A generalized strain energy function using fractional powers: Application to isotropy, transverse isotropy, orthotropy, and residual stress symmetry

Abstract

In this paper, we propose a generalized strain energy density function based on invariants of stretch tensor with arbitrary exponents. We employ polynomial, logarithmic and exponential functions of these invariants to develop the strain energy functions. We also study characteristics and applications of the proposed model for isotropy, transverse isotropy, orthotropy with a special focus on initial/residual stress symmetry. The proposed invariants generalize the existing strain energy potentials constructed with invariants of Cauchy stretch. We construct generalized strain energy functions for initial stress problems using initial stress symmetries. We obtain objectivity of strain energy for initially stressed transversely isotropic solids to derive the invariants and study resulting symmetry. By extending Dunford–Taylor integration based approach and tensor diagonalization approach, we obtain stress through differentiation of anisotropic scalar invariants with respect to a tensor. These approaches are usually applicable for derivative of isotropic tensor functions with respect to tensors. Using initial stress compatibility, we derive constraint equations for material parameters by evaluating the limits of Cauchy stress in the reference configuration. In order to apply the proposed model for initial stress problems, we further investigate bending and unbending of hyperelastic structures. We study bending of a rectangle to a cylinder and unbending of a cylinder to a rectangle in presence of initial/residual stress and observe both magnifying and moderating effects of initial stress for stress distribution and flexural characteristics. Using experimental data we substantiate the proposed model for seat foam, bovine pericardium and rabbit skin which represent compressible isotropic and incompressible orthotropic materials. For demonstration, we use both polynomial and exponential functions of these invariants. Furthermore, we develop a nonlinear finite element computational model for thermo-hyperelastic structure where thermal stress represents the initial stress. We corroborate this stress–strain data with the proposed model for initial stress symmetry and observe nice agreements.