Expermentally-base Vscoelasc Moel for Polyurea, Inclung Pressure an emperature Effects Alreza V. Amrkhz, Grauate Stuent, Center of Excellence for Avance Materals, Department of Mechancal an Aerospace Engneerng, Unversty of Calforna, San Dego, La Jolla, CA 92093-0416 Jon Isaacs, Sr. Development Engneer, Center of Excellence for Avance Materals, Department of Mechancal an Aerospace Engneerng, Unversty of Calforna, San Dego, La Jolla, CA 92093-0416 Sa Nemat-Nasser, Professor, Center of Excellence for Avance Materals, Department of Mechancal an Aerospace Engneerng, Unversty of Calforna, San Dego, La Jolla, CA 92093-0416 ABSRAC he results of a systematc stuy of the vscoelastc propertes of polyurea over broa ranges of stran rates, temperatures, an pressures are presente. Base on the expermental ata an a master curve evelope by Knauss [1] we have prouce a moel for large-eformaton vscoelastc response of ths elastomer. Hgh stranrate ata are obtane usng Hopknson bar experments. We show that the ncluson of pressure senstvty nto the moel successfully reprouces the expermental results. We also present an equvalent smplfe moel that nvolves a fnte number of nternal state varables talore for mplementaton nto explct fnte-element coes. he moel ncorporates the classcal Wllams-Lanel-Ferry (WLF) tme-temperature transformaton [2] an pressure senstvty, n aton to a thermoynamcally soun sspaton mechansm. Fnally we show that usng ths moel for the shear behavor of polyurea along wth the elastc bulk response, one can successfully reprouce the very hgh stran rate pressure-shear expermental results recently reporte by Clfton an Jao [3]. Consttutve moel We have evelope a complete moel for the response of polyurea uner a we range of stran-rate, temperature, an pressure. Most polymers have substantally fferent behavor uner shearng an volumetrc eformatons. In ths case, shear eformaton s moele by a temperature- an pressure-epenent vscoelastc relaxaton moulus. he volumetrc response s also temperature-senstve but s moele as nonlnearly elastc. Here we summarze ths moel; for more etal see [4]. Base on WLF tme-temperature superposton prncple, the shear relaxaton curve of a polymer can be represente for any temperature n the range, as, Here g g 100K t G( t, ) G(, ). a(, g s the glass transton temperature, G ( t, ) s the relaxaton moulus at temperature, g 50K, an (, erence temperature usually taken to be temperature an pressure P to that at the erence contons, (1) s a a s the rato of the relaxaton tme-scale at an zero pressure: Proceengs of the 2006 SEM Annual Conference an Exposton on Expermental an Apple Mechancs
(K) A B(K) C tp (K/GPa) C V (J/mm 3 /K) CE(/K) m(gpa/k) n (GPa) G (GPa) 273-10 107.54 7.2 1.977 10-3 2 10-4 -0.015 4 4.948 0.0224 p 1 p 2 p 3 p 4 q 1 (ms) q 2 (ms) q 3 (ms) q 4 (ms) 0.8458 1.686 3.594 4.342 463.4 0.06407 1.163 10-4 7.321 10-7 able 1. Consttutve parameters of polyurea use n numercal moelng. All parameters except for the heat capacty C V an the coeffcent of thermal expanson CE are efne n the text. ( ) a(, a( ) 10 A( B ) Here A an B are materal propertes, an s the effectve temperature that epens lnearly on pressure, C P. tp he master curve can be approxmate by a Prony seres wth any number of terms, G t / q t G 1 p e. n 1. he tme ntegrals neee for the stress calculaton can then be evaluate at each tme step usng a fnte number of approprate nternal state varables. Otherwse, one woul nee to ntegrate the complete eformaton hstory at each tme step ncrement, whch woul rener numercal calculatons prohbtvely resource consumng. Furthermore the sspate power s also calculate usng the same state varables, W t Here volumetrc eformaton, G t n 1 p q t W s the sspate work an t :, t s the -th nternal varable. he pressure s calculate from the ln J P, J where J s the Jacoban of the eformaton, an s a temperature epenent bulk