Yet another best approximation isotropic elasticity tensor in plane strain

TL;DR

This paper addresses the problem of approximating a general planar anisotropic elasticity tensor by an isotropic counterpart. It introduces a novel two-test macroscopic procedure, using a concentrated couple and a center of dilatation, and fits FEM results to analytical isotropic radial solutions to extract $\kappa$ and $\mu$. The key finding is that the resulting isotropic tensor is not the same as Norris’ best Euclidean or logarithmic fit, challenging the notion of a unique best-fit isotropic tensor. The results show $\kappa^{\rm iso}$ closely tracks $\kappa$ while $\mu^{\rm iso}$ depends strongly on $\mu^*$, with full-field fitting sometimes aligning more with the logarithmic Norris estimate or deviating, highlighting practical implications for homogenization and metamaterial design. The work provides a generalizable framework applicable to any symmetry class and offers a potential route to calibrate micro-scale elasticity tensors in complex continua.

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 Lamé-moduli $μ$, $λ$ (or $μ$ and the bulk-modulus $κ$) 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.

Figures (14)

• Figure 1: Visualization of Norris formula for the Euclidean distance (orange line) and the logarithmic distance (blue line). The contour plot shows the associated best approximated $\mu^{\rm iso}=1$ for given $\mu,\mu^*$ (cubic symmetry) while the bulk modulus $\kappa=\kappa^{\rm iso}$ has no effect. Both approximations coincide for $\mu=\mu^*=1$.
• Figure 2: (Left) Schematic representation of the concentrated couple. (Right) Schematic representation of the center of dilatation.
• Figure 3: Variation of the normalized modulus of displacement $\lVert u \rVert \mu$ for a concentrated couple (unit strength), (Left) cubic material ($\mu^*=0.1\space \mu$, $\lambda=0.29\space \mu$), (Right) isotropic material ($\lambda=0.29\space \mu$).
• Figure 4: Variation of the normalized modulus of displacement $\lVert u \rVert \mu$ for a center of dilatation (unit strength) (Left) Cubic material ($\mu^*=0.1\space \mu$, $\lambda=0.29\space \mu$), (Right) Isotropic material ($\lambda=0.29\space \mu$).
• Figure 5: Instead of applying the load at the origin as done in the analytical solution in Figure \ref{['fig:DM']}, in the numerical approximation we apply the load on a small circle at the center of the domain. It remains to choose the diameter of the circle in relation to the width of the domain. (Left) Schematic drawing of the statically equivalent reactions in the case of the concentrated couple. (Right) Schematic drawing of the statically equivalent reactions in the case of the center of dilatation.