Deformation analysis of functionally graded beams by the direct approach

Transcription

1 Deformton nlyss of functonlly grded bems by te drect pproc Mrce Bîrsn, Holm Altenbc, Tomsz Sdowsk, Vctor Eremeyev, Dnel Petrs To cte ts verson: Mrce Bîrsn, Holm Altenbc, Tomsz Sdowsk, Vctor Eremeyev, Dnel Petrs. Deformton nlyss of functonlly grded bems by te drect pproc. Compostes Prt B: Engneerng, Elsever,, 43 (3), pp <l-8637> HAL Id: l-8637 ttps://l.rcves-ouvertes.fr/l-8637 Submtted on 8 My 3 HAL s mult-dscplnry open ccess rcve for te depost nd dssemnton of scentfc reserc documents, weter tey re publsed or not. Te documents my come from tecng nd reserc nsttutons n Frnce or brod, or from publc or prvte reserc centers. L rcve ouverte plurdscplnre HAL, est destnée u dépôt et à l dffuson de documents scentfques de nveu recerce, publés ou non, émnnt des étblssements d ensegnement et de recerce frnçs ou étrngers, des lbortores publcs ou prvés.

2

3 tree dmensonl problems to muc smpler one dmensonl problems. To dentfy tese mecncl propertes for generl non omogeneous rods, we compre te solutons of extenson, bendng nd torson problems n te drect pproc wt te cor respondng results from te tree dmensonl teory [3,4]. Tus, we obtn te effectve bendng stffness, extensonl stff ness, torsonl rgdty nd oter couplng coeffcents. Also, to determne te effectve ser stffness, we compre te ser vbrtons of rectngulr bems n te two pproces (drect nd tree dmensonl). Tese results re presented n Sectons 3 nd 4 n te cse of sotropc non omogeneous bems wt rb trry cross secton spe. In Sectons 5 nd 6 we consder bems composed of two dfferent non omogeneous mterls, eter ortotropc or sotropc, nd we derve generl formuls for te effectve stffness coeffcents. Tese formuls re expressed n terms of te solutons to some uxlry plne strn boundry v lue problems defned on te cross secton domn. In generl, te solutons of tese uxlry boundry vlue problems re not esy to fnd n closed form, but we present n Secton 7 some specl cses for te geometry/mterl prmeters n wc we cn ob tn te results n closed form. In Secton 8 we employ our nlyt cl modelng to nlyze te deformton of FGM bems mde of metl foms. Te mss densty dstrbuton of te cellulr mterl n te bem s gven by power lw functon of te cross secton coordnte, wle te Young s modulus s expressed by te Gb son Asby formul for closed cell lumnum foms [5]. Fnlly, we verfy our nlytcl modelng by comprng te results ob tned n te drect pproc of FGM bems wt te numercl soluton of vrous bendng problems obtned by fnte element nlyss usng ABAQUS. Te close greement between te nlytcl nd numercl solutons ndctes tt te drect pproc to rods, togeter wt te formuls for te effectve stffness coeffcents derved n ts pper, represent n effcent tool for te nlyss of te deform ton of functonlly grded rods.. Equtons for curved rods n te drect pproc.. Mterl ndependent equtons In ts expostory secton we present te bsc non lner equ tons for bems nd rods, obtned by te drect pproc n [,]. In ts pproc, te tn body s modeled s deformble curve endowed wt trd of rgdly rottng vectors ttced to ec pont. We denote by C te deformble curve n ts reference (n tl) confgurton nd by s te mterl coordnte long C, wc s lso te rclengt prmeter. Te poston of te drec ted curve s descrbed by te poston vector r(s) nd te t tced vectors d (s), =,, 3, lso clled drectors. Te unt vectors d (s) re mutully ortogonl nd tey re cosen suc tt d 3 concdes wt te unt tngent t r, nd d, d belong to te norml plne to te curve C. Te rottons of te t tced trd of drectors descrbe te rottons of te rod s cross sectons durng deformton. Let C be te deformed confgurton of te rod t tme t, wc s crcterzed by te vector felds (see Fg. ) R Rðs; tþ; D D ðs; tþ; ; ; 3; ðþ were R s te poston vector nd D re te drectors fter defor mton. We ve D D j = d j (te Kronecker symbol), but D 3 s not tngent to te curve C,.e. te ntl cross sectons re no longer norml to te mddle curve fter deformton. In ts model t s s sumed tt te cross sectons of te bem do not deform, but tey only rotte wt respect to te mddle curve. Let P(s,t)=D k (s,t) d k (s) be te rotton tensor. We employ trougout te Ensten s summton conventon nd te drect tensor notton n te sense of [6,7]. Greek ndces rnge over te set {,}, wle Ltn ndces tke te vlues {,,3}. Denote by superposed dot te mterl tme dervtve nd by ðþ d. ds Te velocty vector s Vðs; tþ Rðs; _ tþ, nd te ngulr velocty vector x(s,t) s determned by te relton P _ x P,.e. x s te xl vector of te ntsymmetrc tensor P _ P T. Te equtons of moton for rods re N ðs; tþþq F q ðv þ H xþ ; M ðs; tþþr Nðs; tþþq L q ½V H x þðv H þ H xþ Š; ðþ were N s te force vector, M s te moment vector, F nd L re te externl body force nd moment per unt mss, q s te mss densty per unt lengt of C, wle te second order tensors H (s,t) nd H (s,t) re te nert tensors per unt mss. Accordng to [], te tensors H re expressed by H ðs; tþ Pðs; tþ H ðsþ P T ðs; tþ, were H ðsþ re te nert tensors n te reference confg urton, wc re gven by q H ð Þq ~l dx dx ; R q H ½ð Þ Š q ~l dx dx : ð3þ R Fg.. Te deformed confgurton of te rod. Here q s te mss densty n te tree dmensonl rod, s te second order unt tensor, R s te domn of te cross secton n te norml plne, = x d + x d nd l þ n, were R c s te r Rc dus of curvture of te curve C nd n s te prncpl norml unt vector. In te cse of strgt rods, we ve clerly l. In te generl cse of curved rods, snce te dmeter of te rod s muc smller tn R c, we ve jnj, nd tus l > nd l s vlue Rc close to. We note tt H s ntsymmetrc, H s symmetrc. Te felds F nd L ccount lso for te lods ctng on te lterl surfce of tree dmensonl rods. Te vectors of deformton re defned s follows: te vector of extenson ser E R P t, nd te vector of bendng tor son U s gven by P = U P,.e. U s te xl vector of te ntsymmetrc tensor P P T. We lso ntroduce te energetc vectors of deformton E nd U defned by E P T E nd U = P T U [,]. For generl elstc bems, te consttutve ssumptons mply tt te nternl energy densty U s functon of te followng rguments fe ; U g. In our work we consder tt te nternl en ergy s qudrtc functon of ts rguments. Tus, we ve te fol lowng consttutve equtons

