The closest isotropic, cubic and transversely isotropic stiffness and compliance tensor to an arbitrary anisotropic material

Abstract

The aim of this paper is to provide, in the framework of Green elasticity, the closest or nearest fourthorder isotropic, cubic and transversely isotropic elasticity tensors with higher symmetries for a general anisotropic elasticity tensor or any other tensors with lower symmetry. Using a gauge parameter, the procedure is done on a dimensionless form based on different generalized Euclidean distances, namely conventional, log-, and power-Euclidean distance functions. In the case of power-Euclidean distance functions, results are presented for powers of 0.5, 1 and 2. Except for the conventional distance function, the different generalized distance functions adopted in this paper preserve the property of invariance by inversion, meaning that the results for the closest stiffness tensor are also valid for the compliance tensor. Explicit formulations are given for determining the closest isotropic and cubic tensors, where the multiplication tables of the bases are diagonal. More involved coupled equations are given for the coefficients of the closest transversely isotropic elasticity tensors, which can be solved numerically. Two different material cases are studied in the numerical examples, which illustrate the material coefficients and error measures based on the present methods, including the influence from the gauge parameter.