Universal equilibrium solutions

Abstract

In this chapter we study solutions to the equations of continuum mechanics instead of the equations themselves. In particular, our aim will be to obtain general equilibrium solutions to the field equations of continuum mechanics that are independent, in a specific sense, of the material from which a body is composed. Such solutions are of fundamental importance to the practical application of the theory of continuum mechanics. This is because they provide valuable guidance to the experimentalist who would like to design experiments for the determination of a particular material's constitutive relations. Generally, in an experiment it is only possible to control and measure (to a greater or lesser extent) the tractions and displacements associated with the boundary of the body being studied. From this information one would like to infer the stress and deformation fields within the body and ultimately extract the functional form of the material's constitutive relations and the values of any coefficients belonging to this functional form. However, if the interior stress and deformation fields explicitly depend on the functional form of the constitutive relations, then it is essentially impossible to infer this information from a practical experiment. According to Saccomandi [Sac01], a deformation which satisfies the equilibrium equations with zero body forces and is supported by suitable surface tractions alone is called a controllable solution . A controllable solution that is the same for all materials in a given class is a universal solution .