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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2015

Nonlinear Korn inequalities

Résumé

Let Ω be a bounded and connected open subset of Rn with a Lipschitz-continuous boundary Γ, the set Ω being locally on the same side of Γ, and let Θ:Ω→Rn and Φ:Ω→Rn be two smooth enough “deformations” of the set Ω. Then the classical Korn inequality asserts that, when Θ=id, there exists a constant c such that ||v||_{H1(Ω)} ≤ c ||v||_{L2(Ω)} + ||∇v + ∇v^T||_{L2(Ω)} for all v ∈ H1(Ω), where v := (Φ − id) : Ω → Rn denotes the corresponding “displacement” vector field, and where the symmetric tensor field ∇v + ∇v^T : Ω → Sn is nothing but the linear part with respect to v of the difference between the metric tensor fields ∇Φ^T ∇Φ and I that respectively correspond to the deformations Φ and Θ = id. Assume now that the identity mapping id is replaced by a more general orientation-preserving immersion Θ ∈ C1(Ω;Rn). We then show in particular that, given any 1 < p < ∞ and any q ∈ R such that max{1, p} ≤ q ≤ p, there exists a constant C such that ||Φ−Θ||_{W1,p(Ω)} ≤C ||Φ−Θ||_{Lp(Ω)} + ||∇Φ^T ∇Φ−∇Θ^T ∇Θ||_{Lq(Ω)} for all Φ ∈ W1,2q(Ω) that satisfy det∇Φ > 0 almost everywhere in Ω. Such an inequality thus constitutes an instance of a “nonlinear Korn inequality”, in the sense that the symmetric tensor field ∇Φ^T ∇Φ − ∇Θ^T ∇Θ : Ω → Sn appearing in its right-hand side is now the exact difference between the metric tensor fields corresponding to the deformations Φ and Θ. We also show that, like in the linear case, an analogous nonlinear Korn inequality holds, but without the norm ||Φ − Θ||_{Lp (Ω)} in its right-hand side, if the difference Φ−Θ vanishes on a subset Γ0 of Γ with dΓ-meas Γ0 > 0. The key to providing such nonlinear Korn inequalities is a generalization of the landmark “geometric rigidity lemma in H1(Ω)” established in 2002 by G. Friesecke, R.D. James, and S. Muller, as later extended to W1,p(Ω) by S. Conti.

Dates et versions

hal-01301434 , version 1 (12-04-2016)

Identifiants

Citer

Philippe G. Ciarlet, Cristinel Mardare. Nonlinear Korn inequalities. Journal de Mathématiques Pures et Appliquées, 2015, 104, pp.1119-1134. ⟨10.1016/j.matpur.2015.07.007⟩. ⟨hal-01301434⟩
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