The Normal Stress Dıstribution in an Infinite. Elastic Body with a Locally Curved and Hollow. Fiber under Geometrical Nonlinear Statement

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1 Nonlinea Analyi and Diffeential quation Vol no HIKARI td htt://dx.doi.og/0.988/nade The Nomal Ste Dıtibution in an Infinite latic Body with a ocally Cuved and Hollow Fibe unde Geometical Nonlinea Statement K. Simek Alan Yildiz Technical Univeity Faculty of Chemity and Metallugy Deatment of Mathematical ngineeing 400 Itanbul Tukey Coyight 06 K. Simek Alan. Thi aticle i ditibuted unde the Ceative Common Attibution icene which emit uneticted ue ditibution and eoduction in any medium ovided the oiginal wok i oely cited. Abtact The eult of the nomal te ditibution in an elatic body with a locally cuved and hollow fibe ae exounded. The nomal te ditibution i tudied when the body i loaded at infinity by unifomly ditibuted nomal foce in the fibe diection. The invetigation ae caied out in the famewok of the iecewie homogeneou body model with the ue of the thee-dimenional geometically nonlinea exact equation of the theoy of elaticity. The mathematical fomulation of coeonding bounday-value oblem wa given. The bounday fom etubation method i emloyed fo the invetigation. Numeical eult elated to the nomal te ditibution conideed and the influence of aamete on thee tee ae analyzed. The coeonding numeical eult ae eented. Keywod: Fibou comoite te ditibution geometic nonlineaity locally cuving Intoduction It hould be noted that in the tuctue of the unidiectional fibou comoite in many cae fibe have an initial cuving caued by deign facto o caued by action of vaiou facto duing technological ocee ee Tanoolky Yu Jigun and Polyakov (987) Guz (990) Kelly (998) Akbaov and Guz (000) Guz (00). Fo thi uoe within the famewok of a iecewie homogeneou

2 96 K. Simek Alan body model by uing the exact thee-dimenional equation of elaticity theoy in Akbaov and Guz (985) ae given a method fo invetigation of the te tate in unidiectional comoite. In Simek (005) ooed a method to invetigation a locally cuved fibe in geometically nonlinea tatement. Alan and Akbaov (0) conideed the nomal te ditibution with unidiectional locally cuved coveed nanofibe. Alan (04) i analyzed the nomal te ditibution in an elatic body with a locally cuved and hollow nanofibe. Howeve the eult emained within the limit of the fit aoximation. It i obviou that i neceay to inceae the numbe of aoximation added to the olution o a to inceae the enitivity of the numeical eult. In thi ae The Reult of the nomal te ditibution in an elatic body with a locally cuved and hollow fibe ae analyzed. The Method i develoed in uch way a to obtain the nomal te value u to econd aoximation on the fibe and matix inteface. Within the famewok of the thee dimenional geometical nonlinea exact equation of the theoy of elaticity we invetigate the nomal tee in the comoite with unidiectional locally cuved and hollow fibe. We aume that the concentation of fibe in the comoite i mall and any inteaction between them i diegaded. Fomulation of the Poblem In the eent tudy we conide an infinite body containing a ingle locally cuved and hollow fibe with an initial local imefection and we detemine though the agange coodinate Oxx xin the cylindical ytem of coodinate O z aociated with the middle line of the fibe (Fig.). Value elated to matix will be denoted by ue indice to fibe by ue indice (). We uoe that the matix and fibe mateial ae homogeneou iotoic and linee elatic. Moeove we aume that the fibe co ection i on the lane which i eendicula to it axial line i a cloed cicula ing of the contant inne adiu R and oute adiu R. Fo fomulation of the oblem can be een in Akbaov Koke and Simek (005). Figue : The geomety of tuctue and comoite conideed

3 Nomal te dıtibution in an infinite elatic body 97 Solution Method Uing the eie eentation in Akbaov and Guz (000) with eect to mall aamete and doing mathematical maniulation imila that made in Akbaov and Guz (000) we obtain the bounday-value oblem fo detemination of the zeoth and the fit aoximation. Fo the olution method fo zeoth and fit aoximation ee Alan (04). By uing the olution ocedue given in Koke and Simek (006) we obtain the contact condition fo econd aoximation. 0 0 ( i ) ( i ) ( i ) ( i ) ( i) f f z z ( i) ( ) i ( i) ( z z i) f f ( i) ( i) ( i) ( i) ( i) f z z f 0 0 () i z 0 z ( i) z f 0 z ( i) 0 ( i) z () i z u ( i u ) ( i u ) ( i u ) ( i u ) ( i ) ( i) z z z u f f u ( i) u ( i) u ( i) 0 z z f f f 0 0 In q. the following notation i ued: q () q q R t R t q q=0 d ( t ) f ( t ) co R co (t) d (t) Rco dt R dt ( t) d ( t in ) x z co x ( t) ex( ) co( m ) () R dt 0 Simila contact condition can alo obtained fo ubequent aoximation. In thi cae an exact analytical olution i given in Akbaov and Guz (000) Guz (00) Akbaov and Guz (985). Accoding to which at following elation: () we have the

