Exact geometry four-node solid-shell element for stress analysis of functionally graded shell structures via advanced SaS formulation

Abstract

The exact geometry four-node solid-shell element formulation using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the shell N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the use of Lagrange basis polynomials of degree N − 1 in the through-thickness interpolations of displacements, strains, stresses and material properties leads to a very compact form of the SaS shell formulation. The SaS are located at Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. To implement efficient 3D analytical integration, the extended assumed natural strain method is employed. As a result, the proposed hybrid-mixed solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the assumed stress and strain approximations are utilized in the framework of the mixed Hu-Washizu variational formulation. It can be recommended for the 3D stress analysis of thick and thin doubly-curved functionally graded shells because the SaS formulation with only Chebyshev polynomial nodes allows the obtaining of numerical solutions, which asymptotically approach the 3D solutions of elasticity as the number of SaS tends to infinity.