The Importance of Nonlinear Strain Considerations in Belt Cover Indentation Loss

Transcription

1 Whtepaper Belt Conveyor Technology The Importance of Nonlnear Stran Consderatons n Belt Cover Indentaton Loss Energy loss due to rollng resstance of rubber conveyor belts s a major factor n conveyor desgn and ts determnaton s dependent on the propertes of the rubber compound of the belt backng. Vulcanzed, carbon black-flled backng compounds typcally exhbt the so-called Payne effect, where, under cyclc loadng, there s a dependence of the storage and loss modul on the ampltude of the stran. In effect, the stress-stran response of these compounds s hghly nonlnear. Yet n belt system desgn the ndentaton loss calculaton s often based on the unform stran, lnear propertes. In ths study we use the measured propertes of a typcal backng materal, ncludng stran ampltude dependent data, and couple a smple one dmensonal stress/stran materal model wth smple calculaton methods of the ndentaton resstance to show that the ndentaton loss factor can vary consderably wth the stran ampltude mpressed upon the backng under typcal desgn loads. Authors: T.J. Rudolph, Department of Aerospace Engneerng, Iowa State Unversty, Ames, IA, USA A. Recks, Overland Conveyor Co., Lakewood, CO, USA 1/

2 The Importance of Nonlnear Stran Consderatons n Belt Cover Indentaton Loss T. J. Rudolph and A.V. Recks* Abstract: Energy loss due to rollng resstance of rubber conveyor belts s a major factor n conveyor desgn and ts determnaton s dependent on the propertes of the rubber compound of the belt backng. Vulcanzed, carbon black-flled backng compounds typcally exhbt the so-called Payne effect, where, under cyclc loadng, there s a dependence of the storage and loss modul on the ampltude of the stran. In effect, the stress-stran response of these compounds s hghly nonlnear. Yet n belt system desgn the ndentaton loss calculaton s often based on the unform stran, lnear propertes. In ths study we use the measured propertes of a typcal backng materal, ncludng stran ampltude dependent data, and couple a smple one dmensonal stress/stran materal model wth smple calculaton methods of the ndentaton resstance to show that the ndentaton loss factor can vary consderably wth the stran ampltude mpressed upon the backng under typcal desgn loads. Keywords: Belt conveyng, ndentaton loss, rollng resstance, rubber property characterzaton, nonlnear stress-stran, Payne effect, Kraus model. * Thomas J. Rudolph, Department of Aerospace Engneerng, Iowa State Unversty, Ames, IA, USA Al Recks, Overland Conveyor Co., Lakewood, CO, USA Emal: rudolph@astate.edu T. J. Rudolph 1 9/1/11

3 1. Introducton Modern conveyor desgns rely on predctve and analytcal models of the rollng resstance due to ndentaton of dlers nto the belt backng or cover materal. A typcal belt layup ncludes two rubber covers wth a cord ply or steel cable carcass sandwched between, wth the top cover desgned to resst wear and abrason from the carred materals and the bottom to mnmze rollng resstance over the belt system dlers. The ndentaton of the bottom cover by dlers s a prmary source of power loss, and hence an mportant factor n the desgn of the system. The partcular characterstc of rubber compounds, especally those wth carbon black fllers, that produces a nonlnear stress-stran behavor, even for relatvely small strans, s called the Payne effect. Ths phenomenon refers to a dependence of the vscoelastc modul on appled cyclc stran ampltude, where t s observed that for strans above about.1%, the storage modulus decreases as the loss modulus ncreases (cf. refs. [1-8]). As strans experenced by belt covers, even under modest loads, can routnely exceed.1%, and both the storage and loss modul are factors n the ndentaton loss predcton, t s mportant to account for the effect of stran ampltude n any predctve model. In ths study we use the measured vscoelastc propertes of a typcal backng rubber materal and ft that data to an establshed stran ampltude dependent consttutve model. The consttutve model s then ncorporated nto a one dmensonal model of the ndentaton resstance to show that the ndentaton loss factor depends sgnfcantly on the stran levels and hence the loads carred by the belt.. Rubber propertes and materal characterzaton Snce characterzaton of nonlnear vscoelastc materals s an extenson of the lnear theory, we frst revew the lnear methodology. As dsspatve materals are nherently tme dependent, testng n a cyclc mode at varous frequences s a natural way to capture the tme dependence through Fourer analyss. For a snusodal stran hstory of the form ε = ε sn( ωt), where ω s the angular frequency and ε the ampltude, after an ntal transtory state the stress follows the stran n frequency, but at a delayed phase or angle δ. For a lnear vscoelastc materal, the one-dmensonal stress/stran relatonshp s of the form, * ( ω) = E ( ω) ε sn( ωt δ ) σ + (1) where σ denotes the stress, E * ( ω) = E ( ω) + E ( ω ) s the magntude of the complex modulus wth real and magnary components E ( ω) and E ( ω), called the storage and loss modul, respectvely. For vscoelastc materals undergong harmonc deformaton, the loss tangent s determned by tan( δ ) = E / E and s the phase angle of stress followng stran. By testng over a wde temperature range but lmted frequences, the superposton prncple (cf. Wenman, et. al. [9] or Ferry [1]) of lnear vscoelastcty, allows extrapolaton to frequences or load rates consderably beyond those possble n T. J. Rudolph 9/1/11

