fiber-exp-pow-uncoupled

Module: solid

Category: material

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

Parameters

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

Description

The material type for a single fiber family with an exponential-power law, in an uncoupled strain energy formulation, is fiber-exp-pow-uncoupled. Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a stable uncoupled material that acts as a ground matrix, using a uncoupled 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 stress \(\tilde{\boldsymbol{\sigma}}\) for this fibrous material is given by

\[ \tilde{\boldsymbol{\sigma}}=H\left(\tilde{I}_{n}-1\right)\frac{2\tilde{I}_{n}}{J}\frac{\partial\tilde{\Psi}}{\partial\tilde{I}_{n}}\mathbf{n}\otimes\mathbf{n}, \]

where \(\tilde{I}_{n}=\tilde{\lambda}_{n}^{2}=\mathbf{n}_{r}\cdot\mathbf{\tilde{C}}\cdot\mathbf{n}_{r}\) is the square of the fiber stretch, \(\mathbf{n}=\mathbf{\tilde{F}}\cdot\mathbf{n}_{r}/\tilde{\lambda}_{n}\), and \(H\left(.\right)\) is the unit step function that enforces the tension-only contribution. The fiber strain energy density is given by

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

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

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

\[ \lim\limits_{\alpha\to0}\tilde{\Psi}=\frac{\xi}{\beta}\left(\tilde{I}_{n}-1\right)^{\beta}. \]

Note: When \(\beta>2\), the fiber modulus is zero at the strain origin (\(\tilde{I}_{n}=1\)). 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 Mooney-Rivlin ground matrix.

<material id="1" type="uncoupled solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <k>10e3</k>
  <solid type="Mooney-Rivlin">
    <c1>10.0</c1>
    <c2>0</c2>
  </solid>
  <solid type="fiber-exp-pow-uncoupled">
    <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 Mooney-Rivlin ground matrix.

<material id="1" type="uncoupled solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <k>10e3</k>
  <solid type="Mooney-Rivlin">
    <c1>10.0</c1>
    <c2>0</c2>
  </solid>
  <solid type="fiber-exp-pow-uncoupled">
    <ksi>5</ksi>
    <alpha>20</alpha>
    <beta>3</beta>
    <mat_axis type="angles">
        <theta>90</theta>
        <phi>25</phi>
    </mat_axis>
  </solid>
  <solid type="fiber-exp-pow-uncoupled">
    <ksi>5</ksi>
    <alpha>20</alpha>
    <beta>3</beta>
    <mat_axis type="angles">
        <theta>-90</theta>
        <phi>25</phi>
    </mat_axis>
  </solid>
</material>