The behavior of elastomers at high strain rates

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1 Structurs Undr Shock and Impact IX 97 Th bhavior of lastomrs at high strain rats M. S. Hoo Fatt & X. Ouyang Dpartmnt of Mchanical Enginring, Th Univrsity of Akron, Ohio, USA Abstract Tnsil impact xprimnts wr prformd to obtain strss-xtnsion ratio curvs of uniaxial strip spcimns and dynamic forc-xtnsion curvs of thin rubbr shts at impact rats of loading. Rsults from th uniaxial tnsion tsts indicatd that although th rubbr bcam stiffr with incrasing strain rat, th strss-xtnsion ratio curvs rmaind virtually th sam abov 80 s -. Abov this critical strain rat, strngth, fractur strain and toughnss dcrasd with incrasing strain rat. Whn strain rats wr blow 80 s -, th initial modulus, tnsil strngth and braking xtnsion incrasd as th strain rat incrasd. Btwn strain rats of 80 s - and 80 s -, th initial modulus and tnsil strngth incrasd with incrasing strain rat but th xtnsion at brak dcrasd with incrasing strain rat. A hypr-viscolastic constitutiv rlation of intgral form was usd to dscrib th rat-dpndnt matrial bhavior of th rubbr. Th proposd constitutiv quation was implmntd in ABAQUS Explicit via a usr-dfind subroutin and usd to prdict th dynamic rspons of th rubbr shts in th xprimnts. Numrical prdictions for th transint dformation and failur of th rubbr sht wr within 0% of xprimntal rsults. Kywords: lastomrs, high strain rats, hypr-viscolastic constitutiv quation, finit lmnt analysis. Introduction Elastomrs or rubbr-lik matrials ar oftn usd to mitigat damag causd by impulsiv or impact loads bcaus of thir low modulus, high damping and larg braking xtnsion. Rubbr isolation barings, for xampl, protct buildings and bridgs from arthquaks by imposing a layr of low shar modulus btwn th structur and ground thrby allowing latral, almost rigid body motion of th structur. Rubbr shock absorbrs arrst impacting bodis with minimum WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin) doi:0.495/su0600

2 98 Structurs Undr Shock and Impact IX forc transmission by undrgoing a sufficintly larg dflction bfor bringing th body to rst. Thy also rduc larg-amplitud vibrations and minimiz rbound by dissipating nrgy via intrnal damping. Polyura coatings on th insid walls of concrt buildings offr significant protction from xtrior trrorist bomb blasts by inhibiting fragmntation and allowing an othrwis brittl wall to dform undr th blast. Scurity films, which ar tough but pliabl polystr layrs bondd to window glass, rduc glass fragmntation and rtain glass shards whn th window is subjctd to gust wind and impact. Th mchanical bhavior of all th abov-mntiond lastomric structurs is dominatd by larg strain (gratr than 0%), high strain rat (mor than 0 s - ) and nonlinar viscolastic/viscoplastic matrial rspons. Yt tsting and modlling of lastomrs hav bn confind mostly to quasi-static dformation, crp, rlaxation or small strain, linar viscolastic vibration rspons. This papr is concrnd with th dvlopmnt of constitutiv quations for rubbr undr high rats of loading. High spd xprimnts will b conductd to charactriz th tnsil dformation and failur of Styrn Butadin Rubbr at impact rats (-0 3 s - ). A hypr-viscolastic constitutiv rlation of intgral form will thn b dvlopd to dscrib th rat-dpndnt matrial bhavior of th rubbr. Th proposd constitutiv quation will b implmntd in ABAQUS Explicit via a usr-dfind subroutin and usd to prdict th dynamic rspons of th rubbr shts in th xprimnts. Numrical prdictions for th transint dformation and failur of th rubbr sht will b compard to xprimntal rsults. Exprimntal stup Th tnsil impact apparatus shown in Figurs (a) and (b) was dvlopd for obtaining th dformation and fractur charactristics of rubbr-lik matrials undr impact rats [, ]. This apparatus can b usd to obtain both dynamic strss-strain curvs of uniaxial strip spcimns and forc-xtnsion curvs for thin shts. As shown in Figur (a), th Charpy impact pndulum impacts a spcially dsignd slidr bar connctd to two coppr cabls. Th coppr cabls ar dirctd around pullys and attachd to guidd bass with grips that hold opposit nds of th spcimn (s Figur (b)). Each nd of th spcimn is grippd into a guid bas, which also carris load clls and displacmnt transducrs. PCB 00-B0 pizolctric dynamic load clls with a linar rang of N (0-0 lb) ar mountd on ach guidd bas and rcord th impact tnsil forc as th coppr cabls pull on th guidd bas. Th tnsil forc is actually masurd as a comprssiv forc as th pizolctric load clls hit th backsid of rctangular slots. Th xtnsion of th spcimn is rcordd by two RDP D5 Linar Variabl Diffrntial Transformrs (LVDTs), which hav a rang of ±50 mm and ar mountd on ach of th guidd bass. Dsigning th xprimnt so that thr ar qual grip sparation vlocitis on both sids of th spcimn isolats th middl of th spcimn so that th fractur procss in th cntr of th rubbr sht can b capturd with a high-spd camra. WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

