Thermoelastic Buckling Analysis of Power-law, Sigmoid, Exponential FGM Circular Plates with Piezoelectric Actuators

Transcription

1 3rd International Conference on Mecanical, Electronics and Mecatronics Engineering (ICMEME'4) Marc 9-, 4 Abu Dabi (UAE) ermoelastic Buckling Analysis of Power-law, Sigmoid, Exonential FGM Circular Plates wit Piezoelectric Actuators A. R. Korsidvand, J. Sajedi, and M. Javadi Abstract In tis aer, buckling of elastic, circular lates made of functionally graded material wit surface-bounded iezoelectric layers subjected to termal loading ave been investigated. Boundary condition of te late as immovable clamed edge is considered. e material roerties of te FG lates excet oisson s ratios are assumed to vary continuously trougout te tickness direction according to te volume fraction of constituents defined by owerlaw, sigmoid, and exonential function. e nonlinear equilibrium equations are derived based on te classical late teory using variational formulations and ten linear stability equations are used to obtain te critical buckling of solid FG circular late under termal load as uniform temerature rise. e effects of iezoelectric actuators on buckling of late P-, S-, E-FGM are comared. e results are validated wit te known data in te literature. Keywords Classical late teory, Functionally graded material, ermal buckling. M I. INRODUCION ANY studies are reorted on buckling and bending beavior of FGM structures. Axisymmetric bending of functionally graded circular and annular lates is studied by Reddy et al. []. ey resented te solutions for deflections and force and moment resultants based on te first-order late teory in terms of tose obtained using te classical late teory. e buckling analysis of circular ortotroic lates under termal loads are given by Najafizade, and Eslami []. Korsidvand et al. [3] resented buckling analysis of circular FGM late integrated wit iezoelectric layers subjected to tree kinds of termal loadings based on classical late teory. Lane [7] obtained te closed form solution for te termal buckling of functionally graded rectangular simly suorted lates subjected to two tyes of temerature fields; uniform temerature rise and gradient across te tickness of te late, emloying te first-order sear deformation teory. A. R. Korsidvand, Deartment of Mecanical Engineering, Sout eran Branc, Islamic Azad University eran, Iran (corresonding autor to rovide one: ; Ar_korsidvand@azad.ac.ir ). J. Sajedi, is wit te Mecanical Engineering Deartment, Sout eran Branc, Islamic Azad University eran, Iran. ( J_sajedi@azad.ac.ir). M. Javadi is wit te Mecanical Engineering Deartment, Sout eran Branc, Islamic Azad University eran, Iran. ( m_javadi@azad.ac.ir). e resent aer deals wit determination of te stability roblem and resents closed-form solutions for critical buckling temerature of Piezoelectric P-, S-, E-FGM circular late, wic are subjected to uniform temerature rise. Clamed edge boundary condition is assumed for te late. II. DERIVAION OF GOVERNING EQUAIONS Consider a uniform tin circular late made of FGM, as sown in Figure (). o extract formulations, a cylindrical coordinate system is taken in te center of late s middle lane. e FGM rofile across te tickness direction of te late, made of ceramic and metal constituent materials, may be assumed to follow a function form as P-FGM lates as Pr(z) rm rcm ( z )n e value of n, ower law index, equal to zero reresents a fully ceramic late. wo ower law functions S-FGM lates as -z n Pr(z) r m r cm [ - ( ) ] for z / n Pr(z) r z m r cm [ ( ) ] for /z () and exonential function E-FGM lates z Pr(z) AEx[B( )], A Prm, B Ln(Pr (3) c/prm) Were r,r c, r m denote any material roerty of te FGM, metal, and ceramic; suc