A THEORETICAL ANALYSIS FOR STATIC AND DYNAMIC BEHAVIOR OF FUNCTIONALLY GRADED PLATES

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1 Meril Phic nd Mechnic 14 (1) Received: M 11, 1 A THEORETICAL ANALYSIS FOR STATIC AND DYNAMIC BEHAVIOR OF FUNCTIONALLY GRADED PLATES Thr Hine Doudji 1,*, Adeouhed Touni, Lzreg Hdji 1,, Adelziz Hdj Henni 1,, Add Bedi El Ae 1 Univerié In Khldoun, BP 78 Zrour, 14 Tire, Algérie. Loroire de Mériu & Hdrologie, Univerié de Sidi Bel Ae, BP 89 Cié Ben M hidi Sidi Bel Ae, Algérie. *e-mil: doudjih@hoo.fr Arc. Theoreicl formulion, Nvier oluion of recngulr ple ed on new higher order her deformion model re preened for he ic nd dnmic nli of funcionll grded ple (FGP). Thi heor enforce rcion free oundr condiion ple urfce. Sher correcion fcor re no required ecue correc repreenion of rnvere hering rin i given. Unlike n oher heor, he numer of unknown funcion involved i onl four, gin five in ce of oher her deformion heorie. The mechnicl properie of he ple re umed o vr coninuoul in he hickne direcion imple power-lw diriuion in erm of he volume frcion of he coniuen. Numericl illurion concern fleurl ehvior of FG ple wih Mel Cermic compoiion. Prmeric udie re performed for vring cermic volume frcion, volume frcion profile, pec rio nd lengh o hickne rio. Reul re verified wih ville reul in he lierure. I cn e concluded h he propoed heor i ccure nd imple in olving he ic nd dnmic ehvior of funcionll grded ple. 1. Inroducion The concep of funcionll grded meril (FGM) were he fir inroduced in 1984 group of meril cieni in Jpn, ulrhigh emperure rein meril for ircrf, pce vehicle nd oher engineering pplicion. Funcionll grded meril (FGM) re new compoie meril in which he micro-rucurl deil re pill vried hrough non-uniform diriuion of he reinforcemen phe. Thi i chieved uing reinforcemen wih differen properie, ize nd hpe, well inerchnging he role of reinforcemen nd mri phe in coninuou mnner. The reul i microrucure h produce coninuou or mooh chnge on herml nd mechnicl properie he mcrocopic or coninuum level (Koizumi [1]; Hiri nd Chen []). Now, FGM re developed for generl ue rucurl componen in eremel high emperure environmen. Therefore, i i imporn o ud he wve propgion of funcionll grded meril rucure in erm of non-derucive evluion nd meril chrcerizion. Severl udie hve een performed o nlze he mechnicl or he herml or he hermo-mechnicl repone of FG ple nd hell. A comprehenive review i done Tnigw [3]. Redd [4] h nlzed he ic ehvior of funcionll grded recngulr ple ed on hi hird-order her deformion ple heor. Cheng nd Br [5] hve reled he deflecion of impl uppored FG polgonl ple given he fir-order her 1, Iniue of Prolem of Mechnicl Engineering

2 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 111 deformion heor nd hird-order her deformion heor o h of n equivlen homogeneou Kirchhoff ple. The ic repone of FG ple h een inveiged Zenkour [6] uing generlized her deformion heor. In recen ud, Şimşek [7] h udied he dnmic deflecion nd he ree of n FG impl-uppored em ujeced o moving m uing Euler Bernoulli, Timohenko nd he prolic her deformion em heor. Şimşek [8], Benchour e l. [9] nd Adelziz e l. [1] udied he free virion of FG em hving differen oundr condiion uing he clicl, he fir-order nd differen higher-order her deformion em nd ple heorie. The non-liner dnmic nli of FG em wih pinned pinned uppor due o moving hrmonic lod h een emined Şimşek [11] uing Timohenko em heor. The primr ojecive of hi pper i o preen generl formulion for funcionll grded ple (FGP) uing new higher order her deformion ple heor wih onl four unknown funcion. The preen heor ifie equilirium condiion he op nd oom fce of he ple wihou uing her correcion fcor. The hperolic funcion in erm of hickne coordine i ued in he diplcemen field o ccoun for her deformion. Governing equion re derived from he principle of minimum ol poenil energ. Nvier oluion i ued o oin he cloed-form oluion for impl uppored FG ple. To illure he ccurc of he preen heor, he oined reul re compred wih hreedimenionl elici oluion nd reul of he fir-order nd he oher higher-order heorie. In hi ud, new diplcemen model for n nli of impl uppored FGM ple re propoed. The ple re mde of n ioropic meril wih meril properie vring in he hickne direcion onl. Anlicl oluion for ending deflecion of FGM ple re oined. The governing equion re derived from he principle of minimum ol poenil energ. Numericl emple re preened o illure he ccurc nd efficienc of he preen heor compring he oined reul wih hoe compued uing vriou oher heorie.. Prolem Formulion Conider ple of ol hickne h nd compoed of funcionll grded meril hrough he hickne (Fig. 1). I i umed h he meril i ioropic nd grding i umed o e onl hrough he hickne. The plne i ken o e he undeformed mid plne of he ple wih he z i poiive upwrd from he mid plne. Fig. 1. Geomer of recngulr ple compoed of FGM.

