Two-dimensional model for the combined bending,stretching and shearing of shells: A general approach and application to laminated cylindrical shells derived from three-dimensional elasticity

For laminated shells, the displacement field can be approximated in each layer by a third-order Taylor–Young expansion in thickness. Then we are motivated to consider a simplified theory based on the thickness-wise expansion of the potential energy truncated at third-order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field in each layer. These lead to an analytical expression for the two-dimensional potential energy of cylindrical shells in terms of the zero-order displacement field, and its derivatives, that includes non-standard transverse shearing and normal stress energy. As a consequence this potential energy satisfies the stability condition of Legendre–Hadamard which is necessary for the existence of a minimizer.