AN EXACT SOLUTION OF MECHANICAL BUCKLING FOR FUNCTIONALLY GRADED MATERIAL BIMORPH CIRCULAR PLATES

Assocition of Metllurgicl Engineers of Serbi AMES Scientific pper UDC: 6.:66.7/.8 AN EXACT SOLUTION OF MECHANICAL BUCKLING FOR FUNCTIONALLY GRADED MATERIAL BIMORPH CIRCULAR PLATES Jfr Eskndri Jm, Mhmood Khosrvi, Nder Nmdrn Composite Mterils & Technology Center, Tehrn, Irn Received 5.6. Accepted.7. Abstrct Presented herein is the exct solution of mechnicl buckling response of FGM (Functionlly Grded Mteril) bimorph circulr pltes, performed under uniform rdil compression, by mens of the clssic theory nd the non-liner Von-Krmn ssumptions, for both simply supported nd clmped boundry conditions. Mteril properties re ssumed to be symmetric with respect to the middle surfce nd re grded in the thickness direction ccording to simple power lw, in wy tht the middle surfce is pure metl nd the two sides re pure cermic. Using the energy method the non-liner equilibrium equtions re derived nd the stbility equtions hve been used, so s to determine the criticl buckling pressure, considering the djcent equilibrium criterion, nd finlly closed-form solution hs been chieved for it. The effect of different fctors, including thickness to rdius vrition rte of the plte, volumetric percentge of mteril index, nd Poisson's rtio on the criticl buckling compression hve been investigted for two simply supported nd clmped boundry conditions, nd the results chieved re compred with ech other nd the results vilble in the literture. Keywords: Mechnicl buckling nlysis, circulr pltes, functionlly grded mterils, clssic theory. Introduction In the recent yers, functionlly grded mterils (FGMs) hve been widely used in dvnced industries especilly in erospce nd nucler engineering pplictions, nd due to their importnce they hve lwys been ttrctive for designers, nd mny reserch studies hve been performed on this topic. Generlly, FGMs re not homogeneous nd re composed of mixture of cermic nd metl, or mixture of Corresponding Author: Jfr Eskndri Jm, jejm@gmil.com

6 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 different metls. Microscopiclly, FGMs re non-homogenous nd their structurl properties including distribution type nd the size of the phses vry continuously nd homogenously cross the thickness, nd this vrition results in the grdul chnge of mechnicl properties for these mterils []. Stbility nlysis nd studies on the buckling behvior of pltes hve lwys been considered s one of the most importnt subjects in structurl nlysis. The first solution of the stbility problem for pltes ws presented by Bryn [] in 89. In this study, the buckling of circulr plte under uniform rdil loding ws performed for the clmped boundry conditions. Ymki [3] investigted the buckling of nnulr pltes under lodings pplied on the inner nd outer edges, nd showed tht buckling in this stte does not necessrily hppen in the first mode. Timoshenko nd Gere [] studied the buckling nlysis problem of different engineering structures including: columns, frmes, curved bems, pltes nd shells. After tht Almorth nd Brush [5] presented generl nlysis of buckling for columns, pltes, nd shells, nd lso investigted different methods of formultion for the non-liner equilibrium nd stbility equtions. Reddy nd Khdeir [6] studied the buckling nlysis nd free vibrtions of composite lminted rectngulr pltes, using the clssic, the first nd the third order sher theories, under different boundry conditions. In ddition to the nlyticl solution, numericl solution ws done on the bsis of the finite element method in this pper. Results of this study indicte tht the clssic theory overestimtes the nturl frequencies nd criticl buckling lod, nd s the thickness to lterl length rtio increses, this discrepncy increses. Rju et l. [7, 8] investigted the post buckling of the homogenous nd orthotropic circulr pltes with liner thickness vrition, under mechnicl nd therml lodings, in terms of the uniform temperture rise. Therml buckling nlysis of these pltes ws crried out, using finite element theory. Ozkc et l. [9] using finite element method nd tking into ccount the sher effects s the first order sher theory; performed the buckling nlysis nd thickness optimiztion for the circulr nd nnulr pltes. Njfizdeh nd Eslmi [, ] studied buckling nlysis of the one-sided FGM circulr pltes, under different types of therml loding, for the clmped boundry conditions, nd lso for rdil mechnicl compression loding, under simply supported nd clmped boundry conditions. In these studies, the clssic theory ws used nd the nlyticl solutions were provided. Njfizdeh nd Hedyti [] hve performed the therml nd mechnicl buckling nlyses for one-sided FGM circulr pltes, using the first order sher theory nd hve compred the results chieved with the clssic theory. In this study, therml buckling nlysis ws performed only for the clmped boundry conditions, nd the mechnicl buckling nlysis ws performed for both the simply supported nd clmped boundry conditions. Njfizdeh nd Heydri [3, ] hve investigted the buckling nlysis of one-sided FGM circulr pltes, under different therml lodings, for the clmped boundry conditions, nd lso investigted it for uniform rdil compression loding, under simply supported nd clmped boundry conditions. In these studies, the third order sher theory hs been used, nd the results obtined re compred with the results of the clssic nd first order sher theories.

