Exact 3-D piezoelasticity solution for buckling of hybrid cross-ply plates using transfer matrices

Abstract

A three dimensional exact piezoelasticity solution is presented for buckling of simply-supported symmetrically laminated hybrid plates with elastic substrate and piezoelectric layers. Buckling is considered under inplane normal strains in open as well as closed circuit conditions of the piezolayers and actuation potentials for movable and immovable inplane end conditions. A mixed formulation is used to form the governing equations for the buckling mode in terms of eight primary variables: displacements u,v,w, potential ϕ, stresses σ_z,τ_yz,τ_zx and electric displacement D_z. These entities are expanded in double Fourier series that satisfy the end conditions. The governing equations reduce to eight first-order homogeneous ordinary differential equations in z with constant coefficients dependent on the inplane strains and actuation potentials. The solution has eight constants for each layer. The transfer matrix is derived relating the eight primary variables at the top and bottom of a layer. The eight conditions σ_z= τ_yz=τ_zx=0,ϕ/D_z=0 at the top and bottom of the plate are used to set up four homogeneous equations for u,v,w,D_z/ϕ at the bottom. The determinant of their coefficient matrix is set to zero to obtain the buckling strains/potential. Benchmark results are presented for hybrid highly inhomogeneous test plate, cross-ply composite plate and sandwich plates. The effects of thickness parameter, aspect ratio and the electric boundary conditions on the buckling loads are illustrated.