COUPLED THERMO-MECHANICAL ANALYSES OF DYNAMICALLY LOADED RUBBER CYLINDERS 1

Transcription

1 COUPLED THERMO-MECHANICAL ANALYSES OF DYNAMICALLY LOADED RUBBER CYLINDERS Arthur R. Jhnsn and Tzi-Kang Chen Army Research Labratry, MS 24 Analytical and Cmputatinal Methds Branch NASA Langley Research Center Hamptn, VA 268- ABSTRACT A prcedure that mdels cupled therm-mechanical defrmatins f viscelastic rubber cylinders by emplying the ABAQUS finite element cde is described. Cmputatinal simulatins f hysteretic heating are presented fr several tall and shrt rubber cylinders bth with and withut a steel disk at their centers. The cylinders are cmpressed axially and are then cyclically laded abut the cmpressed state. The nn-unifrm hysteretic heating f the rubber cylinders cntaining a steel disk is presented. The analyses perfrmed suggest that the cupling prcedure shuld be cnsidered fr further develpment as a design tl fr rubber degradatin studies. KEY WORDS: Viscelasticity, Structural Analysis, Thermal Analysis. INTRODUCTION. Backgrund Rubber is emplyed t carry large lads in tires, gaskets, and tank track pads. It is als used t prvide damping and system stability in cmplex mechanical systems such as helicpter rtrs. In these applicatins, the rubber is typically stiffened by the additin f carbn black. The filled rubber tends t be a pr cnductr f heat, yet it als exhibits very large hysteretic energy lss during cyclic lading. Als, the mechanical prperties f rubber are strngly dependent n temperature. Faced with the abve issues, designers interested in mdeling the detailed respnse f cmplex-shaped rubber cmpnents need t be able t cmpute the cupled therm-mechanical behavir f rubber. An example f the imprtance f understanding the therm-mechanical respnse f filled rubber is given in a series f papers presented at the "Thirty Secnd Sagamre Army Materials Research Cnference" held a Lake Luzerne, NY in 985 (-4). In these papers, hysteretic heating, therm- This paper is declared a wrk f the U.S. Gvernment and is nt subject t cpyright prtectin in the United States.

2 mechanical degradatin, and fatigue f rubber-cated rad wheels and tank track pads are all discussed. Uncupled therm-mechanical finite element analyses, and sensitivity studies were cnducted with finite element cdes. It was bserved that the viscelastic prperties and the shape f the rubber slid are the mst imprtant factrs in determining temperature rise (). The degradatin studies indicate that the failure f cyclically laded "rubber-like" plyurethane blcks depends n the hard segment transitin temperature (2). Experiments were cnducted which prved that the large strain hysteretic heating rate des nt crrelate with the heating rates predicted using the ppular cmplex mdulus material data (). Als determined is the fact that failure under cyclic lading can be "significantly different frm that btained in cnstant rate testing" (4). These cnclusins suggest that detailed cmputatinal simulatins f large strain dynamic lading f rubber-like slids, perfrmed as part f a material degradatin study, require accurate mdeling f the large strain viscelastic prperties and a cupling f the mechanical and thermal mdels. The purpse f this wrk is t establish and test a cmputatinal tl fr analyzing hysteretic heating in rubber cmpnents. In this paper, large defrmatin rubber viscelasticity and heat transfer finite element mdels are cupled and emplyed t cmputatinally simulate the hysteretic heating f dynamically laded rubber cylinders. The frmulatins fr large strain rubber viscelasticity and heat transfer emplyed in ABAQUS are utlined belw t facilitate the descriptin f the therm-mechanical cupling perfrmed in this study. Detailed infrmatin n these tw frmulatins and their finite element implementatin is available in the ABAQUS Thery Manual (5). Additinal infrmatin n frmulatins fr rubber viscelasticity are available in the literature (6-). Only mderate temperature changes were simulated in this study, s time-temperature superpsitin is nt discussed..2 Rubber Viscelasticity The virtual wrk statement, withut inertial effects included, fr a slid f vlume V and surface area S is: T T δ WI = s : δdv dv = δv t ds + δv f dv [] V S V where δ WI is the internal energy due t the virtual displacement δ v, s is the Cauchy stress, δ D v is the virtual rate f defrmatin, t is the tractin stress vectr acting n S, and f is the bdy frce vectr acting within V. Details n the ntatin used belw are given in the Appendix. Equatin [] is used t build the mechanical finite element equilibrium equatins. The fllwing details describe hw the viscelastic behavir f rubber is apprximated. In the reference cnfiguratin, the strain energy part f Equatin [] is written: δ W Js : δd dv [2] I = v V where V is the vlume V in its reference state. When the slid is rubber, Equatin [2] is expressed as: 2

