Boundary conditions in the presence of structured deformations

Structured deformations are objects conceived to describe geometrically the changes occurring in the internal structure of a continuous body, both at the macroscopic and at submacroscopic levels. Mathematically, a structured deformation is a pair (g, G) of fields over the region occupied by the body in the reference configuration; in the pair, the first item describes the macroscopic deformation, and the second item is a local measure of the part of the deformation not due to structural changes. A structured deformation can be obtained as the limit of a sequence of simple deformations. These are piecewise smooth functions, which have jumps at some singular surface. In the limit, the jumps may either concentrate on a singular surface or diffuse throughout the body. The two possibilities correspond to macroscopic and submacroscopic structural changes, respectively. The presence of the jumps allows the simultaneous convergence of the approximating functions to g and of their gradients to G; because the gradients are local measures of deformation, G is the limit of the local measures of deformation taken away from the singular surfaces, that is to say, ignoring that part of the deformation which, in the limit, determines the structural changes. The equilibrium problem for a body undergoing structured deformations can be studied via energy minimization. The energy of a simple deformation involves two energy densities, a bulk density depending on G and an interfacial density depending on the amplitude of the jumps, and the energy of a structured deformation is defined as the lowest energy which can be attained in the limit by the energies of the simple deformations in an approximating sequence. Unfortunately, an explicit expression for an energy defined in this way is available only in the one-dimensional case. That is why, in this communication, I confine myself to one-dimensional bodies. In an energetic approach, equilibrium configurations are characterized by the non-negativeness of the first variation of the total energy. From this condition, a number of field equations and boundary conditions can be obtained. The field equations include the standard equilibrium equation for an axially loaded bar, a pointwise relation between the axial force and the amplitude of the jumps, and a set of conditions reminiscent the specification of the elastic range and the flow rule in the theory of elastic-plastic bodies. Surprisingly enough, the equations of the equilibrium problem for an elastic-plastic body, determined heuristically long ago on the basis of experimental observation, are re-obtained here as a result of energy minimization, just by considering from the outset deformations more general than those of classical continuum mechanics. I consider two types of boundary-value problem, of traction and of place. For the first problem the boundary conditions are classical, while for the second they take a peculiar form, dictated by the possibility that a fracture occurs exactly at the endpoints of the bar. For both problems I find that, in the presence of structural changes, the global response of the bar is strongly influenced by the analytical shape of the interfacial energy. Here I consider two particular shapes, convex and piecewise concave, and I show that in a stable equilibrium configuration only macroscopic structural changes are possible in the first case, while only submacroscopic structural changes are allowed in the second case. Moreover, in the convex case the force-elongation curves of the two boundary-value problems coincide, while in the piecewise concave case the problem of place has a larger variety of solutions than the problem of traction. As a final subject, I discuss the influence of the type of convergence required when approximating a structured deformation by a sequence of simple deformations. I compare two types of convergence, and I show that the weaker convergence provides some solutions not allowed by the stronger. Among them are the fine mixtures of solid phases in the case of a non-convex bulk energy, and solutions with co-existing macro and submacrostructural changes in the case of a convex interfacial energy. More generally, the class of phenomena which can be studied with the theory of structured deformations may be restricted by the choice of the type of convergence. For example, the free motion of defects inside a body is compatible with the weaker convergence but not with the stronger. This suggests the conclusion that the type of convergence is, in fact, a constitutive assumption on the material of which the body is made. An extension of the above results to higher dimension is for now pure conjecture. A major technical difficulty is that no explicit form of the energy of a structured deformation is available for two- or three-dimensional bodies. Nevertheless, a three-dimensional version of the energy formula does not seem impossible to prove, at least for sufficiently regular structured deformations.