fiber-pow-linear-uncoupled

Module: solid

Category: material

Type string: "fiber-pow-linear-uncoupled"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
k	bulk modulus	0	[P]
pressure_model	pressure_model	0	[]
E	E	0	[P]
lam0	lam0	1	[]
beta	beta	2	[]
fiber			[]

Description

This material type is fiber-pow-linear-uncoupled.

The fiber strain energy density is given by

\[ \tilde{\Psi}_{n}\left(\tilde{I}_{n}\right)=\begin{cases} 0 & \tilde{I}_{n}<1\\ \frac{\xi}{\beta}\left(\tilde{I}_{n}-1\right)^{\beta} & 1\leqslant\tilde{I}_{n}\leqslant I_{0}\\ B\left(\tilde{I}_{n}-I_{0}\right)-E\left(\tilde{I}_{n}^{1/2}-I_{0}^{1/2}\right)+\frac{\xi}{\beta}\left(I_{0}-1\right)^{\beta} & I_{0}<\tilde{I}_{n} \end{cases}\,, \]

where \(I_{0}=\lambda_{0}^{2}\),

\[ \xi=\frac{E}{4\left(\beta-1\right)}I_{0}^{-3/2}\left(I_{0}-1\right)^{2-\beta},\,B=\xi\left(I_{0}-1\right)^{\beta-1}+\frac{E}{2}I_{0}^{-1/2}\,. \]

Example: (Fiber model as specified in a solid mixture)

<solid type="fiber-pow-linear-uncoupled">
  <fiber type="angles">
    <theta>20</center>
    <phi>90</phi>
  </fiber>
  <E>1</E>
  <beta>2.5</beta>
  <lam0>1.06</lam0>
</solid>