DEFORMATION ANALYSIS OF AN INFLATED CYLINDRICAL MEMBRANE OF COMPOSITE WITH RUBBER MATRIX REINFORCED BY TEXTILE CORDS

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1 DEFORMATION ANALYSIS OF AN INFLATED CYLINDRICAL MEMBRANE OF COMPOSITE WITH RUBBER MATRIX REINFORCED BY TEXTILE CORDS Bohdana Marvalová and Tran Huu Nam Summary: We present the orthotropc hyperelastc materal model for numercal smulaton of the loadng of the cylndrcal membrane. The coeffcents of stran energy functon of the hyperelastc orthotropc materal are ftted to the expermental results by the nonlnear least squares method. The components of the deformaton gradent are determned from measured dsplacements of the grd ponts drawn on the cylndrcal surface of the sprng. The stress tensor s calculated from the membrane theory. The deformed shape of the sprng surface s measured from the photographc records. The stran energy functon s expressed n terms of tensoral nvarants wth regard to the assumed materal symmetry. The deformaton of ar-sprng s calculated by solvng the system of fve frst-order ordnary dfferental equatons wth the materal consttutve law and proper boundary condtons.. Introducton The man purpose of authors s the numercal smulaton of nflaton of the composte cylndrcal membrane made of rubber matrx renforced by textle cords. Some orthotropc and transversely hyperelastc consttutve models approprate for such type of materal can be found n lterature. Most of them are represented by stran energy functon n the form of a polynomal, exponental or logarthmc [, 4, 0] functon of stran nvarants regardng the assumed materal orthotropy. However the development of the consttutve theory of ansotropc elastc or vscoelastc materals at fnte strans s stll far to be complete and the publcatons n ths feld are sparse. The consttutve equatons of the transversally sotropc materal n the nonlnear stress and deformaton doman are presented n the papers of Holzapfel and coll. [], Bonet and Burton [] and Verron []. We use the consstent consttutve model of drecton dependent hyperelastc materal presented n papers of Ogden, Holzapfel, Gasser and coll. [, 4] appled by authors to the problem of the mechancal response of arteral walls and of fber renforced compostes at fnte strans. The deformaton feld s generally determned by the fnte element method. However, we use the method of the numercal ntegraton of the system of the ordnary dfferental equatons of problem descrbed by Green and Adkns [9] and recently for Doc. Ing. Bohdana Marvalová, CSc, Ing. Tran Huu Nam, Katedra mechanky a pružnost SF, TU v Lberc, bohda.marvalova@vslb.cz, huunam.tran@vslb.cz

2 sotropc membrane by Guo [4]. We ncorporated nto ths procedure our own orthotropc materal law. Ths method appeared to be qute promsng and we presume to use t for the nverse dentfcaton of materal parameters. Detals on the expermental setup and the experment evaluatons can be found n the prevous papers of authors [5-8].. Deformaton of cylndrcal orthotropc membrane We determne the man geometrc features of the nflated membrane n accordng wth the dervaton n [, 4]. The thn cylndrcal membrane of ar-sprng at Fg. has the ntal radus of md-surface R, and length L. Its ntal wall thckness H s assumed to be unform. The undeformed profle of membrane s descrbed by polar coordnate system, (X, Φ, R). The cylndrcal membrane s nflated by the nternal pressure. X, x F r c(x, φ, r) P S s L l O C(X, Φ, R) R, r R Fg. Undeformed and deformed profle of cylndrcal orthotropc membrane The deformed cylndrcal membrane s referred to the polar coordnate system (x, φ, r). A materal partcle moves durng the deformaton from the poston n the undeformed profle, C(X, Φ, R) to the deformed profle, c(x, φ, r), along ts quas-equlbrum path. We assume the axsymmetrc deformaton, φ Φ. The prncpal stretch n axal and crcumferental drectons, prncpal curvatures and geometrc relatons are ds ds λ =, = r R λ, dr = sn, dx d =, κ =, κ = () ds ds ds r

