Analysis of the second Piola‐Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid‐shell elements

Abstract

SummaryThis paper presents the finite rotation exact geometry four‐node solid‐shell element using the sampling surfaces (SaS) method. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through‐thickness distributions of displacements, strains and stresses leads to a robust higher‐order shell formulation. The SaS are located at only Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. The proposed hybrid‐mixed four‐node solid‐shell element is based on the Hu‐Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated through efficient 3D analytical integration and its explicit form is given. As a result, the proposed exact geometry solid‐shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement‐based solid‐shell elements.