4 testng. Accordng to ths prncple, f T denotes temperature and T a reference temperature, then a materal property at tme t and temperature T s equvalent to that at tme a T t and temperature T, where a T s a shft parameter n tme so that a = T at ( T,T ). A commonly used expresson for a T, applcable to amorphous materals such as rubbers, at temperatures above the glass transton temperature, due to Wllams, Landel and Ferry [11] and referred to as the WLF equaton, s the logarthmc form, log ( ) a T C ( T T ) ( T T ) 1 = () C + where C 1 and C are constants determned emprcally from data. For data taken from cyclcal or frequency controlled stran experments, an emprcal representaton of the shft parameter a T may be determned by overlayng, through shfts n the frequency, the temperature dependent modulus data onto a master curve and software to do ths shftng s usually a part of the DMA nstrumentaton or can be wrtten accordng to varous curve fttng or error mnmzaton algorthms. Varous arrangements of lnear mechancal elements of sprngs and dashpots are classcally used to ft the vscoelastc behavor of eqn. (1). The smple Maxwell model conssts of a sngle sprng (sold) n parallel wth another sprng and a dashpot (flud). A generalzaton of ths, called a generalzed Maxwell, or Wechert model [9], conssts of many such sprngs and dashpots n parallel wth one sprng. For such an arrangement of N elements, the storage and loss modul of eqn. (1) are related to the mechancal element values by the Prony seres, E N ( ω ) = E + E E ( ) = = 1 + ω τ 1 ω τ ωτ ω (3) 1 ω τ N E = 1 + where E and η are the dscrete sprng and dashpot constants and τ = η E are the perods of the varous sprng/dashpot elements. The E values, as a set, are also called the spectrum, or spectral modul, of the storage and loss modul of eqn. (3), respectvely. As a shear mode of deformaton s often used n testng, the equvalent shear storage modulus G ( ω) and loss modulus G ( ω) are convenent to use, nstead of the tensle modul of eqn. (3), wth G ( ω) = E ( ω) 3 and G ( ω) = E ( ω) 3, assumng ncompressblty of the rubber. A computer program, based on least square curve fttng for shftng and overlayng test data at varous temperatures, was wrtten to determne the a T vs. temperature relatonshp. Also, a least squares fttng process determnes constants C 1 and C of eqn. () and a least square, non-negatve fttng process determnes the spectral values of the modul of eqn. (3). Here we determned the spectrum by overlayng the storage modulus wth spectral values placed at half-decade ncrements n the frequency space, whch was suffcent to accurately overlay the data. For a truly lnear vscoelastc materal, the spectral values as determned from the storage modulus should also T. J. Rudolph 3 9/1/11

