Stability of nonhomogeneous aging viscoelastic bodies under dynamic loading

Abstract

One important problem in the mechanics of continua is the stability of a deformable solid body. This problem may be formulated as following: to find restrictions on external forces which guarantee that small perturbations of load cause small perturbations of stress-strain in the body. The stability of a deformable body is a nonlinear problem due to the geometrical nonlinearity of dynamic equations and the physical nonlinearity of material properties [l-3]. There are two fundamental approaches to the study of the stability of deformable bodies. The first approach may be called “bifurcation”. It is used for quasistatic loading processes when the forces of inertia are neglected. Through this approach, the state of the body is stable if this state minimizes the Helmholtz free energy of the system. Analysis of stability consists of the evaluation of the second variation of free energy, and testing conditions on external forces which ensure this functional is positive definite. Some conditions of stability were formulated using the method in [l , 41. Note that the bifurcation approach can be applied to viscoelastic bodies only because of the definition of specific free energy for viscoelastic or viscoelastoplastic materials. The second approach may be applicable to quasistatic and dynamic processes of loading. It consists of the linearization of the equations of movement and constitutive equations, and the direct analysis of the perturbation equations using, e.g. techniques of integral estimates. Linearization in the nonlinear theory of elasticity was developed in [l], whilst the method of integral estimates for determination of critical loads was considered in [4]. Applications of these methods to real engineering problems meet great difficulties because of the variety of types of loading and material properties at finite strains. Hence, we need to introduce some assumptions which simplify the problem of stability. In this paper our interest is focused on the analysis of the stability of nonhomogeneous aging viscoelastic bodies. The model for such materials was formulated in [5] for infinitesimal strains and in [5-71 for the case of finite strains. Aging of materials means that the mechanical properties depend on time because of physical or chemical transformations.