3 δw U U U 2 U vl I = + I : + J I I B δe δε 2 I 2 J V 2 B dv [] where U is the slid's strain energy density functin. In this effrt, a standard plynmial frm f the rubber's strain energy density was emplyed as fllws: U i = ( I ) ( I ) N N j Cij 2 + i j i D + = = i ( + εth) 2i [4] where the cnstants C ij and D i are determined frm measured stress-strain data, and ε th is the A histry integral methd is emplyed in ABAQUS t mdel thermal expansin strain. viscelastic material behavir. In the viscelasticity frmulatin the material cnstants in the strain energy density functin are made time dependent as fllws: C ij ( t) n = C ij g l e l= t τ l and n = k e t τ l ( ) l t D i l= Di [5] where the cnstants g l, k l, and stresses t experimental data (5, 9-2), the cnstants τ l are determined by fitting the analytical frm f relaxatin C ij and D i are the "elastic" cnstants, mentin abve in Equatin [4], determined frm high strain rate lading tests, and the superscript refers t the material's instantaneus respnse.. Heat Transfer. The variatinal statement f the energy balance equatin fr heat transfer, tgether with Furier's law, fr a defrmed slid f vlume V, and surface area S is: δθ x θ x θ ρ δθ dv + k dv = δθ r dv + V du dt V V S δθ q ds [6] where θ is the temperature, U θ is the internal thermal energy, ρ is the density, q is the heat flux per unit area, r is the heat supplied per unit vlume, k is a cnductivity matrix, and δθ is a virtual temperature field satisfying the essential bundary cnditins. Equatin [6] is used t build the transient heat transfer finite element equatins. 2. COUPLED THERMO-MECHANICAL MODEL Finite element discretizatin f the mechanical and thermal variatinal statements, Equatins [] and [6], results in systems f time dependent mechanical and thermal differential equatins. Fr the variatinal statements given abve, the nly therm-mechanical cupling that exists is thrugh the thermal expansin strain, ε th. Additinal therm-mechanical cupling can be apprximated by cmputing the rate f viscelastic energy dissipatin frm the mechanical equatins and inputting that rate int the thermal equatins. ABAQUS internally cmputes and stres a running ttal f the energy dissipated as a functin f time. Internal variables can be

