved as an array with an integer representing the number of the phase (or crystal). - The image below shows an example of such a polycrystal. There are here 10 crystals. The orientations - and volume fractions are given in the file `polycrystal.csv`. The geometry is used to - compute the morphological tensors (see the `python` script). + This geometry is saved as an array and will be used only for the approach based on morphological + tensors. The image below shows an example of such a polycrystal. There are here 10 crystals. The + volume fractions can also be computed on this microstructure ![Geometry of the polycrystal](./img/polycrystal.png){width=50%} + The morphological tensors can be computed by following the instruction of the `README.md` + in this [folder](./downloads/Morphological_tensors.zip). + + The orientations of the slip systems will be given by angles, stored in the file `polycrystal.csv`. + This file also contains, at the end of each line, the volume fraction associated to a crystal. + They are in our case the same as the one computed on the generated microstructure. + The potential $\psi_k^r$ will be of the form \begin{aligned} \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^{k,r}}{n+1}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n+1} @@ -293,7 +299,8 @@ The derivatives of the potential are ## Details of implementation We will mainly present the approach based on self-consistent -scheme, we invite the reader to see the behaviour for the other approach. +scheme, we invite the reader to see the `.mfront` file in the +`MFrontGalleryProject` for the approach based on morphological tensors. ### Headers @@ -346,7 +353,8 @@ located at the same place as the `.mfront` file. #include "TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -We will also generate a polycrystal and compute the tensors via FFT (see the `python` script [here](./downloads/Morphological_tensors.zip)). +Of cours the tensors are computed before via FFT as explained above +(see the folder mentioned above for the computation). ### Sliding systems @@ -398,7 +406,7 @@ morphological tensors, is This matrix will store the morphological tensors. Indeed, in the `InitLocalVariables` code block, we can load these coefficients, -which are in fact situated in the header `` +which are in fact situated in the header `../extra-headers/TFEL/Material/tensors.hxx` ~~~~ {#Begin .cpp .numberLines} DELTA=Delta::get_tensor(); @@ -435,10 +443,12 @@ for (int r=0;r(IM0); Note that here the matrix phase is defined by the elasticity `(k0,mu0)` but this elasticity will play no role in the sequel. -Then, we will add some `SphereDistribution` objects, -each of them representing a grain with its shape (`Sphere`), volume fraction -and elasticity: +Indeed, we will add the grains as `SphereDistribution` objects, +and because the sum of the volume fractions is equal to one, +the matrix will automatically have a fraction equal to zero. + +The grains are added like that: ~~~~ {#Integrator .cpp .numberLines} const auto sphere=Sphere(); @@ -472,10 +484,13 @@ for (int r=0 ; r(KGSC); const auto MSC_star = invert(invert(PSC)- LSC); ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Note here that we isotropize the homogenized stiffness with the function +Note here that we also isotropize the homogenized stiffness with the function `computeKGModuli`, on line 1. This is more consistent with the fact that we used `true` in the `computeSelfConsistent` function. @@ -608,6 +631,8 @@ this line in the `mtest` file: Here, we choose to make vary the reference medium because when `t` increases, `n` increases, and hence the homogenized stiffness increases. +This seems hence more consistent to consider a reference medium whose +stiffness increases also. The optimal value of `r0` is not immediate, but some numerical tests can be carried out. From 03e9cbf6fef027117ba8ca52378d959413ea5452 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Mon, 2 Feb 2026 13:15:46 +0100 Subject: [PATCH 07/25] minor modifs --- docs/web/MassonAffineFormulation.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index b19253970..9c1ef1ca0 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -448,7 +448,7 @@ is a parameter that the user can change. It is necessary because the tensors $\tenseurq M_r$ are not invertible (see also [@bornert_second-order_2001], appendix B.1). In our case, we set it as a `MaterialProperty` that can evolve with time (see `mtest` file below). -We can hence consider $\tenseur L_r$, the inverse of $\tenseur M_r+\kappa\tenseurq I$. +We can hence consider $\tenseurq L_r$, the inverse of $\tenseurq M_r+\kappa\tenseurq I$. ### Self-consistent scheme @@ -486,7 +486,7 @@ for (int r=0 ; r Date: Mon, 2 Feb 2026 13:18:25 +0100 Subject: [PATCH 08/25] minor modifs --- docs/web/gallery.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/docs/web/gallery.md b/docs/web/gallery.md index 15cf4cf1b..865069acc 100644 --- a/docs/web/gallery.md +++ b/docs/web/gallery.md @@ -104,6 +104,8 @@ behaviour. This framework is described `@BehaviourVariable` and hence any behaviour law on each phase. - A tutorial on implementation of Ponte-Castaneda bound (1992) for a non-linear elastic composite is detaile [here](PonteCastaneda1992.html). +- A tutorial on implementation of the [affine formulation](MassonAffineFormulation.html) + from Masson (2000) for a viscoplastic polycrystal. From 78161a0e5b8124b82104de05f28e694d87f348ec Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Mon, 2 Feb 2026 13:20:21 +0100 Subject: [PATCH 09/25] minor modifs --- docs/web/MassonAffineFormulation.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 9c1ef1ca0..06d3ed2b8 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -503,7 +503,7 @@ const auto A_SC=hmSC.mean_strain_localisation_tensors; On line 1, we say we want a tolerance of `1e-6` (this is related to the stopping criterion of the iterative self-consistent algorithm, -see the [doc](./tfel-material.html#homogenization)). +see the [doc](./tfel-material.html#homogenization-of-general-microstructures)). We also consider an isotropic homogenized stiffness with `true`. This is related to the computation of the Hill tensor associated to the spherical inclusions. if using an anisotropic reference medium for computing this tensor, From f039989f2b8b59e11920a73a38ff43083686d1d2 Mon Sep 17 00:00:00 2001 From: antoinecea Date: Tue, 3 Feb 2026 13:45:01 +0100 Subject: [PATCH 10/25] Update docs/web/tfel-material.md Co-authored-by: Thomas Helfer --- docs/web/tfel-material.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/web/tfel-material.md b/docs/web/tfel-material.md index 7cd319823..736bb0015 100644 --- a/docs/web/tfel-material.md +++ b/docs/web/tfel-material.md @@ -1079,7 +1079,7 @@ auto hmSC=computeSelfConsistent<3u,stress>(micro_1,1e-6,true); We note that `computeSelfConsistent` not only takes the microstructure as an argument, but also takes one real (`1e-6`) as -a parameter, which pilots the precision of the result. Indeed, at each iteration +a parameter, which pilots the accuracy of the result. Indeed, at each iteration of the self-consistent iterative algorithm, the function computes the relative difference between the new and the old homogenized stiffness. This relative difference must be smaller than the tolerance given as a parameter. From acefa85e08681c62ce9920666f9f7489edace4af Mon Sep 17 00:00:00 2001 From: Thomas Helfer Date: Tue, 3 Feb 2026 14:01:03 +0100 Subject: [PATCH 11/25] small fixes --- docs/web/MassonAffineFormulation.md | 85 +++++++++++++++++++++++++---- 1 file changed, 75 insertions(+), 10 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 06d3ed2b8..3f10b5a1e 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -8,6 +8,7 @@ colorlinks: true geometry: - margin=2cm papersize: a4 +bibliography: bibliography.bib figPrefixTemplate: "$$i$$" tblPrefixTemplate: "$$i$$" secPrefixTemplate: "$$i$$" @@ -64,23 +65,30 @@ and show the details of the `mfront` file. We consider a polycrystalline material, which means that each phase $r$ is associated to a crystal with corresponding slip systems $\tenseur \mu_k^r$ ($1\leq k\leq K$). The strain rate in each crystal $r$ is given by +\[ \begin{aligned} \dot{\tepsilon}=\sum_k \dot{\gamma}_k^r\, \tenseur \mu_k^r\qquad\text{with}\quad \tenseur\mu_k^r = \dfrac12\left(\tenseur n_k^r\otimes\tenseur m_k^r + \tenseur m_k^r\otimes\tenseur n_k^r\right) \end{aligned} +\] where $\dot{\gamma}_k^r$ is the shear strain rate on the $k^{th}$ slip system of crystal $r$, and is given as a function of the shear stress $\tau_k^r=\tsigma\dbldot\tenseur \mu_k^r$ by means of a potential $\psi_k^r$: +\[ \begin{aligned} \dot{\gamma}_k^r= \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right) \end{aligned} +\] The expressions above show that on phase (or crystal) $r$: +\[ \begin{aligned} \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\text{with}\quad\psi_r \left(\tsigma\right)=\sum_k \psi_k^r\left(\tau_k^r\right) \end{aligned} - +\] Now, the behaviour of the polycrystal is governed by a potential $\psi$: +\[ \begin{aligned} \dot{\tepsilon}=\deriv{\psi_r}{\tsigma}\left(\tsigma\right)\qquad\psi(\tsigma)= \sum_{r=1}^{N}\chi_r\,\psi_r (\tsigma) \end{aligned} +\] where $N$ is the number of phases (or crystals) and $\chi_r$ is characteristic function of phase $r$. In all the sequel, we just note $\tepsilon$ for $\dot{\tepsilon}$. @@ -88,12 +96,13 @@ where $N$ is the number of phases (or crystals) and $\chi_r$ is characteristic f ## The non-linear homogenization problem We impose a strain rate $\overline{\tepsilon}$ to the polycrystal and look for the solution $\tepsilon,\tsigma$ such that: - +\[ \begin{aligned} &\div\,\tsigma=\tenseur 0\\ &\tepsilon\in\Ka(\overline{\tepsilon})\\ &\tepsilon = \deriv{\psi}{\tsigma}\left(\tsigma\right) \end{aligned} +\] where we introduced the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$, depending on the boundary conditions used (in the implementation, we work with periodic boundary conditions). @@ -102,6 +111,7 @@ where we introduced the space of kinematically admissible fields $\mathcal K(\ov ## The affine linearization The idea is to linearize the behaviour around a reference stress $\tsigma^r$: +\[ \begin{aligned} \tepsilon_r(\tsigma)\approx\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma+\tenseur e^r \end{aligned} @@ -109,35 +119,45 @@ The idea is to linearize the behaviour around a reference stress $\tsigma^r$: \begin{aligned} \tenseurq M_r\left(\tsigma^r\right)=\deriv{\tepsilon_r}{\tsigma}\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\tenseur e^r = \tepsilon_r(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r \end{aligned} +\] This affine behaviour can be viewed as a so-called "thermoelastic comparison composite", and this composite can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by +\[ \begin{aligned} \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) \end{aligned} +\] where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@masson_affine_2000] leads to the simple assumption that these reference stresses are the averages of the stress in the thermoelastic composite: +\[ \begin{aligned} \tsigma^r = \langle\tsigma\rangle_r \end{aligned} - where $\tsigma$ here stands for the stress solution of the thermoelastic homogenization problem. +\] +where $\tsigma$ here stands for the stress solution of the thermoelastic homogenization problem. ## Macroscopic stress and tangent operator The macroscopic stress hence can be obtained by the classical relation: +\[ \begin{aligned} \overline{\tsigma}=\sum_rc_r\,\langle\tsigma\rangle_r=\tenseurq C^{\mathrm{eff}}\dbldot\overline{\tepsilon}+\tenseur\tau^{\mathrm{eff}} \end{aligned} - + \] where - \begin{aligned} + \[ +\begin{aligned} \tenseurq C^{\mathrm{eff}}=\sum_rc_r\,\tenseurq A_r\qquad\text{and}\qquad\tenseur\tau^{\mathrm{eff}}=\sum_{r,s}c_r\,\tenseurq B_{rs}\dbldot\tenseur e^s \end{aligned} +\] The tangent operator is obtained by derivation: +\[ \begin{aligned} \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}=\sum_r c_r\,\deriv{\tsigma^r}{\overline{\tepsilon}} \end{aligned} +\] In the expressions of the tangent operator, we can compute the term $\deriv{\tsigma^r}{\overline{\tepsilon}}$ by means of the jacobian matrix, also done and explained in the documentation of the [Implicit DSL](https://thelfer.github.io/tfel/web/implicit-dsl.html#computation-of-the-consistent-tangent-operator). We explain it below. @@ -150,12 +170,13 @@ The resolution consists in \text{Find}\quad\tsigma^r\quad\text{such that}\quad\tsigma^r=\langle\tsigma\rangle_r \] where \(\tsigma\) is the stress field solution of a homogenization problem of the form: +\[ \begin{aligned} &\div\,\tsigma=\tenseur 0\\ &\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\overline{\tepsilon}\right)\\ &\tenseurq M_r=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r \end{aligned} - +\] ## Possible implementations @@ -177,29 +198,40 @@ The macroscopic stress and the tangent operator must also be computed with the f ### Computation via morphological tensors In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\overline{\tepsilon}\right)$, we have, because, $\div\,\tsigma=0$, +\[ \begin{aligned} \tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) \end{aligned} +\] where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives +\[ \begin{aligned} \tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau \end{aligned} +\] and hence (due to the expressions of $\tenseur\tau$ and $\tepsilon$) +\[ \begin{aligned} \tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) \end{aligned} - +\] +\[ \begin{aligned} \tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq \Delta_0\left(\tenseur e^r\right) \end{aligned} +\] with $\tenseurq\delta\tenseurq M_r=\tenseurq M_r-\tenseurq M_0$, $\tenseurq M_0=\tenseurq L_0^{-1}$, and $\tenseurq \Delta_0=\tenseurq L_0-\tenseurq L_0\dbldot\tenseurq \Gamma_0\tenseurq L_0$. We will not resolve this equation exactly, but we will consider the average of the fields: +\[ \begin{aligned} \langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s \end{aligned} +\] where +\[ \begin{aligned} \tenseurq \Delta_{rs} = \sum_j\langle\,\tenseurq\Delta_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r \end{aligned} +\] is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor makes intervene the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). Hence, the $\langle\tsigma\rangle_r$ 's are solution of a linear system. Depending on the number of phases $N$, the computation of the $\tenseurq \Delta_{rs}$ is more or less costly, but it is achieved @@ -217,42 +249,59 @@ $\tenseurq L_{\mathrm{SC}}$ needed for the computation of the macroscopic stress but also the effective polarization $\tenseurq \tau_{\mathrm{SC}}$ which permit to compute the macroscopic stress: +\[ \begin{aligned} \overline{\tsigma} = \tenseurq L_{\mathrm{SC}}\dbldot\overline{\tepsilon}+\tenseur \tau_{\mathrm{SC}} \end{aligned} +\] However, we also need the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ for computing the average of the stress field $\langle\tsigma\rangle_r$ (i.e. the residue). To that extent, we go back to the equation +\[ \begin{aligned} \tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) \end{aligned} +\] and consider averages of the fields: +\[ \begin{aligned} \langle\tepsilon\rangle_r-\overline{\tepsilon}=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s \end{aligned} +\] with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r$. + A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ (i.e. the grains have a spherical shape) so that +\[ \begin{aligned} \langle\tepsilon\rangle_r-\overline{\tepsilon}=-\tenseurq P_0\dbldot\left(\langle\tenseur\tau\rangle_r-\langle\tenseur\tau\rangle\right) \end{aligned} +\] and due to the expression of $\tenseur\tau$: +\[ \begin{aligned} \left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\overline{\tepsilon}\right)=-\left[\langle\tsigma\rangle_r-\overline{\tsigma}\right] \end{aligned} +\] We know that the reference medium, in the self-consistent approach, is identified with the homogenized medium. Hence, we note $\tenseurq P_0=\tenseurq P_{\mathrm{SC}}$, $\tenseurq L_0=\tenseurq L_{\mathrm{SC}}$, and $\tenseurq M_{\mathrm{SC}}^*=\left[\tenseurq P_{\mathrm{SC}}^{-1}-\tenseurq L_{\mathrm{SC}}\right]^{-1}$, so that, with $\langle\tepsilon\rangle_r=\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r$, and $\langle\tsigma\rangle_r=\tsigma^r$ +\[ \begin{aligned} \left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\overline{\tepsilon}\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tsigma^r-\overline{\tsigma}\right] \end{aligned} +\] which gives +\[ \begin{aligned} \tsigma^r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left[\overline{\tepsilon}-\tenseur e^r+\tenseurq