3 where s s the arc length measured from pole (x = 0) to the partcle c(x, φ, r) along the merdan of the deformed profle. S s the length correspondng to s n the undeformed profle. We ntroduce an auxlary varable, the angle of the tangent lne. The radus r and the thckness h of the membrane are wth respect to the deformed confguraton. The radal stretch λ s determned from the ncompressblty constrant λ λ λ () = then h = H λλ () where R and H are the radus and the thckness n the undeformed confguraton.. Consttutve equatons The Cauchy stresses are defned as the partal dervatves of stran energy functon Ψ wth respect to the deformaton [, 4]. We have the followng expressons: Ψ σ σ = λ Ψ σ σ = λ λ ( λ, λ), λ ( λ, λ), If we express Ψ as a functon of the three prncpal stretches Ψ=Ψ(λ, λ, λ ) p*(j ), wth the ndetermnate Lagrange multpler p*, we can express Cauchy stresses [] as (4) Ψ, * σa = p + λa λ a a =,, (5) We assume the sochorc deformaton and we neglect the dsspaton due to rreversble effects. The energy stored n the fbers s assumed n the form of an exponental functon. The free energy functon n two dmensonal problem can be supposed n the form [] µ k Ψ( λ, λ ) = λ +λ +λ λ + exp[k ( λ cos +λ sn ) ], (6) ( ) { } = k where λ and λ are the axal and crcumferental stretches respectvely, and s the angle of the two famles of renforcng fbers. We suppose the renforcng fbers are double-helcally arranged n the matrx materal symmetrcally to the crcumferental drecton. The angle of fbers s supposed to be 48.8 o. The parameters µ and of Ogden s model of rubber [] are µ = 60 kpa, µ =. kpa, µ = -0 kpa, =., = 5, = -. The stress-lke parameter k and the non-dmensonal parameter k are determned from the expermental results and from the D cylndrcal membrane approxmaton.

4 4. Identfcaton of materal parameters The theory of nonlnear membranes has been presented by Green and Adkns [9] and appled to varous nflated structures [4]. The quas-statc equlbrum equatons of problem are d ds ( Tr) dr = T, κt+ κt= p, (7) ds where p s the nner pressure, T and T are the stress resultant forces per unt length n the merdonal and crcumferental drectons. The stress resultant forces n the deformed confguraton are T = hσ, T = hσ, (8) where Cauchy stresses σ and σ are gven by (5). We effectuated several seres of experments of nflaton of cylndrcal ar-sprng wth the varable axal force F and the nner pressure. The Cauchy stress s determned from the equlbrum n Fg. pπ r F σ = (9) πrh Substtutng r = λ R and () nto (9) we obtan σ as where σ = A λ( λ B) (0) pr A = H, B = F π pr We can deduce σ from equlbrum equaton (7). We assume σ = - p after the theory of nflated membrane. F () p σ σ r Fg.

5 After the substtutons nto equatons (4) we obtan set of the nonlnear equatons for the two varables k and k = ( ) µ λ λλ + 4kexp( km ) mλ sn = C+ p µ ( ) λ λλ + 4kexp( km ) mλ cos = p( D+ ) C = κ κ () κ µ λ λ + 4kexp( km ) m( λ cos λ sn ) = pd C + = κ p F where C = λ λr, D = H πrp H pr cos λλ, m = λ cos +λ sn The expermentally measured values of λ and λ n several ponts of the central part of our cylndrcal membrane were substtuted nto the equatons (). Takng the logarthm of () we wll get a set of lnear equatons for the varables lnk and k. The resultng overdetermned system of lnear equatons was solved n Matlab. The parameters were k =4.87e+04 kpa and k = The functon of the Helmholtz energy potental for these parameters s convex. 5. Determnaton of deformaton of cylndrcal membrane After the substtuton of (), (), (5) and (8) nto equatons of equlbrum (7) we get after some smplfcatons the system of fve ordnary dfferental equatons for the prncpal stretches λ and λ, the tangent angle, the coordnate x n the deformed confguraton and the nner pressure p wth respect to the coordnate X of the undeformed confguraton where dλ λsn sn = N A λλ + A λ ( λ B) Q dx A cos cos ( λ B) M R dλ = λ dx R.sn ( λ ) ( ( ) ) d = p + µ λ λλ + dx R A B λ = + 4kexp( km ) mλ cos Aλλ dx = λ.co s, dx dp 0 dx =, ()

