A new mathematical foundation for contact interactions in continuum physics

The investigation of contact interactions, such as traction and heat flux, that are exerted from contiguous bodies across the common boundary is a fundamental issue in continuum physics. However, the traditional theory of stress established by Cauchy and extended by Noll and his successors is insufficient for needs in modern physics where one has to handle lack of regularity that is present in shocks, corners, and contact of deformable bodies. The talk provides a new mathematical foundation to the treatment of contact interactions. Based on mild physically motivated postulates, that essentially differ from those used before, the existence of a corresponding interaction tensor is verified. While in former treatments contact interactions are basically defined on surfaces, here contact interactions are rigorously considered as maps on pairs of subbodies. This allows to define the interaction exerted on a subbody not only, as usual, for sets with a sufficiently regular boundary but for any Borel set (which includes all open and closed sets). In addition to the classical representation of such interactions by means of integrals on smooth surfaces, a general representation using the distributional divergence of the tensor is derived. In the case where concentrations occur the new approach allows a more precise description of contact phenomena than before.