Pressure-induced elastic instabilities are investigated in the prototypic ionic and covalent solids (MgO, CaO, SiO2 and Si) using generalized elastic stability criteria based on the elastic stiffness coefficients (c(ij)) which are determined directly from stress-strain relations. From first-principles computer simulations of the instabilities, we demonstrate the validity and importance of the generalized criteria relative to the conventional criteria in describing the crystal stability under hydrostatic pressure in relation to the real structural transformations. We examine systems for which the two phases can be related by a simple deformation, and in all cases we show that the generalized elastic stiffness coefficient associated with that deformation softens toward the transition. The shear stability criterion (c(44) > 0) bounds the first-order B1-B2 phase transition pressure from above and below in MgO and CaO, suggesting a wide pressure regime of metastability, whereas the tetragonal shear stability criterion ((c(11) - c(12))/2 > 0) predicts precisely the second-order rutile-to-CaCl2 transition in SiO2. The high-pressure elastic behaviour of diamond structure Si is studied in detail. A tetragonal shear instability corresponding to its transformation to the beta-Sn structure should occur in diamond structure Si at a pressure of 101 GPa, compared to the experimental value of 9 to 13 GPa for the transition pressure.
