On stability and instability of stratified elastic solids in a gravity field

Abstract

Stability and instability of the equilibrium of stratified elastic solids in a uniform gravity field are studied by means of analysis of the second variation of total (elastic and gravitational) potential energy of the solid. By stratification is meant a continuous dependence of density and incremental elastic moduli of the material on vertical coordinate. The unperturbed stress state is supposed to be hydrostatic, whereas the incremental shear stiffnesses for perturbing deformations are nonzero. Obtained analytically both the necessary condition and the sufficient condition for stability demonstrate stabilizing effect of shear stiffness. For shear stiffnesses approaching zero, each of the conditions converts into obtained earlier by the authors the necessary and sufficient stability condition for stratified compressible fluids (having zero shear stiffness).The sufficient condition for stability for given stratifications of density and bulk modulus, and arbitrary dimensions of the domain, occupied by the solid, yields semi-infinite range of values of shear stiffnesses ensuring stability of the solid for conditions under which the corresponding stratified compressible fluid would be unstable.The sufficient condition for instability (negation of the necessary condition for stability) yields such a nonzero value of shear stiffness, that for shear stiffnesses below it the solid is surely unstable. This value (the lower bound for the exact critical value) depends on horizontal dimensions of the domain occupied by the solid, and on thickness of the layer where the stability condition for corresponding stratified fluid would be violated.Possible applications of obtained theoretical results are discussed.