3 11 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi.1. Diplcemen field nd rin. The umed diplcemen field i follow: w w uz (,, ) u(, ) z f( z), w w vz (,, ) v (, ) z f( z), (1) wz (,, ) w(, ) w(, ), where u nd v re he mid-plne diplcemen of he ple in he nd direcion, repecivel; w nd w re he ending nd her componen of rnvere diplcemen, repecivel, while f ( z ) repreen hpe funcion deermining he diriuion of he rnvere her rin nd ree long he hickne nd i given : z f z z h h h ( ) 1 ec nh. () I hould e noed h unlike he fir-order her deformion heor, hi heor doe no require her correcion fcor. The kinemic relion cn e oined follow: zk f( z) k, zk f( z) k, zk f( z) k, (3) z z gz ( ), z gz ( ), z, z where u, k w, k w, v, k u v w, k, k w, (4) w, k w,

4 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple z w, z w, gz ( ) 1 f'( z), nd df ( z) f '( z). dz Coniuive relion. In FGM, meril proper grdion i conidered hrough he hickne nd he epreion given elow repreen he profile for he volume frcion. z 1 Pz ( ) P P P h k, (5) where P denoe generic meril proper like modulu, P nd P denoe he proper of he op nd oom fce of he ple repecivel, nd k i prmeer h dice meril vriion profile hrough he hickne. Here, i i umed h module E nd G vr ccording o he equion (5) nd i umed o e conn. The liner coniuive relion re: Q11 Q1 Q1 Q11 z Q44 z, (6) Q z 55 z Q 66 where Q E ( z), Q1 Q11, 1 11 Ez ( ) Q44 Q55 Q66 1. (7).3. Governing equion. The governing equion of equilirium cn e derived uing he principle of virul diplcemen. The principle of virul work in he preen ce ield: h/ d dz q wd, (8) z z z z h/ where i he op urfce nd q i he pplied rnvere lod. Suiuing Eq. (3) nd (6) ino Eq. (8) nd inegring hrough he hickne of he ple, Eq. (8) cn e rewrien : N N N M k M k M k M k M k M k Sz z Sz z d qwd, (9) where

5 114 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi N, N, N 1 h/ M, M, M,, z dz, (1) h/ M, M, M f( z) h/ z z z z S, S, g( z) dz. (1) h/ The governing equion of equilirium cn e derived from Eq. (9) inegring he diplcemen grdien pr nd eing he coefficien u, v, w, nd w zero eprel. Thu one cn oin he equilirium equion ocied wih he preen her deformion heor, N : N u, N N v :, M M M w : q, (11) M M M S S z z w : q. Uing Eq. (6) in Eq. (1), he re reuln of ndwich ple mde up of hree ler cn e reled o he ol rin N A B B M A D D k M B D H k, S A, (1) where,,, M M, M, M, M M, M, M N N N N,,, k k, k, k, k k, k, k, (13), (13) A A A A A , A 66 B B B B B , B 66 D D D D D D 66, (13c)