3 Structurs Undr Shock and Impact IX 99 Charpy Impact Pndulum Slidr Bar Coppr Cabl Pully Rigid Support Slidr bar Rubbr Groovd Support LVDTs Load Clls Groovd Support Coppr Cabl Signal Conditionr Data Acquisition a) Sid viw of Charpy impact hammr. (b) Top viw tnsil impact apparatus. Figur : Exprimntal stup: (a) Sid viw of Charpy impact hammr and (b) Top viw tnsil impact apparatus. Th abov apparatus is usd to obtain strss-xtnsion ratio curvs of strip spcimns and forc-strtch rspons of thin sht spcimns. Th spcimns ar cut from.54 mm thick shts of SBR, which wr providd by th Akron Rubbr Dvlopmnt Laboratory (ARDL). An ASTM D4 strip spcimn [3] is chosn to obtain strss-xtnsion ratio curvs in th xprimnt. Rctangular strips, 6.35 mm wid and 50.8 mm long, ar cut from th SBR shts using a razor and hammr. A lngth of.7 mm is clampd on ithr nd so th ffctiv gomtry of th spcimn is 6.35 mm wid x 5.4 mm long x.54 mm thick. For th sht xprimnts, 50.8 mm x 50.8 mm squars ar cut from th SBR sht. A lngth of.7 mm is clampd into th grips along two opposing dgs so th ffctiv gomtry of th sht spcimn is 5.4 mm long x 50.8 mm wid x.54 mm thick. 3 Exprimntal rsults 3. Dynamic strss-xtnsion ratio curvs from strips Th Cauchy or tru strss-xtnsion ratio tst data ar shown in Figur togthr with analytical prdictions, which will b discussd in a latr sction. Th strain rats quotd in Figur ar th nginring strain rat, i.., vlocity impartd to WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

