Linear elastic anisotropic solid model¶

Overview¶

This model may be used to represents a solid material with a linear elastic anisotropic response to loading. The relationship between stress and strain is typically represented by Hooke’s Law, which for the case of linear elasticity, may be expressed as:

$\sigma_{ij} = C_{ijkl} \epsilon_{kl},$

where $\sigma_{ij}$ is the $ij$ component of the cauchy stress tensor, $\epsilon_{kl}$ is the $kl$ component of the the strain tensor, and $C_{ijkl}$ is the fourth order elastic stiffness tensor.

Hooke’s Law may also be expressed using Voigt notation for the stress and strain tensors as:

$\tensor{\sigma} = \tensor{C} \cdot \tensor{\epsilon},$

or,

$\begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1113} & c_{1112} \\ c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2213} & c_{2212} \\ c_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3313} & c_{3312} \\ c_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2313} & c_{2312} \\ c_{1311} & c_{1322} & c_{1333} & c_{1323} & c_{1313} & c_{1312} \\ c_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2313} & c_{2312} \end{bmatrix} \begin{bmatrix} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33} \\ 2\epsilon_{23} \\ 2\epsilon_{13} \\ 2\epsilon_{12} \end{bmatrix}.$

Variations¶

The application of linear elasticity as presented above is typically restricted to the case of infinitesimal strain. For the case of infinitesimal strain, the above relations are applied directly. For the case of finite strain, the above relations may be slightly modified to an incremental update and rotation:

$\Delta \tensor{\sigma} &= \tensor{C} \cdot \hat{\tensor{D}},\\ \tensor{\sigma}^{n+1} &= \hat{\tensor{R}}( \tensor{\sigma}^{n} + \Delta \tensor{\sigma} ) \hat{\tensor{R}}^T,$

where $\hat{\tensor{D}}$ is the “incremental rate of deformation tensor” and $\hat{\tensor{R}}$ is the incremental rotation tensor, which are typically calculated from the velocity gradient. This extension into finite strain constitutes a hypo-elastic update, and the choice of method to calculate $\hat{\tensor{D}}$ , and $\hat{\tensor{R}}$ determines if the update is objective. One commonly used method is the Hughes-Winget algorithm.

Parameters¶

The following attributes are supported:

Name	Type	Default	Description
defaultDensity	real64	required	Default Material Density
defaultStiffness	real64_array2d	required	Default Elastic Stiffness Tensor in Voigt notation (6x6 matrix)
name	string	required	A name is required for any non-unique nodes

Example¶

<Constitutive>
  <LinearElasticAnisotropic name="shale"
                            defaultDensity="2700"
                            c11=1.0e10 c12=1.1e9  c13=1.2e9  c14=1.3e9 c15=1.4e9 c16=1.5e9
                            c21=2.0e9  c22=2.1e10 c23=2.2e9  c24=2.3e9 c25=2.4e9 c26=2.5e9
                            c31=3.0e9  c32=3.1e9  c33=3.2e10 c34=3.3e9 c35=3.4e9 c36=3.5e9
                            c41=4.0e9  c42=4.1e9  c43=4.2e9  c44=4.3e9 c45=4.4e9 c46=4.5e9
                            c51=5.0e9  c52=5.1e9  c53=5.2e9  c54=5.3e9 c55=5.4e9 c56=5.5e9
                            c61=6.0e9  c62=6.1e9  c63=6.2e9  c64=6.3e9 c65=6.4e9 c66=6.5e9 />
</Constitutive>