In this final chapter of Part I, we discuss the formulation and specification of well-defined problems in continuum mechanics. For simplicity, we restrict our attention to the purely mechanical behavior of materials. This means that, unless otherwise explicitly stated, in this chapter we will ignore thermodynamics. The resulting theory provides a reasonable approximation of real material behavior in two extreme conditions. The first scenario is that of isentropic processes (see Section 6.2.5), where the motion and deformation occurs at such a high temporal rate that essentially no flow of heat occurs. In this scenario the strain energy density function should be associated with the internal energy density at constant entropy. The second scenario is that of isothermal processes (see Section 6.2.5), where the motion and deformation occurs at such a low temporal rate that the temperature is essentially uniform and constant. In this scenario the strain energy density function should be associated with the Helmholtz free energy density at constant temperature.

We start by discussing the specification of initial boundary-value problems in Section 7.1. Then, in Section 7.2 we develop the principle of stationary potential energy. Finally, in Section 7.3 we introduce the idea of stability and ultimately derive the principle of minimum potential energy.

Initial boundary-value problems

So far we have laid out an extensive set of concepts and derived the local balance laws to which continuous physical systems (which satisfy the various assumptions we have made along the way) must conform.