4 00 Structurs Undr Shock and Impact IX spcimn dividd by original gag lngth. Th quasi-static tnsil tst rsult at 0. s - was obtaind using an Instron univrsal tsting machin and is also providd in Figur for comparison. Blow 80 s -, th modulus, tnsil strngth and fractur strain incras with incrasing strain rat. Btwn 80 s - and 80 s -, th sam trnd in th modulus and tnsil strngth occurs but th fractur strain dcrass with highr strain rats. At strain rats abov 80 s -, th modulus rmains virtually th sam but th tnsil strngth and fractur strain dcras with highr strain rats. Th limiting Cauchy strss-xtnsion ratio curv has bn trmd a locking curv. Th load duration for th xprimnt at 80 s - is about 0 ms and th xistnc of a locking curv at this strain rat signifis that thr is insufficint tim for th rlaxation procsss associatd with rat-dpndnt bhaviour to occur. Cauchy Strss (MPa) Extnsion Ratio 390 /s, prdictd 390 /s, tst 80 /s, prdictd 80 /s, tst 50 /s, prdictd 50 /s, tst 30 /s, prdictd 30 /s, tst 80 /s, prdictd 80 /s, tst 4 /s, prdictd 4 /s, tst 0. /s, prdictd 0. /s, tst Figur : Cauchy strss-xtnsion ratio curvs for strips at various strain rats. 3. Dynamic forc-xtnsion curvs from shts Figur 3 shows th xprimntally dtrmind forc-xtnsion rspons of th SBR shts at diffrnt loading rats. Prdictd rsponss using finit lmnt analysis ar also shown in Figur 3 and will b discussd in a latr sction. Th loading rat is dfind as th ratio of grip vlocity to original hight of th sht. Th sht dforms first without any sign of fractur until a hol initiats at th cntr of th sht. Th hol initiats at th pak loads shown in Figur 3, and th forc-xtnsion graphs bcom unstabl whn th hol nlargs (not shown WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

5 Structurs Undr Shock and Impact IX 0 in Figur 3). Th spcimn braks into two parts at th nd of th xprimnt. Th sris of ths vnts and thir rlation with forc-xtnsion rspons curvs hav bn confirmd with th us of a high-spd vido camra. 00 Forc(N ) s/s FEA 400 /s, tst 300 /s, FEA 300 /s, tst 50 /s, FEA 50 /s, tst 75 /s, FEA 75 /s, tst Extnsion (mm) Figur 3: Forc-xtnsion curvs for shts at various loading rats. 4 Hypr-viscolastic constitutiv quation Th dynamic strss-xtnsion ratio curvs in Figur ar boundd btwn th quasi-static and an uppr limiting curv. In th litratur, ths two curvs ar rfrrd to as th quilibrium and instantanous rspons curvs, rspctivly. Assum that th matrial can b dscribd by an lastic spring A in paralll with a Maxwll lmnt B, as shown in Figur 4. Both springs in A and B ar hyprlastic. Th total strss at any tim is σ = σ A + σ B. Th Maxwll lmnt allows th forc in th lastic spring of B to vary with loading rat. Whn th loading rat is vry slow, dformation is govrnd only by A sinc no forc is transmittd through th viscous dampr in B. Th matrial is assumd to b in a fully rlaxd stat and th corrsponding quilibrium rspons can b dscribd by a hyprlastic constitutiv quation. Whn th loading rat is vry high, th viscous dampr in B dos not hav tim to dform and th adjacnt spring forc approachs a limiting valu, as if th dampr wr not prsnt. At vry fast loading rats, th corrsponding instantanous rspons can b dscribd again by two hyprlastic constitutiv quations. In btwn vry slow and fast loading rats, th matrial xhibits tim-dpndnt bhavior and must b dscribd by a hypr-viscolastic thory. WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

6 0 Structurs Undr Shock and Impact IX A B Figur 4: Rhological rprsntation of a hypr-viscolastic matrial. 4. Equilibrium and instantanous hyprlastic rspons A thr-dimnsional hyprlastic constitutiv quation for incomprssibl matrial is W W W σ = p I + ( + I ) b b b () I whr p is an unspcifid hydrostatic prssur, b is th lft Cauchy-Grn tnsor, I is th idntity matrix, W=W (I, I ) is a strain nrgy potntial function, and I and I ar th first and scond invariants of th lft Cauchy-Grn dformation tnsor. For th spcial cas of uniaxial strss σ, on gts W σ = + W λ λ () whr λ is th xtnsion ratio. To obtain a smooth functional fit for both th quilibrium and instantanous hyprlastic rspons curvs, th following nrgy potntial is assumd: α W = µ (I 3) (3) whrµ and α may b non-intgr matrial constants. From Equation (), th uniaxial Cauchy strss-xtnsion ratio rlation bcoms σ α ( I 3) = µα (4) WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