as te modulus of elasticity E and te coefficient of termal exansion., e relations (), () and (3) indicate tat te to surface of te late (z = /) is ceramic-ric wereas te bottom surface (z = /) of te late is metal-ric. Generally, Poisson s ratio is assumed constant across te late tickness. A. Basic Equations e material roerties are assumed to be indeendent of temerature, and te stress and strain relations are linear. e constitutive relations of functionally graded materials in termal environment for te lane-stress condition are written as rr E(z)( rr ) /( ) E(z) (z)(z) /( ) (4) E(z)( rr) /( ) E(z) (z)(z) /( ) (5) () 8

2 3rd International Conference on Mecanical, Electronics and Mecatronics Engineering (ICMEME'4) Marc 9-, 4 Abu Dabi (UAE) r E(z) /( ) r (6) e late is assumed to be comaratively tin, and according to te Love-Kircoff assumtions, sear deformations normal to te late are disregarded. Using te classical late teory (CP), strain comonents at distance z from te middle lane are given in matrix form as [4] rr r u,r(w,r) w,rr r v, r u ( ) z ( ) r w, r w,r r w, v, ) r u, r v, (w, r w,r r w, () () = +z (7) were a comma in subscrit indicates artial differentiation and were rr,, and r are te strain comonents along te r-,, and z- directions, resectively. e stress comonents in lane-stress condition in te late (suerscrit ) are written as following rr Q Q rr(z)(z) Q Q (z)(z) (8) r Q r 44 Were te lane-stress-reduced stiffness are defined as E(z) E(z) Q Q, Q Q Q, Q (9) 44 ( ) And stress comonents in iezoelectric arts of te late are written as following rr c c r c rr (z) c (z) c 44 r c 3 Err e 3 E Ez e total otential energy for iezoelectric FG circular late can be written as follows U U U U U ({ - { }} [c]{ - { }} - {E} [k]{e} r z { - { }} [e]{e})rdzddr rr( rr - (z)(z)) ( - (z)(z)) ]rdzd dr r z r r () (9) ere [ c ],[ k],[ e] are matrix form of elastic, dielectric ermeability, and iezoelectric material coefficients, resectively and are defined as [ c] c c c c c 44,[k] k k k 33,[e] e 3 e 3 () Assuming tat te actuator is oled along te z, and viewing te iezoelectric material as a transversely isotroic material, wic is true for iezoelectric ceramics, many of te arameters in te mentioned matrices will be eiter zero or can be exressed in terms of te oter arameters. In articular, te non-zero coefficients of iezoelectric roerties may be written as c c, c, c 44, e 3 e 3, k k,k 33 () e only non-zero electric field is in te z-direction and te vector of alied electric field can be sown E Ez (3) Considering relations (4) to () and substituting relations () and integrating wit resect to z, te total otential energy is obtained. Alying te Euler equations for total functional of U in Eq. (), equilibrium equations are yield and ten te stability equations of te circular late are derived using te adjacent equilibrium criterion are obtained as [4] N rr,r r (N rr N ) r N r, r N r r N, N r,r (4) e force and moment resultants of late are exressed in terms of te stress comonents troug te tickness as follow N N Nrr N r / a / { } dz a { }dz { } dz M M Mrr M r / a / { } zdz a { }zdz { } zdz (5) Stress resultants can be simlified in te matrix form as {N} {M} were (rn rr w,r N r w, M ),r [AaC] [B] [B] [DLC] () () ( r N w, N r w,r r M r ), (rm rr ),rr (M r ),r ( r M ), {N } {M } {N (E) } {M (E) } (6) 9

