An anisotropic hyperelastic strain energy function based on 21 icosahedron fiber distributions

Abstract

The microscopic Cauchy strain energy for linear elasticity based on the sum of quadratic strain energies due to pair potentials has only 15 material rari-constants. It is shown that the six vectors connecting opposing vertices of a regular icosahedron can be used to develop a strain energy function for general linear elastic anisotropic response with 21 material constants. Specifically, the six strains of material fibers characterized by these vectors are enhanced by 15 fiber distribution strains due to all combinations of distinct pairs of these vectors. These two-vector fiber distributions introduce coupling that is essential to obtaining general anisotropy. The model is generalized for large deformations by replacing the strains with stretches and by using a Fung-type exponential strain energy which couples the responses of the 21 stretches. The resulting nonlinear hyperelastic strain energy function can be used to model the anisotropic hyperelastic response of fibrous tissues.