4 q U U þ N E þ M U þ E A E þ E B U þ U C U ; P T ; P T ; ð4þ were U s sclr, N, M re vectors, nd A, B, C re second order tensors, defned on te reference confgurton. Te structure nd sgnfcnce of te elstcty tensors A, B nd C ve been dscussed n [,]... Structure of consttutve tensors In our study we re nterested to determne te structure of consttutve tensors for bems nd rods mde of functonlly grded mterls. We ssume tt te mterl propertes do not vry long te lengt of te bem, but only cross te cross sec tons. In oter words, tey depend on (x,x ), but not on s. In ec cross secton we cose te drectors d nd d long te prncpl xes of nert. Tus, we ve q x q x ; q x x ; ð5þ R were we denote by f R f dx dx for ny feld f. Te structure of te consttutve tensors cn be determned usng te generlzed teory of tensor symmetry [,8]. In te generl cse of curved rods, te consttutve tensors depend on te geometry of te rod troug te Drboux vector s of te curve C, nd troug te ngle of nturl twstng. Te expressons of A, B nd C for omogeneous curved rods re pre sented n [,]. If we restrct for smplcty to strgt rods wt out nturl twstng, ten we ve s = nd r =. Imposng tt te ortogonl tensor t t belongs to te symmetry group of ny consttutve tensor, we fnd tt for non omogeneous rods A, B nd C ve te followng structures A A d d þ A d d þ A 3 t t þ A ðd d þ d d Þ; B B 3 d t þ B 3 t d þ B 3 d t þ B 3 t d ; C C d d þ C d d þ C 3 t t þ C ðd d þ d d Þ: Remrk. Te structure of te consttutve tensors cn be derved lso n te more generl cse of rods wt nturl twstng. In ts cse, te consttutve coeffcents depend lso on te ngle of nturl twst r(s), nd te expressons correspondng to (6) ve to be supplemented wt ddtonl terms. Our m s to determne te consttutve coeffcents A, C, A, C, B 3 nd B 3 for functonlly grded bems nd rods, n terms of te tree dmensonl elstc propertes. Tese coeffcents de scrbe te effectve stffness propertes of tn bems nd rods. Snce te consttutve coeffcents do not depend on te deform ton, ter expressons cn be derved by comprson of exct solu tons for drected curves wt te results from tree dmensonl elstcty n te frmework of lner teory. In order to relze suc comprson of exct solutons, we re strct ourselves to te lner teory. Let us note tt n te te ory of bems nd rods tere s long trdton of usng lner elstcty to derve one dmensonl bems nd rods teores ncludng some non lner effects. Ts trdton s bsed on te fct tt one cn clculte stffness prmeters of bem or rod on te bse of lner elstcty nd ten use te stffness modul n geometrclly non lner teory of bems nd rods. In deed, te coeffcents of te strn energy densty consdered s te qudrtc functon of strn mesures concde for lner nd for geometrclly non lner teores of bems nd rods. Ts fct s used for exmple n [3,,] were dfferent pproces ð6þ re ppled. In te pper te geometrclly non lner pproc wt pysclly lner consttutve reltons s consdered. Suc teory cn be ppled for stndrd mterl. Excepton s, for exmple, rubber lke mterl for wc te qudrtc form of te strn energy densty s not vld n te cse of lrge deformtons, n generl. Some recent ttempts to pply non ln er elstcty to constructon of one dmensonl teores of bems nd rods re gven for exmple n [9 37]. 3. Lnerzed equtons for drected curves 3.. Geometrcl lnerzton In te lner settng, te dsplcement u(s, t) = R(s, t) r(s) s s sumed to be nfntesml. Also, te rotton tensor cn be repre sented s P = + w, were w(s,t) s te vector of smll rottons. Te feld w, wc s ssumed to be nfntesml, sts fes te reltons w _ x nd w = U. Te vectors of deformton re denoted n te lner cse by e nd j, nd tey re gven by e u þ t w E E ; j w U U : ð7þ Te consttutve Eq. (4) reduce to q Uðe; jþ e A e þ e B j þ j C j; UÞ ; UÞ : ð8þ Te equtons of moton () smplfy to te forms N þ q F q ð u þ H wþ; M þ t N þ q L u H þ H w : ð9þ q To te governng feld Eqs. (7) (9) we djon boundry condtons nd ntl condtons. Let l be te lengt of te rod, so tt te rc lengt prmeter rnge over te ntervl s [, l]. We denote te two endponts by s nd s l for convenence, nd we consder boundry condtons of te type uðs c ; tþ u ðcþ ðtþ or Nðs c ; tþ N ðcþ ðtþ; for c ; ; wðs c ; tþ w ðcþ ðtþ or Mðs c ; tþ M ðcþ ðtþ; for c ; : Te ntl condtons re uðs; Þ u ðsþ; _uðs; Þ v ðsþ; wðs; Þ w ðsþ; _wðs; Þ x ðsþ; were te functons u, w, v, x, s well s u (c), w (c), N (c), M (c) re prescrbed. Te correspondence between te dsplcement nd rotton felds {u,w} for drected curves nd te dsplcement vector u for tree dmensonl rods s estblsed by te followng reltons [] q u þ H w q u ~l; q u H þ H w q ð u Þ~l: ðþ Also, te reltons between te felds {N,M} nd te Cucy stress tensor T from tree dmensonl teory re gven by N t T ; M ðt T Þ: ðþ Tese reltons re useful wen comprng te solutons of some problems n te two dfferent pproces. 3.. Strgt rods In wt follows we restrct our ttenton to strgt rods wtout nturl twstng. In ts cse, we cn cose te Crtesn

5 coordnte frme Ox x x 3 suc tt te curve C s stuted on te xs Ox 3, between te lmts x 3 =,l, nd we ve t d 3 e 3 ; n d e ; d e ; s x 3 ; l ; H ; q H I e e þ I e e þði þ I Þe 3 e 3 ; I q x ; I q x ; were e denote te unt vectors long Ox. To dstngus between te extensonl, torsonl, bendng, nd ser deformton, we decompose te vectors u; w; e; j; N; M; F nd L by te tngent drecton t nd te norml plne (e, e ): u ut þ w; w wt þ t #; e u t þc; j w t þ t # ; N Ft þ Q; M Ht þ t L; F F t t þ F n ; L L t t þ L n ; ðþ wt c = w #. Te vectors w; #; c; Q; L; F n nd L n re ortog onl to t. Here c s te trnsverse ser vector, u s te longtudnl dsplcement, w = w e s te vector of trnsversl dsplcement, w s te torson, # # e s te vector of bendng deformton, F s te longtudnl force, Q = Q e s te vector of trnsversl force, H s te torson moment nd L = L e s te vector of bendng mo ment. Usng te decompostons () nd te structure of consttu tve tensors (6), we remrk tt te consttutve Eq. (8) cn be wrtten n component form s Q A w # þa w # þb3 w ; Q A w # þa w # þb3 w ; F A 3 u B 3 # þb 3# ; H C 3w þb 3 w # þb3 w # ; L C # C # þb 3u ; L C # þc # B 3 u : ð3þ Te consttutve coeffcents re constnts, snce we consder rods mde of non omogeneous mterls wc propertes do not de pend on te xl coordnte s. We observe tt te generl boundry ntl vlue problem for non omogeneous rods does not decouple nto sub problems. Note tt n te cse of omogeneous mterls te generl prob lem decouples nto te extenson torson problem nd te bend ng ser problem, see []. Te reltons of dentfcton () nd (), wrtten for strgt rods, become q w q u ; q u q u 3 ; q q ; q x u 3 # q x u 3 ; # ; w q x u x u ; I I I þ I Q t 3 ; F t 33 ; L x t 33 ; H x t 3 x t 3 ; ð4þ were u nd t j re te components of u nd T, respectvely. Te reltons (4) wll be used to dentfy te correspondng felds n te two pproces (drected curves nd tree dmensonl) Extenson, bendng nd torson n te drect pproc Let us fnd te exct soluton of te problem of extenson, bend ng nd torson of drected curves. We menton tt ts soluton s exct up to rgd body dsplcement nd rotton felds. In te ln er teory te rgd body felds ve te generl form u þ b r; w b, were nd b re rbtrry constnt vectors. Let us determne te equlbrum of strgt rod subjected to n xl force F, torson moment H, nd bendng moments L p pled to bot ends. Te body forces nd moments re bsent. In our cse, te equlbrum equtons correspondng to (9) re Q ðsþ ; F ðsþ ; L ðsþþq ðsþ ; H ðsþ ; s ð; lþ; wle te boundry condtons on te ends of te rods re ð5þ Q ðþ Q ðlþ ; FðÞ FðlÞ F; L ðþ L ðlþ L ; HðÞ HðlÞ H: ð6þ Usng te consttutve Eq. (3) we obtn system of ordnry df ferentl equtons wc yelds te soluton w ðsþ s þ b s; uðsþ 3 s; # ðsþ s; wðsþ b 3 s; ð7þ were te constnts nd b re determned by te lgebrc lner systems C C B 3 L A A B 3 b C C B L 5; 4A A B 3 54b ð8þ B 3 B 3 A 3 3 F B 3 B 3 C 3 H Te force nd moment vector felds correspondng to ts soluton re gven by N Fe 3 ; M L e þ L e þ He 3 : ð9þ Ts soluton wll be used lter for comprson wt tree dmen sonl solutons, n order to dentfy te effectve stffness coeff cents for non omogeneous tn rods. 4. Determnton of consttutve coeffcents for sotropc rods 4.. Deformton of non omogeneous tree dmensonl rods Let us consder tree dmensonl rod wc occupes te do mn B fðx ; x ; x 3 Þjðx ; x ÞR; x 3 ½; lšg. Te cross secton R s rbtrry nd te symmetry reltons (5) re stsfed. Te body B s mde of n sotropc nd non omogenous mterl suc tt te mss densty q nd te Lmé modul k, l re ndependent of te xl coordnte,.e. we ve q q ðx ; x Þ; k kðx ; x Þ; l lðx ; x Þ: We consder te deformton of suc cylnders under te cton of termnl forces nd moments. We ssume tt te body B s n equlbrum, n te bsence of externl body lods nd trctons on te lterl surfces. On te two ends of te cylnder ct resultnt xl force nd resultnt moment. We consder te sme problem s n Secton 3.4, but for multed n te tree dmensonl settng. In vew of te reltons (4) 7 we tke te boundry condtons t 3 ; t 33 F; x t 33 L ; x t 3 x t 3 H for x 3 ; l: ðþ Te soluton of ts tree dmensonl problem for non omoge neous rods s presented n [3] Secton 3.3 nd Secton 3.4, were t s expressed n terms of te solutons to some uxlry plne strn problems. For te ske of completeness nd for lter reference we present tese tree dmensonl results. We denote by u ðþ ; u ðþ nd u ð3þ te solutons of te 3 plne strn problems D ðþ ; D ðþ nd D ð3þ respectvely, defned on te do mn R by D ðcþ : D ð3þ : t ðcþ b;b þðkx cþ ; n R; t ðcþ b n b kx c n t ð3þ b;b þ k ; n R; t ð3þ b n b kn b 3 ðþ were obvously t ðkþ b ku ðkþ q;qd b þ l u ðkþ ;b þ uðkþ b;,(k =,, 3 nd, b =, ) nd n n e s te outwrd unt norml Let u(x,x ) be te soluton of te Neumnn type boundry vlue ðlu ; Þ ; l ; x l ; x n R; n x n ðþ