4 98 K. Simek Alan 0 ()0 0 0 ()0 0 0 u z u z z u u ()0 () ()0 0 ()0 0 ()0 0 ()0 u u 0 0 () ()0 0 ()0 0 ()0 0 ()0 0 z z z z () () Whee in () eeent the elaticity module of the matix fibe eectively. By emloying the olution ocedue decibed in Guz (999) we obtain the following equation fo the fit and econd aoximation ( k )q ( k )q ( k)q ( k)q z ( k )q ( k )q ( k)0 u 0 z z ( k )q ( k )q ( k )q ( k )q z ( k)q ( k)0 u 0 z z ( k )q ( k )q ( k )q ( k)q z z ( k)q ( k)0 uz z 0 q z z (4) Thee equation coincided with the -dimenional lineaized elaticity equation. We can witten the tain-dilacement a follow. ( k )q ( k )q ( k )q u ( k )q ( k )q u u u ( k)q ( k)q ( k )q z z u u ( k)q ( k)q ( k )q ( k )q u u q z ( k )q z ( k )q u z ( k )q uz z u ( k )q ( k )q z To detemine the contact condition and the eq. (4) and (5) fo the econd aoximation we ue the eeentation in Guz (999). (5) q q q u ψ χ θ z ; q q q uθ ψ χ ; θz ( k )

5 Nomal te dıtibution in an infinite elatic body 99 u (λ μ ) (λ μ )Δ (μ σ ) χ z q (k 0 q z (6) The function of q q ψ χ ae detemined by the following equation: q Δ ξ ψ 0 z q Δ ξ Δ ξ χ 0 z z (7) Whee ξ ξ i (k=; i=) given in the following equation: ( k)0 μ σ ξ μ ( k)0 μ σ ξ μ λ μ λ μ σ ( k)0 We aly the following exonential Fouie tanfom with eect to z i.e. f iz ( ) f ( z) e dz (8) We elect the olution to the equation (7) and emloying the Fouie tanfom we can detemine ( k) ( k) ( k) ( k) and function ψ χ A ia ()K (ξ ()K (ξ )inθ ) A ()K (ξ ) coθ (9) ψ χ () () A i A () () ()I (ξ ()I (ξ () () ) ) A A () () ()K (ξ ()K (ξ () () ) inθ ) A () ()I (ξ () ) A () ()K (ξ () ) coθ ψ A ( )K (ξ )inθ χ i[a ( )K (ξ ) A ( )K (ξ )+{A ( )K (ξ ) A ( )K (ξ )}]coθ

6 00 K. Simek Alan () () () () () ψ A ( )I (ξ ) B ( )K (ξ ) inθ (0) () () () () () () () () () χ i A 0 ( )I 0(ξ )+B 0 ( )K 0(ξ ) A 0 ( )I (ξ ) B 0 ( )K (ξ ) A ( )I (ξ ) B ( )K (ξ ) A ( )I (ξ ) B ( )K (ξ () () () () () () () () ) co θ whee I n (x) ae Beel function of a uely imaginay agument and K n (x) ae the Macdonald function. Moeove and ae Fouie tanfom aamete. The unknown contant enteing (9) and (0) ae detemined fom the elation (6) (7) and the contact condition equation given in Alan (04). Thi give the fit and econd aoximation of tageted tee. The following invee Fouie tanfom i ued to find the eal te value fo examle i e iz d () The exeion fo the othe ought fo quantitie ae detemined likewie. Thu by the above-decibed method we detemine comletely the value in the econd aoximation. 4 Numeical Reult and Dicuion By uing the olution ocedue decibed in Akbaov and Guz (000) we obtain the numeical eult fo fit and econd aoximation. In Alan and Akbaov (00) ae given alo thee olution method and mathematical calculation fo fit and econd aoximation. See Alan and Akbaov (00). We will analyze ome numeical eult elated to the nomal te which contitute the matix and fibe inteection uface. In view of coeonding ymmety we conide the ditibution of thee tee only fo x 0 and 0 (Fig. ). If 0 (i.e. if the cuving i abent) the tee coincide with. We intoduce the aamete R bk= hr and () aume that how the influence of geometical non-lineaity on the value of the nomal tee mentioned.

