On constitutive equations for anisotropic nonlinearly viscoelastic solids
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abstract
Many biological, geological and synthetic bodies are anisotropic. In particular, some of these bodies reflect the anisotropy due to fiber reinforcement along a direction or more than one direction. In a recent paper Merodio developed constitutive equations for fiber-reinforced transversely isotropic nonlinearly viscoelastic bodies where the transverse isotropy was a consequence of the presence of a single family of unidirectional reinforcements. It was shown that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants, eight of them associated with the deformation and the orientation of the fiber. Here we provide a corresponding analysis for more general constitutive equations: anisotropic models with two preferred fiber directions, i.e., two families of fiber reinforcements. In this case, we show that the constitutive equations can be expressed in terms of functions of 37 independent invariants. These invariants are analyzed with regard to their properties, e.g., their non-negativity, etc, which have a bearing on the response characteristics of the body. There are 11 coupling invariants arising from the simultaneous existence of both family of fibers. These invariants depend, among other variables, on the relative fiber orientations, and the physical implications of these invariants are discussed. We determine the invariants for two illustrative deformation gradients: (i) a diagonal one aligned with one of the two fiber directions and corresponding to a homogeneous deformation, and (ii) a simple shear deformation along one of the two fiber directions in a plane containing both fibers. The physical significance of all the invariants, in the two different cases, is discussed. The need for simplifying these models to a form that is amenable to analysis and experimental corroboration is also discussed.