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 7 Nei et l. [5] studied the mechnicl buckling of one-sided FGM circulr pltes with vrible thickness, under uniform rdil compression, using finite element technique. The boundry conditions used for the circulr plte were simply supported nd clmped. Jlli et l. [6] performed the therml stbility nlysis for circulr functionlly grded sndwich pltes of vrible thickness nd FGM fce sheets, using pseudo-spectrl method. In this study, the circulr plte is under uniform therml loding, nd is presented ccording to the first order sher theory. FGM pltes cn be of two types, nmely, one-sided FGMs nd FGM bimorphs, bsed on their property vritions cross the thickness, s shown in figure. One-sided FGM pltes hve pure cermic on one side nd pure metl on the other side; nd the mteril properties vry continuously from one side to the other. But in FGM bimorph pltes, both sides re pure cermic nd the mid-plne is pure metl. Figure.FGM plte: () one-sided, (b) bimorph. In this study, the clssic theory nd the Von Krmn ssumptions hve been used to nlyze the mechnicl buckling of FGM bimorph circulr pltes, under uniform rdil compression. In order to determine the non-liner equilibrium equtions, the minimum potentil energy hs been used, nd by ppliction of the perturbtion method, the non-liner equilibrium equtions hve become liner. In order to determine the criticl buckling pressure, the stbility equtions hve been derived, using the djcent equilibrium criterion from the liner equilibrium equtions, nd then re nlyticlly solved for simply supported nd clmped boundry conditions, nd closed-form solution is provided for it, lter on. The effect of different fctors, including thickness to rdius vrition rte of the plte, volumetric percentge of mteril index, nd Poisson's rtio on the criticl buckling pressure hve been investigted for two simply supported nd clmped boundry conditions, nd the results chieved re compred with the results obtined for homogenous nd one-sided FGMs.

8 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 Formultion of the problem Properties of the FGM The circulr plte studied, is composed of FGM bimorph mterils in which the volumetric percentge of cermic nd s result the properties of the mteril re symmetric with respect to the middle surfce, nd vry continuously through the thickness direction, in wy tht the middle surfce is pure metl nd s we pproch the outer surfces, the cermic percentge increses nd t the upper nd lower surfces, i.e. t h z=- nd h z= it becomes pure cermic. Since FGMs re compound of metl nd cermic, properties of these mterils like the modulus of elsticity E, bsed on the clssic liner rule of mixtures re expressed s follows: E = E V + E V () f m m c c Wherein c nd m show the properties of the cermic nd metl, respectively, nd V c nd V m indicte their respective volume frctions [6]. Here, the vritions of the mteril properties like the modulus of elsticity re expressed by the simple power lw s follows [7]: z n h P ( z) = Pm + ( Pc Pm).( ), z h () z n h P ( z) = Pm + ( Pc Pm).( ), z h P shows the mteril properties nd cn be the modulus of elsticity. Vrition of the Poisson's rtio is negligible nd is supposed to be constnt. n is the volume frction index of the FGM, which indictes the compound of the volume frctions of the cermic nd metl cross the thickness, which cn be bigger thn or equl to zero. The zero nd infinity vlues for this index men pure cermic nd pure metl, respectively. Stbility nd equilibrium equtions A circulr FGM bimorph plte with rdius nd thickness h is considered. The displcement field ccording to the clssic theory cn be expressed in the polr coordintes with xisymmetry, s follows [9]: dw U ( r, z ) = u( r) z dr (3) V =, W r, z = w r ( ) ( ) Where U, V, nd W re displcements long r, θ, nd z xes nd u(r), w(r) re displcements of the middle surfce long r nd z directions. The buckling problem is ctegorized in the non-liner geometricl problems, nd in non-liner problems, displcements re in rnge tht the non-liner strindisplcement terms cnnot be disregrded. The strin-displcement reltions bsed on the Von-Krmn ssumptions, nd reltion (3) cn be stted s [9]: du dw U ε r = +, εθ =, γ θ = ε θ = dr dr r r r ()

