A Simplified Formulation of Third-Order Elasticity for Solids of Cubic Symmetry

Summary A notation for third-order elastic constants (TOECs) for solids of cubic symmetry is proposed. Based upon Walpole’s formulation of linear elasticity, we show that there are six distinct sixth-rank tensors of elasticity. This leads naturally to a diagonalizable basis for the six-dimensional space of cubic TOECs. The formulation provides simple relationships between the stiffness coefficients and the compliance coefficients for each of the six TOECs of cubic symmetry. It also provides a natural partition of stress and strain into hydrostatic and deviatoric parts at the linear and quadratic level. Relations for the isotropic Voigt and Reuss averages of cubic TOECs reduce to simple expressions in terms of the proposed six-dimensional basis for cubic TOECs. The six new cubic TOECs map to the set of three isotropic Voigt and Reuss averaged TOECs in subsets of one, two and three. The Voigt and Reuss isotropic TOECs are also the unique minimizers of the “distance” between elastic moduli of isotropic and cubic symmetries, providing a simple means to estimate the orientation averaged elastic constants of cubic crystals. An analysis of 59 cubic crystals reveals that the nonlinear contribution to hydrostatic stress is typically negative, which concurs with material stiffening behavior under hydrostatic pressure. The framework introduces a nonlinear anisotropy measure analogous to Zener’s index and establishes quantitative metrics for assessing departure from isotropy in TOECs.