Benchmark 3D solution and assessment of a zigzag theory for free vibration of hybrid plates under initial electrothermomechanical stresses

A benchmark three-dimensional (3D) exact piezothermoelasticity solution is presented for free vibration of simply supported cross-ply symmetrically laminated hybrid rectangular plates under initial in-plane electrothermomechanical stresses. The solution, based on second Piola stress tensor and nonlinear Lagrangian strains, accounts for the effect of initial transverse normal strain and the two-way electromechanical coupling. Benchmark results are presented for the natural frequencies under initial stresses and the modal distributions of displacements and stresses across the thickness, against which the accuracy of various 2D plate theories for hybrid plates can be assessed. A recently developed coupled 2D zigzag theory for hybrid plates is extended for the vibration under initial stresses. The theory uses a piecewise linear approximation for the thermal and potential fields across sublayers, and combines a global third-order variation with a piecewise linear (zigzag) variation across the layers for the inplane displacements. Using the conditions of continuity of transverse shear stresses at the layer interfaces and the shear traction-free conditions at the top and bottom, the number of primary displacement variables is reduced to five, as in the corresponding smeared third-order theory (TOT). The zigzag theory and the smeared TOT are assessed in comparison with the exact 3D solution for the natural frequencies and mode shapes of simply-supported rectangular plates under initial stresses. The zigzag theory results show excellent agreement with the 3D results when the initial thermal transverse normal strain is neglected, whereas the TOT may yield very inaccurate results depending upon the plate layup.