NONLINEAR SAINT–VENANT COMPATIBILITY CONDITIONS AND THE INTRINSIC APPROACH FOR NONLINEARLY ELASTIC PLATES

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 (Eαβ) 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 ½(∂αηβ + ∂βηα + ∂αη3∂βη3) = 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 = ½(∂αηβ + ∂βηα + ∂αη3∂βη3) ∈ L2(ω) and Fαβ = ∂αβη3 ∈ L2(ω) and that this problem has a solution in a manifold of symmetric matrix fields (Eαβ) and (Fαβ) whose components Eαβ ∈ L2(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.