7 Nomal te dıtibution in an infinite elatic body 0 σ / The Figue how deendencie of on the aamete hr fo vaiou value of m=0 at and () / =50. Thee gah how alo the influence of the aamete m on the ditibution of the nomal tee. It follow fom the gah that maximal value σ / of the nomal tee inceae monotonically with m. We note that when the thickne of fibe i inceaed it i oible to obtain the ame eult a thoe obtained in the cae of the ingle locally cuved fibe in an infinite body in the ame aamete value. Note that unde comeion of the conideed infinite body we aume that the elected value of ae malle than thoe of citical value c which coeond to the mico-buckling of the fibe in the matix in Guz (990) /II m= 0.98 m= m= h/r Fig : Ditibution of the with eect to h /R whee () / = and 0. unde m 0 ; m ; m Figue how the gah of the deendencie among σ / and x / fo h/r= and () / =50. The influence of the value m on the nomal te ditibution illutated by thee gah. It follow fom thee gah that abolute maximal value of the nomal tee σ / inceae monotonically with m. Note that unde comeion of the conideed infinite body we aume that the elected value of ae malle than thoe of c. which coeond to the mico-buckling of the fibe in the matix in Guz (990).

8 0 K. Simek Alan /II m= m= m= X/ Fig : Ditibution of the with eect to x whee () / = and 0. unde m 0 ; m ; m σ / () Table tabulate the value of fo vaiou m and. In thi σ / cae the value of ae calculated unde 0..0.It follow σ / fom the eult given in Table abolute maximal value of inceae () monotonically with inceaing and m. Thi table indicate that the σ / value of the nomal tee inceae with. We aume that i malle than it citical value buckling of the fibe in the matix in Guz (990). c coeonding to mico Table : The value of obtained fo vaiou and () () m A. N. Tenion comeion

9 Nomal te dıtibution in an infinite elatic body 0 Table : (Continued): The value of () obtained fo vaiou and Concluion In eent ae the invetigation wee made within coe of the iecewie homogeneou body model with the ue of the thee-dimenional geometically nonlinea exact equation of the theoy of elaticity. A ingle locally cuved and hollow fibe ha been invetigated. The mathematical fomulation of the coeonding bounday-value oblem wa given. The method i develoed in uch way a to obtain the nomal tee value u to the econd aoximation on the inteface the hollow fibe and matix mateial. Conequently the aim of the invetigation wa to tudy the influence of the nomal te ditibution in an elatic body with a locally cuved and hollow fibe fo the econd aoximation The numeical eult wee eented fo a ingle locally cuved and hollow fibe. A a eult of the numeical invetigation it wa etablihed that: (i) When the adiu of hollow goe to 0 a a limit it wa obtained the ame nomal tee value in a locally cuved fibe in a comoite mateial in ame aamete value. (ii) The ffect of geometical nonlineaity inceae monotonically with a () well a with. (iii) The maximum abolute value of the tee conideed inceae ocillation fequency of the local cuving. The numeical eult obtained agee with well-known mechanical conideation and in ome aticula cae coincide with known eult. Refeence [] Yu.M. Tanoolky I.G. Jigun and V.A. Polyakov Satially-Reinfoced Com-

10 04 K. Simek Alan oite Mateial Handbook Mahinotoyenia Mocow Ruia 987. [] A.N. Guz Failue Mechanic of Comoite Mateial in Comeion Naukova Dumka Ruia 990. [] A. Kelly Comoite Mateial Imediment do wide ue and ome uggetion to ovecome thee Chate in uoean Confeence on Comoite Mateial (CCM-8) Naole-Italy Vol. I [4] S.D. Akbaov A.N. Guz Mechanic of Cuved Comoite Kluwe Academic Publihe Dodecht The Netheland 000. htt://dx.doi.og/0.007/ [5] A.N. Guz On one two-level model in the meomechanic of comeion factue of Cacked Comoite Int. Al. Mech. 9 (00) no htt://dx.doi.og/0.0/a: [6] S.D. Akbaov A.N. Guz Method of Solving Poblem in the Mechanic of Fibe Comoite with Cuved Stuctue Soviet Alied Mechanic 0 (984) htt://dx.doi.og/0.007/bf [7] A.N. Guz Fundamental of the Thee-Dimenional Theoy of Stability of Defomable Bodie Singe New Yok 999. htt://dx.doi.og/0.007/ [8] R. Koke K. Simek and S. D. Akbaov Ste Ditibution in an Infinite latic Body with a ocally Cuved Fibe in a Geometically Nonlinea Statement Mechanic of Comoite Mateial 4 (005) no htt://dx.doi.og/0.007/ [9] S.D. Akbaov R. Koke and K. Simek On the Nomal Stee in the latic Body with a ocally Cuved Fibe unde Geometically Nonlinea Statement Sigma Jounal of ngiineeing and Natual Science 4 (006) no [0] K.S. Alan S.D. Akbaov Analyi of tee in an infinite elatic body with a locally cuved coveed nanofibe Mechanic of Comoite Mateial 47 (0) no htt://dx.doi.og/0.007/ y [] K.S. Alan Nomal Stee in an Infinite latic Body with a ocally Cuved and Hollow Nanofibe CMC-Comute Mateial & Continua 44 (04) no. -. Received: May 8 06; Publihed: June 06