fiber-exp-pow-linear

Module: solid

Category: material

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

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
E	E	0	[]
alpha	alpha	0	[]
beta	beta	3	[]
lam0	lam0	1	[]
fiber			[]

Description

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

The fiber strain energy density is given by

\[ \Psi_{n}=\begin{cases} 0 & I_{n}<1\\ \frac{\xi}{\alpha\beta}\left(\exp\left[\alpha\left(I_{n}-1\right)^{\beta}\right]-1\right) & 1\le I_{n}\le I_{0}\\ B\left(I_{n}-I_{0}\right)-E\left(I_{n}^{1/2}-I_{0}^{1/2}\right)+\frac{\xi}{\alpha\beta}\left(\exp\left[\alpha\left(I_{0}-1\right)^{\beta}\right]-1\right) & I_{n}>I_{0} \end{cases} \]

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

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

and

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

For this material type, the fiber elasticity at the strain origin reduces to zero unless \(\beta=2\). This model reduces to \fiber-pow-linear\ when \(\alpha=0\). Alternatively, in the limit when \(I_{0}=1\), the above parameters reduce to \(\xi=0\) and \(B=E/2\) and the strain energy density takes the quadratic form

\[ \Psi_{n}=\begin{cases} 0 & I_{n}<1\\ \frac{E}{2}\left(I_{n}^{1/2}-1\right)^{2} & I_{n}>1 \end{cases} \]

where \(I_{n}^{1/2}=\lambda_{n}\) is the stretch ratio along the fiber.

Example:

<solid type="fiber-exp-pow-linear">
 <E>1080</E>
 <alpha>1400</alpha>
 <beta>2.73</beta>
 <lam0>1.012</lam0>
 <fiber type="vector">0,0,1</fiber>
</solid>