6 Te exstence of solutons to te bove boundry vlue problems () nd () s proved n [3], Sectons 3. nd 3.4. Ten, te solu ton of our tree dmensonl problem for te lods () s gven by u ^ x 3 sx x 3 þ X3 k ^ k u ðkþ ðx ; x Þ; u ^ x 3 þ sx x 3 þ X3 ^ k u ðkþ ðx ; x Þ; k u 3 ð^ x þ ^ x þ ^ 3 Þx 3 þ suðx ; x Þ; were te constnts s nd ^ re gven by te reltons ð3þ H s nd D j^ j L ; D 3j^ j F: ð4þ D Here te torsonl rgdty D s expressed by D l½x ðx þ u ; Þþx ðx u ; ÞŠ; ð5þ wle te coeffcents D j re gven by D b D 3 ðk þ lþx x b þ kx u ðbþ c;c ; D 33 ðk þ lþþku ð3þ c;c ; ðk þ lþx þ kx u ð3þ c;c ; D 3 ðk þ lþx þ ku ðþ c;c : ð6þ In [3], Secton 3.3, t s sown tt D j = D j nd det (D j ) 33, so tt we cn determne te constnts ^ from te system (4),3. Remrk. If we ntroduce te stress functon v(x,x ) by te reltons v ; lðu ; þ x Þ; v ; lðu ; x Þ; ten te torsonl rgdty s gven by D vðx ; x Þ: ð7þ Te stress functon v cn be obtned from te boundry vlue problem l v ; n R; v ; provded tt te domn R s smply connected. In te cse of mul tply connected cross sectons R, te torson problem s been studed n, e.g., [38,39]. Let us compre now te tree dmensonl soluton (3) wt te soluton (7) obtned n te drect pproc to rods, tkng nto ccount te reltons (5) nd (4). By comprson, t follows tt we ve to dentfy te constnts C 3 D ; A 3 D 33 ; C D ; C D ; C D ; B 3 D 3 ; B 3 D 3 ; B 3 B 3 : ð8þ Tus, from (6) (8) we obtn te followng expressons for te consttutve coeffcents C 3 vðx ;x Þ; A 3 ðk þ lþþku ð3þ c;c ; C C B 3 ðk þ lþx þ kx u ðþ c;c ; C ðk þ lþx þ kx u ð3þ c;c ; B 3 ðk þ lþx þ kx u ðþ c;c ; ðk þ lþx x þ kx u ðþ c;c ; B 3 ; ðk þ lþx þ kx u ð3þ c;c : ð9þ By vrtue of te dentfctons (4) nd (9) we cn verfy tt te felds u, w, w, N nd M clculted for te solutons n te two df ferent pproces concde. Remrk. For te felds # correspondng to te tree dmensonl soluton (3) we obtn from (4), nd (5) te expressons # ^ x 3 s q x u ; ; ; not summed: q x Comprng ts relton wt te feld # from te soluton (7) 3 for drected curves, we see tt we ve to pproxmte q x u ; ; ; not summed; q x were u(x,x ) s te torson functon gven by (). For exmple, n n o te cse wen R s n ellptcl domn R ðx ; x Þj x þ x < b b nd l s constnt, ten we ve uðx ; x Þ x þb x so tt te bove pproxmton s justfed. We remrk tt, due to te ser bendng couplng n te cse of sttc problems, te effectve ser stffness coeffcents A, A nd A cnnot be obtned by nlyzng sttc ser problems nd usng te sme procedure s bove. (For tn bems, te coef fcents A, A, A wll not enter n te ledng order terms of te solutons.) For ts reson, we determne te effectve ser stff ness coeffcents by solvng free vbrton problem. Te neces sty of consderng free vbrton problems for te determnton of effectve ser stffness propertes s lso dscussed n detls n [] Secton 6, nd n [, pp ]. 4.. Ser vbrtons of rectngulr rods Consder tree dmensonl rod wc occupes te domn R ðx ; x ; x 3 Þjx ; ; b x ; b ; x3 ð; lþ, mde of non omogeneous sotropc mterl. Te mterl prmeters k, l nd q re gven functons of (x,x ). Assume tt te mss den sty q s symmetrcl dstrbuton cross te tckness: q (x,x )=q ( x,x ). Te body lods re zero, te lterl surfces x nd x b re trcton free, nd te end boundry condtons re gven by u u nd t 33 for x 3 ; l: ð3þ To determne te ser vbrtons of ts rod, we serc for solu tons u of te form u W cosðxtþ sn p x e 3 ; ð3þ were W s constnt nd x s te lowest nturl frequency. We observe tt ll te boundry condtons re stsfed by te feld (3), nd te equtons of moton reduce to t 3; q u 3, wc by ntegrton wt respect to x gves t 3 Wx cosðxtþ x = q ðx ; x Þ sn p x dx : Usng te consttutve equton for t 3 we get x x lðx ; x Þ p cos p x = q ðx ; x Þ sn p x dx ð3þ We pply te men vlue teorem for te ntegrl n (3) nd we deduce tt tere exsts pont ðx ; x Þ ; x suc tt x q ðx ; x Þ sn p x x dx q ð; x Þ sn p x dx : ð33þ = Substtutng (33) nto (3) nd ntegrtng over R we obtn p x lðx ; x Þ q ð; x Þ : = ð34þ Let us tret te sme problem usng te pproc of drected curves. We consder strgt rod long te Ox 3 xs for wc te rclengt prmeter s (,l). Te externl body lods F nd L re zero. Accordng to (4) nd (3) we ve te followng boundry condtons on te rod ends Note tt te use of sttc nd dynmc problems for dentfcton purposes must result n te sme effectve stffness propertes. Te type of te problem (sttc or dynmc) sould not nfluence te fnl results [9].

