Nonlinear Saint-Venant compatibility conditions for nonlinearly elastic plates

摘要

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 L~2(ω), there exists a vector field (ηi)i=13 with components η_1, η_2∈H~1(ω) and η_3∈H~2(ω) such that 12(12αηβ+12βηα+12αη312βη3)=Eαβ and 12_(αβ)η_3=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=(ηi) being uniquely defined if it belongs to a particular closed subspace V~0(ω) of H~1(ω)×H~1(ω)×H~2(ω), we study the continuity properties of the nonlinear mapping (E, F)∈(L~2(ω))~4×(L~2(ω))~4→η∈V~0(ω) defined in this fashion.