The internal stability of an elastic solid

Abstract

This paper investigates the conditions of elastic stability that set the upper limits of mechanical strength. Following Gibbs, we determine the conditions that ensure stability against reconfigurations that leave the boundary of the system unchanged. The results hold independent of the nature of properties of the loading mechanisms but are identical with those derived previously for a solid in contact with a reservoir that maintains the Cauchy stress. Mechanisms that control the stress in some other way may add further conditions of stability; nonetheless, the conditions of internal stability must always be obeyed and can be consistently used to define the ultimate strength. The conditions of stability are contained in the requirement that Λ_ijkl δ_ijkl δε_kl ≥ 0 for all infinitesimal strains, where Λ_ijkl =½ (B _ijkl + B _klij), and B is the tensor that governs the change in the Cauchy stress t during incremental strain from a stressed state τ:t_ij = T_ij + B _ijkl δε_kl. Since Λ has full Voigt symmetry, it can be written as the 6 × 6 matrix Λ_ij with eigenvalues Λ_α. Stability is lost when the least of these vanishes. The conditions of stability are exhibited for cubic (hydrostatic), tetragonal (tensile) and monoclinic (shear) distortions of a cubic crystal and some of their implications are discussed. Elastic stability and the limits of strength are now being explored through first-principles calculations that increment uniaxial stretch or shear to find the maximum stress. We discuss the nature of this limiting stress and the steps that may be taken to identify orthogonal instabilities that might intrude before it is reached.