Nonlinear stability analysis of pre-stressed elastic bodies

Abstract

This article is concerned with the nonlinear analysis of the stability of thick elastic bodies subjected to finite elastic deformations. The analysis is based on the theory of small elastic deformations superimposed on a finite elastic deformation. Attention is drawn to methods developed in the stability analysis of fluids and of thin shells and plates which are readily applicable to the present circumstances. The state of development of the nonlinear stability analysis of thick elastic bodies is summarized in order to provide a basis for subsequent studies, and some new results relating to the stability of an elastic plate subjected to a pre-stress associated either with uniaxial thrust or with simple shear in the presence of all-round pressure are discussed. Near-critical modes in the neighbourhood of so-called critical configurations are considered to depend on, for example, a slow time variable, and nonlinear evolution equations for the mode amplitudes are derived both in the case of a monochromatic mode and for a resonant triad of modes. The crucial role of the ‘nonlinear coefficient’ in such an equation in the analysis of stability, imperfection sensitivity and localization is highlighted. An efficient (virtual work) method for the determination of this coefficient is described together with an alternative method based on the calculation of the total energy of a monochromatic near-critical mode. The influence of the boundary conditions and of the form of the pre-stress is examined and explicit calculation of the nonlinear coefficient is provided for the two representative pre-stress conditions mentioned in the above paragraph. It is shown, in particular, that a resonant triad of modes has an effect similar to that generated by the presence of a geometrical imperfection. The Appendices gather together for reference certain expressions which are used in the body of the article. These include expressions, not given previously in the literature, for the components of the tensor of third-order instantaneous elastic moduli in terms of the principal stretches of the deformation in respect of a general form of incompressible isotropic elastic strain-energy function.