Existence and uniqueness results for well-posed laminated composite shell models

This paper presents a mathematical analysis of two particular and original generalized formulations also named Ngatcha et al’s generalized formulations (NGFs for short). The NGFs have been recently developed and used to model the mechanical behavior of a laminated composite shell (LCS) (see Composite Structures, 295(4), 115754; https://​doi.​org/​10.​1016/​j.​compstruct.​2022.115754 and Structures,70, 107445; https://​doi.​org/​10.​1016/​j.​istruc.​2024.​107445). The first Ngatcha et al’s Generalized Formulation (NGF) accounts for the shear mechanical coupling effect due to the Gauss tensor or third fundamental form. The second formulation accounts for warping and twisting effects due to a stretching-through-the-thickness kinematic variable. The laminate constitutive equations (LCE) related to these formulations do not rely on complex empirical considerations, in contrast to some recently developed theories for laminated composite shells in the existing literature. It is imperative to undertake a meticulous mathematical analysis of a laminated composite structure within suitable functional spaces prior to its numerical implementation. The absence of rigorous mathematical demonstrations of the existence and uniqueness of a weak solution (in a variational sense) of a LCS model can compromise convergence when a numerical method is employed. In this study, we revisit NGFs and prove that they are mechanically and physically well-posed and justified via several theorems and demonstrations. We introduce variational formulations for laminated shell models utilizing these formulations and we prove the existence and uniqueness of a weak solution through several demonstrations. We propose several new results and new Green formulas for laminated composite thick shell models in an original way, which allows to mathematically justify the NGFs.