| A new class of composite materials known as "functionally gradientmaterials" has been developed in recent years. With the extensive applicationof piezoelectric materials and corresponding structure complicated, the use ofpiezoelectric materials as sensors and actuators for the purpose of control isbecoming more and more important in the design of modern smart structuresand systems. While the piezoelectric materials as sensors and actuators areusually bonded with smart structures, therefore it is significant to research thebending, buckling and vibration of smart structures bonded piezoelectricpatches.1. Problem statementConsidering piezoelectric materials mostly use as controlling and sensitivitypatches in engineering, whose cross section is often regular shape ofrectangular or circle etc. and thickness is far from less than the size ofmid-plane, studying buckling and bending of functionally gradient structuresbonded piezoelectric patches has the value of theory and engineeringapplication based on the classical theory of plate in the engineering problems.The FGM hybrid rectangular plate under current consideration is comprised ofan FGM substrate and piezoelectric films that are perfectly bonded on its topand bottom surfaces as actuators. There will be buckling and bendingdeformations when in-plane and transverse loading are applied to the platerespectively. So the deformations should be controlled to avoid the effect onthe structures. According to inverse piezoelectric effects, the voltages areconverted the equivalent electric loadings applied to the plate. When theequal-amplitude of same sign voltages are applied to the patches on the topand bottom surfaces, the electric field are converted the equivalent membraneforces. When the equal-amplitude of opposite sign voltages are applied tothem, the electric field are converted the equivalent moments. The thicknessof patch is far less than the FGM's, and the influence of glue is notconsidered.The FGM layer is made of a combined ceramic-metal materials and itsproperties obey a power law across the thickness, we can determined asE ( z ) = Ec m ( z h + 1 2 )k+ Em (1)Where and are elastic modulus of metal and ceramic respectively., is material gradient index. According to actual property,supposing their Poisson's ratio is same, then the Poisson's ratio is given asfollowsE mEcEc m = Ec ? Emk( )ν z= ν0 (2)2. Basic equation2.1 Piezoelectric constitutive equation{σ } = [ Q ]{ε } ? [ Q ][ d]{ E} (3)Where is the stress vector, is the strain vector, is the elasticstiffness matrix,{σ }{ε }[ Q][ d ] is the piezoelectric strain constant matrix, is theelectric field vector.{ E}2.2 Geometric equationAccording to small deflection theory of thin plate , the strains of plate aredefined as2 2ε x = ? z ?? x w 2 , ε y = ? z ?? y w2, γ xy= ?2z ??x 2? wy (4)Where w is the mid-plane deflection.2.3 Equilibrium differential equation2.3.1 Equilibrium differential equation of bucklingUnder applied electric field in the polar direction only andequal-amplitude voltages of same sign are applied across the top and thebottom piezoelectric actuator layers, thenEzE z = V hp.The bending moments are given by/22/22/22hx hxhy hyhxy hxyM zdzM zdzM zdzσστ????? =??? =???? =∫∫∫(5)Carrying out the integration through the plate thickness of h and substitutingEqs. (1) (2) (3) (4) into equation (5), we obtain( )2 22 022 22 02210xyxyM Dw wx yM Dw wy xM Dwx yννν??? = ? ??? ?? +????????? = ? ??? ?? +????????? = ? ? ?? ? (6)And substituting Eq. (6) into buckling equilibrium differential equation ofsmall deflection of plate applied transverse loading and mid-plane forces, oneobtain buckling equilibrium differential equation of small deflection offunctionally gradient elastic plate, that is( )2 22 2D? ? w = q x , y + N x ?? x w 2 + N y ?? y w 2+2N xy??x 2?w y (7)If , the equivalent membrane forces induced by electric field aredefined by carrying out the integration through the plate thickness of h byequation (3)d 31 =d32{ } [ ][ ]{ }22EhN = ∫? hQ d Edz (8)Assuming uniform distributed pressure ( p x ,p y) applied in x = 0,a andof rectangular plate respectively and , the resultantmembrane forces in-plane are given byy = 0,b P0p x = py=N x = ? p x ? N xE , N y = ? p y ? N yE ,Nxy= (9)According to small deflection buckling theory that membrane forces keepinvariableness during buckling, substituting Eqs. (8) (9) into equation (7),and introducing the following equation( ) ( )D? 2 ? 