Yet another best approximation isotropic elasticity tensor in plane strain

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.