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 9 ε r ε r k r εθ = εθ + z kθ γ θ γ r rθ k rθ And, the strins of the middle surfce re given by: du dw u εr = +, εθ =, γrθ = dr dr r The curvtures re defined s: dw dw kr =, kθ = ( ), k θ = r dr r dr (5) (6) (7) Regrding the fct tht the mterils re isotropic, by disregrding the stresses in the thickness direction nd tking the Poisson's rtio to be constnt, the strindisplcement reltions re chieved s follows [5]: E σ r = ( ε ) r + νεθ ν E σθ = ( ε ) θ + νε (8) r ν E τrθ = γrθ = ( + ν ) In the bove reltions, E is the modulus of elsticity for the FGM. σ θ nd σ r re norml stresses nd τ rθ is the sher stress in ech point of the plte thickness, with distnce equl to z from the middle surfce. By integrtion of the stress components over the thickness of the plte, one cn find the resultnt forces nd moments in terms of the stress components s follows: ( N, N ) = ( σ, σ ) dz r θ ( M, M ) = ( σ, σ ) zdz r θ h h h h r r θ θ Substituting eqution (8) into eqution (9), the reltionship between the forces nd the moments in terms of the strin components, cn be clculted s: N r ν ε r = A θ ν N εθ () M r ν k r = B θ ν M kθ (9)

5 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 Also, by substtion of equtions (6) nd (7) into equtions (), the reltionship between the forces nd the moments in terms of the displcement components, is given by: du dw u N r = A( + ( ) + ν ) dr dr r u du dw Nθ = A( + ν ( + ( ) )) r dr dr () dw ν dw M r = B( ) dr r dr dw d w Mθ = B( ν ) r dr dr The coefficients A nd B re chieved by integrting the properties through the thickness of the plte s: h E ( z ) ( A, B ) = (, z ) dz ν h Equilibrium equtions for the circulr pltes cn be derived by minimizing the totl potentil energy, or directly writing the equilibrium equtions for one element. Due to xisymmetry, there re no vritions long the circumference, nd only the derivtives with respect to the rdil direction re present in the differentil equtions set. dn r N r Nθ + = dr r (3) d dw d M r dm r dm θ ( rn r ) + = rdr dr dr r dr r dr In order to find the stbility equtions from the non-liner equilibrium equtions, usully the djcent equilibrium criterion is used [5]. This criterion is used to investigte the stbility nd nlyze the buckling behvior of the structures, bsed on the definition of the first nd second equilibrium pths nd the bifurction point. By mking use of this method one cn obtin the bifurction point, through solving the liner differentil equtions. It is in this mnner tht for n equilibrium stte on the first equilibrium pth, the possibility of the existence of n djcent equilibrium form, under the sme loding is investigted. Such n equilibrium form t the djcency of the first equilibrium is the sign of the existence of bifurction point on the equilibrium pth. The intersection point of these pths is clled the bifurction point, nd t such points the equilibrium equtions hve two solutions, ech of which corresponds to one of the two brnches. The equtions necessry for this method re derived from the non-liner equilibrium equtions of the structure, using the perturbtion method, wherein the displcement fields ( uw, ) re replced by ( u + u, w + w ), in which ( u, w ) shows the first equilibrium stte, i.e. the prebuckling stte, nd lso indictes n equilibrium ()

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 5 stte on the first pth, nd ( u, w ) re the infinitesiml displcements in the displcement field. But, ttention should be pid to the fct tht in the prebuckling stte, the plte is not buckled nd, or there is no lterl deflection nd w is equl to zero. Substituting these new fields into the equtions (3), ll the terms which exclude the infinitesiml displcements re dropped out from the resulted equtions. Also, if the increse in the displcement is dequtely smll, only the first order terms of the displcement ( u, w) remin in the equtions, nd the higher order terms re dropped out. Therefore, the resulting stbility equtions re liner nd homogenous equtions in terms of the supposed smll displcements. This recipe is recognized to be used in the determintion of the stbility, which is clled the djcent equilibrium criterion. Therefore, the stbility equtions re expressed s follows: dn r N r Nθ + = dr r () d dw d M r dm r dm θ ( rn r ) + = rdr dr dr r dr r dr Where, in the bove equtions: du u N r = A( + ν ) dr r u du ( ) Nθ = A + ν r dr M M dw ν dw = B( ) dr r dr r dw d w θ = B ν ( ) r dr dr du u ( ) N r = A + ν dr r u du Nθ = A( + ν ) r dr d w ν dw M r B M = ( ) = dr r dr dw d w θ B ν = ( ) = r dr dr (5) (6) Substitution of equtions (5) nd into equtions () furnishes the stbility equtions in terms of the displcement components: d u du u A ( + ) = dr r dr r (7) d dw B w ( rn r ) = rdr dr

