Anisotropic hyperelasticity using a fourth-order structural tensor approach

Abstract

Many soft biological tissues and fibre-reinforced composites possess a microstructure consisting of an isotropic ground substance reinforced by multiple families of fibres. The strain energy density for such anisotropic bodies is expressed here using a set of fourth-order structural tensors. This provides an extension of the conventional Lamé-type representation of isotropic elasticity to anisotropic bodies and is readily extended to finite deformations. The strain energy density function is decoupled into two parts, each associated with a distinct set of fourth-order structural tensors (FOSTs); naturally satisfying objectivity requirements. The model is first established in consideration of anisotropic linear elastic behaviour; deriving the two sets of fourth-order structural tensors that describe the complete mechanical response of a body possessing two non-orthogonal families subjected to infinitesimal strains. Mathematically, the complete set of fourth-order structural tensors required for a given body is directly related to the integrity basis of a set of symmetric second-order tensors that prescribe the underlying microstructure. The fourth-order structural tensors presented within are used to establish a unique decomposition of the Hookean elasticity tensor into two distinct fourth-order Anisotropic Lamé Tensors. The proposed representation of the strain energy density function presents an ability to characterise an incompressible anisotropic body, or inextensible reinforcing fibre family using the material properties of the continuum, rather than an externally imposed numerical kinematic constraint. A hyperelastic model for finite kinematics of anisotropic bodies is also presented which preserves consistency with linear elasticity in the realm of infinitesimal deformations. The material parameters identified for an incompressible anisotropic body and/or inextensible fibre family are maintained at all scales of deformation.