Yet another best approximation isotropic elasticity tensor in plane strain

TL;DR

This paper introduces a novel method for approximating any anisotropic elasticity tensor in plane strain with an isotropic tensor by fitting two simple radial solutions, challenging traditional notions of best-fit isotropic approximations.

Contribution

The paper proposes a new fitting approach based on simple analytic solutions rather than traditional distance measures, providing a different perspective on isotropic approximation.

Findings

The isotropic tensor obtained differs from Norris's logarithmic and Euclidean fits.

The method uses finite element calculations and quadratic error minimization.

It questions the validity of traditional best-fit isotropic tensors for anisotropic materials.

Abstract

For plane strain linear elasticity, given any anisotropic elasticity tensor $\mathbb{C}_{\rm aniso}$, we determine a best approximating isotropic counterpart $\mathbb{C}_{\rm iso}$. This is not done by using a distance measure on the space of positive definite elasticity tensors (Euclidean or logarithmic distance) but by considering two simple isotropic analytic solutions (center of dilatation and concentrated couple) and best fitting these radial solutions to the numerical anisotropic solution based on $\mathbb{C}_{\rm aniso}$. The numerical solution is done via a finite element calculation, and the fitting via a subsequent quadratic error minimization. Thus, we obtain the two Lam\'e-moduli $\mu$, $\lambda$ (or $\mu$ and the bulk-modulus $\kappa$) of $\mathbb{C}_{\rm aniso}$. We observe that our so-determined isotropic tensor $\mathbb{C}_{\rm iso}$ coincides with neither the best logarithmic fit of Norris nor the best Euclidean fit. Our result calls into question the very notion of a best-fit isotropic elasticity tensor to a given anisotropic material.