Available olie at www.sciecedirect.com Procedia Egieerig 48 (0 ) 388 395 MMaMS 0 Fiite Elemet Aalysis of Rubber Bumper Used i Air-sprigs amas Makovits a *, amas Szabó b a Departmet of Mechaical Egieerig, Uiversity of Debrece, Debrece 408, Hugary b Departmet of Robert Bosch Mechatroics, Uiversity of Miskolc, Miskolc 355, Hugary Abstract I this paper a solutio of a cotact problem for large displacemets ad deformatios are aalyzed where a rubber bumper applied i airsprig. he oliear load-displacemet curve is determied. A FEM code writte i FORRAN has bee developed for the aalysis of early icompressible axially symmetric rubber parts. he Hu Washizu type variatioal priciple is formulated for the Mooey-Rivli material model. Stability ad sesitivity aalyses are also ivestigated. 0 he Authors. Published by Elsevier Ltd. 0 Published by Elsevier Ltd.Selectio ad/or peer-review uder resposibility of the Brach Office of Slovak Metallurgical Society at Faculty Selectio of Metallurgy ad/or peer-review ad Faculty uder of Mechaical resposibility Egieerig, of the Brach echical Office Uiversity of Slovak of Košice Metallurgical Ope access Society uder at CC Faculty BY-NC-ND of Metallurgy licese. ad Faculty of Mechaical Egieerig, echical Uiversity of Košice. Keywords: fiite elemet method; rubber; large displacemets; cotact. Itroductio Numbers of papers are devoted to the applicatio of h-versio fiite elemet method for the aalysis of hyperelastic materials [-6]. he applicatios of p-versio of the fiite elemet for geometrically ad physically oliear problems are relatively recet [7-0]. It is well kow that the elastomers are regarded as icompressible or early icompressible materials. he icompressibility is a auxiliary coditio which ca be eforced by the so called mixed method. I this paper the traditioal pealty method ad the mixed formulatio of three fields are implemeted. he later oe is based o the Hu Washizu type variatioal priciple []. he displacemets ad the volumetric chage are approximated idepedetly from the hydrostatic pressure. he same order of approximatio is selected for the hydrostatic pressure ad the volumetric chage. he displacemet fields are approximated by the tesor product space ad its order is higher tha the order of space used for the pressure ad the volumetric chage at each level of polyomial degree p. I egieerig practice high stresses occur i uilateral cotact. he cotact problem is hadled with a simplified pealty approach. he cotactig boudary is approximated by a polygo. he edge of the cotactig elemet is forced to have a straight lie. I practice, two-ode cotact elemets were implemeted. As a umerical example a rubber bumper applied i air-sprig is chose for this problem. he load-displacemet curve for pressure is give by the producer but oly whe the bumper is loaded betwee two flat rigid surfaces. he aim of this research is to obtai the load deflectio curve up to 0% of compressio i realistic workig circumstaces. * Correspodig author. el.: +36-5-45-55; E-mail address: tamas.makovits@eg.uideb.hu 877-7058 0 Published by Elsevier Ltd.Selectio ad/or peer-review uder resposibility of the Brach Office of Slovak Metallurgical Society at Faculty of Metallurgy ad Faculty of Mechaical Egieerig, echical Uiversity of Košice Ope access uder CC BY-NC-ND licese. doi:0.06/j.proeg.0.09.530
amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 389. Large displacemets ad deformatios Usig the Lagragia descriptio i the referece cylidrical coordiate system the motio of cotiua r = r ( r 0, t), i the curret cofiguratio at time t is give by the referece coordiates u is the displacemet. 0 r of the material poits, see i Fig.., where z V 0 t=0 P 0 (r 0,z 0,ϕ 0 ) dv 0 r 0 r r V u 0 =u dv P(r,z,ϕ) t=t Fig.. he motio of cotiua I the subsequet formulatio the deformatio gradiet is multiplicatively decomposed ito a volumetric chagig part r F = () 0 r F ad a volume preservig part Fˆ V F = F Fˆ () V where ad 3 F ˆ = J I, det F ˆ = (3) 3 F V = J F, det F V = det F = J (4) where J is the Jacobia. Usig this decompositio we ca defie the so called right Cauchy-Gree tesor ad its volumetric preservig part C = F F, Cˆ Fˆ Fˆ 3 = = J Cˆ (5) where deotes the traspose of a tesor. he deformatio is expressed by the Gree-Lagrage strai tesor E = ( C I) (6)