4 emplyed t cmpute the energy dissipatin rate frm this running ttal and pass it n t the thermal analysis. 2. Element The eight-nde hybrid axisymmetric CAX8RHT element in ABAQUS was utilized fr the examples given belw. The element uses biquadratic displacement interplatin, and bilinear temperature and pressure interplatin. It als emplys reduced integratin (see Figure.) z N-4 ur, uz, θ u r, u z N-7 N- u r, uz, θ N-8 u r, u z p p p 4 p 2 N-6 u r, u z Nde Integratin pint N- u r, uz, θ N-5 u r, u z N-2 u r, uz, θ Figure. Axisymmetric CAX8RHT element. r The variables emplyed in the thermal element are the temperatures, θ, at the fur crner ndes. The stress element has radial, u r, and axial, u z, displacements at the eight ndes and hydrstatic pressure, p i, variables at each f the fur integratin pints. Heating rates are applied at the fur integratin pints. 2.2 Cupling Prcedure Tw internal variables are intrduced at each f the fur integratin pints in the element. When cnvergence is achieved, at the end f a time increment, the energy dissipated during the time increment is cmputed internally (fr each integratin pint in each element) by ABAQUS. The increment is added t and stred as a running ttal at each integratin pint. The "user subrutine" ptin in ABAQUS was emplyed with the tw internal variables t cmpute the energy dissipatin rate and input the rate t the thermal analysis. At each integratin pint, ne internal variable stres the "running ttal" dissipated energy at the beginning f the time interval. These stred values are used with the ABAQUS "running ttals" at the end f the time interval t cmpute the rates f energy dissipatin acrss the time interval at each integratin pint. The rate f energy dissipatin is stred as the secnd internal variable and is sent frward t the next time interval t serve as a heating rate in the thermal analysis. 2. Cylinder Dimensins, Material Prperties, and Lading T investigate the applicatin f the prcedure described abve, eight cylinders were analyzed fr hysteretic heating. Only mderate temperature changes were being simulated s the elastic material cnstants were nt 4

5 treated as temperature dependent. Having the elastic cnstants independent f temperature simplified the calculatins presented in this effrt, but it is nt a limitatin f the algrithm. 2.. Dimensins There were tw grups f fur cylinders each. One grup cnsisted f unifrm cylinders and the ther grup had steel disks at their centers, see Figure 2. All cylinders had the radius, R =. 282m. There were fur cylinder heights in each grup. The heights were H =.5 m,.75 m,.25 m,. 25 m respectively. The cylinders were cmpressed between steel fixtures. The mdel simulates the case when a lubricant maintains the fixture-rubber interface as frictinless. The internal steel disks were cmpletely attached (bnded) t the rubber. The height and radius f the disks were.25 m, and.4m, respectively. Axisymmetric FE Mdel Internal heat generated by the dissipated viscelastic energy Steel disk Rubber cylinder Heat cnvects t air at uter surface Heat cnducts t fixture at interface Dynamic lad Fixture R H Fixture Figure 2. Rubber cylinder, finite element mesh, fixture, and steel disk Rubber Prperties The rubber energy density was mdeled with a tw-term Mney- Rivlin (6) expansin and with tw terms in each Prny series (see Equatins [4] and [5]). The material cnstants emplyed are representative f a sft rubber and are listed belw ()-(5). The viscelastic cnstants are: C = Pa, C = Pa, and D = Pa and =.2, k =. 2, τ =. s 2 =.2, k2 =. 2, τ 2 =. s g 2 g The thermal material cnstants are: κ =.294 J / C m s Cnductivity ( ) Density ρ =. kg/ m Specific heat c = 29.4 J /( kg C) 6 Expansin a = 8. ( C) 5

6 The film heat transfer cefficients fr the rubber-air and rubber-steel interfaces are J / C m s = 294 J / C ms, respectively. h A = ( ) and ( ) 2.. Steel Prperties h S The elastic cnstants are: Yung s mdulus E = 26.8 Pissn s Rati ν =. GPa The thermal cnstants are: Cnductivity κ = J / ( C m sec) Density ρ = kg/ m Specific heat c = 46. J /( kg C) 6 Expansin a = 2. ( C) 2..4 Lading Each cylinder was subjected t a cnstant cmpressive lad with a superimpsed cyclic lad f sufficient magnitude t prduce large strain hysteresis. The lading was: f ( t) 2 4 sin ( 4.84t) N = [7] where t is the time in secnds. The maximum defrmatin f the tall ( H =.5m ) unifrm cylinder after cycles is shwn in Figure. The mesh refinement near the tp and uter edges is t accmmdate the heat transfer gradients at thse lcatins. z Surface film fr heat cnductin Reference mesh Frictinless bundary Axisymmetric hybrid element fr incmpressible viscelastic material Surface film fr heat cnvectin H/2 CL C Defrmed mesh Figure. Defrmatin f a tall unifrm cylinder (H =.5 m). r The lading and the displacement f the tp f the cylinder are shwn in Figure 4 fr a time interval f. s. As expected, the displacement curve demnstrates viscelastic sftening r "cyclic creep." 6

