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‎documentation/multi_physics/user-guide/material_models/introduction/readme.md‎

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<h2 id ="user_guide_material_models"> Material Models </h2>
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svMultiPhysics supports a number of material models that can be used for simulations of complex solid mechanics.
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The ALE formulation that svMultiPhysics uses (see section on FSI) solves for displacements and stresses in the solid domain. Different materials have different relationships between the stress and the displacement (or strain) which are represented by material models (also known as constitutive models). These material models are required to solve the system of equations for solids.
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One commonly known material model is the linear elastic model, where the Cauchy stress tensor is related to the small strain tensor through a fourth order stiffness tensor. In 1D, the stress and strain are proportional and the constant of proportionality is known as the Young’s modulus.
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$$\boldsymbol{\sigma} = \mathbb{C}\boldsymbol{\epsilon}$$
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where
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$$\boldsymbol{\epsilon} = \frac{1}{2} [\nabla \mathbf{u} + (\nabla \mathbf{u})^T]$$
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This relationship holds well for metals which experience small deformations. Soft biological tissues on the other hand undergo large nonlinear deformations and are better represented by a class of material models called hyperelastic models. The passive material behavior for hyperelastic materials can be described through the strain energy function $$\Psi$$. Various stress measures can be obtained from the strain energy function by taking a tensor derivative. svMultiPhysics uses the 2nd Piola Kirchhoff Stress $$\mathbf{S}$$.
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$$ \mathbf{S} = 2\frac{\partial \Psi}{\partial \mathbf{C}} $$
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where $$\mathbf{C}$$ is the right Cauchy-green tensor. The general constitutive relation for these materials can be written as
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$$ \mathbf{S} = \mathbb{C}:\mathbf{E}$$
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where $$\mathbf{E}$$ is the Green-Lagrange strain tensor. The small strain tensor used for linear elasticity is a linearized form of this strain tensor.
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The strain energy function consists of two parts - the isochoric (volume preserving) and the volumetric parts.
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$$ \Psi = \Psi_{vol} (J) + \Psi_{iso} (\bar{\mathbf{C}})$$
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Biological materials are rubber-like and are modelled as incompressible or nearly incompressible, and this decomposition is helpful for such cases.
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Tissues have extremely complex microstructure due to which we rely on phenomenological material models. svMultiPhysics has a number of options available for material models suitable for different applications.
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‎documentation/multi_physics/user-guide/material_models/material_models_list/readme.md‎

