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

G. M. Kulikov1, S. V. Plotnikova1

Department of Applied Mathematics and Mechanics, Tambov State Technical University, Tambov 392000, Russia, kulikov@apmath.tstu.ru

Computers, Materials & Continua 2011, 23(3), 233-264. https://doi.org/10.3970/cmc.2011.023.233

## Abstract

This paper presents a robust non-linear piezoelectric exact geometry (EG) four-node solid-shell element based on the higher-order 9-parameter equivalent single-layer (ESL) theory, which permits one to utilize 3D constitutive equations. The term EG reflects the fact that coefficients of the first and second fundamental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on a new concept of interpolation 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 objective, 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 analytical 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.

## Keywords

G. M. . Kulikov and S. V. . Plotnikova, "Finite rotation piezoelectric exact geometry solid-shell element with nine degrees of freedom per node," Computers, Materials & Continua, vol. 23, no.3, pp. 233–264, 2011. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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