Finite deformation of thin shells in the context of analytical mechanics of material surfaces

Abstract

In the framework of the direct approach shells are considered as deformable surfaces consisting of particles, and the relations of the theory are obtained with the methods of analytical mechanics. In the present work we assign to each particle five degrees of freedom, namely three translations and two in-plane rotations. The principle of virtual work produces all the relations of the theory of shells: equations of equilibrium, boundary conditions, definition of the force factors and the general form of constitutive equations. Remarkable consistency and clarity is achieved both in the relations of the theory and in the derivation process. A new formulation of the Piola tensors for a shell is suggested in order to transform the equations to the reference configuration. To analyze the effects of buckling or geometric stiffening, we linearize these equations in the vicinity of a pre-deformed configuration. Some new semi-analytical results on buckling and supercritical behavior of an axially compressed cylindrical shell are presented. The correspondence between the equations and the variational formulation is discussed in view of development of efficient numerical procedures for modeling nonlinear deformations of shells. Results of finite element modeling of the nonlinear deformation of a shell structure are discussed in comparison with the fully three-dimensional solution of the problem.