Homogenization of many-body structures subject to large deformations

Philipp Emanuel Stelzig

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 91-123
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Stelzig, Philipp Emanuel. "Homogenization of many-body structures subject to large deformations." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 91-123. <http://eudml.org/doc/272763>.

@article{Stelzig2012,
abstract = {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.},
author = {Stelzig, Philipp Emanuel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; large deformations; contact mechanics; noninterpenetration; many-body structure; cord-belt; polyconvexity; brittle fracture; Γ-convergence; -convergence; Ciarlet-Nečas condition},
language = {eng},
number = {1},
pages = {91-123},
publisher = {EDP-Sciences},
title = {Homogenization of many-body structures subject to large deformations},
url = {http://eudml.org/doc/272763},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Stelzig, Philipp Emanuel
TI - Homogenization of many-body structures subject to large deformations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 1
SP - 91
EP - 123
AB - 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.
LA - eng
KW - homogenization; large deformations; contact mechanics; noninterpenetration; many-body structure; cord-belt; polyconvexity; brittle fracture; Γ-convergence; -convergence; Ciarlet-Nečas condition
UR - http://eudml.org/doc/272763
ER -

References

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