3 3rd International Conference on Mecanical, Electronics and Mecatronics Engineering (ICMEME'4) Marc 9-, 4 Abu Dabi (UAE) ( A ij, B ij, D ij ) z Q ij (z)(,z,z )dz (i, j,,3) (7) ( ) ( ) ( E ) ( E ) Here { N }, { M }, { N }, { M } are te stress resultants due to te alied temerature and electrical field on te late, and tey can be comuted as ) { N ( } {N } } FGM {N Piezo (E) { N } {e e } 3 Eza 3 Eza { N } {E ) E ) } FGM 4 /( 4 /( { N } { (c c (c c } ) E Piezo ) E ) { M ( } {M } } FGM {M Piezo (E) { M } { } { M } { (c c (c c } ) E Piezo ) E { M } {E /( ) E /( ) } FGM 5 5 (8) were / / E / 4 / (z)e(z) dz, E 5 / E z / 6 dz, E / 7 ( ( / E8 a) / dz, E / 9 ( EE 8 E 9 B. ermal Axisymmetric Buckling a )/ z dz (9) Here olar symmetry condition is considered. us, for tis case of discussion te first and second of stability equations (4), based on te dislacement comonents, lead to (E * Lc ) 4 w N w r N w 3 rr () E * ( u r u u u ) r r3 were E *, E*, E* are given as 3 * / () ( E, E *, E * ) /( ) / (,z,z) E(z)dz 3 Referring to Eqs. (6), using te membrane late teory, te rebuckling forces are obtained as Nrr, E E 6 E 7 (E* a c )(u r u E * ( w r w w ) r u ) r a) / z(z)e(z) dz dz ( ) ( ) N rr e 3 Eza, N N e 3 Eza () us, te set of couled stability equations must be solved. For clamed and immovable edge in r- direction, te boundary conditions are exressed as [5] u (r ), w (r ) finite u ( ra) w ( ra) w' ( ra) (3) e solution of Eqs. () is assumed in te form u (r) A J ( r) A Y ( r) A 3 (/ r) A 4 r w (r) A 5 J ( r) A 6 Y ( r) A 7 Lnr A 8 (4) were and are te Bessel functions of first, zero order, and first and second kinds, resectively. Also, to are te integration constants. Using te first and second boundary conditions yields. Satisfying te tird boundary condition of Eqs. (3), A, A 8 J ( a)a, J ( a) (5) 4 5 us, te smallest root is a = It is seen tat for te clamed edge u ( r) A J ( r), w ( r) A 5 ( J ( r) J ( a)) (6) Substituting te exressions (6) into (4), two linear omogeneous equations are obtained as (E* c 3 E * a )A A 5 3 E * A [ 4 (E * c L) (N e E z a )]A 3 rr 3 5 (7) For a nontrivial solution of tese equations, te determinant of coefficient must be set to zero and wen te temerature distribution of te late is a function of tickness direction only, is constant and yields. (N e E /[(E * c L) E * /(E* z ) rr 3 3 (8) for te case of uniform temerature rise, taking a late at temerature, and te temerature may be raised to f were te late buckles. In suc a case, Substituting from Eqs. (8) into Eq. (8) te critical buckling temerature cr is exressed in te form c a )] [(E * E * * cr ( ){ c L) /(E c a )] 3 e E (c c ) z a } /[Q a ] 3 (9) 3

4 3rd International Conference on Mecanical, Electronics and Mecatronics Engineering (ICMEME'4) Marc 9-, 4 Abu Dabi (UAE) / Were Q (z)e(z)dz, / cr f temerature, is larger for circular lates wit inner ure metal layer. and is coefficient of termal exansion of iezoelectric. ABLE I MAERIAL PROPERIES OF PLAE AND PIEZOELECRIC Aluminium Alumina PZ-5 Em 7GPa m 66c m 3.e 6.3 E c 38GPa m c 5c c 7.4e 6.3 c c 69. 5GPa c c 4. 3GPa III. RESULS AND DISCUSSION e 6 c e3 e3 6.4 m In te following, te axisymmetric stability and termal buckling loads of an FG circular late integrated wit iezoelectric layers subjected to uniform temerature rise is derived and summarized in te receding section. o validate te formulations of te resent article, termal buckling loads of te circular late are comared wit tose obtained by Najafizade and Eslami [] for isotroic late. It is clear tat from (9), taking PFGM circular late, te same results is obtained for te omogeneous isotroic full ceramic circular late. e results are obtained tat are identical to tose reorted as in []. Now, consider an FGM circular late integrated wit two iezoelectric layers. e material roerties of iezoelectric, metal (Aluminium), and ceramic (Alumina) constituents are given in ABLE. For tis examle te results for termal buckling loads is lotted in Figs. () to (4). Figure () reresents te critical buckling temerature versus /a for a P-, S-, E-FGM clamed circular late and witout taking iezoelectric layers under uniform temerature