Homogenization of Many-body Structures Subject to Large Deformations and Noninterpenetration

We give a first contribution to the homogenization of many-body structures in terms of large deformation mechanics by considering some idealized many-body structures resembling so-called cordbelts like they are used to reinforce pneumatic tires. We establish and analyze a model for these structures, the subbodies of which we assume to be hyperelastic with some polyconvex energy density and moreover to initially exhibit a brittle bond with their neighbours along contacting surfaces. Kinematically, noninterpenetration of matter is accounted for by the Ciarlet-Nečas condition and we demand deformations to preserve the local orientation. As the structures compose of more and more bodies, we identify homogenization limits of the associated total energies in the sense of Γ-convergence over those special functions of bounded variation complying with the aforementioned kinematic constraints. Depending on the geometry of the considered many-body structures, their homogenization limits are subject to newly found anisotropy or even feature additional kinematic constraints. We furthermore introduce the concept of pre-deformations to provide approximations within the special functions of bounded variation while preserving the natural kinematic constraints of geometrically nonlinear continuum mechanics.