7 Structurs Undr Shock and Impact IX Nonlinar viscolasticity A constitutiv quation for a nonlinar viscolastic, incomprssibl matrial is assumd as follows: whr t u u T p vi + F I + ( II - C) dτ F (5) I σ = p v is an undtrmind hydrostatic prssur, F is th dformation gradint tnsor and C is th right Cauchy-Grn tnsor. Th abov formulation assums that th strss at any tim dpnds on th tim history of th strain nrgy, and it is vry similar to th K-BKZ intgral constitutiv quation in this rspct. Using Equation () to rprsnt th quilibrium strss and Equation (5) to rprsnt th tim-dpndnt strss, on gts th following hypr-viscolastic constitutiv law: σ = F t ( p + p ) v u u I + W I + + I W ( II - C) dτ F T b - W b b + (6) whr p and W rprsnt th undtrmind prssur and strain nrgy potntial corrsponding to th quilibrium rspons, rspctivly. From Sction 4., on W obtains ( I 3) α W = µα and = 0. W assum th following conditions: u = c c ( I 3) I m( t τ) u (7) = 0 (8) whr c and c ar matrial constants, I is th tim rat of chang of th first m t τ is a mmory function. Th mmory function is givn by invariant and ( ) ( t τ) N θi m = m (9) ( t τ) i= i WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

8 04 Structurs Undr Shock and Impact IX whr m i ar wight factors such that m i =, θi ar rlaxation tims and i= N is som chosn intgr. Strss rlaxation may occur on diffrnt lngth or tim scals, and th wight factor will allow on to adjust rlaxation ffcts ovr svral lngth or tim scals. Substituting th abov xprssions into Equation (6) yilds N t ( p pv) µα ( I 3) c ( cm ( t τ)( I 3) I I) d α σ = + I+ b+ F τ F T (0) For th cas of uniaxial tnsion which bgins at t=0, this rducs to σ = µα ( I 3) t α + c [ c m( t τ)( I 3) I ] dτ () 0 Th first trm in th abov xprssion for σ is th quilibrium rspons, whil th scond intgral rprsnts th tim-dpndnt rspons. Equation () is usd to dtrmin th viscolastic matrial constants from th constant strain rat tsts. Dnot th strain rat ε o such that λ = ε 0 t + and λ = ε o. Th intgral in Equation () may b rformulatd in trms of xtnsion ratio by dξ transforming ξ = ε 0 τ + and dτ =, whr ξ is a dummy variabl: ε σ = µα ( I 3) λ 4c α 0 + λ ξ m ξ ε o + 3 ξ c ξ ξ () dξ Th valus for c and c ar found from th instantanous curv, takn from th tsts prformd at 390 s -. At this high strain rat, th loading tim is considrd small compard to th charactristic rlaxation tim and m(t τ). Dnoting th strss of th instantanous curv as σ i, on gts that WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