M_{\mathrm{SC}}^*\dbldot\overline{\tsigma}\right] \end{aligned} +\] which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s$ (we use $\overline{\tsigma}=\tenseurq L_{\mathrm{SC}}\dbldot\overline{\tepsilon}+\tenseur \tau_{\mathrm{SC}}$): +\[ \begin{aligned} \tenseurq A_r&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_{\mathrm{SC}}\right)\\ \sum_s\tenseurq B_{rs}\dbldot\tenseur e^s&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseur \tau_{\mathrm{SC}}-\tenseur e^r\right) \end{aligned} +\] (and this is sufficient to compute the residue). However, we note that the library `tfel::material::homogenization` provides the tensor $\tenseurq A_{\mathrm{SC}}$ via the variable `mean_strain_localisation_tensors`, which relates $\langle\tepsilon\rangle_r$ and $\overline{\tepsilon}$. This tensor is also equal to $\tenseurq A_{\mathrm{SC}}=\tenseurq M_r\dbldot\tenseurq A_r$, due to the expression of $\langle\tsigma\rangle_r$. This will be more simple for the implementation. @@ -280,21 +329,27 @@ All the files are available in the `MFrontGallery` project, [here](https://githu They are in our case the same as the one computed on the generated microstructure. The potential $\psi_k^r$ will be of the form - \begin{aligned} + \[ +\begin{aligned} \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^{k,r}}{n+1}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n+1} \end{aligned} +\] where the strain rate $\dot{\gamma}_0$ and the creep exponent $n\geq 1$ will be chosen identical for all $k,r$, and the resolved shear stress $\tau_0^{k,r}$ depends on $k,r$. The derivatives of the potential are - \begin{aligned} + \[ +\begin{aligned} &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r\\ &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r \end{aligned} +\] and then defining $\overline{\tau}_k^r=\tsigma^r\dbldot\tenseur \mu_k^r$ where $\tsigma^r$ is the reference stress, we have - \begin{aligned} + \[ +\begin{aligned} &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r\\ &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r \end{aligned} +\] ## Details of implementation @@ -529,9 +584,11 @@ in the `computeSelfConsistent` function. For the residues, we work on the strains (because `MFront` is more suited to strain type residues. Hence, we will define the residues as +\[ \begin{aligned} f_{\sigma^r}=\tenseurq M_r\dbldot\left(\tsigma^r-\tenseurq A_r\dbldot\overline{\tepsilon}-\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s\right) \end{aligned} +\] ~~~~ {#Integrator .cpp .numberLines} for (int r=0;r From 4cd2f3cb9b8db57fd8062e98e9942bd2ae55974e Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Tue, 3 Feb 2026 16:12:06 +0100 Subject: [PATCH 12/25] minor modifs --- docs/web/MassonAffineFormulation.md | 12 +++++++++--- docs/web/gallery.md | 5 +++-- 2 files changed, 12 insertions(+), 5 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 3f10b5a1e..1740c89c9 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -398,16 +398,22 @@ slip systems: For the approach based on morphological tensors, we do not need all these headers, but we need to include a file which contains the morphological tensors. -This file is present in a repository `extra-headers/TFEL/Material/` +This file is present in a repository, let's say `extra-headers/TFEL/Material/` located at the same place as the `.mfront` file. ~~~~ {#Begin .cpp .numberLines} @TFELLibraries {"Material"}; @Includes{ -#include "../extra-headers/TFEL/Material/tensors.hxx" +#include "tensors.hxx" #include "TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The compilation hence will be done like that: + +~~~~ {#Begin .sh .numberLines} +mfront -I ../extra-headers/TFEL/Material --obuild --interface=generic Affine_formulation.mfront +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + Of cours the tensors are computed before via FFT as explained above (see the folder mentioned above for the computation). @@ -461,7 +467,7 @@ morphological tensors, is This matrix will store the morphological tensors. Indeed, in the `InitLocalVariables` code block, we can load these coefficients, -which are in fact situated in the header `../extra-headers/TFEL/Material/tensors.hxx` +which are in fact situated in the header `extra-headers/TFEL/Material/tensors.hxx` ~~~~ {#Begin .cpp .numberLines} DELTA=Delta::get_tensor(); diff --git a/docs/web/gallery.md b/docs/web/gallery.md index 865069acc..7685f2dc2 100644 --- a/docs/web/gallery.md +++ b/docs/web/gallery.md @@ -102,11 +102,12 @@ behaviour. This framework is described [here](ExplicitBerveillerZaouiPolyCrystals.html) - [\(\beta\)-rule](BetaRule.html) can also be implemented with `@BehaviourVariable` and hence any behaviour law on each phase. -- A tutorial on implementation of Ponte-Castaneda bound (1992) +- A tutorial on implementation of Ponte-Castaneda bound [@castaneda_new_1992] for a non-linear elastic composite is detaile [here](PonteCastaneda1992.html). - A tutorial on implementation of the [affine formulation](MassonAffineFormulation.html) - from Masson (2000) for a viscoplastic polycrystal. + from Masson et al. [@masson_affine_2000] for a viscoplastic polycrystal. +# References From 62c08bbd7767795c92d4746271230b96d960f4f5 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Wed, 4 Feb 2026 10:58:20 +0100 Subject: [PATCH 13/25] added some links to MFrontGallery --- docs/web/BetaRule.md | 2 +- docs/web/MassonAffineFormulation.md | 26 +++++++++++++--------- docs/web/PonteCastaneda1992.md | 4 ++-- docs/web/Sachs.md | 2 ++ docs/web/Taylor.md | 2 ++ docs/web/bibliography.bib | 15 +++++++++++++ docs/web/downloads/BetaRule.zip | Bin 3696 -> 0 bytes docs/web/downloads/PonteCastaneda1992.zip | Bin 3106 -> 0 bytes docs/web/gallery.md | 4 +++- 9 files changed, 40 insertions(+), 15 deletions(-) delete mode 100644 docs/web/downloads/BetaRule.zip delete mode 100644 docs/web/downloads/PonteCastaneda1992.zip diff --git a/docs/web/BetaRule.md b/docs/web/BetaRule.md index 4ccf3d692..01d9ad198 100644 --- a/docs/web/BetaRule.md +++ b/docs/web/BetaRule.md @@ -192,7 +192,7 @@ The implementation requires the integration of the local behaviours. These are carried out via the `@BehaviourVariable` keyword. The implementation of the local behaviour is explained [here](./MericCailletaudSingleCrystalPlasticity.html). -All files `MericCailletaudSingleCrystalViscoPlasticity.mfront`, `BetaRule.mfront` and `BetaRule.mtest` can be downloaded [here](./downloads/BetaRule.zip). +All files `MericCailletaudSingleCrystalViscoPlasticity.mfront`, `BetaRule.mfront` and `BetaRule.mtest` are available in the `MFrontGallery` project, [here](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization/). For the example, we assume that the composites is made of only 2 phases. diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 1740c89c9..c4a2c5960 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -52,8 +52,7 @@ but with Ponte-Castaneda second-order estimates. Here, the idea is to show that many implementations of that procedure are possible in `MFront`. We show here a first implementation based on the self-consistent scheme, provided by the `namespace` `tfel::material::homogenization`. A second approach which uses morphological tensors computed by FFT, -on a given geometry of polycrystal, is also detailed. This allows to improve the estimates, but -it necessitates an offline computation. +on a given geometry of polycrystal, is also detailed. This tutorial first presents the homogenization problem, recalls the methodology of the affine formulation, presents different kinds of possible implementations, @@ -112,13 +111,11 @@ where we introduced the space of kinematically admissible fields $\mathcal K(\ov The idea is to linearize the behaviour around a reference stress $\tsigma^r$: \[ - \begin{aligned} \tepsilon_r(\tsigma)\approx\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma+\tenseur e^r - \end{aligned} +\] where - \begin{aligned} +\[ \tenseurq M_r\left(\tsigma^r\right)=\deriv{\tepsilon_r}{\tsigma}\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\tenseur e^r = \tepsilon_r(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r - \end{aligned} \] This affine behaviour can be viewed as a so-called "thermoelastic comparison composite", and this composite can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by @@ -661,6 +658,9 @@ which allows to extract (but also to fill) a block from a `tmatrix`. For the tests, we use `mtest`. However, note that the strain we impose is in fact a strain rate. The idea for us is to keep this strain rate constant and to make vary the parameter $n$ with time. +We hence set it as a function which varies exponentially with time +(this is just a choice). + Besides, we choose two different values for $\tau_0^r$. 5 crystals are such that $\tau_0=10 MPa$, and 5 others such that $\tau_0=50 MPa$. @@ -689,6 +689,9 @@ The `.mtest` file is the following: @OutputFilePrecision 14; ~~~~~~~~~~~~~~ +Note that `kap` on line 9 is related to the small compliance +we add to $\tenseurq M_r$ in order to make it invertible. + For the morphological tensor approach, we talked above about the coefficient $r_0$ that allows to adjust the reference medium. This coefficient can hence be set as a material property, and we will add @@ -698,10 +701,8 @@ this line in the `mtest` file: @MaterialProperty 'r0' "1+t"; ~~~~~~~~~~~~~~ -Here, we choose to make vary the reference medium because when `t` -increases, `n` increases, and hence the homogenized stiffness increases. -This seems hence more consistent to consider a reference medium whose -stiffness increases also. +Here, we choose to increase the elasticity the reference medium because its +homogenized stiffness also increases. The optimal value of `r0` is not immediate, but some numerical tests can be carried out. @@ -710,7 +711,10 @@ can be carried out. In this figure we plotted the results when all the $\tau_0^r$ are equal to $10 Mpa$ (to show the heterogeneity) ("FFT 0"), the implementation of the affine approach -with self-consistent scheme (AFF_SC), and with morphological tensors (AFF_MT). +with self-consistent scheme (AFF_SC), and with morphological tensors (AFF_MT). For +the latter, we plot it for a constant $r_0$ (which is, an elastic medium whose +elasticity is $\tenseurq I$), and for a $r_0=1+t$ (which is, an elastic medium +whose elasticity is $(1+t)\tenseurq I$). FFT results are also plotted (FFT). # References diff --git a/docs/web/PonteCastaneda1992.md b/docs/web/PonteCastaneda1992.md index b0cc0617a..28ef2665e 100644 --- a/docs/web/PonteCastaneda1992.md +++ b/docs/web/PonteCastaneda1992.md @@ -152,7 +152,7 @@ And the macroscopic stress must also be computed: The first step of the resolution consists in computing the secant modulus $\mu_0^r$ and it can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the function $f_r$ in an external file (in our example, the function is to simple to do that). -The second step consists in computing the effective energy $W_0^{\mathrm{eff}}$. This will be done via `tfel::material::homogenization`. This `namespace` provides classical mean-field schemes (we use a Hashin-Shtrikman bound in our example below). In another [tutorial](./PonteCastaneda1996.html) we show that we can take into account more specifically the morphology of the composite by providing interaction tensors. +The second step consists in computing the effective energy $W_0^{\mathrm{eff}}$. This will be done via `tfel::material::homogenization`. This `namespace` provides classical mean-field schemes (we use a Hashin-Shtrikman bound in our example below). This could also be done with interaction tensors, providing externally. In another [tutorial](./MassonAffineFormulation.html) we show that we can take into account more specifically the morphology of a polycrystal by providing interaction tensors. The third step consists in the computation of the second-moments. This is also done via `tfel::material::homogenization`. Here again, there are two strategies: analytical computation when possible, and finite difference or automatic differentiation when the analytical derivation is too tedious or even impossible. In our example below, the computation is analytic. @@ -182,7 +182,7 @@ which means that we only need the derivative of the Hashin-Shtrikman bound. This ## Details of implementation -All the files are available [here](./downloads/PonteCastaneda1992.zip) +The file `PonteCastaneda1992.mfront` is available in the `MFrontGallery` project, [here](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization/). For the jacobian, we adopt the `Numerical Jacobian`, which means that the beginning of the `mfront` file reads: diff --git a/docs/web/Sachs.md b/docs/web/Sachs.md index cf7035258..cfb216911 100644 --- a/docs/web/Sachs.md +++ b/docs/web/Sachs.md @@ -94,6 +94,8 @@ The jacobian matrix is: # Implementation in MFront +The files `Plasticity.mfront` and `Sachs.mfront` are available in the `MFrontGallery` project, [here](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization/). + As for Taylor's scheme, we will use `BehaviourVariable` for the integration of local behaviours. The integration of global behavior, which requires solving a non-linear equation, diff --git a/docs/web/Taylor.md b/docs/web/Taylor.md index 561fb9887..5ce135087 100644 --- a/docs/web/Taylor.md +++ b/docs/web/Taylor.md @@ -51,6 +51,8 @@ The macroscopic tangent operator is given by # Implementation in MFront +The files `Plasticity.mfront` and `Taylor.mfront` are available in the `MFrontGallery` project, [here](https://github.com/thelfer/MFrontGallery/tree/master/generic-behaviours/homogenization/). + It is possible to implement this homogenization scheme through a `Behaviour`. This `Behaviour` must call the behaviour laws of each phase. So, an interesting solution is to use a `BehaviourVariable`. diff --git a/docs/web/bibliography.bib b/docs/web/bibliography.bib index 70a3de89d..32eb7a380 100644 --- a/docs/web/bibliography.bib +++ b/docs/web/bibliography.bib @@ -12684,3 +12684,18 @@ @article{willis_bounds_1977 year = {1977}, pages = {185--202}, } + +@article{idiart_thermodynamic_2025, + title = {Thermodynamic potentials for viscoelastic composites}, + volume = {194}, + issn = {00225096}, + url = {https://linkinghub.elsevier.com/retrieve/pii/S0022509624004022}, + doi = {10.1016/j.jmps.2024.105936}, + abstract = {Explicit expressions for the free-energy and dissipation densities of viscoelastic composites at fixed temperature are proposed. The composites are comprised of an arbitrary number of distinct constituents exhibiting linear Maxwellian rheologies and distributed randomly at a length scale that is much smaller than that over which applied loads vary significantly. Central to their derivation is the recognition that any viscous deformation field can be additively decomposed into an irrotational field and a solenoidal field in such a way that variational approximations available for elastic potentials become applicative to viscoelastic potentials. The thermodynamic potentials conform to a generalized standard model with a finite number of effective internal variables with explicit physical meaning. Specific approximations of the Hashin–Shtrikman and the Self-Consistent types are worked out in detail. Under particular circumstances, these approximations may turn out exact. Macroscopic stress–strain relations and intraphase statistics of the stress field up to second order are also provided.}, + language = {en}, + urldate = {2025-12-03}, + journal = {Journal of the Mechanics and Physics of Solids}, + author = {Idiart, Martín I.}, + year = {2025}, + pages = {105936}, +} diff --git a/docs/web/downloads/BetaRule.zip b/docs/web/downloads/BetaRule.zip deleted file mode 100644 index 23f8fd2c750964da3f620dfc16771a0e6859f5c1..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 3696 zcmb_fXEa>x79MT%K7=G%B#53Nf`~+&U@*!sh%gu>8NJsKM6ZL0j2b<95JHqNdWmR3 zCPB35JxFl*zUzL;z29By{Ol!0AP!{!0En510RY$^IAj2_-*Eb90RYzD zaLy2gZ7rSb?Ogr^dkXnCusD5V=W$7p@4UYCrUfTWRlIM&2Xf*!uH&&(xxU|^tjO*R zWAnOrEOqx><67d3NN8teYU&uYt?dnD-VZC#*^uUu9$Tl*Jty|~UdYuc>`$moMyprJkM;~*-KEaxa4IP-vgrZ2)%oqrckgphfcn2gqrvb~MxP_(X zltYlz(mNlPa)K~Ii3yYmLDnS85+M#qOXMAFJlVc=sGqlj2&sr5hCKOVIIW<(6F)Hn z`mK+c%R(YevVU;gLICg8StqO9kU%{Tu6%4;{FDQu2;ChRV{=J=q<{d)6&g%7e+XNk{z4x6 zf?W_(7h&6hL8PU=W_RW(2N|$b@J>3KB07QwXHzi#1Bh^03YE(Xo)NNgF)bgA+UP0~ zx!4pbxWUj2^^`aw36aTPE_iEa;AlC^d-szUyZC(2@ek~}$c(Pej1}e}p()L!=#XhM zBTr*|{ptk@=#2(5Tvx#bF;4)!J;X-!>m=I{V>3sm#p6pA#w4u)w|X_JZFs)%-ZE8x z+UJQCVq2(WsOx=DQ&&X4#jBEvY#xUjr3FvC?WXA|&a?}WH+$AJyMl5nfvp`J8okum zE+^kC_xI>~zg(L%xN7i}ZMGU;yD|bS-yB=@5oe4RYolg0Po$RZVc|giWFRVWjJqb= z!7^VyeXe}^38FlD$3TGh>h_iGJUxnWjk^a4PFvj`IpR!oDer#-plG0MiPdkK%NDZz zzEDGlC)4FLo8nrJa!0y4svj4CSyLG0ZUYyk_<;QLDIT+eIEl!P^FAVX>z^b#)8;)7 zcO{6syRjR6aDib-p$5@qjsPDvyk_a;d9mMW)B{;X%WURiHnv+9BD{}1k4nHTE8oAN z9gjs*W0=2hfq`GsU(y6|8r;8>2w@-C?>+2seoF$CoqiwY6(5YGA~8vJ1;P9$u^HBFKrWZo=y|CMuO`=-j@^2fQw$mwQEg)m^x_P;)!z(RLLB zEcJeOF$gdUG1D)+k*N{Ofe~pHXRCb|oGkfB5Z9`1HhpXWt*YVdNJ-w)$MqEWDMDq4 zO@(=6>2!