Hybrid-Mixed Solid-Shell Element for Stress Analysis of Laminated Piezoelectric Shells through Higher-Order Theories

Abstract

A geometrically exact hybrid-mixed four-node piezoelectric solid-shell element by using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements and electric potentials of these surfaces as basic shell unknowns. The external surfaces and interfaces are also included into a set of SaS because of the variational formulation. Such a choice of unknowns with the consequent use of Lagrange polynomials in the through-thickness approximations of displacements, strains, electric potential and electric field leads to a very compact piezoelectric shell element formulation. To implement the efficient analytical integration throughout the element, the enhanced assumed natural strain (ANS) method is employed. The proposed hybrid-mixed four-node piezoelectric shell element is based on the Hu-Washizu variational equation and exhibits a superior performance in the case of coarse meshes. It could be useful for the three-dimensional (3D) stress analysis of thick and thin doubly-curved laminated piezoelectric shells since the SaS formulation gives the possibility to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach the exact solutions of piezoelectricity as the number of SaS tends to infinity.