Buckling of an elastic half-space with surface imperfections

Abstract

Over the years, linear analysis of the stability of pre-stressed elastic half-spaces has been carried out for various kinds of materials and various forms of pre-stress; see Biot (1965), Nowinski (1969a, b), Willson (1973a, b), Usmani \& Beatty (1974), Chadwick \& Jarvis (1979), Wu (1979, 1980), Reddy (1983), Ogden (1984), Wu \& Cao (1983, 1984), and Dowaikh \& Ogden (1990, 1991). Furthermore, Wu \& Cao (1984) showed that the bifurcation condition for an elastic half-space is in fact the same as those for an infinite space with a crack of arbitrary shape, for a circular disk, and for a semi-infinite strip. A linear stability analysis provides the condition under which a pre-stressed half-space may buckle, but it gives no information about the form or stability of post-buckling states; such information can only be found by a nonlinear analysis. For a pre-stressed half-space, the critical stress for marginal stability is independent of mode numbers, which makes the corresponding weakly nonlinear analysis very different from that for problems where there is usually a preferred buckling mode. A first attempt at finding a post-buckling solution was made by Ogden \& Fu (1996) who looked for a solution in the form of a Fourier summation. Imposition of a solvability condition at second order of a successive approximation yielded an infinite system of quadratic equations for the Fourier amplitudes. However no non-trivial solutions were found for such a system of algebraic equations. To shed some light on this open problem, we consider in this paper the stability of an imperfect elastic half-space, a half-space the surface of which is not flat but has an sinusoidal profile (a modal imperfection). The imperfect half-space is subjected to a uni-axial compression and the evolution of the surface elevation is followed as the compression is increased. It is found that as the compression approaches a critical value, which is smaller than the critical value predicted by the linear theory for a perfect half-space, static shocks begin to develop in the profiles of surface elevation; no stable solutions exist beyond this critical value. These results support the conjecture that post-buckling solutions associated with a pre-stressed half-space without imperfections may contain static shocks (i.e. singular surfaces across which some of the deformation gradients are discontinuous