orthotropic CLE¶

Module: solid

Category: material

Type string: "orthotropic CLE"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`lp11`	lambda_11 tensile	-6.27744e+66	[P]
`lm11`	lambda_11 compressive	-6.27744e+66	[P]
`lp22`	lambda_22 tensile	-6.27744e+66	[P]
`lm22`	lambda_22 compressive	-6.27744e+66	[P]
`lp33`	lambda_33 tensile	-6.27744e+66	[P]
`lm33`	lambda_33 compressive	-6.27744e+66	[P]
`l12`	lambda_12	-6.27744e+66	[P]
`l23`	lambda_23	-6.27744e+66	[P]
`l31`	lambda_13	-6.27744e+66	[P]
`mu1`	mu_1	-6.27744e+66	[P]
`mu2`	mu_2	-6.27744e+66	[P]
`mu3`	mu_3	-6.27744e+66	[P]
`mat_axis`			[]

Description¶

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

This bimodular elastic material is the orthotropic conewise linear elastic material described by Curnier et al. ¹. It is derived from the following hyperelastic strain-energy function:

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

where \(\lambda_{ba}=\lambda_{ab}\) and

\[ \lambda_{aa}\left[\mathbf{A}_{a}^{r}:\mathbf{E}\right]=\begin{cases} \lambda_{+aa} & \mathbf{A}_{a}^{r}:\mathbf{E}\geqslant0\\ \lambda_{-aa} & \mathbf{A}_{a}^{r}:\mathbf{E}<0 \end{cases},\quad a=1,2,3 \]

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=" orthotropic CLE">
  <density>1</density>
  <lp11>13.01</lp11>
  <lp22>13.01</lp22>
  <lp33>13.01</lp33>
  <lm11>0.49</lm11>
  <lm22>0.49</lm22>
  <lm33>0.49</lm33>
  <l12>0.66</l12>
  <l23>0.66</l23>
  <l31>0.66</l31>
  <mu1>0.16</mu1>
  <mu2>0.16</mu2>
  <mu3>0.16</mu3>
</material>

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