Quantitative Homogenization in Nonlinear Elasticity for Small Loads

We study quantitative periodic homogenization of integral functionals in the context of nonlinear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; p ≧ d -growth from below; and regularity of t...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2018-10, Vol.230 (1), p.343-396
Main Authors: Neukamm, Stefan, Schäffner, Mathias
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Elasticity
Fluid- and Aerodynamics
Functionals
Homogenization
Mathematical and Computational Physics
Physics
Physics and Astronomy
Smoothness
Theoretical
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Summary:We study quantitative periodic homogenization of integral functionals in the context of nonlinear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; p ≧ d -growth from below; and regularity of the microstructure), we show that in a neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula. The latter can be expressed with the help of correctors. We prove that the homogenized integrand admits a quadratic Taylor expansion in an open neighborhood of the rotations – a result that can be interpreted as the fact that homogenization and linearization commute close to the rotations. Moreover, for small applied loads, we provide an estimate on the homogenization error in terms of a quantitative two-scale expansion.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-018-1247-z