An intrinsic approach and a notion of polyconvexity for nonlinearly elastic plates

Let ω be a domain in R2. The classical approach to the Neumann problem for a nonlinearly elastic plate consists in seeking a displacement field η=(η _i)∈V(ω)=H ¹(ω)×H ¹(ω)×H ²(ω) that minimizes a non-quadratic functional over V(ω). We show that this problem can be recast as a minimization problem in terms of the new unknowns Eαβ=1/2(∂ _αη _β+∂ _βη _α+∂ _αη ₃∂ _βη ₃)∈L ²(ω) and F _αβ=∂ _αβη ₃∈L ²(ω) and that this problem has a solution in a manifold of symmetric matrices E=(E _αβ) and F=(F _αβ) whose components E _αβ∈L ²(ω) and F _αβ∈L ²(ω) satisfy nonlinear compatibility conditions of Saint-Venant type. We also show that such an "intrinsic approach" naturally leads to a new definition of polyconvexity. © 2011 Académie des sciences.