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Quantitative homogenization in nonlinear elasticity for small loads

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Stefan Neukamm, Mathias Schäffner

Analysis of PDEs · Submitted 23 Mar 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1703.07947 (version 1)
Pith ID6QO5XE5E, Pith's permanent record of this paper and its review
CitationMathias Sch\"affner, Stefan Neukamm. "Quantitative homogenization in nonlinear elasticity for small loads." arXiv:1703.07947 (2017). https://pith.science/paper/6QO5XE5E
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,365 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.73 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.59 · compares with prior results 0.26 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-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 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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. contribution 0.72 0.87 0.04 asserted 0.76
s1 Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-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. result 0.88 0.93 0.43 asserted 1.00
s2 The latter can be expressed with help of correctors. method 0.51 0.45 0.05 asserted 0.95
s3 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. result 0.86 0.95 0.44 asserted 1.00
s4 Moreover, for small applied loads, we provide an estimate on the homogenization error in terms of a quantitative two-scale expansion. result 0.96 0.96 0.15 asserted 1.00

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Original abstract

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-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 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.

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