prescribed trans iso active contraction

Module: solid

Category: material

Type string: "prescribed trans iso active contraction"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
T0	T0	0	[]
mat_axis			[]

Description

The material type for prescribed isotropic active contraction in a plane transverse to a given direction (or fiber), in a solid mixture of unconstrained materials, is prescribed trans iso active contraction. This material must be combined with a stable compressible material that acts as a passive matrix, using a solid mixture container.

In the reference configuration, 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 active stress \(\boldsymbol{\sigma}^{a}\) for this material is given by

\[ \boldsymbol{\sigma}^{a}=J^{-1}T_{0}\left(\mathbf{B}-\mathbf{n}\otimes\mathbf{n}\right)\,, \]

where \(\mathbf{n}=\mathbf{F}\cdot\mathbf{e}_{1}\) is the stretched fiber orientation in the current (deformed) configuration and \(\mathbf{B}=\mathbf{F}\cdot\mathbf{F}^{T}\) is the left Cauchy-Green tensor.

Example:

Isotropic contraction in plane transverse to \(\mathbf{e}_{1}\), in a mixture containing a neo-Hookean solid.

<material id="1" type="solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="neo-Hookean">
    <E>1.0</E>
    <v>0.3</v>
  </solid>
  <solid type="prescribed trans iso active contraction">
    <T0 lc="2">1</T0>
    <mat_axis type="angles">
      <theta>0</theta>
      <phi>90</phi>
    </mat_axis>
  </solid>
</material>