5 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 It is seen tht the stbility equtions derived re independent from ech other. Therefore, in order to determine the criticl buckling pressure, the second eqution hs been used. In the stbility equtions, N r is the rdil prebuckling force, which is determined from the solution of the non-liner equilibrium equtions. So s to this, by substitution of equtions (6) into the first reltion of eqution (3) we hve: d u du u + = (8) dr r dr r Eqution (8) is clled the membrne eqution of plte. This eqution is n ordinry liner differentil eqution, using whose solution one cn determine the displcements nd the in-plne prebuckling lods. The independent vrible of this eqution is u, which by knowing whose vlue, the in-plne lods re clculted. The solution of eqution (8) is s follows: c u = cr + (9) r The mechnicl loding is provided by mens of the ppliction of uniform rdil lod of P in Newton per meter on the edge of the plte, nd in this stte the edge hs the bility to move freely long the rdius direction. Also, due to the symmetry, the displcement t the center must be limited. As result, the boundry conditions for the eqution (9) re stted s follows: (At center) Finite () u = () N r ( ) = P (At edge) By pplying the first boundry condition of () into reltion (9), we hve P c =, nd by pplying the second boundry condition, we hve c =. A ( + ν ) Therefore, reltion (9) will be s follows: rp u = () A ( + ν ) Substitution of eqution () into equtions (6) the prebuckling forces re derived s: N = N = - P () r q Substitution of eqution () into the second stbility eqution yields: d dw B w = ( rp ) rdr dr (3) By one time integrtion of the bove reltion with respect to r we hve: r α + rα + α( β r ) = ()

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 53 Where in the bove: dw P α =, β = dr B (5) Now, new vrible is defined: s = β. r (6) Using which, the eqution () becomes so: d α dα s + s + ( s ) α = ds ds (7) The generl nswer of this eqution is s follows: α = cj( s) + cy( s ) (8) 5 6 Where J nd Y re the Bessel type one nd two functions, nd c5 nd c 6 re the integrtion constnts. At the center of the plte there is r = s =, nd α must go to zero, so s to stisfy the symmetry conditions. Becuse s goes to zero, the Y ( s) function becomes infinity, nd the bove condition mkes it necessry thtc 6 =. In order to stisfy the clmped boundry conditions for the edges of the plte, there should be: ( α ) r = = (9) And, therefore: J ( β ) = (3) The smllest root of eqution (3) is: β = 3.837 (3) By substitution of this vlue into eqution (5), the criticl buckling pressure becomes: B Pcr = β B =.689 (3) The nswer of reltion (8) cn lso be used for the buckling stte of circulr plte, under simply supported boundry conditions. Like the clmped boundry conditions, in order to stisfy the condition existing t the center of the plte, we must hve c6 set to zero. The second condition for the simply supported boundry conditions α ν t r = is d M = + α r B =. Therefore: dr r dj( s ) J( s ) + ν = dr r r= (33)

5 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 Or dj( s ) s + ν J( s) = ds s= α (3) On the other hnd, using the differentition formultion for the Bessel functions: dj J = J (35) ds s Where J is the type one Bessel function nd is of order zero. Therefore: λj ( β) ( ν) J ( β ) = (36) Eqution (36) hs different nswers for different vlues of the Poisson's rtio, which in tble () the smllest root hs been stted for different vlues of ν. For exmple, if the Poisson's rtio isν =.3, ccording to tble () the smllest vlue of β which stisfies eqution (37), is β =.89. Substituting this vlue into eqution (5), the criticl buckling pressure for the simply supported stte is s follows: B Pcr =.978 (36) Tble. Smllest root of eqution (36) for different vlues of Poisson's rtio. ( β) min.8..95 Poisson's rtio..3.98.89..9.5.659 Discussion nd numericl results In this section, the results of the exct solution of the mechnicl buckling of FGM bimorph circulr pltes hs been compred with the results of the simply supported nd clmped boundry conditions, under uniform rdil compression. The FGM is considered to be compound of Aluminum s metl nd Alumin s cermic. The mechnicl properties for Aluminum nd Alumin hve been included in tble (). Tble..Properties of Aluminum nd Alumin, s compound elements of the FGM. Young's (GP)modulus Mteril 7 Aluminum 38 Alumin In tbles (3) nd () the reltions used for determining the criticl buckling pressure for the FGM bimorph, under clmped nd simply supported boundry conditions hve been stted.

