Keywords: laminated piezoelectric shell, finite deformation, solid-shell element, seven-parameter model.

This paper focuses on the development of the finite deformation exact geometry (EG) piezoelectric four-node solid-shell element on the basis of the first-order seven-parameter equivalent single-layer (ESL) theory, which permits the utilization of three-dimensional constitutive equations. The term EG reflects the fact that coefficients of the first and second fundamental forms are taken exactly at every element node. Therefore, no approximation of the reference surface is required. The present ESL theory is based on the new non-linear strain-displacement relationships, which are invariant under arbitrarily large rigid-body shell motions in any convected curvilinear coordinate system. This is due to the fact that displacement vectors of outer and middle surfaces of the shell are introduced and resolved, in contrast with the isoparametric solid-shell element formulation, in the reference surface frame.

It is assumed that the electric potential is linear through the thickness of the piezoelectric layer and all displacement and electric potential degrees of freedom are coupled via constitutive equations. This allows one to formulate the efficient EG four-node solid-shell element for the non-linear analysis of thin laminated piezoelectric shells. To avoid shear and membrane locking and to have no spurious zero energy modes, the assumed strain and stress resultant fields are invoked. As a result, the hybrid EG piezoelectric solid-shell element developed is free of locking and its tangent stiffness matrix possesses a correct rank. Taking into account that displacement vectors of outer and middle surfaces of the shell are resolved in the reference surface frame, the proposed EG solid-shell element formulation has computational advantages compared to the conventional isoparametric solid-shell element formulation, since it reduces the computational cost of numerical integration in the evaluation of the stiffness matrix. That is true because, first, the tangent stiffness matrix derived requires only direct substitutions, i.e., no expensive numerical matrix inversion is required. The latter is unusual for the isoparametric hybrid/mixed shell element formulations. Secondly, we use the efficient three-dimensional analytical integration that permits coarse meshes to be employed. Therefore, the EG four-node solid-shell element developed is robust and promising because of the fact that electric signals generated by sensors are fed into microprocessors to activate a system of piezoelectric actuators in real time.

The performance of the proposed non-linear EG piezoelectric solid-shell element is evaluated employing several benchmark examples extracted from the literature as well as the authors' example. The comparison showed that our results agree closely with those derived through existing finite rotation isoparametric piezoelectric solid-shell elements. However, the EX solid-shell element formulation developed provides the possibility of additionally utilizing very large load increments.

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