>ot~qk?NbC%nEy^9TTeY5iY$n9hM^t=YwK5H{6t?75h(DispVp0t*+Msg zJ}M;1JY?nGjveA1J;f9W9k?29ZJf}e+kaJ;%RL_mKz(Dbo~pSw;nz;lR*giz_CAQJ zYMLgDFfrXA(CY`5WtO|Hx$$B&K2N0&Uh zFb2#@O}c11Z}|S<)~gZKVGGv$jt~kx4greHr5PZEhsh|J_F=^13p{Z18~NI!5}(j7 zHGA3K@eGRIa3@!vkg4S>weLwSqpAhMsNA>-Z@>_lE0xdh)X)?r=8>r|!pX;yE__TI z3}_wCk8rR82b74+l<`kBKBF!M!?(PQHZ5k0ol5%3cQahz0x^yx?3ew9nT{-zCNc%Y zO6Mr_OK2Fn1!jT979$qLF%e!&akC^3o6Mr%A>Zf4{Zotft4x-6ZYa95ByfN}%9x7n zx7FL!Ta_PKJf^n$LTC#+#~gFCs*~}8+RnsQOg|J)!30JJ(k^;GSA4xM?erD0M^@ET zq9z)rK<#UHNC{ITB{OD6+{z~b08X3!UvvN9 zg^rD#6&j`Bi%HLugkZOu#ZG-28wR&8G_9_ zZ3F%6wd*3Ue-iu}%IeWR`}iR!U97wLm|S|&edFcmjtM5=c`b7oQ~Bc`;|j%YZmgvD zRH-)XFs*K#QKAknlsGSyP^DZ8vuk`9iFCz=?o0G?NjJtJi-L`n9dM1~h{MkKoG>OY zCbtoosy@NKK5Z9f%)|tz_O34>;8t$JaTLwW}-1;gLdEK zj8|FWmgCU)013y!cLpHR0%#O%J4=xP(|4MPA4ZN0l!5g<;tF%V z;P(1m(Ay(MipFeFwZ3f$j>q9Nd>4nfg3|>a(CT`~>gv24tX01rV$yD&{gA)>m>-)J z58=i0?5SJt=Y4PR-W=Fjek-p$h6R#$4x6P4)UD|6O&mRISST1>t%ICfO|fBibP1wl zky}G~Ba#jSz+dcX4{m-iszCp^Sph%=jt089L!^0*<&NR$@>+hG+?u|o9}kQ=3%)`5 z$Zp?f-LgSYPCS_itMGCTCi_hJ+E8W0$AZ&S%bh!|Bp3v>caRY*HM=6Er`??5qk?0@ z+#6eyLrL+!%xFMu_3K#0XM&Wb4+isdfzN@;KYOj*Fb>(=I3^kE?^#z zBXLCUK%m?&C6VAK!8jL`w6wh^Ia7UkXK|R0URgBvEoeZmWi(893@}eVJ4K^pKgt2k zb>^X#{O$m%6MiybtC)gkW&Kg&dIYWY0o zE?ZT-?vFn&LZ&sBHNZ>zM338R*YLGYw=9uJ@my z$i92&+ZxBVHH)+S^5VuYAizprG zy8CV1+ga2|C0=np_S(WwNV9{&u7uqRlE_9oUiad(`}f7b4Ek}|-PO_KD{p7p^s=kK zr7`m(Pg3ePId?VSgaYU?6h4rLU#-2ID-|`YX1nv~^O*-cl z=wG1v6yddmRujga%egL_L0&Z4wO>{|FgT86l~nPES`rmX<`VkdepVl+O(QvogYoS9 zKRD9z{RlI3>^Z8?)qtq8Hf!l9Kae)-@=uM7erm$YjYxRf`y@FmOIyl13%iy)ok}h; 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This framework is described - The implementation of [Sachs scheme](Sachs.html) and [Taylor scheme](Taylor.html) show how to implement basic homogenized bounds with any behaviour laws on the phases using `@BehaviourVariable`. +- The implementation of Idiart's scheme [@idiart_thermodynamic_2025] for a + maxwellian viscoelastic composite with oriented ellipsoid is available [here](https://github.com/thelfer/MFrontGallery/blob/master/generic-behaviours/homogenization/Idiart_viscoelastic.mfront) - The description of the implementation of a polycrystal behaviour based on the Berveiller-Zaoui homogenisation scheme using an explicit scheme is available. @@ -103,7 +105,7 @@ behaviour. This framework is described - [\(\beta\)-rule](BetaRule.html) can also be implemented with `@BehaviourVariable` and hence any behaviour law on each phase. - A tutorial on implementation of Ponte-Castaneda bound [@castaneda_new_1992] - for a non-linear elastic composite is detaile [here](PonteCastaneda1992.html). + for a non-linear elastic composite is detailed [here](PonteCastaneda1992.html). - A tutorial on implementation of the [affine formulation](MassonAffineFormulation.html) from Masson et al. [@masson_affine_2000] for a viscoplastic polycrystal. From b6ffeb09cb5e73100f47b96b1c4252c64592464b Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Wed, 4 Feb 2026 11:03:31 +0100 Subject: [PATCH 14/25] minor modif --- docs/web/MassonAffineFormulation.md | 63 +++++++++++++++-------------- 1 file changed, 32 insertions(+), 31 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index c4a2c5960..6e1902bfe 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -28,7 +28,8 @@ eqnPrefixTemplate: "($$i$$)" \newcommand{\te}{\tenseurq{e}} \newcommand{\sigmaeq}{\sigma_{\mathrm{eq}}} \newcommand{\epsiloneq}{\varepsilon_{\mathrm{eq}}} - +\newcommand{\tE}{\tenseur E} +\newcommand{\tSigma}{\tenseur \Sigma} \newcommand{\tepsilonto}{\underline{\epsilon}^{\mathrm{to}}} \newcommand{\tepsilonel}{\underline{\epsilon}^{\mathrm{el}}} @@ -94,16 +95,16 @@ where $N$ is the number of phases (or crystals) and $\chi_r$ is characteristic f ## The non-linear homogenization problem -We impose a strain rate $\overline{\tepsilon}$ to the polycrystal and look for the solution $\tepsilon,\tsigma$ such that: +We impose a strain rate $\tE$ to the polycrystal and look for the solution $\tepsilon,\tsigma$ such that: \[ \begin{aligned} &\div\,\tsigma=\tenseur 0\\ - &\tepsilon\in\Ka(\overline{\tepsilon})\\ + &\tepsilon\in\Ka(\tE)\\ &\tepsilon = \deriv{\psi}{\tsigma}\left(\tsigma\right) \end{aligned} \] -where we introduced the space of kinematically admissible fields $\mathcal K(\overline{\tepsilon})$, depending on the boundary conditions used (in the implementation, we work with periodic boundary conditions). +where we introduced the space of kinematically admissible fields $\mathcal K(\tE)$, depending on the boundary conditions used (in the implementation, we work with periodic boundary conditions). # The affine formulation @@ -121,7 +122,7 @@ The idea is to linearize the behaviour around a reference stress $\tsigma^r$: can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by \[ \begin{aligned} - \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) + \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) \end{aligned} \] where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). @@ -139,7 +140,7 @@ where $\tsigma$ here stands for the stress solution of the thermoelastic homogen The macroscopic stress hence can be obtained by the classical relation: \[ \begin{aligned} -\overline{\tsigma}=\sum_rc_r\,\langle\tsigma\rangle_r=\tenseurq C^{\mathrm{eff}}\dbldot\overline{\tepsilon}+\tenseur\tau^{\mathrm{eff}} +\tSigma=\sum_rc_r\,\langle\tsigma\rangle_r=\tenseurq C^{\mathrm{eff}}\dbldot\tE+\tenseur\tau^{\mathrm{eff}} \end{aligned} \] where @@ -152,11 +153,11 @@ where $\tsigma$ here stands for the stress solution of the thermoelastic homogen The tangent operator is obtained by derivation: \[ \begin{aligned} - \dfrac{\mathrm{d}\overline{\tsigma}}{\mathrm{d}\overline{\tepsilon}}=\sum_r c_r\,\deriv{\tsigma^r}{\overline{\tepsilon}} + \dfrac{\mathrm{d}\tSigma}{\mathrm{d}\tE}=\sum_r c_r\,\deriv{\tsigma^r}{\tE} \end{aligned} \] -In the expressions of the tangent operator, we can compute the term $\deriv{\tsigma^r}{\overline{\tepsilon}}$ by means of the jacobian matrix, also done and explained in the documentation of the [Implicit DSL](https://thelfer.github.io/tfel/web/implicit-dsl.html#computation-of-the-consistent-tangent-operator). We explain it below. +In the expressions of the tangent operator, we can compute the term $\deriv{\tsigma^r}{\tE}$ by means of the jacobian matrix, also done and explained in the documentation of the [Implicit DSL](https://thelfer.github.io/tfel/web/implicit-dsl.html#computation-of-the-consistent-tangent-operator). We explain it below. # Summary and possible implementations @@ -170,7 +171,7 @@ where \(\tsigma\) is the stress field solution of a homogenization problem of th \[ \begin{aligned} &\div\,\tsigma=\tenseur 0\\ -&\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\overline{\tepsilon}\right)\\ +&\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\tE\right)\\ &\tenseurq M_r=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r \end{aligned} \] @@ -180,7 +181,7 @@ where \(\tsigma\) is the stress field solution of a homogenization problem of th The iterative resolution of the non-linear system can be summarized as \[ -\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq M_r= \pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right),\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\tsigma^r = \tenseurq A_r\dbldot\overline{\tepsilon} + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} +\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq M_r= \pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right),\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\tsigma^r = \tenseurq A_r\dbldot\tE + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} \] The first step of the resolution consists in computing the moduli $\tenseurq M_r$ and free strains $\tenseur e^r$. It can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the local behaviour in an external file (in our example, the potential is to simple to do that). @@ -194,33 +195,33 @@ The macroscopic stress and the tangent operator must also be computed with the f ### Computation via morphological tensors -In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\overline{\tepsilon}\right)$, we have, because, $\div\,\tsigma=0$, +In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\tE\right)$, we have, because, $\div\,\tsigma=0$, \[ \begin{aligned} -\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseur\tau\right) \end{aligned} \] where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives \[ \begin{aligned} -\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau +\tsigma=\tenseurq L_0\dbldot\tE-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau \end{aligned} \] and hence (due to the expressions of $\tenseur\tau$ and $\tepsilon$) \[ \begin{aligned} -\tsigma=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) +\tsigma=\tenseurq L_0\dbldot\tE-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) \end{aligned} \] \[ \begin{aligned} -\tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\overline{\tepsilon}-\tenseurq \Delta_0\left(\tenseur e^r\right) +\tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\tE-\tenseurq \Delta_0\left(\tenseur e^r\right) \end{aligned} \] with $\tenseurq\delta\tenseurq M_r=\tenseurq M_r-\tenseurq M_0$, $\tenseurq M_0=\tenseurq L_0^{-1}$, and $\tenseurq \Delta_0=\tenseurq L_0-\tenseurq L_0\dbldot\tenseurq \Gamma_0\tenseurq L_0$. We will not resolve this equation exactly, but we will consider the average of the fields: \[ \begin{aligned} -\langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\overline{\tepsilon} -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s +\langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\tE -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s \end{aligned} \] where @@ -248,7 +249,7 @@ to compute the macroscopic stress: \[ \begin{aligned} -\overline{\tsigma} = \tenseurq L_{\mathrm{SC}}\dbldot\overline{\tepsilon}+\tenseur \tau_{\mathrm{SC}} +\tSigma = \tenseurq L_{\mathrm{SC}}\dbldot\tE+\tenseur \tau_{\mathrm{SC}} \end{aligned} \] @@ -256,13 +257,13 @@ However, we also need the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ for com the stress field $\langle\tsigma\rangle_r$ (i.e. the residue). To that extent, we go back to the equation \[ \begin{aligned} -\tepsilon=\overline{\tepsilon}-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseur\tau\right) \end{aligned} \] and consider averages of the fields: \[ \begin{aligned} -\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s +\langle\tepsilon\rangle_r-\tE=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s \end{aligned} \] with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r$. @@ -270,29 +271,29 @@ with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ (i.e. the grains have a spherical shape) so that \[ \begin{aligned} -\langle\tepsilon\rangle_r-\overline{\tepsilon}=-\tenseurq P_0\dbldot\left(\langle\tenseur\tau\rangle_r-\langle\tenseur\tau\rangle\right) +\langle\tepsilon\rangle_r-\tE=-\tenseurq P_0\dbldot\left(\langle\tenseur\tau\rangle_r-\langle\tenseur\tau\rangle\right) \end{aligned} \] and due to the expression of $\tenseur\tau$: \[ \begin{aligned} -\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\overline{\tepsilon}\right)=-\left[\langle\tsigma\rangle_r-\overline{\tsigma}\right] +\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\tE\right)=-\left[\langle\tsigma\rangle_r-\tSigma\right] \end{aligned} \] We know that the reference medium, in the self-consistent approach, is identified with the homogenized medium. Hence, we note $\tenseurq P_0=\tenseurq P_{\mathrm{SC}}$, $\tenseurq L_0=\tenseurq L_{\mathrm{SC}}$, and $\tenseurq M_{\mathrm{SC}}^*=\left[\tenseurq P_{\mathrm{SC}}^{-1}-\tenseurq L_{\mathrm{SC}}\right]^{-1}$, so that, with $\langle\tepsilon\rangle_r=\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r$, and $\langle\tsigma\rangle_r=\tsigma^r$ \[ \begin{aligned} -\left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\overline{\tepsilon}\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tsigma^r-\overline{\tsigma}\right] +\left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\tE\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tsigma^r-\tSigma\right] \end{aligned} \] which gives \[ \begin{aligned} -\tsigma^r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left[\overline{\tepsilon}-\tenseur e^r+\tenseurq M_{\mathrm{SC}}^*\dbldot\overline{\tsigma}\right] +\tsigma^r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left[\tE-\tenseur e^r+\tenseurq M_{\mathrm{SC}}^*\dbldot\tSigma\right] \end{aligned} \] -which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s$ (we use $\overline{\tsigma}=\tenseurq L_{\mathrm{SC}}\dbldot\overline{\tepsilon}+\tenseur \tau_{\mathrm{SC}}$): +which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s$ (we use $\tSigma=\tenseurq L_{\mathrm{SC}}\dbldot\tE+\tenseur \tau_{\mathrm{SC}}$): \[ \begin{aligned} \tenseurq A_r&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_{\mathrm{SC}}\right)\\ @@ -301,7 +302,7 @@ which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{ \] (and this is sufficient to compute the residue). However, we note that the library `tfel::material::homogenization` provides the tensor $\tenseurq A_{\mathrm{SC}}$ via the variable `mean_strain_localisation_tensors`, which relates -$\langle\tepsilon\rangle_r$ and $\overline{\tepsilon}$. This tensor is also equal to $\tenseurq A_{\mathrm{SC}}=\tenseurq M_r\dbldot\tenseurq A_r$, due to the expression of $\langle\tsigma\rangle_r$. This will be more simple for the implementation. +$\langle\tepsilon\rangle_r$ and $\tE$. This tensor is also equal to $\tenseurq A_{\mathrm{SC}}=\tenseurq M_r\dbldot\tenseurq A_r$, due to the expression of $\langle\tsigma\rangle_r$. This will be more simple for the implementation. # Implementation in MFront @@ -589,7 +590,7 @@ suited to strain type residues. Hence, we will define the residues as \[ \begin{aligned} -f_{\sigma^r}=\tenseurq M_r\dbldot\left(\tsigma^r-\tenseurq A_r\dbldot\overline{\tepsilon}-\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s\right) +f_{\sigma^r}=\tenseurq M_r\dbldot\left(\tsigma^r-\tenseurq A_r\dbldot\tE-\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s\right) \end{aligned} \] @@ -615,24 +616,24 @@ sig=LSC*(eto+deto)+tauSC; ### Tangent operator -The computation of the tangent operator necessitates the computation of $\deriv{\tsigma^r}{\overline{\tepsilon}}$. +The computation of the tangent operator necessitates the computation of $\deriv{\tsigma^r}{\tE}$. To do that, we use a technique widely used in `MFront`. The idea is to write the non-linear system to solve as: \[ \begin{aligned} -\tenseurq M_r\dbldot\left[\tsigma^r - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\overline{\tepsilon} - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)\right]= f_{\sigma^r}(\overline{\tepsilon},\tsigma^1,...,\tsigma^N)=0 +\tenseurq M_r\dbldot\left[\tsigma^r - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)\right]= f_{\sigma^r}(\tE,\tsigma^1,...,\tsigma^N)=0 \end{aligned} \] -and by derivation of $f_{\sigma^r}$ w.r.t. $\overline{\tepsilon}$: +and by derivation of $f_{\sigma^r}$ w.r.t. $\tE$: \[ \begin{aligned} --\tenseurq M_r\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\overline{\tepsilon}}=0 +-\tenseurq M_r\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\tE}=0 \end{aligned} \] where $\tenseurq J_{rs}=\deriv{f_{\sigma^r}}{\tsigma^s}$ stands for the sub-block $(r,s)$ of the Jacobian. Hence \[ \begin{aligned} -\deriv{\tsigma^r}{\overline{\tepsilon}}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq M_s\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) +\deriv{\tsigma^r}{\tE}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq M_s\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) \end{aligned} \] where $\mathbf {iJ}$ is the inverse of the Jacobian. The implementation is From 4e1317a329e70e743be5d7fd5255ecb4460fd20d Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Wed, 4 Feb 2026 13:12:12 +0100 Subject: [PATCH 15/25] minor correction on bindings python --- .../python/tfel/material/MicrostructureLinearHomogenization.cxx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/bindings/python/tfel/material/MicrostructureLinearHomogenization.cxx b/bindings/python/tfel/material/MicrostructureLinearHomogenization.cxx index 055443c4f..946ee1a6c 100644 --- a/bindings/python/tfel/material/MicrostructureLinearHomogenization.cxx +++ b/bindings/python/tfel/material/MicrostructureLinearHomogenization.cxx @@ -71,7 +71,7 @@ requires( HomogenizationScheme computeSelfConsistentScheme( tfel::material::homogenization::elasticity:: ParticulateMicrostructure µ, - const tfel::types::real &precision, + const tfel::types::real &tolerance, bool isotropic, int max_iter_anisotropic_integration, const std::vector> From 