6 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 115 B B B B B D D, D D D B H H, H H H D H 66, (13d) z, z, z, z S S S, A A 44 A 55, (13e) where A ij, B ij, ec., re he ple iffne, defined A11 B11 D11 B11 D11 H 11 1 h/ A1 B1 D1 B1 D1 H1 Q11 1, z, z, f( z), z f( z), f ( z) dz, (14) h/ A66 B66 D66 B66 D66 H 66 1 nd A, B, D, B, D, H A11, B11, D11, B11, D11, H11 hn hn1, (14) A44 A55 Q44 g( z) dz. (14c) Suiuing from Eq. (1) ino Eq. (11), we oin he following equion, Adu Ad u A A d vbd w B B d w B B d w B d w, A d v A d v A A d u B d w B B d w B B d w B d w, B d u B B d u B B d v B d v D d w D D d w D d w D d w D D d w D d w q, B d u B B d u B B d v B d v D d w D D d w D d w H d w H H d wh d w A d w A d w q, (15) (15) (15c) (15d) where d ij, d ijl nd d ijlm re he following differenil operor: d ij, i j d ijl 3, i j l d ijlm 4, di i j l m, (,,, 1,). i i j l m (16)

7 116 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi.4. Ec oluion for impl-uppored FGM ple. Recngulr ple re generll clified in ccordnce wih he pe of uppor ued. We re here concerned wih he ec oluion of Eq. (15 d) for impl uppored FG ple. The following oundr condiion re impoed he ide edge: w v w w N M M w u w w N M M /, /, (17) /, /. (17) To olve hi prolem, Nvier umed h he rnvere mechnicl nd emperure lod, q in he form of doule rigonomeric erie q q in( )in( ), (18) where m /, n /, nd q repreen he ineni of he lod he ple cener. Following he Nvier oluion procedure, we ume he following oluion form for u, v, w nd w h ifie he oundr condiion, i u Uco( )in( ). e i v V in( )co( ). e, i w W in( )in( ). e i w W in( )in( ). e (19) where i he nurl frequenc nd U, V, W, nd W re rirr prmeer o e deermined ujeced o he condiion h he oluion in Eq. (19) ifie governing Eq. (15). Equion (19) reduce he governing equion o he following form: For fleurl nli, C P, () And for virion nli, ([ C] [ G]), () where UVW,,, W,C nd G refer o he fleurl iffne nd m mrice nd o he correponding frequenc. C m11 m G, (1) m 33 m34 m34 m44,

8 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 117 in which: , A A, 1 A1 A66 [ B ( B B ) ], [ B ( B B ) ], A66 A, [( B B ) B ], [( B B ) B ], D11 ( D1 D66 ) D, D11 ( D1 D66 ) D, H11 ( H11 H 66 ) H A55 A44, () m11 m I1, m I I, ( ) m I I, ( ) ( ) m I I where: h / 1 ) ( z), I, I 3, I 4, I 5, I 6 (1, z, z, f ( z), zf ( z), f ( z) I dz, h / z z 1 ) h k ( ) ( C m )( m. (3) 3. Numericl reul nd dicuion The ud h een focued on he ic nd dnmic ehvior of funcionll grded ple ed on he preen new higher order her deformion model. Here ome repreenive reul of he Nvier oluion oined for impl uppored recngulr ple re preened Sic nli. For ic nli he ple re ujeced o doule rigonomeric diriued rnvere lod given :

9 118 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi m n q q in( )in( ), (4) where q repreen he ineni of he lod he ple cener. A funcionll grded meril coniing of Aluminum - Alumin i conidered. The following meril properie re ued in compuing he numericl vlue (Bouzz e l. [1]): Mel (Aluminium, Al): E m 7 GP; Poion' rio.3 ; Cermic (Alumin, Al O 3 ): E c 38 GP; Poion' rio.3. Now, funcionll grded meril coniing of luminum nd lumin i conidered. Young' modulu for luminum i 7 GP while for lumin i 38 GP. Noe h, Poion' rio i eleced conn for oh nd equl o.3. The vriou non-dimenionl prmeer ued re h Ec 1 h Ec h w w(, ), u (,, ) 4 u 4, q q 4 u 3 1 h Ec h h h u (,, ) 4, (,, ), (5) q 6 q h h h h (,, ), (,, ), q 3 q 3 h h h z z(,, ), z z(,, ). q 6 q I i cler h he deflecion incree he ide-o-hickne rio decree. The me reul were oined in mo lierure. In ddiion, he correlion eween he preen new higher order her deformion heor nd differen higher-order nd fir-order her deformion heorie i elihed he uhor in hi recen pper. I i found h hi heor predic he deflecion nd ree more ccurel when compred o he fir nd hird-order heorie. For he ke of compleene, reul of he preen heor re compred wih hoe oined uing new Nvier-pe hree-dimenionll ec oluion for mll deflecion in ending of liner elic ioropic homogeneou recngulr ple. The cener deflecion w nd he diriuion cro he ple hickne of in-plne longiudinl re nd longiudinl ngenil re re compred wih he reul of he 3-D [13] oluion nd re hown in Tle 1 nd Tle. The preen oluion i relized for qudric ple, wih he following fied d: = 1, = 1, E m =E c =E=1, q = 1, ν=.3 nd hree vlue for he ple hickne: h =.1, h =.3 nd h =.1. I i o e noed h he preen reul compre ver well wih he 3-D oluion [13]. All deflecion gin compre well wih he 3-D oluion, nd how good convergence wih he verge 3-D oluion. Tle 1. Cener deflecion of ioropic homogenou ple (k=, E m =E c =E=1 nd /=1). h/ CPT [14] 3D [13], z= SSDPT [6] Preen heor NHPSDT Redd [4]