7 w ; F ; w ; L ð ; Þ; for s ; l: ð35þ In order to study te ser vbrtons, we serc for solutons of te Eqs. (9), (3) of te form # W cosðxtþ; # w ; u w ; ð36þ were W s constnt nd x s te nturl frequency of te rod. In vew of te consttutve Eq. (3), we see tt te boundry cond tons (35) re stsfed. Imposng tt te felds (36) verfy te equ tons of moton (9) we fnd x A I nd A : ð37þ We dentfy te nturl frequences x nd x from (34) nd (36), nd we obtn te expresson of te consttutve coeffcent A s follows: A k lq x AreðRÞ q ð; x Þx wt k p ; ð38þ were te fctor k s smlr to te ser correcton fctor ntro duced frst by Tmosenko [4] n te teory of bems (note tt n te orgnl contrbuton of Tmosenko te vlue s /3). One cn proceed nlogously for te x drecton nd fnd smlr expresson for A. Tese reltons express te trnsverse ser stff ness coeffcents for non omogeneous rectngulr rods. Te vlue gven by (38) wll be verfed n Secton 8, were we consder te bendng of cntlever functonlly grded bems nd mke com prson wt numercl results. Remrks.. In te cse of omogeneous rods, l nd q re constnt, nd from (38) we get te well known formuls [] A A klareðrþ; A : ð39þ Te vlue of te fctor k n relton (39) s been dscussed n [].. In te cse of tn rods, wen q s smoot vrton cross te tckness, we cn employ te pproxmton q ð; x Þ q ðx ; x Þ: Ten, we substtute (4) nto (38) nd fnd A c ð4þ kl q x c AreðRÞ q x c ðc ; not summedþ; A : ð4þ Te smplfed (pproxmte) formuls (4) cn be used to estmte te trnsverse ser stffness for rbtrry non omogeneous rods (not necessrly rectngulr or symmetrcl) n most cses. 5. Bems composed of two dfferent mterls In ts secton we consder bems nd rods mde of two sotro pc nd non omogeneous mterls. Te body B s decomposed n two regons B nd B suc tt B q fðx ; x ; x 3 Þjðx ; x ÞS q ; x 3 ð; lþg. Tus, te cross secton R s decomposed n two domns S nd S wt S \ S = ;, see Fg.. We denote by C te curve of seprton between te domns S nd S nd by C, C te complementry subsets suc S q = C \ C q. Let P ={(x,x,x 3 )j (x,x ) C,x 3 (,l)} be te surfce of seprton of te two mterls. We ssume tt te two mterls re welded togeter long P nd tere s no seprton of mterl long P, so we ve te condtons ½u Š ½u Š ; ½T Š n ½T Š n on P ; ð4þ were n n e s te unt norml of P, outwrd to B. Te not tons [f] nd [f] represent te vlues of ny feld f on P, clculted s te lmts of te vlues from te domns B nd B, respectvely. Let us denote te Lmé modul of te mterl occupyng te do mn B q by k (q) (x,x ) nd l (q) (x,x ), wt (x,x ) S q, q =,. Consder te problem of extenson, bendng nd torson of suc compound tree dmensonl bem, under te resultnt forces nd moments () ctng on te ends. Ts problem s been tre ted n [3, Secton 3.6], nd te exct soluton s expressed n terms of te solutons to some uxlry plne strn problems. Let us de note by u ðþ ; u ðþ nd u ð3þ te solutons of te 3 plne strn prob lems P ðþ ; P ðþ nd P ð3þ respectvely, formulted on te domn R = S [ S [ C by P ðcþ : t ðcþ b;b þðkðqþ x c Þ ; ns q ; t ðcþ b n b k ðqþ x c n on C q; u ðcþ u ðcþ ; tðcþ b n b t ðcþ b n b þðkðþ k ðþ Þx c n on C ; P ð3þ : t ð3þ b;b þ kðqþ ; ns q ; t ð3þ b n b k ðqþ n on C q; u ð3þ u ð3þ ; tð3þ b n b t ð3þ b n b þðkðþ k ðþ Þn on C : ð43þ We lso ntroduce te functon u(x,x ) wc s te soluton of te boundry vlue problem ðl ðqþ u ; Þ ; ½uŠ ½uŠ ; l ðqþ ; x l ðqþ ; x n S q ; @u n x n on C q; þðl ðþ l ðþ Þ x n x n on C : ð44þ Comprng te soluton of te extenson bendng torson problem n te drect pproc gven n Secton 3. wt te soluton of te correspondng tree dmensonl problem presented n [3, Secton 3.6], we deduce (n te sme mnner s n Secton 4.) te followng expressons for te consttutve coeffcents X A 3 ðk ðqþ þ l ðqþ þ k ðqþ u ð3þ c;c Þdx dx ; B 3 B 3 ; q B 3 X q B 3 X C X q C X q q C X C 3 X q q x x x x () x k ðqþ þ l ðqþ þ k ðqþ u ð3þ c;c k ðqþ þ l ðqþ þ k ðqþ u ð3þ c;c ðk ðqþ þ l ðqþ Þx þ k ðqþ u ðþ c;c ðk ðqþ þ l ðqþ Þx þ k ðqþ u ðþ c;c dx dx ; ðk ðqþ þ l ðqþ Þx þ k ðqþ u ðþ c;c dx dx ; dx dx ; dx dx ; (b) Fg.. Te cross-secton of rods composed of two mterls. dx dx ; l ðqþ x ðx þ u ; Þþx ðx u ; Þ dx dx ; ð45þ were te functons u ðkþ ðx ; x Þ re determned by (43) nd u(x,x ) s gven by (44).