270448bc1d6833488c5a46c4e13ea981dd664fbb Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Wed, 4 Feb 2026 13:21:08 +0100 Subject: [PATCH 16/25] added functions in bindings python --- bindings/python/tfel/material/IsotropicModuli.cxx | 2 ++ 1 file changed, 2 insertions(+) diff --git a/bindings/python/tfel/material/IsotropicModuli.cxx b/bindings/python/tfel/material/IsotropicModuli.cxx index c2f8a56e9..fb440e34d 100644 --- a/bindings/python/tfel/material/IsotropicModuli.cxx +++ b/bindings/python/tfel/material/IsotropicModuli.cxx @@ -92,4 +92,6 @@ void declareIsotropicModuli(pybind11::module_& m) { declareKGModuli(m, "KGModuli"); declareYoungNuModuli(m, "YoungNuModuli"); declareLambdaMuModuli(m, "LambdaMuModuli"); + m.def("computeKGModuli", &tfel::material::computeKGModuli); + m.def("computeIsotropicStiffnessTensor", &tfel::material::computeIsotropicStiffnessTensor); } From 8c97d053a1d26464808ddf846306a066a1c3a0e6 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Wed, 11 Feb 2026 14:51:57 +0100 Subject: [PATCH 17/25] updated docs and add class ExtendedPolycrystals --- docs/web/BiphasicLinearHomogenization.md | 8 +- docs/web/MassonAffineFormulation.md | 10 +-- docs/web/tfel-material.md | 12 ++- include/CMakeLists.txt | 2 - .../ExtendedPolyCrystalsSlidingSystems.hxx | 70 ---------------- .../ExtendedPolyCrystalsSlidingSystems.ixx | 81 ------------------- include/TFEL/Material/IsotropicModuli.hxx | 15 ++-- .../Material/PolyCrystalsSlidingSystems.hxx | 30 ++++++- .../Material/PolyCrystalsSlidingSystems.ixx | 56 +++++++++++++ tests/Material/IsotropicModuli.cxx | 18 +++++ 10 files changed, 131 insertions(+), 171 deletions(-) delete mode 100644 include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx delete mode 100644 include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx diff --git a/docs/web/BiphasicLinearHomogenization.md b/docs/web/BiphasicLinearHomogenization.md index a3b731c44..9f0e43432 100644 --- a/docs/web/BiphasicLinearHomogenization.md +++ b/docs/web/BiphasicLinearHomogenization.md @@ -127,13 +127,13 @@ std::array tab_k={k0,ki}; auto mu0=E0/2/(1+nu0); auto mui=Ei/2/(1+nui); std::array tab_mu={mu0,mui}; -auto pa2=computeIsotropicHashinShtrikmanBounds<3u,2u,stress>(tab_f,tab_k,tab_mu); +auto pa2=computeIsotropicHashinShtrikmanBounds<3u,stress>(tab_f,tab_k,tab_mu); auto UB=std::get<1>(pa2); auto kh=std::get<0>(UB); -auto muh=std::get<1>(UB) -kg=KGModuli(kh,muh); -auto Enu=kg.ToYoungNu(); +auto muh=std::get<1>(UB); +kg=tfel::material::KGModuli(kh,muh); }; +auto Enu=kg.ToYoungNu(); E=Enu.young; } ~~~~ diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 6e1902bfe..6983df6ce 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -382,7 +382,7 @@ for the homogenization of the thermoelastic composite: and the header for handling the rotations of the grains with their slip systems: - - ExtendedPolyCrystalsSlidingSystems.hxx + - PolyCrystalsSlidingSystems.hxx ~~~~ {#Begin .cpp .numberLines} @TFELLibraries {"Material"}; @@ -391,7 +391,7 @@ slip systems: #include "TFEL/Material/IsotropicEshelbyTensor.hxx" #include "TFEL/Material/MicrostructureDescription.hxx" #include "TFEL/Material/MicrostructureLinearHomogenization.hxx" -#include "TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +#include "TFEL/Material/PolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For the approach based on morphological tensors, we do not need all these headers, @@ -403,7 +403,7 @@ located at the same place as the `.mfront` file. @TFELLibraries {"Material"}; @Includes{ #include "tensors.hxx" -#include "TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx"} +#include "TFEL/Material/PolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The compilation hence will be done like that: @@ -434,10 +434,10 @@ but the rotations of the grains will be performed later, in the `Integrator` code block: ~~~~ {#Integrator .cpp .numberLines} -using PolyCrystalsSlidingSystems = +using ExtendedPolyCrystalsSlidingSystems = ExtendedPolyCrystalsSlidingSystems, real>; const auto& gs = - PolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems("polycrystal.csv"); + ExtendedPolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems("polycrystal.csv"); ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ### Poperties and variables diff --git a/docs/web/tfel-material.md b/docs/web/tfel-material.md index 736bb0015..8ba9473e6 100644 --- a/docs/web/tfel-material.md +++ b/docs/web/tfel-material.md @@ -433,6 +433,8 @@ to the tolerance `eps`. The homogenization functions are part of the namespace `tfel::material::homogenization`. A specialization for elasticity is defined: `tfel::material::homogenization::elasticity`. +Note that the functionalities below are also available in +the `Python` module (see the doc [here](tfel-python.html#the-tfel.material.homogenization-module)). ## Eshelby, Hill and localisation tensors @@ -608,6 +610,11 @@ not isotropic, it must be provided in the local basis defined by `n_a,n_b`. ## Homogenization schemes for biphasic media +Note that the functionalities below are also available in +the `Python` module (see the doc [here](tfel-python.html#homogenization-schemes-in-biphasic-media)). +See also [here](BiphasicLinearHomogenization.html) a tutorial on the computation of homogenized schemes for biphasic +particulate microstructures. + Different classical mean-field homogenization schemes are implemented for biphasic media. These schemes are introduced by the header `LinearHomogenizationSchemes.hxx`. They only deal with isotropic matrices and locally isotropic inclusions @@ -709,9 +716,6 @@ const auto C_PCW = computePCWScheme(E0,nu0,f,Ei,nui,A_av,D); ~~~~ Here, the three above schemes return `st2tost2` objects. -A tutorial on the computation of homogenized schemes for biphasic particulate microstructures -is available [here](BiphasicLinearHomogenization.html). - ## Homogenization bounds @@ -796,6 +800,8 @@ of the `pair` corresponds to a phase (matrix or spheres). A `ParticulateMicrostructure` object can be created for homogenization of general matrix-inclusion microstructures. +Note that the functionalities below are also available in +the `Python` module (see the doc [here](tfel-python.html#homogenization-of-general-microstructures)). ### Description and construction of a microstructure diff --git a/include/CMakeLists.txt b/include/CMakeLists.txt index 31a4f0c13..0a0b72fb1 100644 --- a/include/CMakeLists.txt +++ b/include/CMakeLists.txt @@ -575,8 +575,6 @@ install_header(TFEL/Material MetallicCFCGenericSlidingSystemsInteractionMatrix.h install_header(TFEL/Material MetallicCFCGenericSlidingSystemsInteractionMatrix.ixx) install_header(TFEL/Material PolyCrystalsSlidingSystems.hxx) install_header(TFEL/Material PolyCrystalsSlidingSystems.ixx) -install_header(TFEL/Material ExtendedPolyCrystalsSlidingSystems.hxx) -install_header(TFEL/Material ExtendedPolyCrystalsSlidingSystems.ixx) install_header(TFEL/Material OrthotropicAxesConvention.hxx) install_header(TFEL/Material OrthotropicAxesConvention.ixx) install_header(TFEL/Material GursonTvergaardNeedleman.hxx) diff --git a/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx b/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx deleted file mode 100644 index 10796a6d3..000000000 --- a/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx +++ /dev/null @@ -1,70 +0,0 @@ -/*! - * \file include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.hxx - * \brief - * \author Thomas Helfer - * \brief 16 mai 2013 - * \copyright Copyright (C) 2006-2018 CEA/DEN, EDF R&D. All rights - * reserved. - * This project is publicly released under either the GNU GPL Licence - * or the CECILL-A licence. A copy of thoses licences are delivered - * with the sources of TFEL. CEA or EDF may also distribute this - * project under specific licensing conditions. - */ - -#ifndef TFEL_MATERIAL_EXTENDEDPOLYCRISTALSSLIDINGSYSTEMS_HXX -#define TFEL_MATERIAL_EXTENDEDPOLYCRISTALSSLIDINGSYSTEMS_HXX 1 - -#include -#include "TFEL/Math/stensor.hxx" -#include "TFEL/Math/vector.hxx" - -namespace tfel { - - namespace material { - - /*! - * \param[in] Np : number of phases - * \param[in] GS : class describing the sliding system of one phase - * \param[in] NumType : numeric type used - */ - template - struct ExtendedPolyCrystalsSlidingSystems { - //! number of sliding systems of the mono crystal - static constexpr unsigned short Nss = GS::Nss; - //! a simple alias - typedef NumType real; - //! a simple alias - typedef tfel::math::stensor<3u, real> StrainStensor; - //! return the uniq instance of the class - static const ExtendedPolyCrystalsSlidingSystems &getPolyCrystalsSlidingSystems( - const std::string &); - //! tensor of directional senses, sorted by phases - tfel::math::vector> mus; - //! \brief volume fractions per phases - tfel::math::vector volume_fractions; - - private: - /*! - * default constructor (declared private~: it can only be used - * by the getExtendedPolyCrystalsSlidingSystems method) - * \param[in] f : file where the angles are declared - */ - ExtendedPolyCrystalsSlidingSystems(const std::string &); - /*! - * copy constructor (declared private to disable it) - */ - ExtendedPolyCrystalsSlidingSystems(const ExtendedPolyCrystalsSlidingSystems &); - /*! - * assignement operator (declared private to disable it) - */ - ExtendedPolyCrystalsSlidingSystems &operator=( - const ExtendedPolyCrystalsSlidingSystems &); - }; - - } // end of namespace material - -} // end of namespace tfel - -#include "./ExtendedPolyCrystalsSlidingSystems.ixx" - -#endif /* TFEL_MATERIAL_EXTENDEDPOLYCRISTALSSLIDINGSYSTEMS_HXX */ diff --git a/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx b/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx deleted file mode 100644 index b682be8b1..000000000 --- a/include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx +++ /dev/null @@ -1,81 +0,0 @@ -/*! - * \file include/TFEL/Material/ExtendedPolyCrystalsSlidingSystems.ixx - * \brief - * \author Thomas Helfer - * \brief 16 mai 2013 - * \copyright Copyright (C) 2006-2018 CEA/DEN, EDF R&D. All rights - * reserved. - * This project is publicly released under either the GNU GPL Licence - * or the CECILL-A licence. A copy of thoses licences are delivered - * with the sources of TFEL. CEA or EDF may also distribute this - * project under specific licensing conditions. - */ - -#ifndef TFEL_MATERIAL_EXTENDEDPOLYCRYSTALSSLIDINGSYSTEMS_IXX -#define TFEL_MATERIAL_EXTENDEDPOLYCRYSTALSSLIDINGSYSTEMS_IXX 1 - -#include -#include "TFEL/Utilities/CxxTokenizer.hxx" - - -namespace tfel { - - namespace material { - - template - const ExtendedPolyCrystalsSlidingSystems - &ExtendedPolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems( - const std::string &f) { - static ExtendedPolyCrystalsSlidingSystems gs(f); - return gs; - } - - template - ExtendedPolyCrystalsSlidingSystems:: - ExtendedPolyCrystalsSlidingSystems(const std::string &f) - : mus(Np), volume_fractions(Np) { - using namespace std; - using namespace tfel::math; - using namespace tfel::utilities; - // the sliding systems of one phase - const auto &gs = GS::getSlidingSystems(); - const real pi = real(4) * std::atan(real(1)); - // reading from the specified file - CxxTokenizer file(f); - file.stripComments(); - CxxTokenizer::const_iterator p = file.begin(); - const CxxTokenizer::const_iterator pe = file.end(); - for (unsigned short i = 0; i != Np; i++) { - this->mus[i].resize(Nss); - tmatrix<3u, 3u, real> drot; - const real psi = CxxTokenizer::readDouble(p, pe) * pi / 180.0; - const real the = CxxTokenizer::readDouble(p, pe) * pi / 180.0; - const real phi = CxxTokenizer::readDouble(p, pe) * pi / 180.0; - const real cospsi = std::cos(psi); - const real costhe = std::cos(the); - const real cosphi = std::cos(phi); - const real sinpsi = std::sin(psi); - const real sinthe = std::sin(the); - const real sinphi = std::sin(phi); - drot(0, 0) = cosphi * cospsi - sinphi * costhe * sinpsi; - drot(0, 1) = cosphi * sinpsi + sinphi * costhe * cospsi; - drot(0, 2) = sinphi * sinthe; - drot(1, 0) = -sinphi * cospsi - cosphi * costhe * sinpsi; - drot(1, 1) = -sinphi * sinpsi + cosphi * costhe * cospsi; - drot(1, 2) = cosphi * sinthe; - drot(2, 0) = sinthe * sinpsi; - drot(2, 1) = -sinthe * cospsi; - drot(2, 2) = costhe; - for (unsigned short j = 0; j != Nss; j++) { - auto &mu = this->mus[i][j]; - mu = gs.mus[j]; - mu.changeBasis(drot); - } - this->volume_fractions[i] = CxxTokenizer::readDouble(p, pe); - } - } - } // end of namespace material - -} // end of namespace tfel - -#endif /* TFEL_MATERIAL_EXTENDEDPOLYCRYSTALSSLIDINGSYSTEMS_IXX */ diff --git a/include/TFEL/Material/IsotropicModuli.hxx b/include/TFEL/Material/IsotropicModuli.hxx index 98a147a51..0beb99f45 100644 --- a/include/TFEL/Material/IsotropicModuli.hxx +++ b/include/TFEL/Material/IsotropicModuli.hxx @@ -61,10 +61,7 @@ namespace tfel::material { requires( tfel::math::checkUnitCompatibility()) struct IsotropicModuli { - IsotropicModuli() = default; - IsotropicModuli(const IsotropicModuli& IM) = default; - virtual IsotropicModuli& operator=( - const IsotropicModuli& IM) = default; + virtual ~IsotropicModuli() = default; virtual YoungNuModuli ToYoungNu() const = 0; virtual LambdaMuModuli ToLambdaMu() const = 0; @@ -78,7 +75,9 @@ namespace tfel::material { : public IsotropicModuli { StressType young; types::real nu; - + + YoungNuModuli() = default; + YoungNuModuli(const YoungNuModuli& ) = default; YoungNuModuli(const StressType& Young, const types::real& Nu) : IsotropicModuli(), young(Young), nu(Nu) {} @@ -112,6 +111,9 @@ namespace tfel::material { : public IsotropicModuli { StressType kappa; StressType mu; + + KGModuli() = default; + KGModuli(const KGModuli& ) = default; KGModuli(const StressType& Kappa, const StressType& Mu) : IsotropicModuli(), kappa(Kappa), mu(Mu) {} @@ -146,6 +148,9 @@ namespace tfel::material { : public IsotropicModuli { StressType lambda; StressType mu; + + LambdaMuModuli() = default; + LambdaMuModuli(const LambdaMuModuli& ) = default; LambdaMuModuli(const StressType& Lambda, const StressType& Mu) : IsotropicModuli(), lambda(Lambda), mu(Mu) {} diff --git a/include/TFEL/Material/PolyCrystalsSlidingSystems.hxx b/include/TFEL/Material/PolyCrystalsSlidingSystems.hxx index 47e663687..83a9aeb79 100644 --- a/include/TFEL/Material/PolyCrystalsSlidingSystems.hxx +++ b/include/TFEL/Material/PolyCrystalsSlidingSystems.hxx @@ -35,7 +35,7 @@ namespace tfel::material { typedef NumType real; //! a simple alias typedef tfel::math::stensor<3u> StrainStensor; - //! return the uniq instance of the class + //! return the unique instance of the class static const PolyCrystalsSlidingSystems& getPolyCrystalsSlidingSystems( const std::string& = ""); //! tensor of directional senses, sorted by phases @@ -57,6 +57,34 @@ namespace tfel::material { */ PolyCrystalsSlidingSystems& operator=(const PolyCrystalsSlidingSystems&); }; + + /*! + * \brief the same class as PolyCrystalsSlidingSystems but with + * volume fraction associated to each grain (fourth item of each line of the file + * given as an argument to the method getPolyCrystalsSlidingSystems) + * \param[in] Np : number of phases + * \param[in] GS : class describing the sliding system of one phase + * \param[in] NumType : numeric type used + */ + template + struct ExtendedPolyCrystalsSlidingSystems { + static constexpr unsigned short Nss = GS::Nss; + typedef NumType real; + typedef tfel::math::stensor<3u, real> StrainStensor; + + static const ExtendedPolyCrystalsSlidingSystems &getPolyCrystalsSlidingSystems( + const std::string &); + tfel::math::vector> mus; + + //! \brief volume fractions per phases + tfel::math::vector volume_fractions; + + private: + ExtendedPolyCrystalsSlidingSystems(const std::string &); + ExtendedPolyCrystalsSlidingSystems(const ExtendedPolyCrystalsSlidingSystems &); + ExtendedPolyCrystalsSlidingSystems &operator=( + const ExtendedPolyCrystalsSlidingSystems &); + }; } // end of namespace tfel::material diff --git a/include/TFEL/Material/PolyCrystalsSlidingSystems.ixx b/include/TFEL/Material/PolyCrystalsSlidingSystems.ixx index 695f407e0..052b05255 100644 --- a/include/TFEL/Material/PolyCrystalsSlidingSystems.ixx +++ b/include/TFEL/Material/PolyCrystalsSlidingSystems.ixx @@ -87,6 +87,62 @@ namespace tfel::material { } } } + + template + const ExtendedPolyCrystalsSlidingSystems + &ExtendedPolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems( + const std::string &f) { + static ExtendedPolyCrystalsSlidingSystems gs(f); + return gs; + } + + template + ExtendedPolyCrystalsSlidingSystems:: + ExtendedPolyCrystalsSlidingSystems(const std::string &f) + : mus(Np), volume_fractions(Np) { + using namespace std; + using namespace tfel::math; + using namespace tfel::utilities; + // the sliding systems of one phase + const auto &gs = GS::getSlidingSystems(); + const real pi = real(4) * std::atan(real(1)); + // reading from the specified file + CxxTokenizer