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<h3 id ="user_guide_material_models"> List of Available Hyperelastic Models </h3>
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Volumetric constitutive models for struct/ustruct equations:
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<table class="table table-bordered" style="width:100%">
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<tr>
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<th> Volumetric Model </th>
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<th> Input Keyword </th>
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</tr>
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<tr>
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<td> Quadratic model </td>
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<td> "quad", "Quad", "quadratic", "Quadratic" </td>
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</tr>
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<tr>
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<td> Simo-Taylor91 model </td>
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<td> "ST91", "Simo-Taylor91" </td>
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</tr>
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<tr>
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<td> Miehe94 model </td>
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<td> "M94", "Miehe94" </td>
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</tr>
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</table>
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Isochoric constitutive models for struct/ustruct equations.
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<table class="table table-bordered" style="width:100%">
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<tr>
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<th> Isochoric Model </th>
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<th> Input Keyword </th>
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</tr>
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<tr>
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<td> Saint Venant-Kirchhoff $$\dag$$ </td>
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<td> "stVK", "stVenantKirchhoff" </td>
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</tr>
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<tr>
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<td> Neo-Hookean model </td>
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<td> "nHK", "nHK91", "neoHookean", "neoHookeanSimo91" </td>
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</tr>
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<tr>
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<td> Holzapfel-Gasser-Ogden model </td>
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<td> "HGO" </td>
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</tr>
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<tr>
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<td> Guccione model </td>
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<td> "Guccione", "Gucci" </td>
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</tr>
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<tr>
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<td> Holzapfel-Ogden model </td>
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<td> "HO", "HolzapfelOgden" </td>
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</tr>
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<tr>
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<td> Holzapfel-Ogden Modified Anisotropy model </td>
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<td> “HO_ma”, “HolzapfelOgden-ModifiedAnisotropy” </td>
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</tr>
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</table>
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$$\dag$$ : These models are not available for ustruct.
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svMultiPhysics has two options for solving the solid equations - struct and ustruct. “Struct” uses a displacement based formulation i.e. the unknowns that we are solving for in each element are displacements. “Ustruct” uses a mixed formulation where the unknowns are displacements and pressures.
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 1px solid #d0d0d0">
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&lt;<strong>Add_equation</strong> type=<i>"struct"</i>&gt; // or "ustruct"
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&lt;<strong>Coupled&gt;</strong> <i>true</i> &lt;/<strong>Coupled</strong>&gt;
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&lt;<strong>Min_iterations&gt;</strong> <i>1</i> &lt;/<strong>Min_iterations</strong>&gt;
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&lt;<strong>Max_iterations&gt;</strong> <i>3</i> &lt;/<strong>Max_iterations</strong>&gt;
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&lt;<strong>Tolerance&gt;</strong> <i>1e-9</i> &lt;/<strong>Tolerance</strong>&gt;
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<br><br>
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/*
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Add constitutive model, output type, solver type, boundary conditions
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*/
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<br><br>
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/&lt;<strong>Add_equation</strong>&gt;
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</div>
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Volumetric Models: These models set the volumetric part of the strain energy function. There is only one material parameter needed in the input file to define this term.
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For a displacement based formulation (“struct”), the volumetric part of the strain energy function is a penalty to allow for small amounts of compressibility (models the material as nearly incompressible).
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$$ \Psi_{vol} = K_p G(J) $$
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where $$ K_p$$ can be interpreted as the bulk modulus. $$G(J)$$ is the penalty function and takes different forms depending on the type of model. Two parameters are p and pl are defined internally to add to the stresses and elasticity tensors. “Struct” , the displacement based formulation calculates these as:
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$$ p = \frac{\partial \Psi_{vol}}{\partial J}$$
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$$ pl = p + J\frac{dp}{dJ}$$

‎documentation/multi_physics/user-guide/material_models/references/readme.md‎

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## References
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<p><a id="ref-1">
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[1] Chandran KB, Rittgers SE, Yoganathan AP. <strong>Biofluid mechanics: the human circulation.</strong> CRC press; 2006 Nov 15. </a></p>
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<p><a id="ref-2"> <a href="https://doi.org/10.1063/1.2772250">
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[2] Boyd, Joshua, James M. Buick, and Simon Green. <strong>Analysis of the Casson and Carreau-Yasuda Non-Newtonian Blood Models in Steady and Oscillatory Flows Using the Lattice Boltzmann Method</strong>. Physics of Fluids 19, no. 9 (September 2007): 093103.</a></a></p>
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<p><a id="ref-3">
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[3] Vedula V, Lee J, Xu H, Kuo CC, Hsiai TK, Marsden AL.<strong> A method to quantify mechanobiologic forces during zebrafish cardiac development using 4-D light sheet imaging and computational modeling.</strong> PLoS computational biology. 2017 Oct 30;13(10):e1005828.</a></p>
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<p><a id="ref-4">
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[4] Mittal R, Seo JH, Vedula V, Choi YJ, Liu H, Huang HH, Jain S, Younes L, Abraham T, George RT. <strong>Computational modeling of cardiac hemodynamics: current status and future outlook.</strong> Journal of Computational Physics. 2016 Jan 15;305:1065-82. </a></p>
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<p><a id="ref-5"> <a href="https://doi.org/10.1115/1.4048032">
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[5] Kong, F., and Shadden, S. C. (2020). <strong>Automating Model Generation for Image-based Cardiac Flow Simulation.</strong> ASME. J Biomech Eng. </a> </a></p>
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<p><br><br><br><br><br></p>

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