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

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 (F_αβ) with components in L²(ω), there exists a vector field (η_i) with components η₁, η₂ H¹(ω) and η₃ H²(ω) such that (∂ _αη_β + ∂_βη _α + ∂_αη₃∂ _βη₃) = E_αβ and ∂_αβη₃ = F_αβ 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 E_αβ = (∂_αη_β + ∂_βη _α + ∂_αη₃∂ _βη₃) L²(ω) and F _αβ = ∂_αβη₃ L²(ω) and that this problem has a solution in a manifold of symmetric matrix fields (E_αβ) and (F _αβ) whose components E_αβ L ²(ω) and F_αβ 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.