file(f); + file.stripComments(); + CxxTokenizer::const_iterator p = file.begin(); + const CxxTokenizer::const_iterator pe = file.end(); + for (unsigned short i = 0; i != Np; i++) { + this->mus[i].resize(Nss); + tmatrix<3u, 3u, real> drot; + const real psi = CxxTokenizer::readDouble(p, pe) * pi / 180.0; + const real the = CxxTokenizer::readDouble(p, pe) * pi / 180.0; + const real phi = CxxTokenizer::readDouble(p, pe) * pi / 180.0; + const real cospsi = std::cos(psi); + const real costhe = std::cos(the); + const real cosphi = std::cos(phi); + const real sinpsi = std::sin(psi); + const real sinthe = std::sin(the); + const real sinphi = std::sin(phi); + drot(0, 0) = cosphi * cospsi - sinphi * costhe * sinpsi; + drot(0, 1) = cosphi * sinpsi + sinphi * costhe * cospsi; + drot(0, 2) = sinphi * sinthe; + drot(1, 0) = -sinphi * cospsi - cosphi * costhe * sinpsi; + drot(1, 1) = -sinphi * sinpsi + cosphi * costhe * cospsi; + drot(1, 2) = cosphi * sinthe; + drot(2, 0) = sinthe * sinpsi; + drot(2, 1) = -sinthe * cospsi; + drot(2, 2) = costhe; + for (unsigned short j = 0; j != Nss; j++) { + auto &mu = this->mus[i][j]; + mu = gs.mus[j]; + mu.changeBasis(drot); + } + this->volume_fractions[i] = CxxTokenizer::readDouble(p, pe); + } + } + + + } // end of namespace tfel::material #endif /* TFEL_MATERIAL_POLYCRYSTALSSLIDINGSYSTEMS_IXX */ diff --git a/tests/Material/IsotropicModuli.cxx b/tests/Material/IsotropicModuli.cxx index 4ecd59b5d..f4a147518 100644 --- a/tests/Material/IsotropicModuli.cxx +++ b/tests/Material/IsotropicModuli.cxx @@ -57,6 +57,8 @@ struct IsotropicModuliTest final : public tfel::tests::TestCase { using namespace tfel::material; const auto tens = 3 * kappa * J + 2 * mu * K; const auto yes_it_is_isotropic = isIsotropic(tens, 2 * eps); + + TFEL_TESTS_ASSERT(yes_it_is_isotropic); @@ -64,6 +66,22 @@ struct IsotropicModuliTest final : public tfel::tests::TestCase { const auto Enu = KG.ToYoungNu(); const auto LG = KG.ToLambdaMu(); const auto Enu_bis = LG.ToYoungNu(); + + tfel::material::LambdaMuModuli lg_ter; + tfel::material::LambdaMuModuli lg_quad(LG); + lg_quad= LG; + lg_ter= LG; + + tfel::material::YoungNuModuli Enu_ter; + tfel::material::YoungNuModuli Enu_quad(Enu_bis); + Enu_quad= Enu_bis; + Enu_ter= Enu_bis; + + tfel::material::KGModuli kg_ter; + tfel::material::KGModuli kg_quad(KG); + kg_quad= KG; + kg_ter= KG; + const auto E = Enu.young; const auto nu = Enu.nu; From 643c12154167e7545218e2288efecd1aa33eef51 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Thu, 12 Feb 2026 18:11:20 +0100 Subject: [PATCH 18/25] updated Affine formulation --- docs/web/MassonAffineFormulation.md | 209 ++++++++++--------- docs/web/downloads/Morphological_tensors.zip | Bin 4801 -> 4813 bytes docs/web/img/Affine_formulation.png | Bin 25680 -> 42094 bytes 3 files changed, 107 insertions(+), 102 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 6983df6ce..5ac2e84ab 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -116,37 +116,51 @@ The idea is to linearize the behaviour around a reference stress $\tsigma^r$: \] where \[ - \tenseurq M_r\left(\tsigma^r\right)=\deriv{\tepsilon_r}{\tsigma}\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\tenseur e^r = \tepsilon_r(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r + \tenseurq M_r\left(\tsigma^r\right)=\deriv{\tepsilon_r}{\tsigma}\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\quad\tenseur e^r = \deriv{\psi_r}{\tsigma}(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r \] - This affine behaviour can be viewed as a so-called "thermoelastic comparison composite", and this composite - can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tsigma$ is given by + For a given set of reference stresses, this affine behaviour can be viewed as a so-called "thermoelastic comparison composite", and this composite + can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tepsilon$ is given by \[ \begin{aligned} - \langle\tsigma\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) + \langle\tepsilon\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) \end{aligned} \] where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). -The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@masson_affine_2000] leads to the simple assumption that these reference stresses are the averages of the stress in the thermoelastic composite: +The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@masson_affine_2000] leads to the simple assumption that these reference stresses are the averages per phase of the stress in the thermoelastic composite: \[ \begin{aligned} \tsigma^r = \langle\tsigma\rangle_r \end{aligned} \] where $\tsigma$ here stands for the stress solution of the thermoelastic homogenization problem. + +Now, if $\tenseurq M_r$ is invertible and by noting $\tenseurq L_r=\tenseurq M_r^{-1}$, +the thermoelastic problem can be written as: +\[ +\begin{aligned} +\div\left[\tenseurq L_r\dbldot\left(\tepsilon-\tenseur e^r\right)\right]=0,\qquad\tepsilon\in\Ka\left(\tE\right) +\end{aligned} +\] +Hence, $\tsigma^r = \langle\tsigma\rangle_r$ is equivalent to $\tsigma^r = \tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)$, which is exactly +\[ +\begin{aligned} +\deriv{\psi_r}{\tsigma}(\tsigma^r) = \langle\tepsilon\rangle_r +\end{aligned} +\] ## Macroscopic stress and tangent operator The macroscopic stress hence can be obtained by the classical relation: \[ \begin{aligned} -\tSigma=\sum_rc_r\,\langle\tsigma\rangle_r=\tenseurq C^{\mathrm{eff}}\dbldot\tE+\tenseur\tau^{\mathrm{eff}} +\tSigma=\sum_rc_r\,\tsigma^r=\sum_rc_r\,\tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)=\tenseurq C^{\mathrm{eff}}\dbldot\tE+\tenseur\tau^{\mathrm{eff}} \end{aligned} \] where \[ \begin{aligned} - \tenseurq C^{\mathrm{eff}}=\sum_rc_r\,\tenseurq A_r\qquad\text{and}\qquad\tenseur\tau^{\mathrm{eff}}=\sum_{r,s}c_r\,\tenseurq B_{rs}\dbldot\tenseur e^s + \tenseurq C^{\mathrm{eff}}=\sum_rc_r\,\tenseurq L_r\dbldot\tenseurq A_r\qquad\text{and}\qquad\tenseur\tau^{\mathrm{eff}}=\sum_{r,s}c_r\,\tenseurq L_r\dbldot\tenseurq B_{rs}\dbldot\tenseur e^s-\sum_{r}c_r\,\tenseurq L_r\dbldot\tenseur e^r \end{aligned} \] @@ -165,14 +179,14 @@ In the expressions of the tangent operator, we can compute the term $\deriv{\tsi The resolution consists in \[ - \text{Find}\quad\tsigma^r\quad\text{such that}\quad\tsigma^r=\langle\tsigma\rangle_r + \text{Find}\quad\tsigma^r\quad\text{such that}\quad\deriv{\psi_r}{\tsigma}(\tsigma^r) = \langle\tepsilon\rangle_r \] -where \(\tsigma\) is the stress field solution of a homogenization problem of the form: +where \(\tepsilon\) is the strain field solution of a homogenization problem of the form: \[ \begin{aligned} -&\div\,\tsigma=\tenseur 0\\ -&\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\in\Ka\left(\tE\right)\\ -&\tenseurq M_r=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r +&\div\,\left[\tenseurq L_r\dbldot\left(\tepsilon-\tenseur e^r\right)\right]=\tenseur 0\\ +&\tepsilon\in\Ka\left(\tE\right)\\ +&\tenseurq L_r=\left[\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\right]^{-1}\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq L_r^{-1}\dbldot\tsigma^r \end{aligned} \] @@ -181,63 +195,67 @@ where \(\tsigma\) is the stress field solution of a homogenization problem of th The iterative resolution of the non-linear system can be summarized as \[ -\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq M_r= \pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right),\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq M_r\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\tsigma^r = \tenseurq A_r\dbldot\tE + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} +\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq L_r= \left[\pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\right]^{-1},\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq L_r^{-1}\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\langle\tepsilon\rangle_r = \tenseurq A_r\dbldot\tE + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} \] The first step of the resolution consists in computing the moduli $\tenseurq M_r$ and free strains $\tenseur e^r$. It can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the local behaviour in an external file (in our example, the potential is to simple to do that). The second step is the computation of the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$. This can be achieved by means of mean-field schemes (the `namespace` `tfel::material::homogenization` provides the good tensors). This can also be done via morphological tensors, computed on a given geometry of polycrystal. We will present below the two possibilities. -We note that the equation relative to the second step is the non-linear equation to solve and will hence correspond to our residue. -The macroscopic stress and the tangent operator must also be computed with the formulas shown above. - -## Computation of tensors Ar and Brs - -### Computation via morphological tensors - -In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995] for the primal version). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tepsilon=\tenseur e+\tenseurq M\dbldot\tsigma\in\Ka\left(\tE\right)$, we have, because, $\div\,\tsigma=0$, +We note that the equation relative to the second step will give our residue. +Indeed, the equation to solve is \[ \begin{aligned} -\tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseur\tau\right) +\deriv{\psi_r}{\tsigma}(\tsigma^r) = \langle\tepsilon\rangle_r = \tenseurq A_r\dbldot\tE + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s \end{aligned} \] -where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives +Hence the residue will be \[ \begin{aligned} -\tsigma=\tenseurq L_0\dbldot\tE-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tenseur\tau\right)+\tenseur\tau +f_{\tsigma^r}=\tepsilon_r\left(\tsigma^r\right)-\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE -\sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) \end{aligned} \] -and hence (due to the expressions of $\tenseur\tau$ and $\tepsilon$) +where \[ \begin{aligned} -\tsigma=\tenseurq L_0\dbldot\tE-\tenseurq L_0\dbldot\tenseurq \Gamma_0\left(\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right)\right)+\tsigma-\tenseurq L_0\dbldot\left(\tenseur e^r+\tenseurq M_r\dbldot\tsigma\right) +\tepsilon_r(\tsigma^r)=\deriv{\psi_r}{\tsigma}(\tsigma^r) \end{aligned} \] + +## Computation of tensors Ar and Brs + +### Computation via morphological tensors + +In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995]). +Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tsigma=\tenseurq L\dbldot\left(\tepsilon-\tenseur e\right)$. We have $\tenseur\tau=\tenseurq\delta\tenseurq L\dbldot\tepsilon-\tenseurq L\dbldot\tenseur e$ with $\tenseurq\delta\tenseurq L=\tenseurq L-\tenseurq L_0$. We have, because, $\div\,\tsigma=0$, \[ \begin{aligned} -\tsigma+\tenseurq \Delta_0\left(\tenseurq\delta\tenseurq M_r\dbldot\tsigma\right)=\tenseurq L_0\dbldot\tE-\tenseurq \Delta_0\left(\tenseur e^r\right) +\tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseur\tau\right) \end{aligned} \] -with $\tenseurq\delta\tenseurq M_r=\tenseurq M_r-\tenseurq M_0$, $\tenseurq M_0=\tenseurq L_0^{-1}$, and $\tenseurq \Delta_0=\tenseurq L_0-\tenseurq L_0\dbldot\tenseurq \Gamma_0\tenseurq L_0$. We will not resolve this equation exactly, but we will consider the average of the fields: +where $\tenseurq \Gamma_0$ is the Green operator associated to the elasticity $\tenseurq L_0$ (this elasticity must be chosen by the user). It gives (due to the expression of $\tenseur\tau$) \[ \begin{aligned} -\langle\tsigma\rangle_r + \sum_s\tenseurq \Delta_{rs}\dbldot\tenseurq\delta\tenseurq M_s\dbldot\langle\tsigma\rangle_s= \tenseurq L_0\dbldot\tE -\sum_s\tenseurq \Delta_{rs}\dbldot\tenseur e^s +\tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseurq\delta\tenseurq L\dbldot\tepsilon-\tenseurq L\dbldot\tenseur e\right) +\end{aligned} +\] +We will not resolve this equation exactly, but we will consider the average of the fields: +\[ +\begin{aligned} +\langle\tepsilon\rangle_r + \sum_s\tenseurq \Gamma_{rs}\dbldot\tenseurq\delta\tenseurq L_s\dbldot\langle\tepsilon\rangle_s= \tE +\sum_s\tenseurq \Gamma_{rs}\dbldot\tenseurq L_s\dbldot\tenseur e^s \end{aligned} \] where \[ \begin{aligned} -\tenseurq \Delta_{rs} = \sum_j\langle\,\tenseurq\Delta_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r +\tenseurq \Gamma_{rs} = \sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r \end{aligned} \] -is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor makes intervene the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). -Hence, the $\langle\tsigma\rangle_r$ 's are solution of a linear system. -Depending on the number of phases $N$, the computation of the $\tenseurq \Delta_{rs}$ is more or less costly, but it is achieved -at the beginning, once and for all. +is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor depends on the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). +Depending on the number of phases $N$, the computation of the $\tenseurq \Gamma_{rs}$ is more or less costly, but it is achieved +at the beginning, once and for all. By inverting the system above, we directly obtain the expressions of $\tenseurq A_r$ and $\tenseurq B_{rs}$. -We note that the reference medium can be adjusted with a scalar coefficient. Indeed, if the user wants to choose -a reference medium $r_0=10$ times stiffer, it is possible by dividing $\tenseurq M_0$ by $r_0$. It will be also necessary -to change $\tenseurq \Delta_{rs}$ and to multiply it by $r_0$. +We note that the reference medium can be chosen in this approach. A natural approach is to take a $\tenseurq L_0$ isotropic near $\tenseurq L^{\mathrm{hom}}$. ### Computation via a self-consistent scheme @@ -254,7 +272,7 @@ to compute the macroscopic stress: \] However, we also need the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ for computing the average of -the stress field $\langle\tsigma\rangle_r$ (i.e. the residue). To that extent, we go back to the equation +the strain field $\langle\tepsilon\rangle_r$ (i.e. the residue). To that extent, we go back to the equation \[ \begin{aligned} \tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseur\tau\right) @@ -266,7 +284,6 @@ and consider averages of the fields: \langle\tepsilon\rangle_r-\tE=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s \end{aligned} \] -with $\tenseurq\Gamma_{rs}=\sum_j\langle\,\tenseurq\Gamma_0(\chi_s\,\tenseur s_j)\otimes\tenseur s_j\,\rangle_r$. A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ (i.e. the grains have a spherical shape) so that \[ @@ -274,35 +291,35 @@ A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to con \langle\tepsilon\rangle_r-\tE=-\tenseurq P_0\dbldot\left(\langle\tenseur\tau\rangle_r-\langle\tenseur\tau\rangle\right) \end{aligned} \] -and due to the expression of $\tenseur\tau$: +and due to the expression of $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$: \[ \begin{aligned} \left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\tE\right)=-\left[\langle\tsigma\rangle_r-\tSigma\right] \end{aligned} \] We know that the reference medium, in the self-consistent approach, is identified with the homogenized medium. Hence, we note -$\tenseurq P_0=\tenseurq P_{\mathrm{SC}}$, $\tenseurq L_0=\tenseurq L_{\mathrm{SC}}$, and $\tenseurq M_{\mathrm{SC}}^*=\left[\tenseurq P_{\mathrm{SC}}^{-1}-\tenseurq L_{\mathrm{SC}}\right]^{-1}$, so that, with $\langle\tepsilon\rangle_r=\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r$, and $\langle\tsigma\rangle_r=\tsigma^r$ +$\tenseurq P_0=\tenseurq P_{\mathrm{SC}}$, $\tenseurq L_0=\tenseurq L_{\mathrm{SC}}$, and $\tenseurq M_{\mathrm{SC}}^*=\left[\tenseurq P_{\mathrm{SC}}^{-1}-\tenseurq L_{\mathrm{SC}}\right]^{-1}$, so that, with $\langle\tsigma\rangle_r=\tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)$, \[ \begin{aligned} -\left(\tenseurq M_r\dbldot\tsigma^r+\tenseur e^r-\tE\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tsigma^r-\tSigma\right] +\left(\langle\tepsilon\rangle_r-\tE\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)-\tSigma\right] \end{aligned} \] which gives \[ \begin{aligned} -\tsigma^r=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left[\tE-\tenseur e^r+\tenseurq M_{\mathrm{SC}}^*\dbldot\tSigma\right] +\langle\tepsilon\rangle_r=\left(\tenseurq I+\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_r\right)^{-1}\dbldot\left[\tE+\tenseurq M_{\mathrm{SC}}^*\dbldot\left(\tSigma+\tenseurq L_r\dbldot\tenseur e^r\right)\right] \end{aligned} \] which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s$ (we use $\tSigma=\tenseurq L_{\mathrm{SC}}\dbldot\tE+\tenseur \tau_{\mathrm{SC}}$): \[ \begin{aligned} -\tenseurq A_r&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_{\mathrm{SC}}\right)\\ -\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s&=\left(\tenseurq