Exact 2D Solution and Assessment of a 1D Zigzag Theory for Vibration of Thermally Stressed Hybrid Beams

ABSTRACT

A benchmark two-dimensional (2D) piezothermoelasticity solution is presented for buckling and free vibration of simply supported symmetrically laminated hybrid piezoelectric beams under initial stresses due to thermoelectric load. The solution, based second Piola stress tensor and nonlinear Lagrangian strains, accounts for the effect of initial transverse normal strain and the two-way piezoelectric coupling. Benchmark results are presented for the buckling temperature, natural frequencies under initial thermal stresses and modal distributions of displacements and stresses across the thickness, against which the accuracy of 1D beam theories for hybrid beams can be assessed. A recently developed coupled 1D zigzag theory for hybrid beams is extended for vibration under initial thermal stresses. The theory uses a piecewise linear approximation for the thermal and potential fields across sublayers, an approximation for the deflection, which explicitly accounts for the transverse normal strain due to thermal and electric fields and a layerwise (zigzag) variation for the axial displacement. The transverse shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are enforced to reduce the number of primary displacement variables to three. The new 1D theory is assessed in comparison with the exact 2D solution for the natural frequencies and mode shapes of beams and panels under initial thermoelectric load. The zigzag theory results show excellent agreement with the 2D results when the initial transverse normal strain is neglected.