7 Lad Displacement Time (s) Figure 4. Viscelastic sftening f a tall unifrm cylinder (H =.5 m).. COMPUTED TEMPERATURE DISTRIBUTIONS A number f analyses were perfrmed t evaluate the nnlinear elastic, the viscelastic, and the transient heat transfer finite element algrithms separately. Hand calculatins and finite difference calculatins verified that the algrithms functined crrectly. In the preliminary calculatins, the cupling prcedure was used with prperties representative f track pad materials and the rate at which the temperature increased was similar t rates measured fr tests n track pad materials (2). The thermal bundary cnditins did nt significantly affect the temperature fields cmputed. The fllwing analyses were perfrmed, emplying the cupling prcedure described abve, t investigate the heating f sft rubber cylinders underging large strain dynamic defrmatins.. Unifrm Cylinders The cyclic lad given by Equatin [7] was applied t the fur unifrm cylinders described abve. The temperatures, at the center f the cylinders as a functin f time fr the first 2 s f lading are presented in Figure 5. The frictinless rubber-fixture surface allws the strains t be unifrm in the cylinder. Rubber is a pr cnductr f heat and the fact that the cylinders heated nearly unifrmly regardless f shape (tall r shrt) was expected..2 Cylinders Cntaining a Disk It is difficult t estimate hysteretic heating in rubber slids f cmplex shape because cupled therm-mechanical analyses are needed in regins f high strain gradients. The grup f cylinders cntaining disks were cyclically laded t bserve the ability f the cupling prcedure t predict the distributin f viscelastic heating in a rubber slid with high strain gradients. The results btained are reasnable. Figure 6 shws the meshes n the 7

8 reference and defrmed shapes fr the tall cylinder. The temperature distributin in the tall blck after 2 s f dynamic lading is shwn in Figure 7. Temperatures at pints A, B, C, and D in Figure 7 are pltted as a functin f time in Figure 8. The uter radial end f the internal disk (pint C) is predicted t heat much faster than the rest f the cylinder. Small scillatins in the cmputed heating rates were bserved in these cmputatins. The cause f these scillatins has nt been determined. 2. Fur curves verlapped Each cylinder heated at nearly the same rate Time (s) Figure 5. Temperature as a functin f time at the center f each unifrm cylinder. z Surface film fr heat cnductin Frictinless bundary Reference mesh Surface film fr heat cnvectin Defrmed mesh Steel disk CL Figure 6. Defrmatin f a tall cylinder with an internal disk (H =.5 m). r 8

9 z A B CL r Steel disk C D Figure 7. Temperature distributin in a tall cylinder with an internal disk (H =.5 m) C D A Time (s) Figure 8. Temperature as a functin f time fr a tall cylinder with an internal disk (H =.5m). B The results were similar fr the shrter cylinders. The temperature distributin in the shrtest cylinder is presented in Figure 9. As in the case f the tall cylinder, the regin f high strain gradient lcated near the uter radial end f the internal disk (pint C) has the mst rapid rise in temperature (see Figure.) 9

10 A B z r CL Steel disk Figure 9. Temperature distributin in a shrt cylinder with an internal disk (H =.25 m). C D H =.25 m C A B D Time (s) Figure. Temperature as a functin f time fr a shrt cylinder with an internal disk (H =.25 m). 4. CONCLUSIONS Accurate predictins f the strain and temperature distributins in rubber cmpnents, emplyed in dynamically laded structures, are required t perfrm degradatin studies. A prcedure which cuples a viscelastic stress analysis with a heat transfer analysis was described. The ABAQUS finite element cde was emplyed t demnstrate the prcedure. Bth the stress and the thermal analyses are valid fr large strains and displacements. A user subrutine was written t track dissipatin f energy acrss time intervals and determine heating rates. The heating rate data is passed frward ne time interval in the prcedure.