M_r+\tenseurq M_{\mathrm{SC}}^*\right)^{-1}\dbldot\left(\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseur \tau_{\mathrm{SC}}-\tenseur e^r\right) +\tenseurq A_r&=\left(\tenseurq I+\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_r\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_{\mathrm{SC}}\right)\\ +\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s&=\left(\tenseurq L_{\mathrm{SC}}^*+\tenseurq L_r\right)^{-1}\dbldot\left(\tenseur \tau_{\mathrm{SC}}+\tenseurq L_r\dbldot\tenseur e^r\right) \end{aligned} \] -(and this is sufficient to compute the residue). However, we note that the library `tfel::material::homogenization` provides -the tensor $\tenseurq A_{\mathrm{SC}}$ via the variable `mean_strain_localisation_tensors`, which relates -$\langle\tepsilon\rangle_r$ and $\tE$. This tensor is also equal to $\tenseurq A_{\mathrm{SC}}=\tenseurq M_r\dbldot\tenseurq A_r$, due to the expression of $\langle\tsigma\rangle_r$. This will be more simple for the implementation. +where $\tenseurq L_{\mathrm{SC}}^*=\left(\tenseurq M_{\mathrm{SC}}^*\right)^{-1}$. +And this is sufficient to compute the residue. However, we note that the library `tfel::material::homogenization` provides +the tensor $\tenseurq A_r$ via the variable `mean_strain_localisation_tensors`. Moreover, the library also provides $\tenseur \tau_{\mathrm{SC}}$ if we specify the polarizations on each phase. We hence only have to compute $\tenseurq L_{\mathrm{SC}}^*$. # Implementation in MFront @@ -329,23 +346,23 @@ All the files are available in the `MFrontGallery` project, [here](https://githu The potential $\psi_k^r$ will be of the form \[ \begin{aligned} - \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^{k,r}}{n+1}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n+1} + \psi_k^r \left(\tau\right)= \dfrac{\dot{\gamma}_0\tau_0^k}{n+1}\left(\dfrac{|\tau|}{\tau_0^k}\right)^{n+1} \end{aligned} \] - where the strain rate $\dot{\gamma}_0$ and the creep exponent $n\geq 1$ will be chosen identical for all $k,r$, and the resolved shear stress $\tau_0^{k,r}$ depends on $k,r$. + where the strain rate $\dot{\gamma}_0$ and the creep exponent $n\geq 1$ will be chosen identical for all $k,r$, and the resolved shear stress $\tau_0^k$ depends on $k$. The derivatives of the potential are \[ \begin{aligned} - &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r\\ - &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\tau|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r + &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r\\ + &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k}}\left(\dfrac{|\tau|}{\tau_0^{k}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r \end{aligned} \] and then defining $\overline{\tau}_k^r=\tsigma^r\dbldot\tenseur \mu_k^r$ where $\tsigma^r$ is the reference stress, we have \[ \begin{aligned} - &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k,r}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n-1}\tenseurq N_k^r\\ - &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k,r}}\right)^{n}\tenseur\mu_k^r + &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n-1}\tenseurq N_k^r\\ + &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r \end{aligned} \] @@ -440,9 +457,9 @@ const auto& gs = ExtendedPolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems("polycrystal.csv"); ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -### Poperties and variables +### Properties and variables -The state variables are the reference stresses $\tsigma^r$ on each phase. +The state variables are the stresses $\tsigma^r$ on each phase. ~~~~ {#Begin .cpp .numberLines} @StateVariable Stensor sigma[Np]; @@ -460,7 +477,7 @@ Note that among several local variables, one of them, for morphological tensors, is ~~~~ {#Begin .cpp .numberLines} -@LocalVariable std::array,Np>,Np> DELTA; +@LocalVariable std::array,Np>,Np> GAMMA; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This matrix will store the morphological tensors. Indeed, in the @@ -468,7 +485,7 @@ This matrix will store the morphological tensors. Indeed, in the which are in fact situated in the header `extra-headers/TFEL/Material/tensors.hxx` ~~~~ {#Begin .cpp .numberLines} -DELTA=Delta::get_tensor(); +GAMMA=Gamma::get_tensor(); ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ### Affine modulus and free-strain @@ -481,18 +498,21 @@ using namespace tfel::math; Stensor4 LSC; Stensor e[Np]; Stensor dpsi_dsigma[Np]; +auto tau0k = tau1; for (int r=0;rint(Nss/12)){tau0k=tau1;} + else{tau0k=tau2;} const auto Nkr = gs.mus[r][k]^gs.mus[r][k]; const auto taukr = gs.mus[r][k] | (sigma[r]+dsigma[r]); - const auto puisn_1 = pow(abs(taukr)/tau0[r], nexp-1); - const auto fac= nexp*gamma0/tau0[r]*puisn_1; + const auto puisn_1 = pow(abs(taukr)/tau0k, nexp-1); + const auto fac= nexp*gamma0/tau0k*puisn_1; M[r]+=fac*Nkr; - const auto puisn = puisn_1*abs(taukr)/tau0[r]; + const auto puisn = puisn_1*abs(taukr)/tau0k; const auto sgn= taukr< 0 ? -real(1) : real(1); dpsi_dsigma[r]+=sgn*gamma0*puisn*gs.mus[r][k]; } @@ -502,6 +522,9 @@ for (int r=0;r(KGSC); -const auto MSC_star = invert(invert(PSC)- LSC); +const auto LSC_star = invert(PSC)- LSC; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Note here that we also isotropize the homogenized stiffness with the function @@ -585,22 +608,15 @@ in the `computeSelfConsistent` function. ### Residues and jacobian -For the residues, we work on the strains (because `MFront` is more -suited to strain type residues. Hence, we will define the residues as - -\[ -\begin{aligned} -f_{\sigma^r}=\tenseurq M_r\dbldot\left(\tsigma^r-\tenseurq A_r\dbldot\tE-\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s\right) -\end{aligned} -\] +Here is the residue: ~~~~ {#Integrator .cpp .numberLines} for (int r=0;r(A,6*r,0)=Ar; auto KGAr = computeKGModuli(Ar); - fsigma[r] = M[r]*(sigma[r]+dsigma[r])-Ar*(eto+deto); - fsigma[r]-= M[r]*(invert(M[r]+MSC_star))*(MSC_star*tauSC - e[r]); + fsigma[r] = dpsi_dsigma[r]-Ar*(eto+deto); + fsigma[r]-= invert(L[r]+LSC_star)*(tauSC + L[r]*e[r]); } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -621,19 +637,19 @@ To do that, we use a technique widely used in `MFront`. The idea is to write the non-linear system to solve as: \[ \begin{aligned} -\tenseurq M_r\dbldot\left[\tsigma^r - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)\right]= f_{\sigma^r}(\tE,\tsigma^1,...,\tsigma^N)=0 +\tepsilon_r\left(\tsigma^r\right) - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)= f_{\sigma^r}(\tE,\tsigma^1,...,\tsigma^N)=0 \end{aligned} \] and by derivation of $f_{\sigma^r}$ w.r.t. $\tE$: \[ \begin{aligned} --\tenseurq M_r\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\tE}=0 +-\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\tE}=0 \end{aligned} \] where $\tenseurq J_{rs}=\deriv{f_{\sigma^r}}{\tsigma^s}$ stands for the sub-block $(r,s)$ of the Jacobian. Hence \[ \begin{aligned} -\deriv{\tsigma^r}{\tE}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq M_s\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) +\deriv{\tsigma^r}{\tE}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) \end{aligned} \] where $\mathbf {iJ}$ is the inverse of the Jacobian. The implementation is @@ -662,9 +678,8 @@ constant and to make vary the parameter $n$ with time. We hence set it as a function which varies exponentially with time (this is just a choice). -Besides, we choose two different values for $\tau_0^r$. -5 crystals are such that $\tau_0=10 MPa$, and 5 others -such that $\tau_0=50 MPa$. +Besides, we choose $\tau_1=5 MPa$, and for $\tau_2$ +we try $\tau_2=5MPa$ and $\tau_2=1MPa$. The `.mtest` file is the following: @@ -674,10 +689,10 @@ The `.mtest` file is the following: @ExternalStateVariable 'Temperature' {0 : 1000,1:1000}; @MaterialProperty 'nexp' "1+0.9*(exp(2.4*t)-1)"; -@MaterialProperty 'tau1' 10.; -@MaterialProperty 'tau2' 50.; +@MaterialProperty 'tau1' 5.; +@MaterialProperty 'tau2' 1.; -@MaterialProperty 'kap' "0.001"; +@MaterialProperty 'kap' "0.05"; @ImposedStrain 'EXX' {0 : 1, 1 : 1}; @ImposedStrain 'EYY' {0 : -1, 1 : -1}; @@ -693,30 +708,20 @@ The `.mtest` file is the following: Note that `kap` on line 9 is related to the small compliance we add to $\tenseurq M_r$ in order to make it invertible. -For the morphological tensor approach, we talked above -about the coefficient $r_0$ that allows to adjust the reference medium. -This coefficient can hence be set as a material property, and we will add -this line in the `mtest` file: - -~~~~ {#Affine_formulation .mtest .numberLines} -@MaterialProperty 'r0' "1+t"; -~~~~~~~~~~~~~~ - -Here, we choose to increase the elasticity the reference medium because its -homogenized stiffness also increases. -The optimal value of `r0` is not immediate, but some numerical tests -can be carried out. - -![Effective resolved shear stress as a function of the parameter $n$](./img/Affine_formulation.png) +![Axial macroscopic stress as a function of the parameter $n$](./img/Affine_formulation.png) -In this figure we plotted the results when all the $\tau_0^r$ are equal to $10 Mpa$ -(to show the heterogeneity) ("FFT 0"), the implementation of the affine approach -with self-consistent scheme (AFF_SC), and with morphological tensors (AFF_MT). For -the latter, we plot it for a constant $r_0$ (which is, an elastic medium whose -elasticity is $\tenseurq I$), and for a $r_0=1+t$ (which is, an elastic medium -whose elasticity is $(1+t)\tenseurq I$). +In this figure we plotted the results when $\tau_2$ is equal to $5 MPa$ +and when it is equal to $1MPa$. +We plot the implementation of the affine approach +with self-consistent scheme (AFF_SC), and with morphological tensors (AFF_MT). FFT results are also plotted (FFT). +We see that when $\tau_0^k$ is the same for all gliding systems, +the self-consistent scheme is in better agreement with FFT results +than the other approach. However, when two gliding systems have a resolved +shear stress 5 times smaller, the macroscopic stress decreases, and the +morphological approach gives results in better agreement with FFT computation. +The self-consistent scheme is too stiff. # References diff --git a/docs/web/downloads/Morphological_tensors.zip b/docs/web/downloads/Morphological_tensors.zip index e229115ea40d277047bb8a0d326e874e55c88a08..55e3e995cdf3185dd01202e5999a4c366040bd8d 100644 GIT binary patch delta 1119 zcmX@8dRCP;z?+#xgaHH`C;Cj})sf8R?90pnVh~+g!Og(P(jmmaz#_&n`2)Me<`U*L z?1GnSePX~`fW#9v1_p)6hTM`9JNWC*JnseT!_dpXur}nr|7`<-x^@4%q9-idvLthX z6n~?_#@>xiH&h-lu3W9PdZ$o%r|`{p3!MMF->tIL*ZcO{#L5k|_WR$T6Mz49W@MjT z2m4IT6rUxP%Xm&?&N|A)s|Ctj!NDNoE%#pb}2t(=C-#d8drFp+IPS7 zz%T-A@J z7w<%lwsRglFUj@LM~zRnxp$rEktQ3pv)fl3RX@h^N3Vf@w&3-X*0WDsf1@w`hDEW= z;f>e+Nxv%ozIuIZY3a)&bw#hHAHS6O@z{n7i`R1NDwJ~ToqRu`%xK$#u8rLDmhGup zF{#IWTj|0r+Mgv=FBJwv#mat`k@Q_{vaPRv_8VihHumo2YXdw*U+j3#8&SA4EBA|3 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16 Feb 2026 10:13:14 +0100 Subject: [PATCH 19/25] update MAsson formulation --- docs/web/MassonAffineFormulation.md | 2 +- docs/web/PonteCastaneda1992.md | 13 +++++++------ 2 files changed, 8 insertions(+), 7 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 5ac2e84ab..35c13acca 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -509,7 +509,7 @@ for (int r=0;r(k_m,mu0); From 27a277de0516e17f1a761796ea01a84f052bc68e Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Fri, 27 Feb 2026 17:48:13 +0100 Subject: [PATCH 20/25] added second moment in python homogenization --- .../python/tfel/material/LinearHomogenizationSchemes.cxx | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/bindings/python/tfel/material/LinearHomogenizationSchemes.cxx b/bindings/python/tfel/material/LinearHomogenizationSchemes.cxx index f4fafea92..60de9cabd 100644 --- a/bindings/python/tfel/material/LinearHomogenizationSchemes.cxx +++ b/bindings/python/tfel/material/LinearHomogenizationSchemes.cxx @@ -14,6 +14,7 @@ #include #include #include "TFEL/Material/LinearHomogenizationSchemes.hxx" +#include "TFEL/Material/HomogenizationSecondMoments.hxx" template requires(tfel::math::checkUnitCompatibility< @@ -233,4 +234,10 @@ void declareLinearHomogenizationSchemes(pybind11::module_& m) { return homogenization::elasticity::computeOrientedPCWScheme( IM, f, IMi, n_a, a, n_b, b, c, D); }); + m.def("computeMeanSquaredEquivalentStrain", &homogenization::elasticity::computeMeanSquaredEquivalentStrain, + pybind11::arg("IsotropicModuli_of_the_matrix"), + pybind11::arg("volume_fraction"), + pybind11::arg("IsotropicModuli_of_the_inclusion"), + pybind11::arg("squared_of_hydrostatic_macro_strain"), + pybind11::arg("squared_of_equivalent_macro_strain")); } From 2d98206e4672b5f94fb3ae4fd943265fa39e2f94 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Fri, 6 Mar 2026 16:10:11 +0100 Subject: [PATCH 21/25] update doc on affine formulation --- docs/web/MassonAffineFormulation.md | 335 ++++++------------ docs/web/img/Affine_formulation.png | Bin 42094 -> 25297 bytes docs/web/tfel-material.md | 2 +- .../MicrostructureLinearHomogenization.ixx | 24 +- 4 files changed, 114 insertions(+), 247 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index 35c13acca..d4d3b8553 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -50,13 +50,12 @@ We present here an implementation of the affine formulation [@masson_affine_2000 for the homogenization of a viscoplastic polycrystal, example which is treated in [@bornert_second-order_2001] but with Ponte-Castaneda second-order estimates. -Here, the idea is to show that many implementations of that procedure are possible in `MFront`. -We show here a first implementation based on the self-consistent scheme, provided by the `namespace` -`tfel::material::homogenization`. A second approach which uses morphological tensors computed by FFT, -on a given geometry of polycrystal, is also detailed. +Here, the idea is to show that an implementation of that procedure based +on morphological tensors computed by FFT, +on a given geometry of polycrystal. This tutorial first presents the homogenization problem, recalls the methodology -of the affine formulation, presents different kinds of possible implementations, +of the affine formulation, presents different the possible implementations, and show the details of the `mfront` file. # The viscoplastic polycrystal @@ -119,13 +118,27 @@ The idea is to linearize the behaviour around a reference stress $\tsigma^r$: \tenseurq M_r\left(\tsigma^r\right)=\deriv{\tepsilon_r}{\tsigma}\left(\tsigma^r\right)=\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\qquad\text{and}\quad\tenseur e^r = \deriv{\psi_r}{\tsigma}(\tsigma^r)-\tenseurq M_r\left(\tsigma^r\right)\dbldot\tsigma^r \] For a given set of reference stresses, this affine behaviour can be viewed as a so-called "thermoelastic comparison composite", and this composite - can be homogenized. This leads to a linear problem, so that the average on phase $r$ of the solution $\tepsilon$ is given by + can be homogenized. This leads to a linear problem of the form +\[ +\begin{aligned} +&\div\, \left(\tenseurq L_r\dbldot\left(\tepsilon-\tenseur e^r\right)\right)=\tenseur 0\\ +&\tepsilon\in\Ka\left(\tE\right)\\ +&\tenseurq L_r=\left[\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\right]^{-1}\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq L_r^{-1}\dbldot\tsigma^r +\end{aligned} +\] + + Resolving this problem gives the average on phase $r$ of the strain solution $\tepsilon$ as \[ \begin{aligned} \langle\tepsilon\rangle_r = \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE + \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) \end{aligned} \] - where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). + where $\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)$ and $\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)$ can be obtained by a homogenization procedure (mean-field scheme, FFT...). The stresses are obtained by substracting the strain $\tenseur e^r$ and multiplying by the moduli $\tenseurq L_r=\left(\tenseurq M_r\right)^{-1}$, if it exists: + \[ + \begin{aligned} + \langle\tsigma\rangle_r = \tenseurq L_r\left(\tsigma^r\right)\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE + \sum_s \tenseurq L_r\left(\tsigma^r\right)\dbldot\tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) - \tenseurq L_r\left(\tsigma^r\right)\dbldot\tenseur e^r\left(\tsigma^r\right) + \end{aligned} +\] The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\leq N$). A discussion in [@masson_affine_2000] leads to the simple assumption that these reference stresses are the averages per phase of the stress in the thermoelastic composite: \[ @@ -135,26 +148,13 @@ The last question is the choice of the reference stresses $\tsigma^r$ ($1\leq r\ \] where $\tsigma$ here stands for the stress solution of the thermoelastic homogenization problem. -Now, if $\tenseurq M_r$ is invertible and