Two-sided estimates in the stability problem for compressed elastic blocks

Abstract

We theoretically study stability problems for the equilibria of homogeneously compressed block-shaped nonlinearly elastic bodies of arbitrary proportions under certain specially kinematic boundary conditions on two out of three pairs of faces. The study is based on the static energy criterion for stability (instability) in small. (The perturbations are small with respect to states with large initial strains and stresses.) The properties of the material under arbitrary strains are determined by the proposed and investigated family of elastic potentials, which generalize the Mooney-Rivlin potential to the case of orthotropy and compressibility. Since there are no known exact solutions of three-dimensional stability problems (which give necessary and sufficient conditions for stability), we solved the problems on necessary and sufficient conditions separately. The necessary conditions for stability (the upper bounds for the critical values of the loading parameter) were sought by the traditional method of kinematic hypotheses on the basis of the proposed new kinematic hypothesis related to the fields-extremals of the variational problem for the Korn constant. The sufficient conditions for stability (the lower bounds for the critical values of the loading parameter) were sought by the Holden method on the basis of specific values of the Korn constant (the solutions of the above-mentioned variational problem) without which the Holden method does not give any results. The finally obtained upper and lower bounds were compared with each other for varied geometric and rigidity parameters of bodies. It was found that the used modifications of well-known methods for studying the stability (instability) for any thicknesses can in principle be used and are sufficiently efficient.