5 reproduce the loss modulus through the second of eqns. (3), and for the low stran data, ths s the case as observed below. For dsspatve materals lke rubber compounds, the macro-mechancal propertes storage and loss modul, ncludng the effect of stran are usually measured dynamcally wth a DMA (Dynamc Mechancal Analyzer). The dynamc modul are usually determned by testng n a one-dmensonal, stran-controlled mode, lke unaxal tenson, pure bendng or pure shear. Typcal tests are performed n a cyclc mode, whether n tenson, bendng or shear, and conssted of sweeps through fxed frequences of about.1 rad/s to about 1 rad/s and at fxed temperatures rangng from about 7 o C to about + 7 o C. The pure shear mode s accomplshed by twstng a relatvely thn and narrow specmen and s generally preferable to tensle or bendng modes so as to mnmze nertal effects at hgher frequences. Ths twstng mode of testng also has the advantage of producng a homogeneous stran feld wthn the specmen, whch s very mportant for testng at hgher stran levels where all elements of the test cycle through the same stran levels. Ths would not be the case for bendng modes, where the stran level n a bendng cycle would vary wth dstance from the neutral axs. For ths study, a specmen (approxmately 4x1.5x7 mm) of the rubber materal taken from the backng of a typcal belt materal was sent to and tested by an ndependent laboratory [9] usng a Rheometrcs Scentfc RDSII nstrument to determne the low stran and hgher stran vscoelastc propertes n a twstng, pure shear mode. At low stran levels, testng was performed over a temperature range of 7 o C to + 75 o C n about 1 degree ncrements and over frequences of. 178 to 1 rad/s. Tests were also performed at varous stran levels up to 6 % at fve temperatures and at a fxed frequency of 1 rad/s to characterze the non-lnear, stran dependent behavor. Fgure 1 shows the data of the test for the low stran modul as taken through harmonc frequency sweeps at fxed temperatures and at strans well below the non-lnear threshold (approxmately.%). Tests were also made at hgher strans (up to 6%) to T. J. Rudolph 4 9/1/11

6 determne the effectve modul at these hgher stran levels as wll be dscussed n Sec. 3. Storage and loss modul [Pa] Frequences n rad/s Temperature [ o C] Fg. 1: Storage and loss modul data for temperatures and frequences tested. Fgures and 3 show the master curves and shft parameter a T for the rubber tested and developed at a reference temperature of o C, whch s where a T equals 1 of Fgure 3. Shown also n the table of Fgure 3 are the values of the WLF constants of eqn. () for ths partcular rubber compound. T. J. Rudolph 5 9/1/11

7 G' and G'' [Pa] Frequency ω [1/s] Fg. : Master curve of the shfted data and Prony seres (spectral) ft to the shfted data Shft coeffcent a T WLF constants: C 1 = 5., C = Temperature [ o C] Fg. 3: Frequency-temperature shft coeffcent a T and WLF curve ft. T. J. Rudolph 6 9/1/11

8 In addton to the modul and the shft factor assocated wth the master curves, an mportant property drectly related to the energy lost n a deformaton cycle s the loss tangent tan( δ ) = E / E = G / G, whch for ths materal, s readly determned from Fgure and s shown n Fgure 4 as a functon of frequency. We observe that for ths materal there s a lower plateau of tan( δ ). 1 at the lower frequency range (hgher temperatures) and peaks at nearly tan( δ ). 5 for a frequency of about 1 8 rad/s. As the rollng resstance of the belt backng over an dler s drectly related to the energy absorbed by the backng materal n the ndentaton deformaton process, the loss tangent relates drectly to the rollng resstance. Thus dependng on the rate (frequency) and temperature, there could be as much as a fvefold varaton n the ndentaton resstance for ths materal. The above deas pertan to a materal characterzed as a lnear vscoelastc one, but can also form the bass for a nonlnear materal by determnaton of approxmately equvalent modul and loss tangent, as dscussed n the followng secton tan(δ).4.3 Spectral curve ft Frequency ω [1/s] Fg. 4: Loss tangent and spectral representaton of the data. 3. Stran Dependence of the Vscoelastc Modul Estmated compresson stran levels nto the backng materal due to ndentaton can exceed 5 to 1% (Rudolph [13]) and, at these stran levels, carbon black-flled rubber compounds are known to exhbt the nonlnear characterstc called the Payne effect, where the storage modulus decreases sgnfcantly wth stran ampltude n cyclc, stran appled tests. As such, the lnear relatonshp between stress and stran that defnes T. J. Rudolph 7 9/1/11

