Nonlinear saint-venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

13 Scopus Citations
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Original languageEnglish
Pages (from-to)2293-2321
Journal / PublicationMathematical Models and Methods in Applied Sciences
Volume23
Issue number12
Online published15 Mar 2013
Publication statusPublished - Nov 2013

Abstract

Let ω be a simply connected planar domain. First, we give necessary and sufficient nonlinear compatibility conditions of Saint-Venant type guaranteeing that, given two 2 × 2 symmetric matrix fields (E αβ) and (Fαβ) with components in L2(ω), there exists a vector field (ηi) with components η1, η2 H1(ω) and η3 H2(ω) such that (∂ αηβ + ∂βη α + ∂αη3βη3) = Eαβ and ∂αβη3 = Fαβ in ω for α, β = 1, 2. Second, we show that the classical approach to the Neumann problem for a nonlinearly elastic plate can be recast as a minimization problem in terms of the new unknowns Eαβ = (∂αηβ + ∂βη α + ∂αη3βη3) L2(ω) and F αβ = ∂αβη3 L2(ω) and that this problem has a solution in a manifold of symmetric matrix fields (Eαβ) and (F αβ) whose components Eαβ L 2(ω) and Fαβ L2(ω) satisfy the nonlinear Saint-Venant compatibility conditions mentioned above. We also show that the analysis of such an "intrinsic approach" naturally leads to a new nonlinear Korn's inequality. © 2013 World Scientific Publishing Company.

Research Area(s)

  • nonlinear Korn inequality, Nonlinear plate theory, Saint-Venant compatibility conditions