11 The therm-mechanical heating f tall and shrt unifrm rubber cylinders (withut internal disks) was cmputed with the cupling prcedure. The viscelastic material prperties emplyed are valid fr large strain defrmatins f sft rubber. The time dependent strains in the cylinders were unifrm, and unifrm heating was cmputed. The therm-mechanical heating f tall and shrt rubber cylinders with internal steel disks was als studied. The internal steel disks prvided high strain gradients within the rubber cylinders and nnunifrm hysteretic heating is bserved. The analyses perfrmed suggest that the cupling prcedure shuld be cnsidered fr further develpment as a design tl fr rubber degradatin studies. An integrated effrt in which the large strain viscelastic material prperties, the heat transfer prperties, and the heating rates due t cyclic lading are all determined is recmmended. 5. REFERENCES. Lesuer, D. R., Gldberg, A., and Patt, J., in Elastmers and Rubber Technlgy, Eds. R. E. Singler and C. A. Byrne, Library f Cngress 86-66, pp Mead, J. L., Singh, S., Rylance, D. K., and Patt, J., in Elastmers and Rubber Technlgy, Eds. R. E. Singler and C. A. Byrne, Library f Cngress 86-66, pp Mead, J. L., and Pattie, E. R., in Elastmers and Rubber Technlgy, Eds. R. E. Singler and C. A. Byrne, Library f Cngress 86-66, pp McKenna, G. B., Bullman, G. W., Flynn, K. M., and Patt, J., in Elastmers and Rubber Technlgy, Eds. R. E. Singler and C. A. Byrne, Library f Cngress 86-66, pp ABAQUS Thery Manual, Versin 5.8, Hibbitt, Karlssn & Srensen Inc., 8 Main St., Pawtucket, RI, Ward, I. M., Mechanical Prperties f Slid Plymers, Jhn Wiley and Sns, Malvern, L. E., Intrductin t the Mechanics f a Cntinuus Medium, Prentice-Hall, Ogden, R. W., Nn-Linear Elastic Defrmatins, Ellis Hrwd Limited, Jhnsn, A. R., Quigley, C. J., and Mead, J. L., Rubber Chemistry and Technlgy, 67(5), 94 (994).. Quigley, C. J., Mead, J., and Jhnsn, A. R., Rubber Chemistry and Technlgy, 68(), 2 (995).. Hill, S. A., NASA TM-894, February Chen, T., NASA TM-2-22 (ARL-TR-226), May Clark, S. K. and Ddge, R. N., NASA Cntractr Reprt 629, Pitts, D. R. and Sissm, L. E., Heat Transfer, McGraw-Hill Bk C. Library f Cngress

12 APPENDIX The fllwing ntatin is included t assist the reader with the descriptin f the ABAQUS finite element algrithm fr rubber viscelasticity. X i (i =,2,) crdinates f a material pint in the reference cnfiguratin. x i (i =,2,) crdinates f a material pint in the defrmed cnfiguratin. ( X,t) x = x vectr mapping between the reference and defrmed cnfiguratins. x F = X defrmatin gradient. J = det F determinate f F which measures vlume change. ( ) F = J F defrmatin gradient scaled fr vlume change. T B = F F left Cauchy Green strain tensr. I = tr( B) first strain invariant (adjusted fr vlume). ( I ) ( B B) ) 2 I2 = 2 tr secnd strain invariant (adjusted fr vlume). δ u displacement. u δ L = δ x gradient f displacement. T δ D = δl + δl 2 rate f defrmatin. ( ) δ D v rate f defrmatin cmputed using virtual displacement δ v. A: B scalar prduct f matrices A and B. vl δε = I : δd vlumetric strain rate. vl δe = δd δε I deviatric strain rate. p = I : s pressure stress (hydrstatic). 2