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2.4.4 Transversely Isotropic Hyperelasticity
Transverse isotropy can be introduced by adding a vector field representing the material preferred direction explicitly into the strain energy . We require that the strain energy depends on a unit vector field , which describes the local fiber direction in the undeformed configuration. When the material undergoes deformation, the vector may be described by a unit vector field . In general, the fibers will also undergo length change. The fiber stretch, , can be determined in terms of the deformation gradient and the fiber direction in the undeformed configuration, Also, since is a unit vector, The strain energy function for a transversely isotropic material, is an isotropic function of and . It can be shown  that the following set of invariants are sufficient to describe the material fully: The strain energy function can be written in terms of these invariants such that The second Piola-Kirchhoff can now be obtained in the standard manner: In the transversely isotropic constitutive models described in Chapter 5↓ it is further assumed that the strain energy function can be split into the following terms: The strain energy function represents the material response of the isotropic ground substance matrix, represents the contribution from the fiber family (e.g. collagen), and is the contribution from interactions between the fibers and matrix. The form (2.4.4-7) generalizes many constitutive equations that have been successfully used in the past to describe biological soft tissues e.g. [53, 56, 57]. While this relation represents a large simplification when compared to the general case, it also embodies almost all of the material behavior that one would expect from transversely isotropic, large deformation matrix-fiber composites.