Nonlinear Kirchhoff--Love shell models derived from the Ciarlet--Geymonat energy: modelling and existence of minimizers

Abstract

Starting from a three-dimensional model based on the Ciarlet--Geymonat energy, we derive nonlinear shell models within the classical elasticity theory of compressible isotropic materials. The Neo-Hookean term involving the norm of the deformation gradient leads to an energy depending on the first, the second, and the third fundamental forms of the deformed midsurface. The coefficients appearing in the resulting shell models depend on the classical Lamé coefficients of the three-dimensional material, on the thickness of the shell, and on the mean and Gaussian curvatures of the reference configuration. This shows that the behavior of the shell is influenced not only by the elastic coefficients but also by the initial geometry of the three-dimensional thin body. Since a purely asymptotic derivation may lead to nonlinear terms for which the lower semicontinuity of the resulting functionals is not clear, we combine the asymptotic reduction through the thickness with positive-weight quadrature rules for selected purely volumetric contributions. After deriving the models, we establish variational well-posedness in the sense of existence of minimizers. More precisely, we prove coercivity and weak lower semicontinuity of the reduced functionals and establish the existence of minimizers in appropriate Sobolev spaces. A key ingredient is a polyconvexity concept for shells together with compensated compactness results identifying the weak limits of the area-weighted mean and Gaussian curvatures. An important consequence of the convexity analysis is that Models I and II require no additional restriction on the thickness beyond the local geometric regularity condition ensuring the regularity of the shell parametrization. Only Model III, in which the quadratic volumetric contribution is treated by a Taylor expansion, requires an additional explicit thickness condition in our existence result.