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Subsubsection 4.13.2.3: Prescribed Isotropic Active Contraction Up Subsection 4.13.2: Contraction in Mixtures of Unconstrained Materials Section 4.14: Viscous Fluids

#### 4.13.2.4 Prescribed Fiber Stress

An active fiber stress, based on a Hill formulation, can be added via the material “active fiber stress”. This material must be combined with a stable compressible material that acts as a passive matrix, using a “solid mixture” container as described in Section 4.1.3.22↑. The stress is given by,
Here, with a the unit vector describing the fiber direction in the spatial frame, is the 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 parametersand 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="solid mixture">
<mat_axis type="local">0,0,0</mat_axis>
<solid type="neo-Hookean">
<E>1.0</E>
<v>0</v>
</solid>
<solid type="active fiber stress">
<smax>150</smax>
<a lc="1">0.1</a>
<stl type="math">
$(l-1)^2$
</stl>
</solid>
</material>

An example defining the stl parameter via a point list.
<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</v>
</solid>
<solid type="active fiber stress">
<smax>150</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>

Subsubsection 4.13.2.3: Prescribed Isotropic Active Contraction Up Subsection 4.13.2: Contraction in Mixtures of Unconstrained Materials Section 4.14: Viscous Fluids