by noting $\tenseurq L_r=\tenseurq M_r^{-1}$, -the thermoelastic problem can be written as: -\[ -\begin{aligned} -\div\left[\tenseurq L_r\dbldot\left(\tepsilon-\tenseur e^r\right)\right]=0,\qquad\tepsilon\in\Ka\left(\tE\right) -\end{aligned} -\] -Hence, $\tsigma^r = \langle\tsigma\rangle_r$ is equivalent to $\tsigma^r = \tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)$, which is exactly -\[ -\begin{aligned} -\deriv{\psi_r}{\tsigma}(\tsigma^r) = \langle\tepsilon\rangle_r -\end{aligned} -\] ## Macroscopic stress and tangent operator The macroscopic stress hence can be obtained by the classical relation: \[ \begin{aligned} -\tSigma=\sum_rc_r\,\tsigma^r=\sum_rc_r\,\tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)=\tenseurq C^{\mathrm{eff}}\dbldot\tE+\tenseur\tau^{\mathrm{eff}} +\tSigma=\sum_rc_r\,\tsigma^r=\sum_rc_r\,\langle\tsigma\rangle_r=\tenseurq C^{\mathrm{eff}}\dbldot\tE+\tenseur\tau^{\mathrm{eff}} \end{aligned} \] where @@ -179,12 +179,12 @@ In the expressions of the tangent operator, we can compute the term $\deriv{\tsi The resolution consists in \[ - \text{Find}\quad\tsigma^r\quad\text{such that}\quad\deriv{\psi_r}{\tsigma}(\tsigma^r) = \langle\tepsilon\rangle_r + \text{Find}\quad\tsigma^r\quad\text{such that}\quad\tsigma^r = \langle\tsigma\rangle_r \] -where \(\tepsilon\) is the strain field solution of a homogenization problem of the form: +where \(\tsigma\) is the stress field solution of a homogenization problem of the form: \[ \begin{aligned} -&\div\,\left[\tenseurq L_r\dbldot\left(\tepsilon-\tenseur e^r\right)\right]=\tenseur 0\\ +&\div\, \left(\tenseurq L_r\dbldot\left(\tepsilon-\tenseur e^r\right)\right)=\tenseur 0\\ &\tepsilon\in\Ka\left(\tE\right)\\ &\tenseurq L_r=\left[\derivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\right]^{-1}\qquad\text{and}\qquad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq L_r^{-1}\dbldot\tsigma^r \end{aligned} @@ -195,36 +195,23 @@ where \(\tepsilon\) is the strain field solution of a homogenization problem of The iterative resolution of the non-linear system can be summarized as \[ -\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq L_r= \left[\pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\right]^{-1},\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq L_r^{-1}\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\langle\tepsilon\rangle_r = \tenseurq A_r\dbldot\tE + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s} +\begin{aligned} +&\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{analytic / finite difference}\\ \\ \\\text{\texttt{@BehaviourVariable} / directly provided}}}{\tenseurq L_r= \left[\pderivdeux{\psi_r}{\tsigma}\left(\tsigma^r\right)\right]^{-1},\quad\tenseur e^r=\deriv{\psi_r}{\tsigma}\left(\tsigma^r\right)-\tenseurq L_r^{-1}\dbldot\tsigma^r}\qquad\rightarrow\qquad \underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{mean-field scheme / morphological tensors}\\ \\ \\\texttt{tfel::material::homogenization}}}{\langle\tsigma\rangle_r = \tenseurq L_r\dbldot\tenseurq A_r\dbldot\tE + \sum_s \tenseurq L_r\dbldot\tenseurq B_{rs}\dbldot\tenseur e^s - \tenseurq L_r\dbldot\tenseur e^r}\\ +&\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\underset{\substack{\\ \\ \\ \Downarrow\\ \\ \\\text{Analytical Jacobian / Numerical Jacobian}\\ \\ \\\texttt{NewtonRaphson NumericalJacobian}}}{f_{\tsigma^r}=\tsigma^r-\langle\tsigma\rangle_r} +\end{aligned} \] The first step of the resolution consists in computing the moduli $\tenseurq M_r$ and free strains $\tenseur e^r$. It can be done analytically (as below) or via finite difference or automatic differentiation. Moreover, for both strategies, we could use a `BehaviourVariable`, defining the local behaviour in an external file (in our example, the potential is to simple to do that). -The second step is the computation of the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$. This can be achieved by means of mean-field schemes (the `namespace` `tfel::material::homogenization` provides the good tensors). This can also be done via morphological tensors, computed on a given geometry of polycrystal. We will present below the two possibilities. +The second step is the computation of the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$. This can be achieved by means of mean-field schemes (the `namespace` `tfel::material::homogenization` provides the good tensors). This can also be done via morphological tensors, computed on a given geometry of polycrystal. We will present this second possibility below. + +We note that the equation relative to the third step will give our residue. -We note that the equation relative to the second step will give our residue. -Indeed, the equation to solve is -\[ -\begin{aligned} -\deriv{\psi_r}{\tsigma}(\tsigma^r) = \langle\tepsilon\rangle_r = \tenseurq A_r\dbldot\tE + \sum_s \tenseurq B_{rs}\dbldot\tenseur e^s -\end{aligned} -\] -Hence the residue will be -\[ -\begin{aligned} -f_{\tsigma^r}=\tepsilon_r\left(\tsigma^r\right)-\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE -\sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right) -\end{aligned} -\] -where -\[ -\begin{aligned} -\tepsilon_r(\tsigma^r)=\deriv{\psi_r}{\tsigma}(\tsigma^r) -\end{aligned} -\] ## Computation of tensors Ar and Brs -### Computation via morphological tensors +### The Lippmann-Schwinger equation discretized In fact, the thermoelastic problem can be rewritten as a Lippmann-Schwinger equation (see [@willis_bounds_1977;@castaneda_effect_1995]). Let us write $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$ with $\tsigma=\tenseurq L\dbldot\left(\tepsilon-\tenseur e\right)$. We have $\tenseur\tau=\tenseurq\delta\tenseurq L\dbldot\tepsilon-\tenseurq L\dbldot\tenseur e$ with $\tenseurq\delta\tenseurq L=\tenseurq L-\tenseurq L_0$. We have, because, $\div\,\tsigma=0$, @@ -253,73 +240,31 @@ where \] is what we call a morphological tensor or an interaction tensor, and can be computed by FFT or FEM before the `mfront` integration. This tensor depends on the Green operator $\tenseurq \Gamma_0$, relative to the elasticity $\tenseurq L_0$, and a basis of symmetric second-order tensors $(\tenseur s_1,...,\tenseur s_d)$ ($d=6$ in 3D). Depending on the number of phases $N$, the computation of the $\tenseurq \Gamma_{rs}$ is more or less costly, but it is achieved -at the beginning, once and for all. By inverting the system above, we directly obtain the expressions of $\tenseurq A_r$ and $\tenseurq B_{rs}$. - -We note that the reference medium can be chosen in this approach. A natural approach is to take a $\tenseurq L_0$ isotropic near $\tenseurq L^{\mathrm{hom}}$. - -### Computation via a self-consistent scheme - -The `tfel::material::homogenization` will carry out the homogenization -with the self consistent scheme. The library provides the homogenized stiffness -$\tenseurq L_{\mathrm{SC}}$ needed for the computation of the macroscopic stress, -but also the effective polarization $\tenseurq \tau_{\mathrm{SC}}$ which permit -to compute the macroscopic stress: +at the beginning, once and for all. +After inversion of the linear system below, whose unknowns are the $\langle\tepsilon\rangle_s$, we obtain \[ \begin{aligned} -\tSigma = \tenseurq L_{\mathrm{SC}}\dbldot\tE+\tenseur \tau_{\mathrm{SC}} +\langle\tepsilon\rangle_r= \sum_t\tenseurq T_{rt}\dbldot\left[\tE +\sum_s\tenseurq \Gamma_{ts}\dbldot\tenseurq L_s\dbldot\tenseur e^s\right] \end{aligned} \] - -However, we also need the tensors $\tenseurq A_r$ and $\tenseurq B_{rs}$ for computing the average of -the strain field $\langle\tepsilon\rangle_r$ (i.e. the residue). To that extent, we go back to the equation -\[ -\begin{aligned} -\tepsilon=\tE-\tenseurq \Gamma_0\left(\tenseur\tau\right) -\end{aligned} -\] -and consider averages of the fields: +where $\tenseurq T_{rt}$ is the linear operator issued from the linear resolution. +and we directly obtain the expressions of $\tenseurq A_r$ and $\tenseurq B_{rs}$: \[ \begin{aligned} -\langle\tepsilon\rangle_r-\tE=-\sum_s\tenseurq \Gamma_{rs}\dbldot\langle\tenseur\tau\rangle_s +&\tenseurq A_r= \sum_t\tenseurq T_{rt}\\ +&\tenseurq B_{rs}= \sum_t\tenseurq T_{rt}\dbldot\tenseurq \Gamma_{ts}\dbldot\tenseurq L_s \end{aligned} \] -A classical approximation (see Eq. (3.17) of [@castaneda_effect_1995]) is to consider that $\tenseurq\Gamma_{rs}=\delta_{rs}\tenseurq P_r-c_s\,\tenseurq P_0$, with $\tenseurq P_r=\tenseurq\Gamma_{rr}$ the Hill tensor associated to $\tenseurq L_0$ and the inclusion shape of phase $r$, and $\tenseurq P_0$ the Hill tensor associated to $\tenseurq L_0$ and a spherical inclusion. This is when considering an isotropic distribution of inclusions (otherwise $\tenseurq P_0$ must be modified). For our polycrystal, we can also consider that $\tenseurq P_r=\tenseurq P_0$ (i.e. the grains have a spherical shape) so that -\[ -\begin{aligned} -\langle\tepsilon\rangle_r-\tE=-\tenseurq P_0\dbldot\left(\langle\tenseur\tau\rangle_r-\langle\tenseur\tau\rangle\right) -\end{aligned} -\] -and due to the expression of $\tenseur\tau=\tsigma-\tenseurq L_0\dbldot\tepsilon$: -\[ -\begin{aligned} -\left[\tenseurq P_0^{-1}-\tenseurq L_0\right]\dbldot\left(\langle\tepsilon\rangle_r-\tE\right)=-\left[\langle\tsigma\rangle_r-\tSigma\right] -\end{aligned} -\] -We know that the reference medium, in the self-consistent approach, is identified with the homogenized medium. Hence, we note -$\tenseurq P_0=\tenseurq P_{\mathrm{SC}}$, $\tenseurq L_0=\tenseurq L_{\mathrm{SC}}$, and $\tenseurq M_{\mathrm{SC}}^*=\left[\tenseurq P_{\mathrm{SC}}^{-1}-\tenseurq L_{\mathrm{SC}}\right]^{-1}$, so that, with $\langle\tsigma\rangle_r=\tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)$, -\[ -\begin{aligned} -\left(\langle\tepsilon\rangle_r-\tE\right)=-\tenseurq M_{\mathrm{SC}}^*\dbldot\left[\tenseurq L_r\dbldot\left(\langle\tepsilon\rangle_r-\tenseur e^r\right)-\tSigma\right] -\end{aligned} -\] -which gives -\[ -\begin{aligned} -\langle\tepsilon\rangle_r=\left(\tenseurq I+\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_r\right)^{-1}\dbldot\left[\tE+\tenseurq M_{\mathrm{SC}}^*\dbldot\left(\tSigma+\tenseurq L_r\dbldot\tenseur e^r\right)\right] -\end{aligned} -\] -which gives directly the expressions of $\tenseurq A_r$ and $\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s$ (we use $\tSigma=\tenseurq L_{\mathrm{SC}}\dbldot\tE+\tenseur \tau_{\mathrm{SC}}$): +### Choice of reference medium + +We note that the reference medium can be chosen in this approach. A natural approach is to take a $\tenseurq L_0$ isotropic near $\tenseurq L^{\mathrm{hom}}$. To that extent we will make use of the fact that for an elasticity $\tenseurq C_1=\alpha_1\tenseurq C_0$ the strain Green operator $\tenseurq \Gamma_1$ relative to $\tenseurq C_1$ is given by \[ -\begin{aligned} -\tenseurq A_r&=\left(\tenseurq I+\tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_r\right)^{-1}\dbldot\left(\tenseurq I + \tenseurq M_{\mathrm{SC}}^*\dbldot\tenseurq L_{\mathrm{SC}}\right)\\ -\sum_s\tenseurq B_{rs}\dbldot\tenseur e^s&=\left(\tenseurq L_{\mathrm{SC}}^*+\tenseurq L_r\right)^{-1}\dbldot\left(\tenseur \tau_{\mathrm{SC}}+\tenseurq L_r\dbldot\tenseur e^r\right) -\end{aligned} +\tenseurq \Gamma_1=\dfrac{1}{\alpha_1}\tenseurq \Gamma_0 \] -where $\tenseurq L_{\mathrm{SC}}^*=\left(\tenseurq M_{\mathrm{SC}}^*\right)^{-1}$. -And this is sufficient to compute the residue. However, we note that the library `tfel::material::homogenization` provides -the tensor $\tenseurq A_r$ via the variable `mean_strain_localisation_tensors`. Moreover, the library also provides $\tenseur \tau_{\mathrm{SC}}$ if we specify the polarizations on each phase. We hence only have to compute $\tenseurq L_{\mathrm{SC}}^*$. +Hence, we will compute the morphological tensors $\tenseurq \Gamma_{rs}$ for a given elasticity $\tenseurq C_0$, and we can change the reference medium by multiplication of $\tenseurq C_0$ by $\alpha_1$ and division of $\tenseurq \Gamma_0$ by $\alpha_1$. +Moreover, in our case, it makes sense to take a reference medium $\tenseurq C_0=2\mu_0\tenseurq K$. # Implementation in MFront @@ -354,65 +299,41 @@ All the files are available in the `MFrontGallery` project, [here](https://githu The derivatives of the potential are \[ \begin{aligned} - &\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r\\ - &\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k}}\left(\dfrac{|\tau|}{\tau_0^{k}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r + &\tepsilon_r(\tsigma)=\deriv{\psi_r}{\tsigma}=\sum_k \deriv{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseur\mu_k^r=\sum_k\mathrm{sgn}(\tau)\,\dot{\gamma}_0\left(\dfrac{|\tau_k^r|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r\qquad\text{with}\quad \tau_k^r = \tsigma\dbldot\tenseur \mu_k^r\\ + &\tenseurq M_r(\tsigma)=\derivdeux{\psi_r}{\tsigma}=\sum_k \derivdeux{\psi_k^r}{\tau}\left(\tau_k^r\right)\,\tenseurq N_k^r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k}}\left(\dfrac{|\tau_k^r|}{\tau_0^{k}}\right)^{n-1}\tenseurq N_k^r,\qquad\text{with}\quad\tenseurq N_k^r=\tenseur \mu_k^r\otimes\tenseur \mu_k^r \end{aligned} \] and then defining $\overline{\tau}_k^r=\tsigma^r\dbldot\tenseur \mu_k^r$ where $\tsigma^r$ is the reference stress, we have \[ \begin{aligned} - &\tenseurq M_r=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n-1}\tenseurq N_k^r\\ - &\tenseur e^r=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r + &\tenseurq M_r(\tsigma^r)=\sum_k \dfrac{n\,\dot{\gamma}_0}{\tau_0^{k}}\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n-1}\tenseurq N_k^r\\ + &\tenseur e^r(\tsigma^r)=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r \end{aligned} \] ## Details of implementation -We will mainly present the approach based on self-consistent -scheme, we invite the reader to see the `.mfront` file in the -`MFrontGalleryProject` for the approach based on morphological tensors. - ### Headers -For the jacobian, we adopt the `Numerical Jacobian`, so that -the beginning of the `mfront` file reads: +The beginning of the `mfront` file reads: ~~~~ {#Begin .cpp .numberLines} @DSL ImplicitII; -@Behaviour Affine_formulation; +@Behaviour Affine_tensors; @Author Martin Antoine; -@Date 24 / 01 / 26; -@Description{"Affine formulation for homogenization of a viscoplastic polycrystal, Masson et al. 2001."}; +@Date 06 / 03 / 26; +@Description{"Affine formulation for homogenization of a viscoplastic polycrystal, Masson et al. 2001., with morphological tensors"}; @UseQt false; -@Algorithm NewtonRaphson_NumericalJacobian; -@PerturbationValueForNumericalJacobianComputation 1e-10; +@Algorithm NewtonRaphson; @Epsilon 1e-14; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Moreover, we need to include header files, those -for the homogenization of the thermoelastic composite: - - - IsotropicEshelbyTensor.hxx - - MicrostructureDescription.hxx - - MicrostructureLinearHomogenization.hxx - -and the header for handling the rotations of the grains with their +Moreover, we need to include a header file for handling the rotations of the grains with their slip systems: - PolyCrystalsSlidingSystems.hxx -~~~~ {#Begin .cpp .numberLines} -@TFELLibraries {"Material"}; -@Includes{ -#include "TFEL/Material/IsotropicModuli.hxx" -#include "TFEL/Material/IsotropicEshelbyTensor.hxx" -#include "TFEL/Material/MicrostructureDescription.hxx" -#include "TFEL/Material/MicrostructureLinearHomogenization.hxx" -#include "TFEL/Material/PolyCrystalsSlidingSystems.hxx"} -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -For the approach based on morphological tensors, we do not need all these headers, -but we need to include a file which contains the morphological tensors. +and a file which contains the morphological tensors. This file is present in a repository, let's say `extra-headers/TFEL/Material/` located at the same place as the `.mfront` file. @@ -420,13 +341,14 @@ located at the same place as the `.mfront` file. @TFELLibraries {"Material"}; @Includes{ #include "tensors.hxx" +#include "TFEL/Material/IsotropicModuli.hxx" #include "TFEL/Material/PolyCrystalsSlidingSystems.hxx"} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The compilation hence will be done like that: ~~~~ {#Begin .sh .numberLines} -mfront -I ../extra-headers/TFEL/Material --obuild --interface=generic Affine_formulation.mfront +mfront -I ../extra-headers/TFEL/Material --obuild --interface=generic Affine_tensors.mfront ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Of cours the tensors are computed before via FFT as explained above @@ -452,7 +374,7 @@ code block: ~~~~ {#Integrator .cpp .numberLines} using ExtendedPolyCrystalsSlidingSystems = - ExtendedPolyCrystalsSlidingSystems, real>; + ExtendedPolyCrystalsSlidingSystems, real>; const auto& gs = ExtendedPolyCrystalsSlidingSystems::getPolyCrystalsSlidingSystems("polycrystal.csv"); ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -482,7 +404,7 @@ morphological tensors, is This matrix will store the morphological tensors. Indeed, in the `InitLocalVariables` code block, we can load these coefficients, -which are in fact situated in the header `extra-headers/TFEL/Material/tensors.hxx` +which are in fact situated in the header `tensors.hxx` ~~~~ {#Begin .cpp .numberLines} GAMMA=Gamma::get_tensor(); @@ -532,103 +454,58 @@ appendix B.1). In our case, we set it as a `MaterialProperty` that can evolve with time (see `mtest` file below). We can hence consider $\tenseurq L_r$, the inverse of $\tenseurq M_r+\kappa\tenseurq I$. -### Self-consistent scheme - -The homogenization of the thermoelastic composite -will be carried out with `tfel::material::homogenization::elasticity`. -We first construct our `ParticulateMicrostructure`. -We instantiate it by doing: +### Computation of Ar and Brs -~~~~ {#Integrator .cpp .numberLines} -using namespace tfel::material::homogenization::elasticity; -const auto IM0=tfel::material::KGModuli(k0,mu0); -auto micro=ParticulateMicrostructure<3u,stress>(IM0); -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The computation of tensors Ar and Brs is as follows -Note that here the matrix phase is defined by the elasticity `(k0,mu0)` -but this elasticity will play no role in the sequel. -Indeed, we will add the grains as `SphereDistribution` objects, -and because the sum of the volume fractions is equal to one, -the matrix will automatically have a fraction equal to zero. - -The grains are added like that: ~~~~ {#Integrator .cpp .numberLines} -const auto sphere=Sphere(); -std::vector polarisations; -const auto zero = Stensor::zero(); -polarisations.resize(Np+1, zero); - -for (int r=0 ; r(sphere,frac[r],L[r]); - micro.addInclusionPhase(distribution); - polarisations[r+1]=-L[r]*e[r]; +tmatrix<6*Np,6*Np,real> MAT; +tmatrix<6*Np,6*Np,real> G; +tmatrix<6*Np,6,real> E; +for (int r=0;r(E,6*r,0)=I; + for (int s=0;s(MAT,6*r,6*s) =1/r0*GAMMA[r][s]*dL; + map_derivative(G,6*r,6*s) =1/r0*GAMMA[r][s]*L[s]; + } + map_derivative(MAT,6*r,6*r) +=I; } -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -The shape of each grain is spherical, and we provide the stiffness tensors `L[r]` -to the constructor `SphereDistribution`. -We also put a polarisation on each grain: it is $-\tenseurq L_r\dbldot\tenseur e^r$. -Note that the `ParticulateMicrostructure` in fact contains `Np` phase: a matrix, -and `Np` grains. Be careful with the indices: `polarisation[0]` is associated to -the matrix. - -The self-consistent scheme is performed afterwards: - -~~~~ {#Integrator .cpp .numberLines} -const auto hmSC=computeSelfConsistent<3u,stress>(micro,1e-6,true,0,polarisations); -LSC = hmSC.homogenized_stiffness; -const auto tauSC = hmSC.effective_polarisation; -const auto A_SC=hmSC.mean_strain_localisation_tensors; +TinyMatrixInvert<6*Np,real>::exe(MAT); +A = MAT*E; +tmatrix<6*Np,6*Np,real> B = MAT*G; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -On line 1, we say we want a tolerance of `1e-6` (this is related to -the stopping criterion of the iterative self-consistent algorithm, -see the [doc](./tfel-material.html#homogenization-of-general-microstructures)). -We also consider an isotropic homogenized stiffness with `true`. This is related -to the computation of the Hill tensor associated to the spherical inclusions. -if using an anisotropic reference medium for computing this tensor, -the integration of the behaviour would be too slow. Hence, we make the choice -of an isotropization. -Moreover, the `0` is just related to the number of anisotropic integration, -but because of `true`, this has no effect. +### Residues, jacobian and macroscopic stress -We will need the following tensors: - -~~~~ {#Integrator .cpp .numberLines} -auto KGSC = computeKGModuli(LSC); -const auto PSC = computeSphereHillPolarisationTensor(KGSC); -const auto LSC_star = invert(PSC)- LSC; -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Note here that we also isotropize the homogenized stiffness with the function -`computeKGModuli`, on line 1. This is more consistent with the fact that we used `true` -in the `computeSelfConsistent` function. - -### Residues and jacobian - -Here is the residue: +Here is the computation of the residue, the jacobian and +of macroscopic stress: ~~~~ {#Integrator .cpp .numberLines} +Chom=Stensor4::zero(); for (int r=0;r(A,6*r,0)=Ar; - auto KGAr = computeKGModuli(Ar); - fsigma[r] = dpsi_dsigma[r]-Ar*(eto+deto); - fsigma[r]-= invert(L[r]+LSC_star)*(tauSC + L[r]*e[r]); + Ar[r]=map_derivative(A,6*r,0); + Chom+=frac[r]*L[r]*Ar[r]; } -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -There is no jacobian because we use the `NumericalJacobian`. Here, -an analytical jacobian would have been tedious, although not impossible. -The computation of the macroscopic stress is simple: +for (int r=0;r(B,6*r,6*s); + tau_eff+=frac[r]*L[r]*Brs*e[s]; + fsigma[r]-=L[r]*Brs*e[s]; + } + dfsigma_ddsigma(r,r)=I; +} -~~~~ {#Integrator .cpp .numberLines} -sig=LSC*(eto+deto)+tauSC; +sig=Chom*(eto+deto)+tau_eff; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The jacobian here is approximative and given on line 15. + ### Tangent operator @@ -637,25 +514,28 @@ To do that, we use a technique widely used in `MFront`. The idea is to write the non-linear system to solve as: \[ \begin{aligned} -\tepsilon_r\left(\tsigma^r\right) - \tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)= f_{\sigma^r}(\tE,\tsigma^1,...,\tsigma^N)=0 +\tepsilon_r\left(\tsigma^r\right) - \tenseurq L_r\left(\tsigma^r\right)\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tE - \sum_s \tenseurq B_{rs}\left(\tsigma^1,...,\tsigma^N\right)\dbldot\tenseur e^s\left(\tsigma^s\right)+\tenseurq L_r\left(\tsigma^r\right)\dbldot\tenseur e^r\left(\tsigma^r\right)= f_{\sigma^r}(\tE,\tsigma^1,...,\tsigma^N)=0 \end{aligned} \] and by derivation of $f_{\sigma^r}$ w.r.t. $\tE$: \[ \begin{aligned} --\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\tE}=0 +-\tenseurq L_r\left(\tsigma^r\right)\dbldot\tenseurq A_r\left(\tsigma^1,...,\tsigma^N\right)+\sum_s\tenseurq J_{rs}\dbldot\deriv{\tsigma^s}{\tE}=0 \end{aligned} \] where $\tenseurq J_{rs}=\deriv{f_{\sigma^r}}{\tsigma^s}$ stands for the sub-block $(r,s)$ of the Jacobian. Hence \[ \begin{aligned} -\deriv{\tsigma^r}{\tE}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) +\deriv{\tsigma^r}{\tE}=\sum_s\mathbf {iJ}_{rs}\dbldot\tenseurq L_s\left(\tsigma^s\right)\dbldot\tenseurq A_s\left(\tsigma^1,...,\tsigma^N\right) \end{aligned} \] where $\mathbf {iJ}$ is the inverse of the Jacobian. The implementation is ~~~~ {#tangent .cpp .numberLines} @TangentOperator{ + for (int r=0;r(A,6*r,0)=L[r]*Ar[r]; + } tmatrix<6*Np,6*Np,real> iJ = jacobian; TinyMatrixInvert<6*Np,real>::exe(iJ); dsigma_deto=iJ*A; @@ -713,15 +593,10 @@ we add to $\tenseurq M_r$ in order to make it invertible. In this figure we plotted the results when $\tau_2$ is equal to $5 MPa$ and when it is equal to $1MPa$. -We plot the implementation of the affine approach -with self-consistent scheme (AFF_SC), and with morphological tensors (AFF_MT). +We plot the implementation of the affine approach with morphological tensors (AFF). FFT results are also plotted (FFT). -We see that when $\tau_0^k$ is the same for all gliding systems, -the self-consistent scheme is in better agreement with FFT results -than the other approach. However, when two gliding systems have a resolved -shear stress 5 times smaller, the macroscopic stress decreases, and the -morphological approach gives results in better agreement with FFT computation. -The self-consistent scheme is too stiff. +We see that the +morphological approach gives results in good agreement with FFT computation. # References diff --git a/docs/web/img/Affine_formulation.png b/docs/web/img/Affine_formulation.png index d24502f3f0131db947271e89151cf3ea35e0f89c..b21782853ef633ddc1183ddd2972e1c0f37dd9ce 100644 GIT binary patch literal 25297 zcmdqJcT|+w*ELw8h>ZfR5)?#KK*=Q_QG$t7D51z%L~@jzgNi5_Bq&*+$U#X(D1y>} zWQipiCFh(em{ayU@AsRvzWL+(X6B#i)xBtns^_`)KIfjZ&))miJ0%4f3UYdK1Oh=J zC;JGEK#<@Nh!dMT1R8>L)1}9(^19N#L>mj-Wc)3(9zDy*3ruR*)?Zldk1q{ z8-5<%+dTYS*UTIp?HojrNbCQ82am12Dbj?sG!U-xmz}JZ0|G&1Nc`_amSmLCX{G)Sr#*8@_Bvto;J_$Sp69P{L^F#};mibY*hdkE z<+fYA^LKBVns6y(_OQcP>K?j%)PtW0rH^M|x``he&UAF|%QJ+pUkzi^wJVlC33nycOn{Lu-mrN2?&2+r 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zS#`Oa1UWKlEOXE?dDsLh1{~Uw9ed0MO}gWlqmMIYC5%QlcxCg+QEOe6?5hmo1!J_+ zj}SA5@quSCBmkM%EFLKHtQl32l~$x{Ozzduu>&cb559yFT`trIoDTXHRX8WBv+bxm zpKT63pe#50Kt{bEQtv8I;cDsMBuDSxDgS7GMVl)x>IKZW3ex`@)nJo*zzV+)CO8qp z{?*ddIF_%!e?R=RMO9@HhxvK8*~ro%uh1TV+>7Bv&=Y~fX({dP?Kd1U0Z$vL^iGC< z3l%cZmPb5yqQ3y2NjTwj&N7R*T(^{|pgPzV-kg}@xx@XX7isNGquq^*+fw3QLrFIb z9E8)Z5`9-F?hmP}HZ?J!!^fL;DQa8T)Nj`JnX_mANKF+0Y!$jt-r-p}%mglaKUZJW zd%0OcA7?!WU^!vGXqp`53dQ@-R&vAb9Sj!dLo4CW;0svwo>5FEsrwFlG|^yJZ-0g= zNQ98qCgnxXE%m+)P)%l=Bun+;&7bQc>Y|mhC^~v7J|i5|(dk3uf+dSJsFCD|^vl5; zbPo;=e$>^~XKy&33(Ua*GMUWJ;3Qe^uC_PP7hr=s^1pi@!tQ9i<915Xv7n{*)!-`t`3ZSQe6;u(C0); } const auto tau0 = polarisations_[0]; - HomogenizationScheme h_s; + auto Chom = C0; auto tau_eff = tau0; @@ -86,9 +86,7 @@ namespace tfel::material::homogenization::elasticity { tau_eff += fi * tfel::math::transpose(Ai) * (taui - tau0); localisators.push_back(Ai); } - h_s.homogenized_stiffness = Chom; - h_s.effective_polarisation = tau_eff; - h_s.mean_strain_localisation_tensors = localisators; + HomogenizationScheme h_s={.homogenized_stiffness = Chom,.effective_polarisation = tau_eff,.mean_strain_localisation_tensors = localisators}; return h_s; }; @@ -114,7 +112,6 @@ namespace tfel::material::homogenization::elasticity { } const auto tau0 = polarisations_[0]; const auto f0 = micro.get_matrix_fraction(); - HomogenizationScheme h_s; auto Chom = C0; auto tau_eff = tau0; std::vector> localisators = {}; @@ -144,9 +141,7 @@ namespace tfel::material::homogenization::elasticity { Chom += fi * (Ci - C0) * Ai; tau_eff += fi * transpose(Ai) * (taui - tau0); } - h_s.homogenized_stiffness = Chom; - h_s.effective_polarisation = tau_eff; - h_s.mean_strain_localisation_tensors = localisators; + HomogenizationScheme h_s={.homogenized_stiffness = Chom,.effective_polarisation = tau_eff,.mean_strain_localisation_tensors = localisators}; return h_s; }; @@ -165,22 +160,21 @@ namespace tfel::material::homogenization::elasticity { const auto np = micro.get_number_of_phases(); const auto f0 = micro.get_matrix_fraction(); const auto C0 = micro.get_matrix_elasticity(); - + const auto polarisations_ = internals::initialize_polarisation(polarisations, np); - - HomogenizationScheme h_s; auto tau_eff = tfel::math::stensor::zero(); - std::vector> localisators = {}; + std::vector> localisators = {}; auto Chom = C0; auto Chom_ = C0; real rel_err = tolerance + 1; + while (rel_err > tolerance) { std::vector> localisators_try = {}; tfel::math::st2tost2 A0 = f0 * tfel::math::st2tost2::Id(); for (std::size_t i = 0; i < np - 1; i++) { - auto phasei = micro.get_inclusionPhase(i); + auto phasei=micro.get_inclusionPhase(i); auto fi = (*phasei).fraction; tfel::math::st2tost2 Ai; if (isotropic) { @@ -228,9 +222,7 @@ namespace tfel::material::homogenization::elasticity { auto Ai = localisators[i + 1]; tau_eff += fi * transpose(Ai) * taui; } - h_s.homogenized_stiffness = Chom; - h_s.effective_polarisation = tau_eff; - h_s.mean_strain_localisation_tensors = localisators; + HomogenizationScheme h_s={.homogenized_stiffness = Chom,.effective_polarisation = tau_eff,.mean_strain_localisation_tensors = localisators}; return h_s; } From 06da6d25d09c5295eb2b25b7d0a55c223dc0aa64 Mon Sep 17 00:00:00 2001 From: MARTIN Antoine Date: Fri, 6 Mar 2026 16:31:08 +0100 Subject: [PATCH 22/25] updated doc on affine formulation --- docs/web/MassonAffineFormulation.md | 21 ++++++++++++++++----- docs/web/img/Affine_formulation.png | Bin 25297 -> 26122 bytes 2 files changed, 16 insertions(+), 5 deletions(-) diff --git a/docs/web/MassonAffineFormulation.md b/docs/web/MassonAffineFormulation.md index d4d3b8553..125ef62cb 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -259,12 +259,13 @@ and we directly obtain the expressions of $\tenseurq A_r$ and $\tenseurq B_{rs}$ ### Choice of reference medium -We note that the reference medium can be chosen in this approach. A natural approach is to take a $\tenseurq L_0$ isotropic near $\tenseurq L^{\mathrm{hom}}$. To that extent we will make use of the fact that for an elasticity $\tenseurq C_1=\alpha_1\tenseurq C_0$ the strain Green operator $\tenseurq \Gamma_1$ relative to $\tenseurq C_1$ is given by +We note that the reference medium can be chosen in this approach. A natural approach is to take a $\tenseurq L_0$ isotropic near $\tenseurq L^{\mathrm{hom}}$. To that extent we will make use of the fact that for an elasticity $\tenseurq C_1=r_0\tenseurq C_0$ the strain Green operator $\tenseurq \Gamma_1$ relative to $\tenseurq C_1$ is given by \[ -\tenseurq \Gamma_1=\dfrac{1}{\alpha_1}\tenseurq \Gamma_0 +\tenseurq \Gamma_1=\dfrac{1}{r_0}\tenseurq \Gamma_0 \] -Hence, we will compute the morphological tensors $\tenseurq \Gamma_{rs}$ for a given elasticity $\tenseurq C_0$, and we can change the reference medium by multiplication of $\tenseurq C_0$ by $\alpha_1$ and division of $\tenseurq \Gamma_0$ by $\alpha_1$. -Moreover, in our case, it makes sense to take a reference medium $\tenseurq C_0=2\mu_0\tenseurq K$. +Hence, we will compute the morphological tensors $\tenseurq \Gamma_{rs}$ for a given elasticity $\tenseurq C_0$, and we can change the reference medium by multiplication of $\tenseurq C_0$ by $r_0$ and division of $\tenseurq \Gamma_0$ by $r_0$. + +Moreover, in our case, it makes sense to take a reference medium $\tenseurq C_0=2\mu_0\tenseurq K$. Afterwards, we will update the elasticity by taking $\tenseurq C_1=2\mu_{\mathrm{hom}}\tenseurq K$, where $\mu_{\mathrm{hom}}$ is equal to the shear modulus given by the projection of $\tenseurq C_{\mathrm{hom}}$ on $\tenseurq K$, where $\tenseurq C_{\mathrm{hom}}$ is the homogenized tensors obtained at previous step of the Newton-Raphson algorithm. # Implementation in MFront @@ -478,6 +479,10 @@ A = MAT*E; tmatrix<6*Np,6*Np,real> B = MAT*G; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +Here we see that we multiply the morphological tensors by `1/r0` in order to adapt the reference medium. +The `r0` will be updated after, depending on the `Chom` obtained. +The `dL` also is impacted and written as `L[r]-r0*L0`. + ### Residues, jacobian and macroscopic stress Here is the computation of the residue, the jacobian and @@ -490,6 +495,10 @@ for (int r=0;raanLBsp&Ye4RnR(~&eV%j9-g~XJ*ZQrse*3(YmzBiFrNl);L&JY0C9Z&mhF*?_ zc5VyjJbW|u;$#dy?%O<6v3Z6vu(8*%(npigv9UBm*_avY(%b1Lt 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125ef62cb..77cb4752f 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -311,6 +311,10 @@ The derivatives of the potential are &\tenseur e^r(\tsigma^r)=\sum_k\left(\mathrm{sgn}(\overline{\tau}_k^r)-n\right)\,\dot{\gamma}_0\left(\dfrac{|\overline{\tau}_k^r|}{\tau_0^{k}}\right)^{n}\tenseur\mu_k^r \end{aligned} \] +We see here that the modulus $\tenseurq M_r$ is not invertible. Hence, in the following, we add a small tensor to $\tenseurq M_r$ in order to make it invertible (see also appendix B.1 of [@bornert_second-order_2001]): +\[ +\tenseurq L_r=\left[\tenseurq M_r+\kappa\tenseurq I\right]^{-1} +\] ## Details of implementation @@ -448,10 +452,8 @@ for (int r=0;r Date: Fri, 13 Mar 2026 16:00:24 +0100 Subject: [PATCH 24/25] suppressed .zip --- docs/web/downloads/Morphological_tensors.zip | Bin 4813 -> 0 bytes docs/web/img/distributions_PCW.pdf | Bin 161825 -> 0 bytes 2 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 docs/web/downloads/Morphological_tensors.zip delete mode 100644 docs/web/img/distributions_PCW.pdf diff --git a/docs/web/downloads/Morphological_tensors.zip b/docs/web/downloads/Morphological_tensors.zip deleted file mode 100644 index 55e3e995cdf3185dd01202e5999a4c366040bd8d..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 4813 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77cb4752f..72a7eafce 100644 --- a/docs/web/MassonAffineFormulation.md +++ b/docs/web/MassonAffineFormulation.md @@ -282,9 +282,7 @@ All the files are available in the `MFrontGallery` project, [here](https://githu ![Geometry of the polycrystal](./img/polycrystal.png){width=50%} - The morphological tensors can be computed by following the instruction of the `README.md` - in this [folder](./downloads/Morphological_tensors.zip). - + The morphological tensors were computed using a `python` script with `numpy.fft`. The orientations of the slip systems will be given by angles, stored in the file `polycrystal.csv`. This file also contains, at the end of each line, the volume fraction associated to a crystal. They are in our case the same as the one computed on the generated microstructure.

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