Energy-Ratio-Based Measure of Elastic Anisotropy

Many materials are anisotropic. However, there is no widely accepted measure for characterizing the degree of elastic anisotropy. Here, assuming that the limiting case of extreme anisotropy should possess a positive semidefinite stiffness matrix, we propose three criteria to evaluate measures of anisotropy and show that the existing measures in the literature do not satisfy all of the proposed criteria. We then introduce a new measure of anisotropy based on the maximum strain energy ratio that is universally applicable to all material systems. The proposed measure is helpful for understanding the properties and behaviors of materials. Furthermore, this measure can be easily generalized to situations involving multiple fields and nonlinearity.

Figure 1

(a) $A_{\text{Zhang}\text{−}\text{Rychlewski}}$ and $A_{\text{Nye}\text{−}\text{Zheng}}$ of selected stiffness matrices with increasing condition number. (b) $A_{\text{energy ratio}}$, $A_{\text{Ranganathan}\text{−}\text{Starzewski}}$, and $A_{\text{Kube}}$ for the same stiffness matrices.

Figure 2

$A_{\text{energy ratio}}$, $A_{\text{Ranganathan}\text{−}\text{Starzewski}}$, and $A_{\text{Kube}}$ for 29 specific crystalline materials.

Figure 3

(a) and (b) The contour plots of total-energy-ratio-based anisotropy degrees for graphene and h-BN with respect to the principal strain state $\widehat{\mathit{ϵ}}$, respectively. (c) and (d) The anisotropy degrees of tangent stiffness tensors for graphene and h-BN under uniaxial tension along the armchair (colored in blue) and zigzag (colored in red) directions, respectively.