The internal stability of an elastic solid

This paper investigates the conditions of elastic stability that set the up per limits of mechanical strength. Following Gibbs, we determine the condit ions that ensure stability against reconfigurations that leave the boundary of the system unchanged. The results hold independent of the nature of pro perties of the loading mechanisms but are identical with those derived prev iously for a solid in contact with a reservoir that maintains the Cauchy st ress. 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 s trength. The conditions of stability are contained in the requirement that lambda (ijkl) delta epsilon (ij) delta epsilon (kl) greater than or equal t o 0 for all infinitesimal strains, where lambda (ijkl) = 1/2 (B-ijkl + B-kl ij), and B is the tensor that governs the change in the Cauchy stress t dur ing incremental strain from a stressed state tau : t(ij) = tau (ij) + B-ijk l delta epsilon (kl). Since lambda has full Voigt symmetry, it can be writt en as the 6 x 6 matrix lambda (ij) with eigenvalues lambda (alpha). Stabili ty is lost when the least of these vanishes. The conditions of stability ar e exhibited for cubic (hydrostatic), tetragonal (tensile) and monoclinic (s hear) distortions of a cubic crystal and some of their implications are dis cussed. Elastic stability and the limits of strength are now being explored through first-principles calculations that increment uniaxial stretch or s hear to find the maximum stress. We discuss the nature of this limiting str ess and the steps that may be taken to identify orthogonal instabilities th at might intrude before it is reached.