A new spectral representation of the strain energy function for linear anisotropic elasticity with a generalization for damage

Abstract

A new spectral representation of the strain energy function for linear anisotropic elasticity is proposed in the form of a sum of 6 scalar eigen-stiffnesses times the squares of their associated scalar eigen-strains. Since this new representation is general, it is no less or more general than the standard representation. However, the spectral form reveals fundamental coupled eigen-modes of deformation and energy associated with general anisotropic response. Specifically, each eigen-strain is the inner product of the strain tensor with its symmetric eigen-tensor. The 6 eigen-strains are linearly independent and the 6 eigen-tensors are also linearly independent. Moreover, unlike the eigenvectors of a symmetric matrix, distinct pairs of eigen-tensors are not all orthogonal. The 15 constants that define the eigen-tensors together with the 6 eigen-stiffnesses form 21 independent material constants that characterize the fundamental physics of material anisotropy. These 21 constants are invariant to the orientation of the base vectors fixed in the material. This spectral representation also simplifies the restrictions on these material constants which ensure that the strain energy is a positive-definite function of strain. In addition, a simple damage model is proposed based on this spectral representation and a generalization for finite deformations is briefly discussed.