2 w = q x ,y ? P + βV ? 2w (10)2.3.2 Equilibrium differential equation of bendingUnder applied electric field in the polar direction only andequal-amplitude voltages of opposite sign are applied across the top and thebottom piezoelectric actuator layers, thenEzE z = V hp. The electric field areconverted the equivalent moments. The influences of deformation of theelastic plate on the electric field are not considered.Similar to the developing of buckling equilibrium differential equation, then2 2*2 2 0202 2*2 2 02020111xyxyM E w wMxyM E w wMy xME wx yννννν??? = ? ? ??? ?? + ?????????? = ? ? ??? ?? + ?????????? = ? + ?? ? (11)Here M * is the equivalent moments induced by electric field .Substituting Eq. (11) into the equilibrium differential equation of classicalsmall deflection plate, that is( )22 2?? Mx 2 x + 2 ?? x M? y xy + ??My 2y= ?q x ,y (12)where q ( x ,y) is transverse distributed loading.Then small deflection bending equilibrium differential equation of FGMbonded piezoelectric patches is given as follows( )D? 2 ? 2 w = q x ,y + q* (13)where ?? 2 M * = q*, that is called equivalent transverse loading induced byelectric field. If electric field is uniform in-plane, then q * = ?? 2 M* = 0。3. Solving equation3.1 Solving buckling equationBoundary conditions of simply supported FGM rectangular platex = 0,a:2w = 0, ?? xw2=0 (14a)y = 0,b:2w = 0, ??y w2=0 (14b)when transverse loading , considering the condition ofz-direction electric field and mid-plane forces together, Suppose bucklingdeflection function satisfied boundary condition (14a) (14b) is given byq ( x ,y )=01 1( , ) mnsin sinm nw x y Am x n ya b∞ ∞π π= == ∑∑(15)Substituting Eq. (15) into buckling equilibrium differential equation (10),we obtain( ) ( )Dπ 2?? m a 2 + n b 2?? = P +βV (16)Obviously, when m=n=1, (P + βV )attain minimum , therefore criticalbuckling loading is obtained, that is( ) ( )22p + β V cr = Daπ2 ??1 +ab ?? (17)3.2 Solving bending equationTaking into account linearity small deflection theory and transforming thequestion into solving the superposition of (a),(b) and (c) by usingsuperposition principle.(a): A simply supported rectangular thin plate is considered, whose lengthand width is a and b respectively and apply uniform loads and boundaryconditions are given by( )2222w x =± a= 0, ??? ??xw???x=±a=0.(18a)( )2222w y =± b= 0, ??? ??yw???y=±b=0.(18b)Suppose bending deflection function satisfied boundary conditions (18a)(18b) is given by1,3 1,3( , ) mncos cosm nw x y Am x n ya b∞ ∞π π= == ∑ ∑(19)Substituting Eq. (19) into Eq. (13) and applying triangle function's orthogonalproperty, we yield06 1,3 1,32 222 2a( , )16cos cosm nm x n yw x yq a bD mn ma bnπ ππ∞ ∞= ==∑ ∑ ??? +??? (20)(b): A simply supported rectangular thin plate is considered, whose lengthand width is a and b respectively and bending moment ( )M y= f x isapplied in2y = ± b and the boundary conditions are given by( )2222w x =± a= 0, ??? ??xw???x=±a=0.(21a)( )( )222220,( ).yby y bybwM D wfy=±=±=±== ? ??? ????? =x(21b)Suppose bending deflection function satisfied the boundary condition (21a)is given by1,3( , ) m( )cosmw x y Y ym xa∞π== ∑(22)Substituting Eq. (22) into Eq. (13) and applying triangle function's propertyand the boundary condition (21b) we yield( ) [2 21 13 31,3( , ) 2 1tanh cosh sinh coscoshmyb m mm mw x yM a m y m y m y m xD m a a a a= π ∑=∞ ? + α?α × απ ? π π???π(23)(c): A simply supported rectangular thin plate is considered, whose lengthand width is a and b respectively and bending moment ( )M x= f y isapplied in 2x= ± a and the boundary conditions are given by( )2222w y =± b= 0, ??? ??yw??? y=±b=0. (24a)( )( )222220,( ).xax x axawM D wfx=±=±=±== ? ??? ????? =y(24b)Suppose bending deflection function satisfied boundary condition (24a) isas follows1,3( , ) m( )cosmw x y X xm yb∞π== ∑(25)Similar to the developing of (b) and yields( ) [2 21 13 31,3( , ) 21tanh cosh sinh coscoshmxc m mm mw x yM b m x m x m x m yD m b b b b= π ∑=∞ ? + β?β × βπ ? π π???π(26)Accordingly we may obtain the deflection as followsw( x , y ) = wa ( x , y ) + wb ( x , y ) + wc ( x , y) (27)At last, we discuss the buckling and bending control of functionallygradient rectangular thin plate bonded piezoelectric patches by numericalexamples. These results show that the structure stability and bendingdeflection can be controlled effectively by means of adjusting the voltageamplitude and direction applied in actuators. |
PDF Full Text Request