Shenoy

Module: solid

Category: material

Type string: "Shenoy"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
mu	mu	0	[]
k	k	0	[]
Ef	Ef	0	[]
lam_c	lam_c	1	[]
lam_t	lam_t	0	[]
n	n	5	[]
m	m	1	[]

Description

This material implements the constitutive model by Wang et al. ¹, which proposes a mechanism for long-range force transmission in fibrous matrices enabled by tension-driven alignment of fibers.

The strain-energy density function is given by,

\[ W=W_{\text{b}}+W_{\text{f}}, \]

where:

\[ \begin{aligned}W_{\text{b}} & =\frac{\mu}{2}\left(\overline{I}_{1}-3\right)+\frac{\kappa}{2}\left(J-1\right)^{2},\\ W_{\text{f}} & =\stackrel[a=1]{3}{\sum}f\left(\lambda_{a}\right), \end{aligned} \]

and \(f\) is defined via its derivative,

\[ \frac{\partial f}{\partial\lambda_{a}}\left(\lambda_{a}\right)=\begin{cases} 0, & \lambda_{a}<\lambda_{1}\\ \frac{E_{f}\left(\frac{\lambda_{a}-\lambda_{1}}{\lambda_{2}-\lambda_{1}}\right)^{n}\left(\lambda_{a}-\lambda_{1}\right)}{n+1}, & \lambda_{1}\leq\lambda_{a}<\lambda_{2}\\ E_{f}\left[\frac{\lambda_{2}-\lambda_{1}}{n+1}+\frac{\left(1+\lambda_{a}-\lambda_{2}\right)^{m+1}-1}{m+1}\right], & \lambda_{a}\geq\lambda_{2} \end{cases} \]

Finally, the parameters \(\lambda_{1}\) and \(\lambda_{2}\) are given as follows.

\[ \lambda_{1} =\lambda_{c}-\lambda_{t}/2 \lambda_{2} =\lambda_{c}+\lambda_{t}/2 \]

Example:

<material id="1" name="Material1" type="Shenoy">
  <density>1</density>
  <mu>0.7692</mu>
  <k>1.667</k>
  <Ef>134.6</Ef>
  <lam_c>1.02</lam_c>
  <lam_t>0.255</lam_t>
  <n>5</n>
  <m>30</m>
</material>

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1. Wang et. al., Biophysical Journal, 107, 2014, pp:2592 - 2603 ↩