6 ( λ B) ( ( ) ) Q= p+ µ λ λλ + R A λ = = + 4kexp( km ) mλ cos Aλλ M = µ + ( ) λ λλ + λ ( ) + 8kexp( km ) λ sn sn + + λ k m m N= µ λλ + k km λ λ km + λ = ( ) 8 exp( ) sn cos ( ) (4) (5) (6) We solved the set of dfferental equatons () by the shootng method n Matlab wth the boundary condton for 0 0 λ and λ determned from the experments. The results are at the Fg. where calculated stretches and deformed profle of membrane s compared wth expermental one. Deformed profle of a cylndrcal membrane - experment - calculated Fg. Results of deformatons of cylndrcal membrane of ar-sprng 6. Conclusons The deformatons of the nonlnear composte membrane were determned expermentally. The problem of the dentfcaton of the materal parameters was solved. The proposed stran energy functon was mplemented nto the calculus of deformatons of the cylndrcal membrane of ar-sprng. The deformatons were determned by numercal soluton the system of ordnary dfferental equatons based on the membrane theory. The method wll be used for

7 the nverse dentfcaton of materal parameters of the nflatable structures namely ar-sprngs. 7. Acknowledgement Ths work was realzed n the framework of the project MŠMT CEZ: MSM Interakce vbrozolačního objektu s člověkem a okolním prostředím." Fnancal support was provded by the Czech Mnstry of Educaton, Youth and Sports. 8. References [] Marvalová, B., Nam, T. H., Identfcaton of materal parameters and deformaton analyss of an nflated cylndrcal membrane of composte wth rubber matrx renforced by textle materal cords, proc. of 9 th nt. conference STRUTEX 00, Lberec, [] Bonet, J., Burton, A.J., A smple orthotropc, transversely sotropc hyperelastc consttutve equaton for large stran computatons, Comput. Methods Appl. Mech. Engrg. 6, 998, [] Holzapfel, G.A., Gasser, T.C., Ogden, R.W., A new consttutve framework for arteral wall mechancs and a comparatve study of materal models, J. of Elastcty, Nov., 000. [4] Guo, X., Large deformaton analyss for a cylndrcal hyperelastc membrane of rubberlke materal under nternal pressure, Rubber chemstry and technology, Vol. 74, 00, [5] Marvalová, B., Expermentální určení elastckých vlastností materálu válcové pryžové pneumatcké pružny, Proc. of VIII. Int. Conf. on the Theory of Machnes and Mechansms, IFTOM, Sept. 000, Lberec, [6] Marvalova, B., Urban, R., Identfcaton of orthotropc hyperelastc materal propertes of cord-rubber cylndrcal ar-sprng, proc. of 9 th nt. conference EAN 00, Tabor, 5-0. [7] Marvalova, B., Urban, R., Expermental analyss of deformaton and stress of nonlnear orthotropc hyperelastc membrane, proc. of 40 th conf. EAN 00, Praha, [8] Marvalová, B., Urban, R., Identfcaton of orthotropc hyperelastc materal propertes of cord-rubber cylndrcal ar-sprng, proc. of EUROMECH Colloquum 40, Prague, Czech Republc, October, -5, 00 [9] Green, A.E., Adkns, J.E., Bolšje upruge deformac nelnejnaja mechanka splošnoj stredy, Moskva, mr, 965. [0] Alena Pozvlova., Consttutve modellng of hyperelastc materals usng the logarthmc descrpton. PhD. Thess. Czech Techncal Unversty n Prague. [] Chevaugeon, N., Verron, E., Peseux, B., Fnte element analyss of nonlnear transversely sotropc hyperelastc membranes for thermoformng applcatons, Proc. of Europ. Congr. on Comput. Meth. n Appl. Sc.& Engrg. ECCOMAS 000, Barcelona.