Estimation of ultimate safe loads in elastic stability theory

Abstract

Ultimate safe load estimates are derived for general isotropic materials. Conditions sufficient to assure stability of equilibrium in the dead load traction boundary value problem are obtained from an energy type criterion by application of certain inequalities and by subsequent separation of the incremental shear strain components from the others. Specific estimates are provided for stability of the undistorted states of every isotropic elastic material, and it is shown that the inequalities based upon the estimates derived here imply those given elsewhere; in the natural state of vanishing stress and strain the conclusions are coincident with the classical inequalities for the shear and bulk moduli. General sufficient conditions for stability of equilibrium of an incompressible Mooney-Rivlin body are derived. Simple shear and simple extension analyses for isotropic Hadamard elastic materials reveal that several classical relations can be obtained for this class of materials in the theory of finite strain, and explicit formulae for determination of certain elastic constants and familiar moduli are provided. Sufficient conditions for stability for every such Hadamard elastic material are obtained by special application of the general formulae developed earlier in the work, and particular results for a compressible Mooney-Rivlin type material are given. It is found that these include as a special case our earlier results for the familiar incompressible Mooney-Rivlin body. It happens also that several inequalities usually obtained from certain requirements that assure resonable material response to the loading, but which otherwise have been unrelated to stability, appear here as natural sufficient conditions for stability in the dead load problem. Finally, several of the results are applied to study the problem of stability of an Euler column, and physical implications of the analyses are discussed.