The elastic behavior and stability of elemental crystals are studied in the neighborhood of a stable equilibrium state, also called a phase, at finite pressure p. It is shown that two kinds of elastic constants are needed to describe elasticity under pressure. One set, designated as c(ij),i,j = 1-6, determines stability or lack of it; another set, designated as c(ij)(p), describes the linear relation between small additional stresses and strains added to the crystal in equilibrium at p. The stress-strain coefficients c(ij)(p) differ from previous formulations of the stress-strain relations by Barron and Klein (1965 Proc. Phys. Soc. 85 523) and Wallace (1972 Thermodynamics of Crystals (New York: Wiley)), who give c(ij) as stress-strain coefficients. Hence we were led to verify the use of the c(ij)(p) using a first-principles numerical calculation example for face centered cubic Al at 1500 kbar. The Gibbs free energy G of the crystal under pressure is shown to provide both a simple definition of equilibrium and an efficient way to calculate all the elastic constants of a general crystal. A computer program finds stable phases by making jumps in structure from an arbitrary initial structure; the jumps converge to minima of G with respect to the structure. In the calculation, 21 elastic constants are evaluated from a special set of G values and the 6 × 6 elastic constant matrix is tested for stability.