Calculus of Variations and Geometric Measure Theory

P. E. Stelzig

Homogenization of many-body structures subject to large deformations

created by stelzig on 01 Apr 2010
modified on 21 Sep 2010


Accepted Paper

Inserted: 1 apr 2010
Last Updated: 21 sep 2010

Journal: ESAIM Control Optim. Calc. Var.
Pages: 29
Year: 2010


We give a first contribution to the homogenization of many-body structures that are exposed to large deformations and obey the noninterpenetration constraint. The many-body structures considered here resemble cord-belts like they are used to reinforce pneumatic tires. We establish and analyze an idealized model for such many-body structures in which the subbodies are assumed to be hyperelastic with a polyconvex energy density and shall exhibit an initial brittle bond with their neighbors. Noninterpenetration of matter is taken into account by the Ciarlet-Ne\v{c}as condition and we demand deformations to preserve the local orientation. By studying Gamma-convergence of the corresponding total energies as the subbodies become smaller and smaller, we find that the homogenization limits allow for deformations of class special functions of bounded variation while the aforementioned kinematic constraints are conserved. Depending on the many-body structures' geometries, the homogenization limits feature new mechanical effects ranging from anisotropy to additional kinematic constraints. Furthermore, we introduce the concept of predeformations in order to provide approximations for special functions of bounded variation while preserving the natural kinematic constraints of geometrically nonlinear solid mechanics.

Keywords: Homogenization, Large deformations, Contact mechanics, Noninterpenetration, Many-body structure, Cord-belt, Polyconvexity, Brittle fracture, Gamma-convergence