Some models for initially stressed and initially strained structurally anisotropic incompressible materials

Abstract

In this paper, we determine the constitutive relations for structurally anisotropic materials from a reference with initial strain or initial stress. Using reference independence of strain energy density, we first deduce the relationship of various initial strain measures with material symmetry and determine the resulting structural tensors. We use these structural tensors to derive the necessary invariants for initially strained materials. We also Examine the polyconvexity of the derived invariants and determine constitutive relations for the initially strained Ogden model. Next, we determine the constitutive relation for a combination of initial stress-texture symmetry which may be a more popular approach of recent interest. We deduce four reference-independent constitutive relations for initially stressed transversely isotropic incompressible materials. The first two strain energy functions are linear in the left Cauchy stretch and its cofactor, respectively, in the stress-free state. In the absence of initial stress, the third strain energy function involves all standard invariants for transversely isotropy. To deduce this strain energy in the presence of initial stress, we invert a fourth-order tensor and perform several involved tensor calculations. Finally, we obtain the constitutive relation for an initially stressed transversely isotropic limited chain extensible (LCE) model. To determine the initial stretch in the preferred texture direction in terms of initial stress, we utilize the geometrical interpretation of the Almansi strain measure.