Buckling Analysis Of Simply-supported Piezoelectric Functionally Gradient Rectangular Thin Plate

In recent years, graded piezoelectric materials with structure and performance varied equally through the space were produced to substitute the uniform piezoelectric materials. It has been widely used because of its excellent performance. At one time, more and more study is focused on its mechanical behavior and has gained several research results. 1 Problem statement and basic equationConsider a piezoelectric functionally gradient rectangular thin plate which consists of n layers with each having different mechanical and piezoelectric properties and is subjected to applied electric field along the thickness direction (z-direction).1.1 Physical properties modelAssuming the electroelastic properties vary with a power form of thickness coordinate variables, and a modified classical laminate theory involving piezoelectric coupling terms is employed, the physical properties of the ith lamina is given bywhere F denotes certain physical properties of piezoelectric FGM; A,B denote the physical properties of the two piezoelectric materials which compose the piezoelectric FGM; p denotes the grads exponent.1.2 Basic equation1.2.1 constitutive equation {σ} = [Q ]{ε} - [ e ]{ E} (2) Where {σ}is the stress vector; {ε}is the strain vector; { E} is the electric field vector; [C ] is the elastic stiffness matrix; [ e ] is piezoelectric modulus matrix.1.2.2 Geometric equation According to the big deflection theory of classical thin plate, the strain of plate is defined as Where u, v, w are the displacements of the mid-plane of a plate along the x-,y-,z-direction.1.2.3 Stress resultants and stress couples expressions Stress resultants and stress couples are defined as Where {σ} = (σ_x ,σ_y ,τ_xy)^T; h_i-1, hi are the z-coordinates of the bottom and top of the ith lamina. From Eqs. (2) to (4) we can obtain as1.2.4 Buckling equation of piezoelectric FGM Substituting (5) into equilibrium equation of bending, and assumingνi=ν0,(i = 1,2...n )we can expurgate the displacement u and v, and get the equilibrium equation of plate, then by using critical equilibrium methods, the buckling equation of piezoelectric FGM thin plate is derived, as2 Critical voltage of FGM plate under uniform electric field Consider a simply-supported rectangular thin plate, in this case the boundary conditions are x =0: w|- =0, w|-, xx=0Supposed buckling deflection function satisfied boundary condition (8) which is given by w = c sin( mπx / a )sin( nπy / b) (9)Substituting (9) into (7),we haveIn multi layer FGM, the electric in each layer can be expessed asWhere V is applied voltage, d lis the thickness of the lth lamina,∈i is the dielectric constant of ith lamina.Moreover, substituting (11) ,(6) in to (10) ,we can obtain Where , the coefficientWe can get the minimum of V , when m=n=1, the critical voltage of buckling is obtained asIf the displacement at mid-plane of the plate is not discussed, the equation of buckling is reduced toFinally, we discussed the influence of the geometrical size of plate, functionally gradient index and displacement at mid-plane of plate on the critical voltage by numerical example. It is found that the effects of the gradient index are significant , so we should pay more attention to checking with the buckling intensity in designing and applying piezoelectric functionally gradient materials.