10 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 119 Tle. Diriuion of ree cro he hickne of ioropic homogenou ple (E m =E c =E=1; / =1 nd k=). (,,- z) (,,- z) z 3D SSDPT NHPSDT Redd 3D SSDPT NHPSDT Redd [13] [6] preen [4] [13] [6] preen [4] h/ In Tle 3, he effec of volume frcion eponen on he dimenionle ree nd diplcemen of FGM qure ple (/h = 1) i given. Thi le how comprion eween reul for ple ujeced o uniform or inuoidl diriued lod, repecivel. A i i well known, he uniform lod diriuion lw over predic he diplcemen nd ree mgniude. A he ple ecome more nd more mellic, he difference incree for deflecion w nd in-plne longiudinl re while i decree for in-plne norml re. I i imporn o oerve h he ree for full cermic ple re he me h for full mel ple. Thi i ecue he ple for hee wo ce i full homogeneou nd he ree do no depend on he modulu of elici. Reul in Tle 4 hould erve enchmrk reul for fuure comprion. Tle 4 nd 5 compre he deflecion nd ree of differen pe of he FGM qure ple (/=1, k=) nd FGM recngulr ple (=3, k=).the deflecion decree he pec rio / incree nd hi irrepecive of he pe of he FGM ple. All heorie (SSDPT, PSDPT nd NHPSDT) give he me il re nd for full cermic ple (k =). In generl, he il re incree wih he volume frcion eponen k. The rnvere her re for FGM ple ujeced o diriued lod. The reul how h he rnvere her ree m e indiinguihle. A he volume frcion eponen incree for FGM ple, he her re will incree nd he full cermic ple give he mlle her ree.

11 1 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi Tle 3. Effec of volume frcion eponen nd loding on he dimenionle ree nd diplcemen of FGM qure ple (/h=1). k Theor w z z NHPSDT(preen) cermic SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) mel SSDPT [6] Redd [4] Tle 4. Comprion of normlized diplcemen nd ree of FGM qure ple (/=1) nd k=. /h Theor w z z NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4]

12 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 11 Tle 5. Comprion of normlized diplcemen nd ree of FGM recngulr ple (=3) nd k=. /h Theor w z z NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] NHPSDT(preen) SSDPT [6] Redd [4] Figure nd 3 how he vriion of he cener deflecion wih he pec nd ide-ohickne rio, repecivel. The deflecion i mimum for he mellic ple nd minimum for he cermic ple. The difference incree he pec rio incree while i m e unchnged wih he incree of ide-o-hickne rio. One of he min inference from he nli i h he repone of FGM ple i inermedie o h of he cermic nd mel homogeneou ple (ee lo Tle 4). I i o e noed h, in he ce of herml or comined lod nd under cerin condiion, he ove repone i no inermedie. w 8,5 8, 7,5 7, 6,5 6, 5,5 5, 4,5 4, 3,5 3,,5, 1,5 1,,5, -,5 cermic k=1 k= k=5 k=1 mel /h=1,5 1, 1,5,,5 3, / Fig.. Dimenionle cener deflecion (w) funcion of he pec rio (/) of FGM ple.

