Homogenization of many-body structures subject to large deformations

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 18; no. 1; pp. 91 - 123
• Main Author: Stelzig, Philipp Emanuel
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.01.2012
• Subjects: 35B27; 49J45; 74B20; 74G65; 74M15; 74Q05; 74R10; brittle fracture; Calculus of variations; contact mechanics; cord-belt; Homogenization; Kinematics; large deformations; many-body structure; Mathematical models; Mathematics; Mechanics; noninterpenetration; polyconvexity; Studies; Textiles; Tires; Γ-convergence
• ISSN: 1292-8119; 1262-3377
• DOI: 10.1051/cocv/2010052

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-Nečas condition and we demand deformations to preserve the local orientation. By studying Γ-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.