Keywords: layerwise mixed formulation, least-squares formulation, finite element model, piezoelectric layers, composite layers.

A new layerwise mixed finite element model is developed based on a least-squares formulation for the coupled electromechanical static analysis of multilayered plates with piezoelectric and composite layers. The model assumes a layerwise variable description for displacements, transverse stresses and in-plane strains, along with the electrostatic potential, transverse electric displacement and in-plane electric field components, taken as independent variables. This original choice for the layerwise mixed formulation is intended to ensure the a priori and complete fulfilment of the requirements that compatibility and equilibrium conditions at the layers interfaces pose on both mechanical and electrical fields, namely the interlaminar C⁰ continuity of all chosen independent variables. This is demonstrated by the three-dimensional exact solutions developed by Heyliger [1]. In addition, this model preserves the benefits of least-squares formulation as shown by previous works [2], specifically, it is able to by-pass inf-sup conditions, and leads to a symmetric positive definite system of linear equations. Moreover, this model takes full advantage of the mixed formulation since post-computation requires no numerical differentiation.

Numerical applications are shown for assessment of the present model predictive capabilities by a thorough comparison with the three-dimensional exact solutions. In particular, a simply supported (PVDF/90/0/90/PVDF) piezoelectric composite plate is considered with different side-to-thickness ratios, namely a/h = 4, 10, 100 i.e. thick, moderately thick and thin plates, under an applied sinusoidal load or surface potential. Actually, the extensive number of results shown by the present model used 3×3 elements of fourth-order in-plane and 5 layers of either third- or fourth-order z-expansion through the layer thickness. Altogether, it is demonstrated that the layerwise mixed least-squares model is able to predict highly accurate results for all the mechanical and electrical variables, in very good agreement with the three-dimensional exact solutions, for both cases of applied load or potential, and for all side-to-thickness ratios. In fact, convergence of the predicted results to the three-dimensional exact solutions is verified by through the thickness refinement (and by in-plane refinement also, although not included here). For thick plates in particular, it is shown that through the thickness refinement, from third- to fourth-order z-expansion through the layer thickness, is indeed useful to achieve more accurate through-thickness predictions. Furthermore, it should be emphasized that the present layerwise mixed least-squares model is shown to be insensitive to shear locking, similarly to previous models [2].

1

P.R. Heyliger, "Exact solutions for simply supported laminated piezoelectric plates", Journal of Applied Mechanics, 64, 299-306, 1997. doi:10.1115/1.2787307

2

F. Moleiro, C.M. Mota Soares, C.A. Mota Soares, J.N. Reddy, "Layerwise mixed least-squares finite element models for static and free vibration analysis of multilayered composite plates", Composite Structures, 92(9), 2328-2338, 2010. doi:10.1016/j.compstruct.2009.07.005

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