8 Remrks. Te bove results (45) lso old wen te dstrbuton of te mterl n te bem s suc tt te seprton curve C s closed curve ncluded n R, see Fg. b. In ts cse we ve C C = ;, os = C [ = C, nd te boundry vlue problems (43), (44) keep te sme forms.. Te results of ts secton cn be extended to te cse wen te bem B s composed of n (n P ) non omogeneous nd sotro pc mterls wt dfferent mecncl propertes. 6. Ortotropc nd non-omogeneous mterls Let us consder next bems nd rods mde of ortotropc nd non omogeneous mterls. Te tree dmensonl consttutve equtons for suc mterls re t ¼ c e þ c e þ c 3e 33 ; t ¼ c e þ c e þ c 3e 33 ; t 33 ¼ c 3e þ c 3e þ c 33e 33 ; t 3 ¼ c 44e 3 ; t 3 ¼ c 55e 3 ; t ¼ c 66e ; ð46þ were te consttutve coeffcents c j depend on (x,x ) R. Our m s to determne te effectve stffness coeffcents from te drect pproc n terms of c j (x,x ). In ts purpose, we con sder te extenson, bendng nd torson of te bem B due to te termnl lods ().Ts tree dmensonl problem s been solved n [3, Secton 4.], wt te elp of some uxlry plne strn problems defned on te domn R, wc re recorded be low. We desgnte by u ðkþ ðx ; x Þ te solutons of te plne strn problems Q ðkþ ; k ; ; 3, gven by Q ðcþ : Q ð3þ : t ðcþ b;b þðc 3x c Þ ; n R; t ðcþ b n b c 3 x c n t ð3þ b;b þ c 3; n R; t ð3þ b n b c 3 n ð47þ Te subscrpt =, s not summed n te reltons (47). Te tor son functon u(x,x ) s determned by te boundry vlue problem ðc 55 u ; Þ ; þðc 44 u ; Þ ; c 55; x c 44; x n R; c 55 u ; n þ c 44u ; n c 55 x n c 44 x n ð48þ By dentfcton of te tree dmensonl soluton from [3, Sec ton 4.], wt te soluton (7) (9) n te drect pproc we get te followng effectve stffness coeffcents A 3 c 33 þ c 3 u ð3þ ; þ c 3u ð3þ ; ; B 3 B 3 ; B 3 x ðc 33 þ c 3 u ð3þ ; þ c 3u ð3þ ; Þ; B 3 x ðc 33 þ c 3 u ð3þ ; þ c 3u ð3þ ; Þ; C x ðc 33 x þ c 3 u ðþ ; þ c 3u ðþ ; Þ; C x c 33 x þ c 3 u ðþ ; þ c 3u ðþ ; ; C x c 33 x þ c 3 u ðþ ; þ c 3u ðþ ; ; C 3 c 44 x ðx þ u ; Þþc 55 x ðx u ; Þ: ð49þ In vew of te dentfctons (49) one cn sow tt te felds u, w, w, N nd M correspondng to te solutons n te two pproces concde. Remrk. Ts metod cn be ppled lso for bems composed of two dfferent ortotropc mterls. Usng te nottons ntroduced n te begnnng of Secton 5, we ssume tt te non omogeneous ortotropc mterl wc occupes te domn B q s te const tutve coeffcents j ðx ; x Þ. If we employ te sme procedure s n Secton 5 nd compre wt te results of [3, Secton 4.], ten we obtn te followng expressons for te effectve stffness coeffcents X A 3 33 þ cðqþ 3 uð3þ ; þ cðqþ 3 uð3þ ; dx dx ; B 3 B 3 ; q B 3 X q B 3 X C X q C X q S q q C X C 3 X q S q x x q S q 33 þ cðqþ 3 uð3þ ; þ cðqþ 3 uð3þ ; dx dx ; x 33 þ cðqþ 3 uð3þ ; þ cðqþ 3 uð3þ ; dx dx ; 33 x þ 3 uðþ ; þ cðqþ 3 uðþ ; dx dx ; x ð 33 x þ 3 uðþ ; þ cðqþ 3 uðþ ; Þdx dx ; S q x 33 x þ 3 uðþ ; þ cðqþ 3 uðþ ; dx dx ; 44 x ðx þ u ; Þþ 55 x ðx u ; Þ dx dx ; ð5þ were u ðkþ ðx ; x Þ; k ; ; 3, re te solutons of te tree plne strn problems t ðcþ b;b þ cðqþ 3 x c n S q ; t ðcþ b ; n b 3 x cn on C q ; u ðcþ u ðcþ ; tðcþ b n b t ðcþ b n b þ cðþ 3 c ðþ 3 x c n on C ; ð5þ t ð3þ b;b þ cðqþ 3; n S q ; t ð3þ u ð3þ u ð3þ ; tð3þ b n b b n b t ð3þ b 3 n on C q ; n b þ cðþ 3 c ðþ 3 n on C : ð5þ In te reltons (5) nd (5) te subscrpt =, s not summed. Te torson functon u(x,x ) pperng n (5) s te soluton of te followng boundry vlue problem 55 u ; þ 44 u ; ; ; 55; x 44; x n S q ; 55 u ; n þ cðqþ 44 u ; n 55 x n 44 x n on C q; ½uŠ ½uŠ on C ; c ðþ 55 u ; n þ cðþ 44 u ; n c ðþ 55 u ; n þ cðþ 44 u ; n þ c ðþ 55 c ðþ 55 x n c ðþ 44 c ðþ 44 x n on C : ð53þ Te reltons (5) for te consttutve coeffcents re vld lso n te cse wen C s closed curve ncluded n R. Moreover, tese formuls cn be extended to te cse of bems composed of n df ferent ortotropc mterls (n P ). 6.. Trnsverse ser stffness To determne te trnsverse ser stffness coeffcents A, A nd A for ortotropc non omogeneous rods, we consder te problem of ser vbrtons of rectngulr rods formulted n Sec ton 4.. Assume tt q s symmetrcl dstrbuton n te x drecton: q (x,x )=q ( x,x ). We serc for soluton n te form (3). Ten te boundry condtons (3) re stsfed nd te equtons of moton reduce to c 55 ðx ; x Þ p cos p x x x = q ðx ; x Þ sn p x dx

9 Insertng ere te relton (33) nd ntegrtng over R we fnd te lowest nturl frequency p x c 55 ðx ; x Þ ð54þ q ð; x Þ : On te oter nd, we solve te sme problem by te drect p proc nd we fnd te rod s nturl frequency x gven by (37). We dentfy x x nd from reltons (37) nd (54) we obtn A k c 55q x AreðRÞ q ð; x Þx ; A : ð55þ To determne A, one cn proceed nlogously n te x drecton. Remrks. If we dmt te pproxmton (4) ten we deduce A A kc 55 q x AreðRÞ q x ; kc 44 q x AreðRÞ q x : ð56þ were p stnds for te vlue of te fctor k. In most cses, tese formuls re pplcble for ortotropc non omogeneous rods wt rbtrry cross secton propertes (not necessrly rectn gulr or symmetrcl).. Consder te cse of non omogeneous rods composed of two dfferent ortotropc mterls: n te regon B c of te body we ve te mss densty q (c) (x,x ) nd te consttutve coef fcents c ðcþ j ðx ; x Þ; c ;. Eqs. (55) nd (56) for trnsverse ser stffness coeffcents remn vld lso n ts cse, wt te specfctons c j X c q x X c c ðcþ j S c dx dx ; q S c q ðcþ x dx dx : X c S c q ðcþ dx dx ; ð57þ Te extenson of formuls (56) nd (57) to te cse of rods com posed of n ortotropc mterls s lso possble. 7. Specl cses nd exmples 7.. Non omogeneous rods wt constnt Posson rto Let us consder te cse wen te rod s mde of n sotropc mterl wt constnt Posson rto m. Te Young s modulus E s n rbtrry functon of (x,x ) nd te spe of cross secton R s rbtrry. Ts type of mterl s of prctcl nterest nd t s been studed n mny works, see e.g. [4]. In ts cse te solu tons u ðkþ ðx ; x Þ of te problems D ðkþ ; k ; ; 3, defned by () ve smple form u ðþ m x x ðþ ; u mx x ; u ðþ mx x ; u ðþ m x x ð3þ ; u mx ; u ð3þ mx : ð58þ Ten, from (9) we obtn te followng expressons for te effec tve stffness coeffcents A 3 Eðx ;x Þ; C x x Eðx ;x Þ; C x Eðx ;x Þ; C x Eðx ;x Þ; B 3 x Eðx ;x Þ; B 3 x Eðx ;x Þ; B 3 : ð59þ Te consttutve coeffcents C 3, A, A nd A keep te sme form s n te generl cse, gven by (9) nd (38). Remrk. In te cse of omogeneous sotropc rod,.e. wen E s lso constnt, from (59) nd (5) we obtn te well known formuls A 3 EAreðRÞ; C Ex ; C Ex ; C ; B 3 B 3 : In vew of (9) te torsonl rgdty C 3 for smply connected cross sectons s gven by C 3 l/ðx ; x Þ wt D/ nr; / on@r: Te effectve trnsverse ser coeffcents re gven by (39). Te bove expressons of te effectve stffness coeffcents for omoge neous nd sotropc drected curves ve been presented n [,]. 7.. Crculr rod composed of two mterls For rods composed of two dfferent sotropc nd non omoge neous mterls we use te nottons nd developments of Secton 5. Tencross secton of te rod os decomposed s R S [ S, were S ðx ; x Þj < x þ x < b nd S ðx ; x Þjx þ x <. Te frst mterl occupes te regon S (,l) nd s te Lmé modul k ðþ ðx ; x Þ k r m ; l ðþ ðx ; x Þ l r m ; q r x þ x ; ðx ; x ÞS ; ð6þ were m >, k nd l re constnts. Ts knd of nomogenety s been nvestgted n mny works, e.g. [4,4]. We denote by k m ðk þl nd E l ð3k þl Þ Þ k þl. Te second mterl occupes te re gon S (,l) nd ts elstc propertes re descrbed by E ðþ ðx ; x Þ EðrÞ; m ðþ ðx ; x Þ m ðconstntþ; ðx ; x ÞS ; ð6þ were E(r) s n rbtrry gven functon of r. In order to use te results presented n Secton 5 we ve to solve te plne strn problems P ðkþ gven by (43) nd te bound ry vlue problem (44) for te torson functon. In our cse, we ob serve tt tese problems dmt te followng solutons u ðþ u ðþ m ðx x Þ; uðþ u ðþ m x x ; u ð3þ m x : ð6þ Insertng tese functons nto te generl results (45) we fnd te effectve stffness coeffcents for ts compound rod p C 3 r þ m 3 EðrÞdr þ pl d m ; A 3 p reðrþdr þ E c m ; C C p r 3 EðrÞdr þ E d m ; C ; B 3 B 3 ; ð63þ were we ve denoted by c m nd d m te expressons ( ( m b m for m 4 m b 4 m for m 4 c m m ; d m 4 m : logðb=þ for m logðb=þ for m 4 ð64þ Let us fnd lso te trnsverse ser stffness coeffcents A nd A. Assume tt te mss densty functon q (x,x ) s gven by q q ðx ; x Þ r m for ðx ; x ÞS ; ð65þ qðrþ for ðx ; x ÞS were q > s constnt nd q(r) s n rbtrry functon. Ten, usng te results (56), (57) speclzed for sotropc mterls we fnd te expressons