390 amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 where I is the uit tesor. he strai eergy fuctios are defied by this specific decompositio W ( J, Cˆ) = U ( J ) + Wˆ ( Cˆ), (7) where U (J ) deotes the strai eergy due to volumetric chage ad W ˆ ( C ˆ) is the strai eergy determied by volume preservig deformatio. he fuctio U (J ) is ofte defied i the papers i the simpliest way where κ is the bulk modulus. he fuctio ˆ ( C ˆ) Wˆ ( Cˆ) U ( J ) = κ J, (8) ( W is give for the Mooey-Rivli material ) μ ( Iˆ 3) + ( ˆ I μ I 3) (9) = 0 0 II where μ 0 ad μ 0 are the material costats, Î I ad Cauchy-Gree tesor defied as Î II are the first ad secod ivariats of the uimodular right ˆ ˆ ˆ ˆ I I = C + C + C33, Iˆ ( ˆ ˆ ˆ II = I I C C) (0) I the referece coordiate system, the II. Piola-Kirchhoff stress tesor for the rubber is give by Wˆ ( Cˆ) S = + pjc () C where p is the hydrostatic pressure. I the curret cofiguratio the Cauchy stress tesor ca be calculated as = J F S F. () 3. Cotact kiematics Let us cosider a system which cosists of two bodies. Oe is elastic siged by, the other is a rigid body siged by see i Fig.., where A c is the cotact regio. V A c A c h V u Q Q c Fig.. Kiematics i the cotact regio
amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 39 I this work ormal cotact is assumed, where Q ad Q. he g ormal gap ca be defied as g c is the ormal uit vector of the cotact surfaces through two poit pairs = g ( u ) = u h, r Ac, (3) + where h is the iitial gap ad u is the ormal compoet of the displacemet vector. here ca be two cases, oe is cotact, whe there is gap (o cotact) betwee the two bodies if where p is the cotact pressure. Both cases 4. Variatioal priciples ad FEM discretizatio g 0 p 0, (4) = g 0 p = 0, (5) p g = 0, r Ac. (6) he mixed formulatio based o the Hu Washizu type variatioal priciple [], the three fields are idepedet from each other, Π hw ˆ ( u, J, p) = W ( Cˆ) dv + U ( J ) dv p( J J ) dv g da ext ( u) V + + Π V V γ, (7) Ac where J is the volumetric chage, J is evaluated by equatio (4) makig use of the displacemets u. he γ is the pealty parameter which ca be metioed as a virtual sprig stiffess called as a Wikler-type cotact problem. Sice equatio (7) is a oliear oe we should liearize it itroducig the icremets of the appropriate variables. Before the FEM discretizatio we itroduce the Gree-Lagrage strai tesor (6) ad the eergy cojugated II. Piola- Kirchhoff stress tesor (). he icremet i the Gree-Lagrage strai tesor Δ E is decoupled ito liear ad oliear parts Δ E = ΔE + ΔE. (8) Meridia sectio is discretized by two-dimesioal rectagular elemets see i Fig. 3. L NL z 4 8 η 7 9 5 3 η 4 7 3 6 ξ 8 6 9 r (-,) (-,-) 5 (,) (,-) Fig. 3. Nie-ode two-dimesioal isoparametric elemet
39 amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 I case of mixed formulatio o each elemet the displacemet fields are approximated by the so called tesor product space usig Legedre polyomials [] u = = N iui, i = 4 + 4( p ) + ( p ), (9) where N i is the shape fuctio, u i are the displacemet parameters, is the umber of shape fuctios which oliearly depeds o the level of approximatio p. he volumetric chage ad the hydrostatic pressure are approximated by lower order of polyomials tha the displacemet J 0 ( p = ) = a + a r + a z, (0) p 0 After the discretizatio we obtai the formula for the Newto-Raphso iteratio ( p = ) = b + b r + b z. () K Δu = f ext f it = Δf, () where K is the tagetial stiffess matrix, Δ u is the displacemet icremet ad Δ f is the ubalaced load vector. A p-versioal fiite elemet code has bee developed for the aalysis of early icompressible materials. he approximatio order for the displacemet is p =. 5. Numerical example A rubber bumper i a air-sprig (see i Fig. 4) is aalyzed by the FEM code based o the theory discussed above. Fig. 4. he air-sprig he load-displacemet curve for pressure is kow from the producer but oly whe the bumper is loaded betwee two flat rigid surfaces. A parameter optimizatio is eeded to fid the material parameters. he measuremet ad the code were compared, see i Fig. 5.
amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 393 Fig. 5. he load-displacemet curve A umerical aalysis is also eeded to obtai the optimal iput parameters for the fiite elemet aalysis such as mesh desity ad the so called pealty parameter. he effect of the mesh desity chage was compared at two discrete poits (7 ad 4 mm compressio). he results of the mesh desity aalysis ca be see i Fig. 6. he 8x8 mesh gives good correlatio to the more fied mesh ad its time demad is much more cost efficiet. he mesh desity versus the volume ratio is evaluated, see i Fig. 6. (a) (b) Fig. 6. Numerical aalyses for (a) mesh desity ad (b) volume ratio I examiatio of a rubber part it is very importat to observe how the icompressibility coditio is fulfilled which depeds o the applied pealty parameter. I Fig. 7. the umerical stability aalysis calculated with differet pealty parameters at mesh desity 8x8 (acceptace limit is 99,9%) ad the displacemet field calculated by the FEM code ca be see. (a) (b) Fig. 7. Results of (a) pealty parameter aalysis ad (b) the code
394 amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 After the umerical stability aalysis the code parameters are determied from the results. I able the most importat fiite elemet settigs ca be see. able. Fiite elemet settigs Material parameters Mooey-Rivli parameter ( μ 0 ) 0,63N/mm Mooey-Rivli parameter ( μ 0 ) 0,575N/mm Fiite elemet code Pealty parameter ( κ ) 000 u mm Icremet ( Δ ) Supports (top ad below) vulcaized he rubber bumper may cotact with two cotact regios i workig circumstaces. he mechaical model of the cotact problem ad the results of the code for displacemet ca be see i Fig. 8a ad Fig 8b, respectively. z RUBBER SEEL (a) r (b) Fig. 8. Rubber bumper s (a) mechaical model ad (b) displacemet at 3mm compressio he load-displacemet curve calculated by the FEM code ca be see i Fig. 9. Fig. 9. Load-displacemet curve of the rubber bumper i workig circumstaces
amas Makovits ad amas Szabó / Procedia Egieerig 48 ( 0 ) 388 395 395 6. Summary A solutio of a cotact problem for large displacemets ad deformatios are aalyzed where a rubber bumper applied i air-sprig is ivestigated by determiig the oliear curve of load versus displacemets. A FEM code which is able to hadle cotact writte i FORRAN has bee developed for the aalysis of early icompressible axially symmetric rubber parts. Later a shape optimizatio problem will be doe if the stability aalysis ad also the compariso with commercial software are adequate. Ackowledgemets he described work was carried out as part of the ÁMOP-4../B-0/-00-0008 project i the framework of the New Hugaria Developmet Pla. he realizatio of this project is supported by the Europea Uio, co-fiaced by the Europea Social Fud. Refereces [] Malkus, D.S., 980. Fiite Elemets with Pealties i Noliear Elasticity, Iteratioal Joural for Numerical Methods i Egieerig 6, p. -36. [] Swaso, S.R., Christese, L.W., Esig, M., 985. Large Deformatio Fiite Elemet Calculatios for Slightly Compressible Hyperelastic Materials, Computers & Structures, No. /, p. 8-88. [3] Ziekiewicz, O.C., Qu, S., aylor, R.L., Nakazawa, S., 986. he Patch est for Mixed Formulatios, Iteratioal Joural for Numerical Methods i Egieerig 3, p. 873-883. [4] Sussma,., Bathe, K.J., 987. A Fiite Elemet Formulatio for Noliear Icompressible Elastic ad Ielastic Aalysis, Computers & Structures 6, No. /, p. 357-409. [5] Simo, J.C., aylor, R.L., 99. Quasi-icompressible Fiite Elasticity i Pricipal Streches. Cotiuum Basis ad Numerical Algorithms, Computatioal Methods i Applied Mechaics ad Egieerig 85, p. 73-30. [6] Gadala, M.S., 99. Alterative Methods for the Solutio of Hyperelastic Problems with Icompressibility, Computers & Structures 4, No., p. -0. [7] Düster, A., Hartma, S., Rak, E., 99. p-fem applied to Fiite Isotropic Hyperelastic Bodies, Computatioal Methods i Applied Mechaics ad Egieerig 85, p. 547-566. [8] Nádori, F., Páczelt, I., Szabó,., 003. Aalysis of Icompressible Materials with p-versio Fiite Elemets, microcad 003 Iteratioal Scietific Coferece, p. -6. [9] Hartma, S., Neff, P., 003. Polycovexity of Geeralized Polyomial-type Hyperelastic Strai Eergy Fuctios for Near-icompressibility, Iteratioal Joural of Solids ad Structures40., p. 767-79. [0] Szabó, Makovits,., 004. FEM Computatios of Hyperelastic Materials, microcad 004 Iteratioal Scietific Coferece, p. 79-84. [] Boet, J., Wood, R.J., 997. Noliear Cotiuum Mechaics for Fiite Elemet Aalysis, Cambridge Uiversity Press. [] Szabó, B.A., Babuska, I., Chayapathy, B.K., 989. Stress Computatio for Nearly Icompressible Materials by the p-versio of the Fiite Elemet Method, Iteratioal Joural for Numerical Methods i Egieerig 8., p. 75-90.