On Korn's first inequality with non-constant coefficients

Patrizio Neff

doi:10.1017/S0308210500001591

Abstract

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In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det Fp(x) ≥ μ+ > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korn's first inequality which is just our result with Fp = 11. With slight modifications, we are also able to treat forms of the type

Crossref Citations

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Neff, Patrizio 2003. Deformation and Failure in Metallic Materials. Vol. 10, Issue. , p. 251.

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Neff, Patrizio 2005. Local existence and uniqueness for a geometrically exact membrane-plate with viscoelastic transverse shear resistance. Mathematical Methods in the Applied Sciences, Vol. 28, Issue. 9, p. 1031.

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Neff, Patrizio Pauly, Dirk and Witsch, Karl-Josef 2011. A canonical extension of Kornʼs first inequality to H(Curl) motivated by gradient plasticity with plastic spin. Comptes Rendus. Mathématique, Vol. 349, Issue. 23-24, p. 1251.

Pompe, Waldemar 2011. Counterexamples to Korn’s inequality with non-constant rotation coefficients. Mathematics and Mechanics of Solids, Vol. 16, Issue. 2, p. 172.

Klawonn, Axel Neff, Patrizio Rheinbach, Oliver and Vanis, Stefanie 2011. FETI-DP domain decomposition methods for elasticity with structural changes:P-elasticity. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 45, Issue. 3, p. 563.

Neff, P. Pauly, D. and Witsch, K.-J. 2012. On a canonical extension of Korn’s first and Poincaré’s inequalities to H(CURL). Journal of Mathematical Sciences, Vol. 185, Issue. 5, p. 721.

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Lankeit, Johannes Neff, Patrizio and Pauly, Dirk 2013. Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity. PAMM, Vol. 13, Issue. 1, p. 361.

Lankeit, Johannes Neff, Patrizio and Pauly, Dirk 2013. Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity. Comptes Rendus. Mathématique, Vol. 351, Issue. 5-6, p. 247.

Lankeit, Johannes Neff, Patrizio and Pauly, Dirk 2013. Uniqueness of integrable solutions to $${\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}$$ for integrable tensor coefficients G and applications to elasticity. Zeitschrift für angewandte Mathematik und Physik, Vol. 64, Issue. 6, p. 1679.

Bîrsan, Mircea and Altenbach, Holm 2013. On the Cosserat model for thin rods made of thermoelastic materials with voids. Discrete and Continuous Dynamical Systems - Series S, Vol. 6, Issue. 6, p. 1473.

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On Korn's first inequality with non-constant coefficients

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