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220.127.116.11 Transversely Isotropic Mooney-Rivlin
The material type for transversely isotropic Mooney-Rivlin materials is “trans iso Mooney-Rivlin”. The following material parameters must be defined:
|<c1>||Mooney-Rivlin coefficient 1||[P]|
|<c2>||Mooney-Rivlin coefficient 2||[P]|
|<c3>||Exponential stress coefficient||[P]|
|<c4>||Fiber uncrimping coefficient||[ ]|
|<c5>||Modulus of straightened fibers||[P]|
|<lam_max>||Fiber stretch for straightened fibers||[ ]|
|<fiber>||Fiber distribution option|
This constitutive model can be used to represent a material that has a single preferred fiber direction and was developed for application to biological soft tissues [46, 47, 58]. It can be used to model tissues such as tendons, ligaments and muscle. The elastic response of the tissue is assumed to arise from the resistance of the fiber family and an isotropic matrix. It is assumed that the uncoupled strain energy function can be written as follows: Here and are the first and second invariants of the deviatoric version of the right Cauchy Green deformation tensor and is the deviatoric part of the stretch along the fiber direction ( , where is the initial fiber direction), and is the Jacobian of the deformation (volume ratio). The function represents the material response of the isotropic ground substance matrix and is the same as the Mooney-Rivlin form specified above, while represents the contribution from the fiber family. The strain energy of the fiber family is as follows: where is the exponential integral function. The resulting fiber stress is evaluated from Here, and are the Mooney-Rivlin material coefficients, lam_max ( is the stretch at which the fibers are straightened, scales the exponential stresses, is the rate of uncrimping of the fibers, and is the modulus of the straightened fibers. is determined from the requirement that the stress is continuous at .
This material model uses a three-field element formulation, interpolating displacements as linear field variables and pressure and volume ratio as piecewise constant on each element .
The fiber orientation can be specified as explained in Section 4.1.1↑. Active stress along the fiber direction can be simulated using an active contraction model. To use this feature you need to define the active_contraction material. This material has a ascl property that takes an optional attribute, lc, which defines the loadcurve. There are also several options:
|<ascl>||Activation scale factor (default=0)|
|<ca0>||Intracellular calcium concentration (default=1)|
|<camax>||Maximum peak intracellular calcium concentration (default=ca0)|
|<beta>||tension-sarcomere length relation constant|
|<l0>||No tension sarcomere length|
|<refl>||Unloaded sarcomere length|
|<Tmax>||Isometric tension under maximal activation at camax (default=1)|
This example defines a transversely isotropic material with a Mooney-Rivlin basis. It defines a homogeneous fiber direction and uses the active fiber contraction feature.
<material id="3" type="trans iso Mooney-Rivlin"> <c1>13.85</c1> <c2>0.0</c2> <c3>2.07</c3> <c4>61.44</c4> <c5>640.7</c5> <k>100.0</k> <lam_max>1.03</lam_max> <fiber type="vector">1,0,0</fiber> <active_contraction> <ascl lc="1">1</ascl> <ca0>4.35</ca0> <beta>4.75</beta> <l0>1.58</l0> <refl>2.04</refl> </active_contraction> </material>