10 p 3 reðrþ A A dr þ l 6b þ m c m r 3 qðrþdr þ q d m rqðrþdr þ q c m : ð66þ 7.3. Ortotropc crculr rod Let us consder n ortotropc rod wt cross secton R ðx ; x Þjx þ x <. We ssume tt te consttutve coeff cents stsfy q c j c j e rr ; r x þ x ; ð67þ were r > nd c j re constnts. Let us ntroduce te nottons E E e rr ; E c 33 c 3m c 3m ; m c 3 c c 3 c d ; m c 3 c c 3 c d ; d c c c : ð68þ Te solutons u ðkþ ðx ; x Þ of te problems Q ðkþ gven by (47) re n ts cse u ðþ m x m x ðþ ; u m x x ; u ðþ m x x ; u ðþ u ð3þ m x ; u ð3þ m x ; m x m x ; ð69þ wle te torson functon u(x,x ) wc solves te boundry v lue problem (48) s uðx ; x Þ c 55 c 44 c 44 þ x x : c 55 ð7þ Tus, n vew of (49) nd (69) nd (7) we fnd te effectve stffness coeffcents C 3 4pc 44 c 55 c 44 þ c 55 C C pe r 3 e rr dr; A 3 pe re rr dr; r 3 e rr dr; C ; B 3 B 3 : ð7þ Assume tt te mss densty of te rod s of te form q ðx ; x Þ q e rr, were q > s constnt. Ten, from reltons (56) we obtn te effectve trnsverse ser stffness coeffcents p 3 A 3 c 55 r 3 e rr dr; A p 3 3 c 44 r 3 e rr dr: 8. Functonlly grded bems mde of metl foms 8.. Dstrbuton of te mterl propertes ð7þ Te mecncl propertes of cellulr solds ve been pre sented n te books [5,43]. In ts secton we nlyze rectngulr bems mde of metl foms. Te cross secton domn s gven by R ðx ; x Þjx ; ; b x ; b. We consder tt te porous mterl s functonlly grded n te x drecton, suc tt te mss densty q of te fom s gven s functon of x by te power lw N ; ð73þ qðx Þ q m þðq s q m Þ jx j were q s s te densty of te bulk (mtrx) mterl, q m s te mn mum vlue of te densty of te fom, nd N s n exponent. Ts type of functonlly grded porous mterls s been studed n te cse of pltes n [43,44]. To express te Young modulus E of te fom we use te formul ndcted by Gbson nd Asby [5] j qðx Þ Eðx Þ E s ; ð74þ q s were E s s te Young modulus of te bulk mterl. In wt follows, we consder closed cell lumnum foms, for wc te exponent j s gven by j =, nd te Posson rto s ssumed to be constnt Es wt te vlue m =.3 [5]. Let us denote by G s te ser ðþmþ modulus of te bulk mterl. Te vrtons of q nd E s functons of x, s gven n (73) nd (74), re depcted n Fg. 3 for severl vl ues of te exponent N. Let us clculte te effectve stffness coeffcents for ts func tonlly grded porous bem. Snce te Posson rto s ssumed constnt, we cn use te relton (59), n conjuncton wt (73) nd (74), to derve te extensonl nd bendng stffness coeffcents A 3 be s r þ N þ b 3 C E s r þ N þ b 3 C E s r þ 6 N þ 3 rð rþþ N þ ð rð rþþ N þ ð rð rþþ 3 N þ 3 ð rþ ; B 3 B 3 ; rþ ; C ; rþ ; ð75þ were we denote by r te rto r qs. Te effectve ser stffness cn be clculted from te reltons (38). We nsert te expresson for q from (73) nto (38) nd obtn A kbg s r þ rð rþþ N þ N þ ð rþ ; ðbþq A kbg s s qððx ÞÞ r þ 3 N þ 3 ð rþ r þ rð rþþ N þ N þ ð rþ ; ð76þ were, ccordng to (33), q((x )) s gven by qððx ÞÞ pb cos px q m x qðfþ sn pf dfdx : ð77þ Usng te expresson (73) n (77) nd mkng some mtemtcl clcultons, we get qððx ÞÞ bq s r þ r N þ þð rþj N ; ð78þ were we ve denoted by " Nþ p # p p J N p p N ðþ p N þ ðcosxþ p N p N ðxþsnx dx : In te lst relton, te polynoml functon p N (x) s gven by p N ðxþ Nx N NðN ÞðN Þx N 3 þ NðN Þ ðn 4Þx N 5 P m ð Þ NðN Þ ðn Þx N ; p ðxþ ; p ðxþ x; were m N s te gretest nteger not exceedng N. Fnlly, f we substtute (78) nto (76) we fnd r þ 3 A kbg s Nþ3 ð r þ Nþ þ J N rþ ð rþ r þ rð rþþ N þ N þ ð rþ : ð79þ Te formul (79) represent te exct expresson for te effectve ser stffness, clculted on te bss of (38). On te oter nd, f we employ te pproxmte relton (4) nsted of (38), ten we deduce te followng smplfed (pproxmte) expresson for A

11 N= N= N=3 N=6 N= Fg. 3. Te dstrbutons of densty q nd Young modulus E for dfferent vlues of N. For te lumnum fom we tke q m 5 kg m 3 ; q s 7 kg m 3 ; E s 7 GP, nd te tckness =.5 m. () (b) Fg. 4. () Cntlever bem wt unform dstrbuted lod q. (b) Cntlever bem wt concentrted end force P. Fg. 5. Cross-secton of te FGM bem nd dstrbuton of Young s modulus. ea kbg s r þ 3 ð rþ Nþ3 r þ ð rþ r þ rð rþþ N þ N þ ð Nþ rþ : ð8þ compre te nlytcl solutons wt te results obtned by f nte element nlyss. 8.. Cntlever bems Let us use te effectve stffness coeffcents for FGM porous bems determned prevously to solve some bendng problems nd Consder cntlever bem mde of functonlly grded closed cell lumnum fom subject to bendng nd ser under

