uncoupled active fiber stress¶

Module: solid

Category: material

Type string: "uncoupled active fiber stress"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`k`	bulk modulus	0	[P]
`pressure_model`	pressure_model	0	[]
`smax`	smax	0	[P]
`activation`	activation	0	[]
`stl`			[]
`stv`			[]
`mat_axis`			[]

Description¶

An active fiber stress, based on a Hill formulation, can be added via the material uncoupled active fiber stress. This material must be combined with a stable compressible material that acts as a passive matrix, using a uncoupled solid mixture container as described in uncoupled solid mixture. The stress is given by,

\[ \boldsymbol{\sigma}^{a}=J^{-1}T\left(\tilde{\lambda}\right)\mathbf{\mathrm{dev}A}. \]

Here, \(\mathbf{A=a\otimes a}\),with a the unit vector describing the fiber direction in the spatial frame, \(\tilde{\lambda}\) is the deviatoric fiber stretch, \(J\) is the jacobian of the deformation, and

\[ T\left(\tilde{\lambda}\right)=s_{max}a\left(t\right)s_{TL}\left(\tilde{\lambda}\right)s_{TV}\left(\dot{\tilde{\lambda}}\right) \]

The parameters \(s_{TL}\) and \(s_{TV}\) are functions that need to be defined in place. There are currently two ways of defining these functions, either via a mathematical expression or a list of sample points. An example is given below. If these parameters are omitted, they are replaced by the constant \(1\) in the equation for the stress above.

Example 1:

An example defining the stl parameter via a mathematical expression.

<material id="1" type="uncoupled solid mixture">
  <k>100.0</k>
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="Mooney-Rivlin">
    <c1>1.0</c1>
    <c2>0</c2>
  </solid>
  <solid type="uncoupled active fiber stress">
    <smax>1.5</smax>
    <a lc="1">0.1</a>
    <stl type="math">
      <math>(l-1)^2</math>
    </stl>
  </solid>
</material>

Example 2:

An example defining the stl parameter via a point list.

<material id="1" type="uncoupled solid mixture">
  <k>100.0</k>
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="Mooney-Rivlin">
    <c1>1.0</c1>
    <c2>0</c2>
  </solid>
  <solid type="uncoupled active fiber stress">
    <smax>1.5</smax>
    <a lc="1">0.1</a>
    <stl type="point">
      <interpolate>linear</interpolate>
      <points>
        <pt>0,0</pt>
        <pt>1,0</pt>
        <pt>2,1</pt>
      </points>
    </stl>
  </solid>
</material>