13 1 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi w 5,5 5, 4,5 4, 3,5 3,,5, 1,5 1,, /h cermic k=1 k= k=5 k=1 mel /= 1 Fig. 3. Dimenionle cener deflecion (w) funcion of he ide-o-hickne rio (/h) of FGM qure ple. Figure 4 nd 5 depic he hrough-he-hickne diriuion of he her ree z nd z ; he in plne longiudinl nd norml ree nd, nd he longiudinl ngenil re in he FGM ple under he uniform lod. The volume frcion eponen of he FGM ple i ken k = in hee figure. Diincion eween he curve in Fig. 5 nd 6 i oviou. A rin grdien incree, he in homogeneiie pl greer role in re diriuion clculion. The hrough-he-hickne diriuion of he her ree z nd z re no prolic nd he ree incree he pec rio decree. I i o e noed h he mimum vlue occur z., no he ple cener in he homogeneou ce /=.5 /=1 /= /h=1 k= ,4 -,,,,4 z/h Fig. 4. Vriion of longiudinl ngenil re ( ) hrough-he hickne of FGM ple for differen vlue of he pec rio.

14 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 13,6,5,4 z,3,,1, -,4 -,,,,4 z/h /=.5 /=1 /= /h=1 k= Fig. 5. Vriion of rnverl her re ( z ) hrough-he hickne of FGM ple for differen vlue of he pec rio.,8,6 /=.5 /=1 /= /h=1 k= z,4,, -,4 -,,,,4 z/h Fig. 6. Vriion of rnverl her re ( z ) hrough-he hickne of FGM ple for differen vlue of he pec rio. A ehiied in Fig. 7 nd 8, he in-plne longiudinl nd norml ree, nd, re compreive hroughou he ple up o z.155 nd hen he ecome enile. The mimum compreive ree occur poin on he oom urfce nd he mimum enile ree occur, of coure, poin on he op urfce of he FGM ple. However, he enile nd compreive vlue of he longiudinl ngenil re, (cf. Fig. 4), re mimum poin on he oom nd op urfce of he FGM ple, repecivel. I i cler h he minimum vlue of zero for ll in-plne ree ; nd occur z.153 nd hi irrepecive of he pec nd ide-o-hickne rio.

15 14 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi /h=5 /h=1 /h= /=1 k= ,4 -,,,,4 z/h Fig. 7. Vriion of in-plne longiudinl re ( ) hrough-he hickne of FGM ple for differen vlue of he ide -o-hickne rio /=1 /= /=3 /h=1 k= ,4 -,,,,4 z/h Fig. 8. Vriion of in-plne norml re ( ) hrough-he hickne of FGM ple for differen vlue of he pec rio. Finll, he ec mimum deflecion of impl uppored FGM qure ple re compred in Fig. 9 for vriou rio of module, Em/Ec (for given hickne, /h = 1). Thi men h he deflecion re compued for ple wih differen cermic mel miure. I i cler h he deflecion decree moohl he volume frcion eponen decree nd he rio of mel-o-cermic module incree.

16 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple 15 4,5 4, w 3,5 3,,5, 1,5 k=1 k= k=3 k=5 k=1 /h=1 /=1 1,,5,1,,3,4,5 Em/Ec Fig. 9. The effec of niorop on he dimenionle mimum deflecion (w) of FGM ple for differen vlue of k. 3.. Dnmic nli. The ccurc of he preen heor i lo inveiged hrough free virion nli of FG ple. The meril properie ued in he preen ud re: Mel (Aluminium, Al): Em 7 Gp, Poion' rio.3, m = 7 kg/m 3 ; Cermic (Alumin, Al O 3 ): Ec 38 Gp, Poion' rio.3, c = 57 kg/m 3. Tle 6. Comprion of fir hree nurl frequencie of Al/Al O 3 FG qure ple for vriou /h rio *). Mode N / h Source 1 % error % error 3 % error 5 NHPSDT(preen) UNSDT [15] Redd [4] HSDT [16] 3D [17] NHPSDT(preen) UNSDT [15] Redd [4] HSDT [16] 3D [17] *) NHPSDT(preen) UNSDT [15] Redd [4] HSDT [16] 3D [17] h E m m , n=

17 16 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi Severl prmeer re vried nd heir dnmic ehvior i udied. The fir hree nurl frequencie for he fundmenl virion mode of m=n=1 of qure Al / Al O 3 FG ple re compred wih he correponding reul of 3D nli Vel e l [17] in Tle 6. The Tle 6 lo preen he reul oined Mung' heor [15] nd Redd' HSDT [4]. Tle 7 preen he effec of power lw inde on dimenionle frequenc. From hee le i i eviden h he preen heor predic reul more ccurel hn he oher model when compred wih 3D elici oluion. Tle 7. Effec of power lw inde on fundmenl frequencie of Al/Al O 3 FG qure ple *). Power lw inde n n NHPSDT(preen) UNSDT [15] Redd [4] HSDT [16] 3D [17] % % % % Source n 1 error error 3 error 5 error *) NHPSDT(preen) UNSDT [15] Redd [4] HSDT [16] 3D [17] h E m m , / h , Fréquence fondmenle 1,,8,6,4, n= n=.5 n=1 n= n=5 n=1, /h Fig. 1. Dimenionle Frequenc m h funcion of ide o E m hickne rio / h for vriou power lw inde of FGM qure ple.

