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5.3.5 Transversely Isotropic Hyperelastic
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 [109, 82, 84]. 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 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). The function represents the material response of the isotropic ground substance matrix, while represents the contribution from the fiber family. The strain energy of the fiber family is as follows: Here, 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 , It also follows that where
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 .