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Homogenization of many-body structures subject to largedeformations

Published online by Cambridge University Press:  23 December 2010

Philipp Emanuel Stelzig
Philipp Emanuel Stelzig*
Affiliation:
Lehrstuhl für numerische Mathematik und Steuerungstheorie, Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching bei München, Germany. stelzig@ma.tum.de Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo (TN), Italy
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Abstract

We give a first contribution to the homogenization of many-body structures that areexposed to large deformations and obey the noninterpenetration constraint. The many-bodystructures considered here resemble cord-belts like they are used to reinforce pneumatictires. We establish and analyze an idealized model for such many-body structures in whichthe subbodies are assumed to be hyperelastic with a polyconvex energy density and shallexhibit an initial brittle bond with their neighbors. Noninterpenetration of matter istaken into account by the Ciarlet-Nečas condition and we demand deformations to preservethe local orientation. By studying Γ-convergence of the correspondingtotal energies as the subbodies become smaller and smaller, we find that thehomogenization limits allow for deformations of class special functions of boundedvariation while the aforementioned kinematic constraints are conserved. Depending on themany-body structures’ geometries, the homogenization limits feature new mechanical effectsranging from anisotropy to additional kinematic constraints. Furthermore, we introduce theconcept of predeformations in order to provide approximations for special functions ofbounded variation while preserving the natural kinematic constraints of geometricallynonlinear solid mechanics.

Keywords

Homogenizationlarge deformationscontact mechanicsnoninterpenetrationmany-body structurecord-beltpolyconvexitybrittle fractureΓ-convergence

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 91 - 123
Copyright
© EDP Sciences, SMAI, 2010

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