18 A heoreicl nli for ic nd dnmic ehvior of funcionll grded ple Effec of ide o hickne rio (/h), pec rio (/) nd modulu rio (E m /E c ) on fundmenl frequencie re how in Fig. 1, 11 nd ,,8 Fréquence fondmenle,6,4, n= n=1 n= n=5 n=1,,,5 1, 1,5,,5 3, / Fig. 11. Dimenionle Frequenc m h funcion of pec rio / E m for vriou power lw inde of FGM qure ple / h 5.,5 Fréquence fondmenle,4,3 n= n=1 n= n=5 n=1,,1,,3,4,5 E m /E c Fig. 1. Dimenionle Frequenc m h funcion of modulu rio E m / Ec E m for vriou power lw inde of FGM qure ple / h 5.

19 18 T. Hine Doudji, A. Touni, L. Hdji, H.H. Adelziz, E.A. Add Bedi 5. Concluion Thi pper preen, new higher order her deformion model i propoed o nlze he ic nd dnmic ehvior of funcionll grded ple. Nvier oluion for fleure nd free virion nli of FG ple re preened. The ree nd diplcemen re compued for ple wih Mel Cermic miure nd i i een h he repone i inermedie o h of mel nd cermic. Hence he grdien in meril properie pl vil role in deermining he repone of FGM ple i m e concluded h he preen model provide eer eime for he deflecion nd ree hn h of generlized her Deformion Theor [15] nd ver cloe o he oluion oined wih h of Redd higher order model [4]. All comprion udie demonred h he deflecion nd ree oined uing he preen new higher order her deformion heorie (wih four unknown) nd oher higher her deformion heorie (wih five unknown) re lmo idenicl. Thi model i lo ued for predicing fundmenl frequencie of FG ple. The influence pled ple pec rio, ide o hickne rio nd modulu rio re udied. The preen model provide reul in ecellen greemen wih he ville reul nd give eer eime hn he oher cceped model [6-15] when compred wih 3D elici oluion. The eenion of he preen heor i lo enviged for generl oundr condiion nd ple of more generl hpe. In concluion, i cn e id h he propoed heor NHPSDT i ccure nd imple in olving he ic ehvior of FGM ple. Reference [1] M. Koizumi // Cermic Trncion, Funcionll Grdien Meril 34 (1993) 3. [] T. Hiri, L. Chen // Meril Science Forum (1999) 59. [3] Y. Tnigw // Appl. Mh. Mech. 48 (1995) 87. [4] J.N. Redd // Inernnionl Journl of Numericl Mehod in Engineering 68 () 643. [5] Z.Q. Cheng, R.C. Br // Arch. Mech. 5 () 143. [6] A.M. Zenkour // Applied Mhemicl Modelling 3 (6) 67. [7] M. Şimşek // Compo. Sruc. 9 (1) 94. [8] M. Şimşek // Nucler Engineering nd Deign 4 (1) 697. [9] A. Benchour, T. Hine Doudji, H. Ai Amne, A. Touni, S.A. Mefh // Compoie B: Engineering 4 (11) [1] H.H. Adelziz, H. Ai Amne, I. Mech, L. Boumi, A. Touni, E.A. Add Bedi // Chinee journl of eronuic 4 (11) 434. [11] M. Şimşek // Compo. Sruc. 9 (1) 53. [1] M. Bouzz, A. Touni, E.A. Add Bedi, M. Meguenni // Advnced Srucure Meril B 15 (11) 669. [13] H. Werner // Commun. Numer. Mehod Eng. 15 (1999) 95. [14] S.P. Timohenko, S. Woinowk-Krieger, Theor of Ple nd Shell (McGrw-Hill, New York, 1959). [15] H. Mung // Compoie Srucure 84 (8) 13. [16] K.N. Trung, Krm S, Gu Bonne // Compoie Srucure 83 (8) 5. [17] S.S. Vel, R.C. Br // Journl of Sound nd Virion 7(3) (4) 73.