fiber-exp-pow

Module: solid

Category: material

Type string: "fiber-exp-pow"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
alpha	alpha	0	[]
beta	beta	2	[]
ksi	ksi	0	[P]
mu	mu	0	[P]
lam0	lam0	1	[]
fiber			[]

Description

The material type for a single fiber family with an exponential-power law is fiber-exp-pow. Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a stable compressible material that acts as a ground matrix, using a solid mixture container.

The fiber is oriented along the unit vector \(\mathbf{e}_{1}\), where \(\left\{ \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right\}\) are orthonormal basis vectors representing the local element coordinate system when specified, or else the global Cartesian coordinate system. The Cauchy stress for this fibrous material is given by

\[ \boldsymbol{\sigma}=H\left(I_{n}-I_{0}\right)\frac{2I_{n}}{J}\frac{\partial\Psi}{\partial I_{n}}\mathbf{n}\otimes\mathbf{n}, \]

where \(I_{n}=\lambda_{n}^{2}=\mathbf{n}_{r}\cdot\mathbf{C}\cdot\mathbf{n}_{r}\) is the square of the fiber stretch, \(\mathbf{n}=\mathbf{F}\cdot\mathbf{n}_{r}/\lambda_{n}\), \(I_{0}=\lambda_{0}^{2}\) is the square of the stretch threshold for the tensile response (\(\lambda_{0}=1\) by default) and \(H\left(.\right)\) is the unit step function that enforces the tension-only contribution. The fiber strain energy density is given by

\[ \Psi=\frac{\xi}{\alpha\beta}\left(\exp\left[\alpha\left(I_{n}-I_{0}\right)^{\beta}\right]-1\right)\,, \]

where \(\xi>0\), \(\alpha\geqslant0\), and \(\beta\geqslant2\).

Note: In the limit when \(\alpha\to0\), this expressions produces a power law,

\[ \lim\limits_{\alpha\to0}\,\Psi=\frac{\xi}{\beta}\left(I_{n}-I_{0}\right)^{\beta} \]

Note: When \(\beta>2\), the fiber modulus is zero at the strain origin (\(I_{n}=I_{0}\)). Therefore, use \(\beta>2\) when a smooth transition in the stress is desired from compression to tension.

Example 1: Single fiber oriented along \(\mathbf{e}_{1}\), embedded in a neo-Hookean ground matrix.

<material id="1" type="solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="neo-Hookean">
    <E>1000.0</E>
    <v>0.45</v>
  </solid>
  <solid type="fiber-exp-pow">
    <ksi>5</ksi>
    <alpha>20</alpha>
    <beta>3</beta>
    <mat_axis type="angles">
        <theta>0</theta>
        <phi>90</phi>
    </mat_axis>
  </solid>
</material>

Example 2:

Two fibers in the plane orthogonal to \(\mathbf{e}_{1}\), oriented at ±25 degrees relative to \(\mathbf{e}_{3}\), embedded in a neo-Hookean ground matrix.

<material id="1" type="solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="neo-Hookean">
    <E>1000.0</E>
    <v>0.45</v>
  </solid>
  <solid type="fiber-exp-pow">
    <ksi>5</ksi>
    <alpha>20</alpha>
    <beta>3</beta>
    <fiber type="angles">
      <theta>90</theta>
      <phi>25</phi>
    </fiber>
  </solid>
  <solid type="fiber-exp-pow">
    <ksi>5</ksi>
    <alpha>20</alpha>
    <beta>3</beta>
    <fiber type="angles">
      <theta>-90</theta>
      <phi>25</phi>
    </fiber>
  </solid>
</material>