Mixed schemes of finite element method for non-standard boundary value problems of the nonlinear theory of thin elastic shells

Variational statements of equilibrium problems for a thin elastic shell within a geometrically nonlinear theory of mean bending for different types of principal boundary conditions, including non-classical, are given. Two classes of shell models are considered: (1) a geometrically nonlinear equilibrium model for an anisotropic shell of a material obeying generalized Hooke's law; (2) a geometrically and physically nonlinear equilibria model for a shallow shell. Sufficient conditions for their generalized solvability in the corresponding Sobolev spaces are derived. In the case of the non-shallow shell the implicit function theorem is used. In the case of the shallow shell the variation problem is investigated using the generalized Weierstrass principle. Mixed finite element methods based on the use of the second derivatives of the deflection as auxiliary variables are constructed for approximate solutions of these problems. Sufficient conditions for solvability of the corresponding discrete problems are obtained. The convergence of approximate solutions is investigated. Accuracy estimates in the case of sufficiently smooth solutions of the original problems are given. Additional conditions are proposed for the problem on the shallow shell to ensure of implementation of the inequalities of the type of strong monotonicity and Lipschitz-continuity of the differential operator.