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 55 Tble 3. Reltions for determintion of the criticl buckling pressure for the FGM bimorph under clmped boundry conditions. Type of loding Clmped Uniform rdil compression B Pcr =.689 Tble. Reltions for determintion of the criticl buckling pressure for the FGM bimorph, under simply supported boundry conditions. Poisson's rtio Simply supported B Pcr = 3.39. B Pcr = 3.6687. B Pcr = 3.9379.3 B Pcr =.978. B Pcr =.87.5 B Pcr =.69 In order to determine the ccurcy of the results chieved from solution of the stbility equtions for the circulr plte, under clmped nd simply supported boundry conditions, first the constnt of the power lw is set to zero, so tht the FGM is turned into homogenous mteril, nd the dimensionless mechnicl prmeter λ is clculted, nd the results obtined re compred with those of the other references. By comprison with the results of the previous reserches, it is observed tht in the mechnicl loding the dimensionless prmeter of buckling λ is determined from the reltion (38). In this regrd, ν, D, E c, nd P cr re Poisson's rtio, bending stiffness, modulus of elsticity of the cermic, nd criticl rdil compression loding pplied on the edge in terms of Newton per meter. P λ = cr (38) D 3 Eh c D = (39) ν ( ) In tble (5), the dimensionless prmeter of mechnicl buckling for the homogenous circulr plte (n=) with clmped nd simply supported boundry conditions, nd the Poisson's rtio of.3 hs been chieved for uniform rdil compression, nd the results hve been compred with those of the other references.

56 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 Tble 5. Dimensionless Prmeter of buckling for homogenous circulr plte (n=), under clmped nd simply supported boundry conditions, with the Poisson's rtio of.3 h/ Reference...5...68.68.68 Present.689.689 9 9 9 C. [].68.68.68.68.68.68.68.68 [].68.68 S.S. b Present [] [].978..5.978..5.978..5.978..5.978..5 C. refers to clmped edge. b S.S. refers to simply supported edge. In tble (6), the dimensionless prmeter of mechnicl buckling for the homogenous circulr plte with clmped nd simply supported boundry conditions, under uniform rdil compression nd different vlues of Poisson's rtio hs been chieved, nd the results hve been compred with those of the other references. In the results, it is seen tht the increse of the Poisson's rtio increses the buckling prmeter. Tble 6. Dimensionless prmeter of mechnicl buckling for homogenous circulr plte, under simply supported boundry conditions, nd different Poisson's rtios. References Poisson's rtio...3..5 Present 3.39 3.6687 3.9379.978.87.69 [ ] 3.39 3.6687 3.9379.978.87.69 [5] 3.389 3.668 3.937.99.8 - After verifiction of the results of stbility equtions, the investigtion of the effects of different fctors on the criticl buckling pressure for the FGM bimorph is performed. Figures () nd (3) show vritions of the criticl buckling pressure of the FGM bimorph in terms of the thickness to rdius rtio of the plte, for different vlues of the volume frction index, under clmped nd simply supported boundry conditions, with the Poisson's rtio of.3. Considering the results, it is seen tht the increse of h / increses the buckling prmeter for the FGM bimorphs, nd the reson is tht s the plte gets thicker, its stiffness increses nd s result its criticl buckling compression increses, too.