12 Tble Comprson of results for cntlever FGM bem wt unform lod. N d FEM (mm) d exct (mm) d pprox (mm) D (%) Exct teoretcl model Approxmte teoretcl model.6.4 Exct teoretcl model Approxmte teoretcl model Fg. 6. Error D n terms of te exponent N, for te mxmum deflecton of cntlever FGM bem wt unform lod. te followng lods: () unformly dstrbuted force q ctng n te x drecton; or (b) concentrted end force P ctng n te x drec ton. We denote by l te lengt of te bem (see Fg. 4). Te nlytcl solutons of tese problems cn esly be derved from te one dmensonl governng dfferentl equtons of d rected rods presented n Secton 3. For te mxmum deflecton d of te bem we obtn te well known reltons d d ql Pl þ l A 4C A þ l 3C!! for unformly dstrbuted force q; for concentrted end force P; ð8þ were te vlues of te effectve ser stffness A nd bendng stffness C for FGM porous bems re gven by (79) (or te pprox mte form (8)) nd (75), respectvely. Te teoretcl predctons (8) wll be compred wt numercl solutons obtned by te f nte element metod. Te cross secton of te bem s te dmensons =5mm nd b = 5 mm (see Fg. 5), te lengt s l = m, nd te closed cell lumnum fom s crcterzed by te mterl prmeters q m 5 kg m 3 ; q s 7 kg m 3 ; E s 7 GP. We ve clcu lted te mxmum deflecton of te bem numerclly, usng te softwre ABAQUS. To descrbe ts functonlly grded structure, te bem domn s been dvded nto lyers ortogonl to te x drecton. Ec lyer s ssumed to ve constnt mterl prm eters E nd q, wc stsfy te power lws (73) nd (74) stepwse. Fg. 7. Error D n terms of te exponent N, for te mxmum deflecton of cntlever FGM bem wt end lod. For te problems presented ere number of 64 or 8 lyers s suffcent. Te clculton s been performed usng 3D sell ele ments nd very dense mes. Te fnte elements ve been tken squre, wt one element per lyer tckness. We denote by d FEM te mxmum deflecton clculted by fnte element nlyss, let d exct be te teoretcl vlue of te mxmum deflecton gven by (8) wt te exct formul (79) for A, nd d pprox be te teoretcl vlue gven by (8) wt te pproxmte formul (8). We clculte te reltve error D by te relton D d FEM d exct mnðd FEM ; d exct Þ : Fg. 8. Tree-pont bendng of FGM bem. Tble Comprson of results for cntlever FGM bem wt concentrted end lod. N d FEM (mm) d exct (mm) d pprox (mm) D (%)

13 Tble 3 Comprson of results for FGM bem n tree-pont bendng. N d FEM (mm) d exct (mm) d pprox (mm) D (%) Exct teoretcl model Approxmte teoretcl model between.7% nd.5%. Fg. 7 presents te percentge of reltve error, for te exct nd pproxmte solutons, wt respect to te numercl one. From Fgs. 6 nd 7 we notce tt te exct teoretcl model g ven by (79) s slgtly better tn te pproxmte one (n te lest squre sense). Moreover, we see tt te pproxmte teoretcl model (8) yelds results n good greement wt te numercl nd exct solutons, nd t s te dvntge of smplcty Tree pont bendng of functonlly grded bem Fg. 9. Error D n terms of te exponent N, for te mxmum deflecton of FGM bem n tree-pont bendng. () For te bendng of cntlever bem by unformly dstrbuted force q 5kNm, we ve employed 64 lyers. Te com prson of te results s presented n Tble, for te vlues of te exponent N =,,...,. We cn observe very good greement between te nlytcl nd te numercl results, snce te errors rnge between % to %. Te percentge of reltve error D s plotted n Fg. 6, n terms of te exponent N. (b) For te bendng of te bem by concentrted end force P = 5 kn, we ve employed 8 lyers. Te concentrted force s been dvded nto equl prts ctng n te nodes long te wole edge of te bem. Ts procedure reduces te concentrton of stress n te numercl soluton. Te comprson between te nlytcl nd te fnte element solutons s sown n Tble. Te errors D re very smll: Let us consder te functonlly grded bem descrbed prev ously n reltons (73) (8) subjected to tree pont bendng. A concentrted centrl force P = 5 kn cts t te md spn of te bem (x 3 = l/) n te x drecton, nd te end edges x 3 =, l re smply supported, see Fg. 8. Te nlytcl soluton of ts bendng problem cn be derved from te equtons gven n Secton 3. For te mxmum deflecton d of te bem, we get!! d exct Pl 4 þ l A C ; d pprox Pl 4 þ l ea C ; ð8þ were te effectve bendng stffness C s gven by (75), wle te effectve ser stffness A s te exct expresson (79), nd e A s te pproxmte form (8). To obtn te mxmum deflecton d FEM by fnte element nlyss, we use 8 lyers to dvde te bem domn. Tble 3 sows te comprson of te teoretcl nd numercl solutons, togeter wt te reltve error D. InFg. 9 we plot te reltve er ror wt respect to te numercl soluton, for N =,...,. We ob serve tt te errors rnge between.9% nd.9%, dependng on te vlue of N. Te spe of te bem n te deformed confgurton s de pcted n Fg. for N =, 5,, n bot numercl nd teoretcl pproces. Te results re n very good greement, so tt te curves for te nlytcl nd numercl solutons re very close n Fg.. Indeed, ccordng to Tble 3, te reltve errors for te Fg.. Deflecton of te FGM bem under tree-pont bendng: numercl nd nlytcl results, for te vlues of te exponent N =, 5,.

14 Fg.. Dstrbuton of norml nd ser stresses n te cross-secton of te FGM bem, for x 3 = l/4. mxmum deflectons d for N=, 5,, re.8%,.3%, nd respec tvely.9%. Let us present some results bout te stress stte n te FGM bem. For te cross secton of te bem crcterzed by te xl coordnte x 3 = l/4, te dstrbutons of te norml stress t 33 nd ser stress t 3 versus te tckness coordnte x re obtned by te fnte element nlyss nd depcted n Fg.. On te oter nd, te nlytcl soluton of ts tree pont bendng problem n te drect pproc yelds te followng trns versl force Q nd bendng moment L, clculted t te xl coordnte x 3 = l/4: Q P ; L Pl 8 : ð83þ Accordng to (4) 7,9, te correspondence between Q, L nd te tree dmensonl stress stte s gven by Q b t 3 dx ; L b x t 33 dx : ð84þ Ten, we cn compre te teoretcl predctons (83) wt te numercl soluton n te form of te resultnts (84). As expected, te greement between te two pproces s very good: for te bendng moment L te reltve error s n te rnge.5.7%; for te trnsversl force Q te reltve error s bout.3% (for every exponent N =,...,). 9. Conclusons In ts pper we ve employed te teory of drected curves to nvestgte te mecncl bevor of non omogeneous, com poste, nd functonlly grded bems. Te structure of te const tutve tensors nd te form of te lner consttutve equtons ve been estblsed n Sectons, 3, nd re presented n te reltons (6) nd (3). We determne te effectve stffness coeff cents v comprson wt tree dmensonl elstcty sttc nd free vbrton solutons n Sectons 4 6. Tus, for non omo geneous sotropc bems we fnd te formuls (9) nd (38), wle for composte bems mde of two dfferent mterls we ve te effectve stffness propertes (45). For ortotropc non omoge neous bems, te effectve ser stffness s expressed by (55), (56), nd te effectve bendng stffness, extensonl stffness, tor sonl rgdty nd couplng coeffcents re gven by (5). In Secton 7 we pply tese generl formuls to determne te effectve stffness propertes of some specl functonlly grded bems, suc s ortotropc bems wt exponentl dstrbuton lw, or composte crculr bems wt power lw dstrbuton of mterl propertes. In Secton 8 we consder rectngulr functonlly grded bems mde of metl foms. Usng te Gbson Asby formul (74) for te Young modulus of closed cell lumnum foms, combned wt te power lw dstrbuton of mss densty (73), we fnd te effectve stffness coeffcents n te form (75) nd (79). In vew of tese re sults, we deduce te nlytcl bem lke solutons for te bendng of FGM cntlever bem subjected to unform nd end lodngs n Secton 8., nd for FGM bem n tree pont bendng n Secton 8.3. Te teoretcl predctons re n good greement wt numer cl results obtned by fnte element nlyss. Ts comprson wt fnte element solutons represents vl dton of our nlytcl modelng concernng te effectve stffness propertes of FGM bems. Neverteless, our pproc s muc more generl nd t cn be used to nlyze te mecncl proper tes of vrous functonlly grded rods, wt dfferent geometrcl nd mterl crcterstcs. Acknowledgments Te utors cknowledge fundng from te E.U. FP7 Progrmme FP7 REGPOT 9 under Grnt Agreement No , nd from te Pols Mnstry of Scence nd Hger Educton, Grnt No. 47 /7, PR UE//7. In ddton, te frst utor (M.B.) ws supported by te Alexnder von Humboldt Foundton, wle te second utor (H.A.) by te Germn Reserc Foundton (Grnt AL34/33 ) nd te Jpnese Socety for te Promoton of Scence (ID No. RC 5). Te fourt utor (V.A.E.) cknowledges sup port from te Germn Reserc Foundton (Grnt AL34/33 ). References [] Trbuco L, Vño JM. Mtemtcl modellng of rods. In: Crlet PG, Lons JL, edtors. Hndbook of numercl nlyss, vol. 4. Amsterdm: Nort Hollnd; 996. p [] Svetltsky VA. Sttcs of rods. Berln: Sprnger;.

