TC nonlinear orthotropic¶

Module: solid

Category: material

Type string: "TC nonlinear orthotropic"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`k`	bulk modulus	0	[P]
`pressure_model`	pressure_model	0	[]
`c1`	c1	0	[]
`c2`	c2	0	[]
`beta`	beta		[]
`ksi`	ksi		[]
`mat_axis`			[]

Description¶

The material type for the tension-compression nonlinear orthotropic material is TC nonlinear orthotropic.

This material is based on the following uncoupled hyperelastic strain energy function ¹:

\[ \Psi\left(\mathbf{C},\lambda_{1},\lambda_{2},\lambda_{3}\right)=\tilde{\Psi}_{iso}\left(\mathbf{\tilde{C}}\right)+\sum\limits_{i=1}^{3}\tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)+U\left(J\right)\,. \]

The isotropic strain energy \(\tilde{\Psi}_{iso}\) and the dilatational energy \(U\) are the same as for the Mooney-Rivlin material. The tension-compression term is defined as follows:

\[ \tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)=\begin{cases} \xi_{i}\left(\tilde{\lambda}_{i}-1\right)^{\beta_{i}} & \tilde{\lambda}_{i}>1\\ 0 & \tilde{\lambda}_{i}\leqslant1 \end{cases}\quad\xi_{i}\geqslant0\quad\left(\text{no sum over }i\right)\,. \]

The \(\tilde{\lambda}_{i}\) parameters are the deviatoric fiber stretches of the local material fibers:

\[ \tilde{\lambda}_{i}=\left(\mathbf{a}_{i}^{0}\cdot\mathbf{\tilde{C}}\cdot\mathbf{a}_{i}^{0}\right)^{1/2}\,. \]

The local material fibers are defined (in the reference frame) as an orthonormal set of vectors \(\mathbf{a}_{i}^{0}\). As with all uncoupled materials, this material uses the three-field element formulation.

A complete example for this material follows.

<material id="7" name="cartilage" type="TC nonlinear orthotropic">
  <c1>1.0</c1>
  <c2>0.0</c2>
  <k>100</k>
  <beta>4.3,4.3,4.3</beta>
  <ksi>4525, 4525, 4525</ksi>
  <mat_axis type="local">0,0,0</mat_axis>
</material>

Ateshian, Gerard A, Ellis, Benjamin J, and Weiss, Jeffrey A, "Equivalence between short-time biphasic and incompressible elastic material responses", J Biomech Eng 129, 3 (2007), pp. 405-12. ↩