Two-dimensional (2D) exact piezoelasticity and one-dimensional coupled zigzag theory solutions are presented for vibration of initially stressed simply-supported cross-ply symmetrically laminated hybrid piezoelectric beams under axial strain and actuation potentials. In the 2D exact solution, the coupled governing equations for the vibration mode are derived using Fourier series. Using transfer matrix approach and the boundary conditions, homogeneous equations are set up for the variables at the bottom. The determinant of their coefficient matrix is set to zero to obtain the natural frequency. An efficient coupled zigzag theory is developed for vibration of initially stressed hybrid beams. A piecewise linear approximation of the potential field, an approximation for the deflection to account for the piezoelectric strain and a combination of global third-order variation and layer-wise linear variation for the axial displacement are employed. The conditions of absence of shear tractions at the top and bottom and the conditions of continuity of transverse shear stress at the layer interfaces are exactly satisfied. The governing equations are derived from extended Hamilton’s principle. Comparison of natural frequencies of beams and panels of different configurations with the exact 2D piezoelasticity solution establish that the present zigzag theory is generally very accurate for moderately thick beams. The first-order and third-order shear deformable theories, which are also assessed, are found in some cases to yield poor results even for thin beams.
Resolving the two-dimensional axial next-nearest-neighbor Ising model using transfer matrices
