Nonlinear Saint-Venant compatibility conditions for nonlinearly elastic plates

Let ω be a simply-connected planar domain. 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 (ηi)i=13 with components η₁, η₂∈H¹(ω) and η₃∈H²(ω) such that 12(1\2αηβ+1\2βηα+1\2αη31\2βη3)=Eαβ and 1\2_αβη₃=F_αβ in ω for α, β=1, 2, the left-hand sides of these equations arising naturally in nonlinearly elastic plate theory. Such a vector field η=(η_i) being uniquely defined if it belongs to a particular closed subspace V⁰(ω) of H¹(ω)×H¹(ω)×H²(ω), we study the continuity properties of the nonlinear mapping (E, F)∈(L²(ω))⁴×(L²(ω))⁴→η∈V⁰(ω) defined in this fashion. © 2011 Académie des sciences.