# Homogenization of many-body structures subject to large deformations

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

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

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topStelzig, 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/221914>.

@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},

month = {2},

number = {1},

pages = {91-123},

publisher = {EDP Sciences},

title = {Homogenization of many-body structures subject to large deformations},

url = {http://eudml.org/doc/221914},

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

DA - 2012/2//

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/221914

ER -

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