moulus, ) ( ) m( ). ( Splt Hopknson-bar experments We have performe a seres of splt-hopknson bar experments on polyurea uner varous contons. o verfy the moel scusse above, a selecte set of these experments s use here. he tests presente here were all performe at an effectve engneerng stran rate of 3000±400/s. he summary of the expermental parameters s gven n able 2. All 4 tests are performe usng a 12.7mm splt-hopknson bar (maragng steel bars). For the confne tests, the sample s ftte nse a steel cylnrcal tube of 17.8mm OD an 26mm length. he Cauchy stress must be estmate for the unconfne test snce the ameter of the sample changes wth ncreasng axal loa. Snce uner the low pressures observe n the unconfne tests, polyurea s nearly ncompressble, we calculate the ameter an the Cauchy stress assumng sochorc eformaton. he resultng loang stressstran curves are shown n Fgure 1. From the ntal part of the unloang curves, one observes that, for the confne tests, the unloang follows essentally the same stress-stran path as that of the loang. In the unconfne case, however, the stress s release faster than the accumulate stran. hs stran s not permanent though an, n all cases, the sample regane ts ntal length after the test was complete. FEM moelng of a pressure-shear test he consttutve moel has been use to smulate one of the pressure-shear tests performe at Brown Unversty an ocumente n [3]. A steel flyer plate ( flyer = 6.991mm) mpacts at velocty V 0 =112.6m/s a sanwch structure that conssts of a front steel plate ( front = 2.896mm), a thn layer of elastomer ( elastomer = 0.11mm), an a rear steel (2) (3) (4) (5) (6) (7) Proceengs of the 2006 SEM Annual Conference an Exposton on Expermental an Apple Mechancs
plate ( rear = 7.041mm). All of the plates are algne at a =18 angle wth respect to the velocty recton an have ameter D = 60mm; see Fgure 2. Upon mpact, elastc waves are create that travel normal to the surface of the mpact. hese are: a longtunal compresson wave (hgh velocty) an a shear wave (low velocty). he mpact parameters are set such that the steel plates reman elastc. he longtunal pressure wave reaches the elastomer layer frst an loas t to a maxmum stress after a few reverberatons. Both normal an transverse partcle veloctes are measure on the back surface of the rear plate usng optcal methos. Name Confnement Dameter (mm) Length(mm) Effectve Stran Rate (/s) emperature (K) UC No 6.17 1.78 3400 294 CL Yes 12.7 5.08 2600 273 CR Yes 12.7 5.08 2800 294 CH Yes 12.7 5.08 2800 333 able 2. Hopknson-bar experments setup. he center of the whole structure, consstng of the flyer, front, an rear plates an the elastomer layer s moele wth three-mensonal elements usng the elastc propertes of steel an the nonlnear vscoelastc user-efne consttutve subroutne for the polyurea. he bounary contons are prescrbe such that the materal s confne laterally but allow for shear eformaton. We constrane the top an bottom noes to have the same splacement egrees of freeom; see Fgure 2. hs mantans a fxe lateral menson an hence the confnng pressure s apple automatcally by the fnte-element solver. At the same tme the element can be sheare laterally. he propagaton of a fnte ampltue elastc shear wave n a unaxally pre-strane layer of elastomer has been scusse n [5]. he expecte partcle velocty on the back surface of the rear plate for an elastc wave woul consst of steppe rses that fnally converge to the mpact transverse velocty regarless of the stffness of the elastomer; see Fgure 3. Nether of these two propertes s observe n the measure transverse velocty by Clfton an Jao [3]. Instea, there s a sngle jump at the begnnng, followe by a graual rse n the velocty. We sought to reprouce these characterstcs usng our vscoelastc moel. he parameters are as lste n able 1 except for the ones ncate n the graph an a lnear bulk moulus of elastcty of =22.5GPa. he long-tme shear moulus, G, an the pressure-senstvty parameter, C tp, are mofe to examne ther effects. Snce we are prmarly ntereste n the shear behavor of the elastomer uner stress, a lnearly elastc moel s use for the bulk response to smplfy the calculaton. Fgure 3 shows that the vscoelastc moel properly captures the qualtatve behavor seen n the experment. By changng the long-tme shear moulus an the pressuresenstvty parameter we are able to reprouce the expermental ata wth reasonable accuracy. Cauchy Sterss (GPa) 0.060 0.040 0.020 UC: Expermental Data UC: Moel Cauchy Stress (GPa) 0.600 0.500 0.400 0.300 0.200 CL: Experment CL: Moel CR: Experment CR: Moel CH: Experment CH: Moel 0.100 0.000 0.00 0.10 0.20 0.30 0.40 Logarthmc Stran 0.000 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Logarthmc Stran Fgure 1. [Left] he unconfne Hopknson-bar test results for polyurea at =273K, an the consttutve-moel result, val up to 8% stran; the stress s estmate base on the lateral expanson precte by the moel. [Rght] Confne Hopknson-bar test an the consttutve-moel results. Proceengs of the 2006 SEM Annual Conference an Exposton on Expermental an Apple Mechancs
V 0 Measurement of normal an transverse partcle veloctes Flyer Front Rear V 0 v 1 Polyurea u 1 =u 2 v 1 =v 2 v 2 u 1 u 2 Fgure 2. Schematcs of the pressure shear experment an the FEM moel. [op] Flyer plate mpacts the front plate at velocty V 0, creatng normal an transverse elastc waves that travel an loa the polyurea layer an eventually the back plate. [Mle] he elements along the center lne passng through the plates an polyurea layer are moele usng LS-DYNA [6]. [Bottom] he constrane central elements. Normalze Velocty 1 0.75 0.5 0.25 ransverse Velocty (m/s) 14 12 10 8 6 4 2 Expermental Data Gnf=22, Ctp=7.2 Gnf=22, Ctp=6 Gnf=30, Ctp=6 Gnf=40, Ctp=5 Gnf=60, Ctp=5 Gnf=60, Ctp=5.5 0 0 0.5 1 1.5 2 Normalze me 0 3 3.5 4 4.5 5 5.5 6 me (µs) [Shfte for smultaneous arrval] Fgure 3. [Left] he profle of the normalze transverse partcle velocty (ve by V 0 sn) on the back surface of the rear plate for a fully elastc materal. he tme s normalze through vng by (l/(v 0 sn)), where l s the thckness of the elastomer. [Rght] he profle of the transverse partcle velocty as measure an calculate on the back surface of the rear plate. he sol curve epcts the expermental results [3] an other curves show the varous possble responses by varyng two parameters: the equlbrum shear moulus, G (n MPa), an the pressure-senstvty parameter, C tp (n K/GPa). Proceengs of the 2006 SEM Annual Conference an Exposton on Expermental an Apple Mechancs
Acknowlegements he authors wsh to thank Ro Clfton an ong Jao for sharng wth us the ata from ther pressure-shear experments an Wolfgang Knauss for provng us wth the relaxaton master curve of polyurea. hs work was supporte by ONR N00014-03-M-0172 uner Dr. Roshy Barsoum s program. References [1] Knauss, W. Vscoelastc materal characterzaton relatve to consttutve an falure response of an elastomer Interm Report to the Offce of Naval Research, GALCI, Pasaena, CA, 2003. [2] Wllams, M. L., Lanel, R. F., Ferry, J. D., he emperature Depenence of Relaxaton Mechansms n Amorphous Polymers an Other Glass-formng Lqus J. Am. Chem. Soc., Vol. 77, 3701-7, 1955. [3] Clfton, R., Jao,. Hgh stran rate response of elastomers n: Presentaton to ERC ACD Workshop, Cambrge, MA, 2004. [4] Amrkhz, A. V., Isaacs, J., McGee, J., an Nemat-Nasser, S. An expermentally-base vscoelastc consttutve moel for polyurea, nclung pressure an temperature effects submtte to Phlosophcal Magazne, 2006 (n revew). [5] Nemat-Nasser S., Amrkhz, A. V. Fnte ampltue shear wave n pre-stresse thn elastomers Wave Moton Vol. 43, 20-8, 2005. [6] Hallqust, J. O., LS-DYNA heoretcal Manual, LSC, Lvermore, CA, 1998. Proceengs of the 2006 SEM Annual Conference an Exposton on Expermental an Apple Mechancs