Cauchy’s stress theorem for stresses represented by measures

A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin & Martins [12]. The main result of the paper says that unbounded Cauchy fluxes are in one-to-one correspondence with tensor valued measures with weak divergence a vector valued measure. Unavoidably, the force transmitted by a surface generally cannot be defined for all surfaces but only for almost every translation of the surface. Also conditions are given guaranteeing that the transmitted force is represented by a measure. These results are proved by using a new homotopy formula for tensor valued measure with weak divergence a vector valued measure.