Prev Subsubsection 4.13.1.3: Uncoupled Prescribed Isotropic Active Contraction Up Subsection 4.13.1: Contraction in Mixtures of Uncoupled Materials Subsection 4.13.2: Contraction in Mixtures of Unconstrained Materials Next
4.13.1.4 Prescribed Fiber Stress
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 Section 4.1.2.14↑. The stress is given by,
Here, with a the unit vector describing the fiber direction in the spatial frame, is the deviatoric fiber stretch, J is the jacobian of the deformation, and
The parameters are defined below.
| smax | scale factor for defining maximum stress | [P] |
| a | activation level | |
| stl | Function defining the stress-stretch relatioship for the material | |
| stv | Function defining the stress-stretch rate relatioship for the material. |
The parameters and 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 parametes are omitted, they are replaced by the constant 1 in the equation for the stress above.
Example:
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>
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>