rise. e mecanical boundary condition at te edge of te late is assumed to be clamed suorted. Here, te Curie temerature is an imortant arameter for te alications of ferroelectrics. For te PZ ceramics, te ase above Curie temerature is araelectric and also non-iezoelectric (isotroic). If te iezoelectric roerties are used in alications, te material cannot be exosed to te temeratures above te Curie temerature to reserve ferroelectric roerties. It is recommended by PZ manufacturers not to use PZ ceramics above.5 c. e Curie temeratures c for commercially roduced PZ s are usually between 5 and 36 C [6]. Figures (3) and (4) are art of Fig. o under te limit of oc C, for PFGM and SFGM late integrated wit two iezoelectric layers wic is magnified. Figure (3) sows te curve associated wit PFGM corresond wit te data reresented [3]. As it can be seen, te effect of iezoelectric layers to increase te termal bucking IV. CONCLUSIONS In te resent article, te buckling analysis of Piezoelectric FG circular late is derive based on te classical late teory. Boundary condition of te late is taken to be clamed. Plate is subjected to uniform temerature rise. e termal buckling caacity of circular lates as closed-form solution is resented. It is concluded tat:. Voltage variation of iezoelectric layers does not ave significant influence on te buckling of an. P-,S-,E-FGM lates.. e critical buckling temerature is reduced wen volume fraction index increases, as te late becomes more metal-ric. 3. A comarison between termal buckling curves of owerlaw, sigmoid, exonential FGM circular lates in uniform temerature rise, indicate tat for a certain value of / a termal buckling caacity of circular late made of P-FGM is better tan S-FGM. ACKNOWLEDGMEN e resent researc work was suorted by Islamic Azad University - Sout eran Branc. REFERENCES [] J. N. Reddy, C.M. Wang, and S. Kitiorncai, Axisymmetric bending of functionally graded circular and annular lates. Eur. J. Mec. A/solids No. 8, , 999. [] M. M. Najafizade, M.R. Eslami, ermoelastic stability of ortotroic circular lates. Journal of ermal Stresses, Vol. 5. No., ,. [3] A.R. Korsidvand, M. Jabbari, and M.R. Eslami, ermoelastic buckling analysis of functionally graded circular lates integrated wit iezoelectric layers, Journal of ermal Stresses, Vol. 35, ,. [4] D.O. Brus, B.O. Almort, Buckling of bars, lates and sells. McGraw-Hill, New York, 975. [5] J. N. Reddy, Mecanics of laminated comosite lates and sells, teory and analysis. CRC Press, New York, 4. [6] J. ic, J. Erart, E. Kittinger,, and J. Prvratsk, Fundamentals of Piezoelectric Sensorics, Mecanical, Dielectric, and ermodynamical Proerties of Piezoelectric Materials, New York, Sringer,. [7] W., ermal buckling of a simly suorted moderately tick rectangular FGM late, Comos. Struct., Vol. 64,. -8, 4. Amad Reza Korsidvand was born in eran, Iran, in 969. He received PD degree in, in mecanical engineering - solid mecanics from Science and Researc Branc, IAU, eran, IRAN. Now, e is member of faculty of mecanical engineering deartment, Sout eran Branc, eran, IRAN. His researc interests include Solid Mecanics, ermo- Elasticity, Comosite Materials, and Stability. Dr. Korsidvand is member of IACSI, IISRO and e as more tan eigt ublised aers in International Journals. 3

5 3rd International Conference on Mecanical, Electronics and Mecatronics Engineering (ICMEME'4) Marc 9-, 4 Abu Dabi (UAE) Fig. : Buckling temerature versus /a for a P-, S-,E-FGM circular clamed late under uniform temerature rise. Fig 3: e effective range of iezoelectric layers versus /a on buckling temerature a circular PFGM clamed late. Fig. 4: e effective range of iezoelectric layers versus /a on buckling temerature a SFGM circular clamed late. 3