Interactions on manifolds and the construction of material structure

A theory of surface interactions on manifolds is presented as a framework for force and stress theory pertaining to materials with micro-structure. The dependence of surface forces on the normal in the case of Euclidean geometry is replaced by dependence on the tangent hyperplane and its orientation. Odd quantities are used to model interactions that satisfy Newton's law of action and reaction. Stresses are defined as vector-valued forms of order m-1, where m is the dimension of the body manifold. It is shown how a stress field induces surface forces for the various bodies. The special case of a scalar-valued balance law is presented with some additional detail. Finally, we consider the way material structures are induced by balance laws.