Finite Rotation Piezoelectric Exact Geometry Solid-Shell Element with Nine Degrees of Freedom per Node

Abstract

This paper presents a robust non-linear piezoelectric exact geometry (EG) four-node solid-shell element based on the higher-order 9-parameter equiva- lent single-layer (ESL) theory, which permits one to utilize 3D constitutive equa- tions. The term EG reflects the fact that coefficients of the first and second fun- damental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on a new concept of interpo- lation surfaces (I-surfaces) inside the shell body. We introduce three I-surfaces and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows us to represent the finite rotation piezoelectric higher-order EG solid-shell element formulation in a very compact form and to utilize in curvilinear reference surface coordinates the strain-displacement relationships, which are ob- jective, that is, invariant under arbitrarily large rigid-body shell motions. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed displacement-independent strain and stress resultant fields are introduced. In this connection, the Hu-Washizu variational equation is invoked. To implement the ana- lytical integration throughout the element, the modified ANS method is applied. As a result, the present finite rotation piezoelectric EG solid-shell element formulation permits the use of coarse meshes and very large load increments.