15 [3] Hodges DH. Nonlner composte bem teory. Progress n stronutcs nd eronutcs, vol. 3. Reston: Amercn Insttute of Aeronutcs nd Astronutcs Inc.; 6. [4] Meuner N. Recursve dervton of one-dmensonl models from treedmensonl nonlner elstcty. Mt Mec Solds 8;3:7 94. [5] Berdcevsky VL. Vrtonl prncples of contnuum mecncs. II: pplctons. Hedelberg: Sprnger; 9. [6] Sures S, Mortensen A. Functonlly grded metls nd metl-cermc compostes. : termomecncl bevour. Int Mter Rev 997;4(3):85 6. [7] Wng CM, Reddy JN, Lee KH. Ser deformble bems nd sells. Amsterdm: Elsever;. [8] Snkr BV. An elstcty soluton for functonlly grded bems. Compos Sc Tecnol ;6(5): [9] Snkr BV, Tzeng JT. Terml stresses n functonlly grded bems. AIAA J ;4(6):8 3. [] Ckrborty A, Goplkrsnn S, Reddy JN. A new bem fnte element for te nlyss of functonlly grded mterls. Int J Mec Sc 3;45(3): [] Ckrborty A, Goplkrsnn S. A spectrlly formulted fnte element for wve propgton nlyss n functonlly grded bems. Int J Solds Struct 3;4():4 48. [] Gunt G, Belouettr S, Crrer E. Anlyss of FGM bems by mens of clsscl nd dvnced teores. Mec Adv Mter Struct ;7(8):6 35. [3] Cossert E, Cossert F. Téore des corps déformbles. A. Hermn et Fls, Prs; 99. [4] Ercksen JL, Truesdell C. Exct teory of stress nd strn n rods nd sells. Arc Rton Mec Anl 958;(): [5] Green AE, Ngd PM. Non-soterml teory of rods, pltes nd sells. Int J Solds Struct 97;6:9 44. [6] Green AE, Ngd PM. On terml effects n te teory of rods. Int J Solds Struct 979;5: [7] Antmn SS. Nonlner problems of elstcty. Seres ppled mtemtcl scences, 7. New York: Sprnger; 995. [8] Rubn MB. Cossert teores: sells, rods, nd ponts. Dordrect: Kluwer Acdemc Publsers.;. [9] ln PA. Mecncs of deformble drected surfces. Int J Solds Struct 976;: [] ln PA. Nonlner teory of tn rods. In: Indetsev DA, Ivnov EA, Krvtsov AM, edtors. Advnced problems n mecncs, vol.. St. Petersburg: Problems Mec. Eng. R.A.S. Publ.; 6. p [] ln PA. Appled mecncs teory of tn elstc rods (n Russn). Petersburg: Poltekn. Unv. Publ., St.; 7. [] Altenbc H, Numenko K, ln PA. A drect pproc to te formulton of consttutve equtons for rods nd sells. In: Petrszkewcz W, Szymczk C, edtors. Sell structures: teory nd pplctons. London: Tylor nd Frncs; 6. p [3] Iesßn D. Clsscl nd generlzed models of elstc rods. Boc Rton - London - New York: Cpmn & Hll/CRC Press; 9. [4] Iesßn D, Scl A. On te deformton of functonlly grded porous elstc cylnders. J Elst 7;87: [5] Gbson LJ, Asby MF. Cellulr solds: structure nd propertes. Cmbrdge sold stte scence seres. Cmbrdge: Cmbrdge Unversty Press; 997. [6] Lure AI. Teory of elstcty. Berln: Sprnger; 5. [7] Lebedev LP, Cloud MJ, Eremeyev VA. Tensor nlyss wt pplctons n mecncs. New Jersey: World Scentfc;. [8] ln PA. Appled mecncs foundtons of sell teory (n Russn). Petersburg: Poltekn. Unv. Publ., St.; 6. [9] elenn AA, ubov LM. Te non-lner teory of te pure bendng of prsmtc elstc solds. J Appl Mt Mec ;64(3): [3] ubov LM. Exct nonlner teory of tenson nd torson of elcl sprngs. Dokldy Pys ;47(8):63 6. [3] Yu W, Hodges DH, Volovo V, Cesnk CES. On Tmosenko-lke modelng of ntlly curved nd twsted composte bems. Int J Solds Struct ;39(9):5. [3] Mrgo J-J, Meuner N. Herrcy of one-dmensonl models n nonlner elstcty. J Elst 6;83(): 8. [33] Mor MG, Müller S. Convergence of equlbr of tree-dmensonl tn elstc bems. Proc Royl Soc Ednburg: Sect A Mt 8;38: [34] Scrd L. Asymptotc models for curved rods derved from nonlner elstcty by C-convergence. Proc Royl Soc Ednburg: Sect A Mt 9;39:37 7. [35] Irsck H, Gerstmyr J. A contnuum mecncs bsed dervton of Ressner s lrge-dsplcement fnte-strn bem teory: te cse of plne deformtons of orgnlly strgt Bernoull Euler bems. Act Mec 9;6:. [36] Humer A, Irsck H. Onset of trnsent vbrtons of xlly movng bems wt lrge dsplcements, fnte deformtons nd n ntlly unknown lengt of te reference confgurton. AMM 9;89(4): [37] Irsck H, Gerstmyr J. A contnuum-mecncs nterpretton of Ressner s non-lner ser-deformble bem teory. Mt Comput Model Dynm Syst ;7():9 9. [38] Muskelsvl NI. Some bsc problems of te mtemtcl teory of elstcty. Gronngen: Noordoff; 953. [39] Solomon L. Élstcté Lnére. Prs: Msson; 968. [4] Tmosenko SP. On te correcton for ser of te dfferentl equton for trnsverse vbrtons of prsmtc bems. Pl Mg 9;4: [4] Lomkn VA. Teory of nonomogeneous elstc bodes (n Russn). MGU, Moscow; 976. [4] Lekntsk SG. Elementry solutons of two specl problems of equlbrum of nonomogeneous cylnders (n Russn). Investgtons on elstcty nd plstcty, vol. 6. Petersburg: Izd. Lenngrd Unv., St.; 967. [43] Altenbc H, Öcsner A. Cellulr nd porous mterls n structures nd processes. CISM courses nd lectures, vol. 5. Wen NewYork: Sprnger;. [44] Altenbc H, Eremeyev VA. Drect pproc-bsed nlyss of pltes composed of functonlly grded mterls. Arc Appl Mec 8;78: