Geometrically Exact Four-Node Piezoelectric Solid-Shell Element

Abstract

This paper presents the geometrically exact assumed stress-strain four-node piezoelectric solid-shell element with six displacement degrees of freedom per node on the basis of the first-order equivalent single-layer theory. The proposed piezoelectric laminated shell formulation is based on the objective strain-displacement relationships, written in curvilinear reference surface coordinates, and generalizes the authors' purely mechanical finite element formulation. The mechanical and piezoelectric degrees of freedom are coupled via constitutive equations and the electric potential is assumed to be linear through the thickness of the piezoelectric layer. To overcome shear and membrane locking and have no spurious zero energy modes, the assumed strain and stress resultant fields are invoked. In order to circumvent thickness locking, the ad hoc modified laminate stiffness matrix, corresponding to the generalized plane stress condition, is employed. The elemental stiffness matrix has six zero eigenvalues and requires only direct substitutions. Moreover, it is evaluated by applying the 3D analytical integration that is economical and allows using extremely coarse meshes.

Keywords:

• assumed stress-strain piezoelectric shell element
• laminated shell
• geometrically exact four-node solid-shell element
• mechanical finite element formulation

Acknowledgments

The authors thank Prof. A. Benjeddou for his comments on numerical examples, which helped to improve a paper.