9 Structurs Undr Shock and Impact IX 05 σ i = µα ( I 3) α λ 4c ξ ξ c ξ ξ dξ (3) A FORTRAN program is writtn to prform numrical intgration and find appropriat valus for c and c. With c = MPa and c =.5, th instantanous rspons can b prdictd. Ths two valus ar thn usd in Equation () and anothr FORTRAN program is writtn to find rlaxation tims. Two rlaxation tims ar ndd to captur th data at low and high strain valus: m = 0.09, θ = 0.005s, m = 0.9, and θ = s. Th analytical solution is compard to tst data in Figur. Both th quilibrium and instantanous rspons curvs ar prdictd vry wll, but thr ar slight discrpancis btwn th analytical solution and tst data. Th worst fit is at 4 s -, whr th analytical solution is about % undr tst data. Improvmnts can b mad by considring mor rlaxation tims but this would lad to a mor complicatd data fitting procdur. Th rlaxation constants givn abov ar dmd sufficintly accurat for this study. 5 Finit lmnt analysis of rubbr shts Th thr-dimnsional, hypr-viscolastic quation proposd in Equation (0) is incorporatd in ABAQUS Explicit via a usr-dfind matrial subroutin, VUMAT. Only th uppr right sid (quartr modl) of th sht is modlld du to symmtry. Fifty C3D8R (continuum, thr-dimnsional lmnts with 8-nod rducd intgration points) lmnts ar usd to modl th quartr sht. Displacmnt at constant loading rat is applid to th sid of th plat, as it was don in th xprimnts. Numrical tim intgration, using a simpl trapzoidal rul, is prformd to calculat th strss at ach tim stp in th VUMAT subroutin. Th total rsultant forc to pull th sht is plottd with rspct to sht xtnsion (displacmnt) for th various loading rats and compard to th tst data in Figur 3. Th FEA prdictions of th forc-xtnsion rspons of th shts compar vry wll to tst rsults. Thy ar within 0% of xpctd rsults. 6 Conclusions High spd xprimnts wr conductd to charactriz th dformation and failur of Styrn Butadin Rubbr at impact rats. Dynamic tnsil strssstrain curvs of uniaxial strip spcimns and forc-xtnsion curvs of thin shts wr obtaind from a Charpy tnsil impact apparatus. Rsults from th uniaxial tnsion tsts indicatd that although th rubbr bcoms stiffr with WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

10 06 Structurs Undr Shock and Impact IX incrasing strain rat, th strss-strain curvs rmaind virtually th sam abov 80 s -. A hypr-viscolastic constitutiv rlation of intgral form was proposd to dscrib th rat-dpndnt matrial bhavior of th rubbr. Mmory functions wr obtaind from th uniaxial tst data. Two charactristic rlaxation tims, 5 ms at 9% of th tim and 0.5 ms at 9% of th tim, wr usd to fit th proposd constitutiv quation to th data. Th proposd constitutiv quation was implmntd in ABAQUS Explicit via a usr-dfind subroutin and usd to prdict th dynamic rspons of th rubbr shts in th xprimnts. Numrical prdictions for th transint dformation and failur of th rubbr sht compard wr within 0% of th xprimntal rsults. Futur rsarch will focus on th dynamic failur critria of rubbr. Acknowldgmnts This rsarch was supportd by NSF Grant CMS Th authors would lik to thank th Akron Rubbr Dvlopmnt Laboratory for supplying SBR matrial. Rfrncs [] Bkar, I., Hoo Fatt, M.S. & Padovan J., Dformation and Fractur of Rubbr undr Tnsil Impact Loading, Tir Scinc and Tchnology Journal, 30(), pp , 00. [] Hoo Fatt, M.S. & Bkar, I., High-Spd Tsting and Matrial Modling of Unfilld Styrn Butadin Vulcanizats at Impact Rats, Journal of Matrials Scinc, 39(3), pp , 004. [3] Annual Book of ASTM Standard, Vol. 09.0, pp. 4, 998. WIT Transactions on Th Built Environmnt, Vol 87, 006 WIT Prss ISSN (on-lin)

Dynamic Modelling of Hoisting Steel Wire Rope. Da-zhi CAO, Wen-zheng DU, Bao-zhu MA *

Dynamic Modelling of Hoisting Steel Wire Rope. Da-zhi CAO, Wen-zheng DU, Bao-zhu MA * 17 nd Intrnational Confrnc on Mchanical Control and Automation (ICMCA 17) ISBN: 978-1-6595-46-8 Dynamic Modlling of Hoisting Stl Wir Rop Da-zhi CAO, Wn-zhng DU, Bao-zhu MA * and Su-bing LIU Xi an High