Geometric non-locality in multi-scale models of manifold-valued microstructures

We develop a geometric framework for representing spatial non-locality in two-scale continuum descriptions of microstructured bodies. Starting with continua endowed with spin-type microstructures, we construct approximation formulas for response functionals that involve microstructural descriptor fields taking values on geodesically complete Riemannian manifolds. This setting provides a mathematically rigorous and physically motivated approach to overcome, in terms of the derived approximations, some foundational problems that arise when considering non-local interactions in finitely extended bodies. The results generalize Coleman and Noll’s theorems on the approximation of constitutive functionals.