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 57 5.5 x 8 5.5 3.5 3.5.5.5 n= n=. n=.5 n= n=5 n=...3..5.6.7.8.9. h/ Figure.Vritions of criticl buckling pressure with respect to h/ rtio, for different vlues of n under clmped boundry conditions, withν =.3. 6 x 7 8 6 n= n=. n=.5 n= n=5 n=...3..5.6.7.8.9. h/ Figure 3.Vritions of criticl buckling pressure with respect to h /, for different vlues of n under simply supported boundry conditions, withν =.3. Figures () nd (5) show the vritions of the criticl buckling pressure P cr of the FGM bimorph with respect to the volume frction index, for different vlues of the thickness to rdius rtio of the plte, under clmped nd simply supported boundry conditions, with the Poisson's rtio of.3, respectively. By the increse of the volume frction index, the criticl buckling pressure is continuously decreses, nd in n = (pure cermic) will hve its mximum vlue. The reson for this is tht the increse of the volume frction cuses the increse of the metl volume frction in the plte, nd s the stiffness of cermic is more thn tht of metl, the totl stiffness of the plte is reduced, nd therefore the buckling strength of the plte is reduced.

58 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 7 x 7 6 5 h/=.5 h/=. h/=.3 3 3 5 6 7 8 9 n Figure.Vritions of the criticl buckling pressure with respect to n for different vlues of h / under clmped boundry conditions, with ν =.3.8 x 7.6...8.6.. h/=.5 h/=. h/=.3 3 5 6 7 8 9 n Figure 5.Vritions of criticl buckling temperture with respect to n, for different vlues of h /, under simply supported boundry conditions, withν =.3 Figures (6) nd (7) show vritions of the criticl buckling pressure with respect to the thickness to rdius rtio of the plte, for different vlues of volume frction index, under clmped nd simply supported boundry conditions, with the Poisson's rtio of.3, for the two mterils of one-sided FGM (reference ) nd FGM bimorph (present model). It is observed in the results tht the mechnicl buckling strength of the FGMs bimorphs is more thn one-sided FGMs, nd this difference becomes more evident in thicker pltes.

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 59 x 8 5.5 3.5 3.5.5.5 n=.,fgm Bimorph n=.,one sided FGM n=.5,fgm Bimorph n=.5,one sided FGM n=5,fgm Bimorph n=5,one sided FGM...3..5.6.7.8.9. h/ Figure 6.Vritions of the criticl buckling pressure with respect to h / for different vlues of n under clmped boundry conditions, with ν =.3 for two FGMs. x 8 5.5 3.5 3.5.5.5 n=.,fgm Bimorph n=.,one sided FGM n=.5,fgm Bimorph n=.5,one sided FGM n=5,fgm Bimorph n=5,one sided FGM...3..5.6.7.8.9. h/ Figure 7.Vritions of the criticl buckling pressure with respect to h/ for different vlues of n under simply supported boundry conditions, with ν =.3 for two FGMs. Figures (8) nd (9) show vritions of the criticl buckling pressure with respect to the volume frction index, for different vlues of thickness to rdius rtio of the plte, under clmped nd simply supported boundry conditions, with the Poisson's rtio of.3, for the two mterils of one-sided FGMs (reference ) nd FGM bimorphs. In the one-sided FGMs nd FGM bimorphs, under clmped nd simply supported boundry conditions, the criticl buckling pressure continuously decreses by the increse of the volume frction index.

6 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 5.5 x 8 5.5 3.5 h/=.,fgm Bimorph h/=.,one sided FGM h/=.7,fgm Bimorph h/=.7,one sided FGM 3.5.5.5 3 5 6 7 8 9 n Figure 8.Vritions of the criticl buckling pressure with respect to n for different vlues of h/ under clmped boundry conditions, with ν =.3 for two FGMs. 6 x 7 h/=.,fgm Bimorph h/=.,one sided FGM h/=.7,fgm Bimorph h/=.7,one sided FGM 8 6 3 5 6 7 8 9 n Figure 9.Vritions of the criticl buckling pressure with respect to n for different vlues of h/ under simply supported boundry conditions, with ν =.3 for two FGMs. Figures () nd () show vritions of the criticl buckling pressure with respect to the thickness to rdius rtio of, for different vlues of the volume frction index nd Poisson's rtio, under clmped nd simply supported boundry conditions for the FGM bimorphs. It is observed in the results tht in FGM bimorph circulr pltes, under simply supported nd clmped boundry conditions, s the Poisson's rtio increses, the criticl buckling strength increses. Figures () nd (3) present the vritions of the criticl buckling pressure for h / =., nd different vlues of the volume frction index nd Poisson's rtio, under clmped nd simply supported boundry conditions for the FGM bimorphs. It is observed in the results tht in FGM

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 6 bimorphs, the criticl buckling strength, under clmped boundry conditions is more thn three times tht of simply supported boundry conditions. The bove result is lso observed for homogenous mterils []. x 8 5.5 3.5 3.5.5.5 v=.5 v=. v=.3 v=. v=. v=...3..5.6.7.8.9. h/ Figure.Vritions of the criticl buckling pressure for different vlues of h /, nd ν nd n= under clmped boundry conditions. x 7 6 8 6 v=.5 v=. v=.3 v=. v=. v=...3..5.6.7.8.9. h/ Figure.Vritions of the criticl buckling pressure for different vlues of / h, nd ν nd n= under simply supported boundry conditions.

