Skip to content

cubic CLE

Module: solid

Category: material

Type string: "cubic CLE"

Parameters

Name Description Default Units
density density 1 [M/L^3]
lp1 lp1 -6.27744e+66 [P]
lm1 lm1 -6.27744e+66 [P]
l2 l2 -6.27744e+66 [P]
mu mu -6.27744e+66 [P]
mat_axis []

Description

The material type for a conewise linear elastic (CLE) material with cubic symmetry is cubic CLE.

This bimodular elastic material is specialized from the orthotropic conewise linear elastic material described by Curnier et al. 1, to the case of cubic symmetry. It is derived from the following hyperelastic strain-energy function:

\[ \Psi_{r}=\mu\mathbf{I}:\mathbf{E}^{2}+\sum\limits_{a=1}^{3}\frac{1}{2}\lambda_{1}\left[\mathbf{A}_{a}^{r}:\mathbf{E}\right]\left(\mathbf{A}_{a}^{r}:\mathbf{E}\right)^{2}+\frac{1}{2}\lambda_{2}\sum\limits_{\begin{array}{c} b=1\\ b\ne a \end{array}}^{3}\left(\mathbf{A}_{a}^{r}:\mathbf{E}\right)\left(\mathbf{A}_{b}^{r}:\mathbf{E}\right)\,, \]

where

\[ \lambda_{1}\left[\mathbf{A}_{a}^{r}:\mathbf{E}\right]=\begin{cases} \lambda_{+1} & \mathbf{A}_{a}^{r}:\mathbf{E}\geqslant0\\ \lambda_{-1} & \mathbf{A}_{a}^{r}:\mathbf{\,}.E<0 \end{cases} \]

Here, \(\mathbf{E}\) is the Lagrangian strain tensor and \(\mathbf{A}_{a}^{r}=\mathbf{a}_{a}^{r}\otimes\mathbf{a}_{a}^{r}\), where \(\mathbf{a}_{a}^{r} (a=1,2,3)\) are orthonormal vectors aligned with the material axes. This material response was originally formulated for infinitesimal strain analyses; its behavior under finite strains may not be physically realistic.

Example: 

<material id="1" type="cubic CLE">
  <density>1</density>
  <lp1>13.01</lp1>
  <lm1>0.49</lm1>
  <l2>0.66</l2>
  <mu>0.16</mu>
</material>

  1. Curnier, A., Qi-Chang, He, and Zysset, P., "Conewise linear elastic materials", J Elasticity 37, 1 (1994), pp. 1-38. ↩