6 Metll. Mter. Eng. Vol 9 () 3 p. 5-63 6 x 8 5 v=.5 v=.3 v=. 3 3 5 6 7 8 9 n Figure.Vritions of the criticl buckling pressure for different vlues of n, nd ν nd h/ =.under clmped boundry conditions. x 8.8.6. v=.5 v=.3 v=...8.6.. 3 5 6 7 8 9 n Figure 3.Vritions of the criticl buckling pressure for different vlues of n nd ν, nd h/ =.under simply supported boundry conditions.

Jm et l. - An exct solution of mechnicl buckling for functionlly grded mteril... 63 Conclusion In the current study, the equilibrium equtions hve been nlyticlly solved, nd closed-form solution is presented for the determintion of the criticl buckling pressure in the FGM bimorph circulr pltes, nd the results chieved re stted s follows: Criticl buckling pressure in FGM bimorph circulr pltes, under simply supported nd clmped boundry conditions, increses by the increse of the thickness to rdius rtio of the plte. Criticl buckling pressure in FGM bimorph circulr pltes, under simply supported nd clmped boundry conditions, decreses continuously by the increse of the volume frction index of the plte. Criticl buckling pressure in FGM bimorph circulr pltes, under simply supported nd clmped boundry conditions, increses by the increse of the Poisson's rtio of the plte. Mechnicl buckling strength in FGM bimorphs is higher thn one-sided FGMs. Mechnicl buckling strength of the FGM bimorphs, under clmped boundry conditions is more thn three times tht of the simply supported boundry conditions. References [] M. Koizumi, Cerm. Trns. Functionlly Grdient Mterils (993) 3-. [] GH. Bryn, In: Proceeding of the London Mthemticl Society, Vol., 89, pp. 5. [3] N. Ymki, J. Appl. Mech. 5 (958) 67-73. [] S.P.Timoshenko, J.M. Gere, Theory of elstic stbility, New York,McGrw- Hill, 96. [5] D.O. Brush, B.O. Almroth, Buckling of brs, pltes nd shells, McGrw-Hill, New York, 975. [6] J.N. Reddy, A.A. Khdeir, AIAA Journl 7 () (989) 88-87. [7] K.K. Rju, G.V. Ro, Comput. Struct. (5) (985) 969-97. [8] K. K. Rju, G. V. Ro, Comput. Struct. 58 (3) (996) 655-658. [9] M. Ozkc, N. Tysi, F. Kolcu, Eng. Struct. 5 (3) 8-9. [] M.M. Njfizdeh, M.R. Eslmi, AIAA Journl (7) (). -5. [] M.M. Njfizdeh, M.R. Eslmi, Int. J. Mech. Sci. () 79-93. [] M.M. Njfizdeh, B. Hedyti, J. Therm. Stresses 7 () 857-8. [3] M.M. Njfizdeh, H.R. Heydri, Eur. J. Mech. A. Solids 3 () 85-. [] M.M. Njfizdeh, H.R. Heydri, Int. J. Mech. Sci. 5 (8) 63-6. [5] M.H. Nei, A. Msoumi nd A. Shmekhi, J. Mech. Eng. Sci, (7). - 7. [6] S.K. Jlli, M.H. Nei, A. Poorsolhjouy, Mter. Design 3 () 755-763. [7] Shi-Rong Li, Jing-Hu Zhng, Yong-Gng Zho, Thin Wlled Struct. 5 (7) 58-536. [8] Chi Shyng-Ho, Chung Yen-Ling, Int. J. Solids Struct. 3 (6) 3657-367. [9] Ugurl,A.C, Stresses in pltes nd shells, WCB/McGrw-Hill edition, in English- nd ed, 998. [] R.Q. Xu, Y. Wng, W.Q. Chen, Comput. Struct. 7 (5) 69-73.