9 the modul n eqns. (1) and (3) s no longer strctly applcable. Also, n rubber compounds, the loss modulus generally ncreases wth stran ampltude to a maxmum at some stran level, and then decreases back to near the low stran plateau as observed by Veweg, et. al. [6,7] and Wang [6]. To accurately predct the ndentaton resstance, we must then ncorporate the nonlnear effects of stran ampltude nto the stress response nto the model of rollng resstance. Kraus [14] model of the Payne effect s a nonlnear consttutve equaton that has been successfully appled to carbon-flled rubbers, and we follow that approach here snce t s easly ncorporated nto smple models of the ndentaton and deformaton of rollng contact. Kraus model s phenomenologcally based, but the macro-mechancal behavor s determned by a small number of parameters that can be determned by onedmensonal, cyclc stran tests. A one-dmensonal test requres then a testng mode that s homogeneous,.e., all elements of the specmen under test must go through the same stress/stran cycle. Then, n addton to the frequency/temperature sweeps to determne the low stran modul of prevous secton, addtonal tests were performed on the same materal at fve temperatures well above the glass transton temperature and at strans rangng from about.4% to 6% and all at a constant frequency - 1 rad/s. The data (modul) from those tests are shown n Fgure 5 as a functon of stran at the fve temperatures. As observed the storage modulus drops by nearly a factor of three from the low stran modulus to that at 6%, whle the loss modulus ncreases slghtly wthn that same stran range. There s also the expected rse n the loss modulus at strans n the range of about 1 to 6%. These test results are smlar to those of Veweg, et.al.[6,7], Wang [8] or Ulmer [4] and for whch Kraus s model can be readly appled. T. J. Rudolph 8 9/1/11

10 .5 3 x 17 - C C 5 C 45 C Loss Modulus G'' [Pa] Shear stran [%] Fg. 5: Storage and loss modul data at varous temperatures. A normalzaton of the data of Fgure 5 s frst determned by selecton of a threshold or reference stran value for whch the modul plateau, whch by nspecton would be about γ =.1 %. Dvson of the modul of each temperature curve by the modulus at ths stran, thus defnng G and G for each, produces Fgure 6. T. J. Rudolph 9 9/1/11

11 Loss - G''/G'' o Nornalzed modul G'/G' and G''/G'' o o Storage - G'/G' o - C C 5 C 45 C Shear stran [%] Fg. 6: Storage and loss modul data normalzed to the small stran values. On Fgure 6 we observe the farly close overlay of the temperature curves, especally for the storage modulus. As such, t would not be napproprate to presume that, for such materals, the modul are nearly temperature, and therefore frequency, ndependent. Ths assumpton s consstent wth Veweg, et.al. [6] for carbon flled rubbers, and we follow t here. To ft the analytcal model of Kraus to the data of Fgure 6 we follow the notaton and approach of Veweg, et. al. [7] and Lon and Kardelky [15], where the nonlnear stran dependent modul, dervng from Kraus model, are wrtten n the form, ( γ ) ( G G ) +, ( γ ) 1+ ( γ γ ) m G = G ( G G )( γ γ ) 1+ ( γ γ ) m G = G + (4) where G and G are the storage modul at small strans and large stran, respectvely, and m s an adjustable parameter. By these equatons, the Kraus model s a fourparameter model for each modulus, wth the common parameter m. But Drozdov and Dorfmann [5] observe, amongst other thngs regardng the parameters, that whle G, G, etc., are farly well predcted from expermental data n dynamcs tests wth fxed frequency, but that fttng data for the storage and loss modul separately from eqns. (4) results n dfferent values of the exponent m. They also observe that n fttng carbonflled rubber data to the storage modulus results n a value of the exponent m. 6. m T. J. Rudolph 1 9/1/11

12 Thus n the fttng of these equatons to the data of Fgure 6 we allow a dfferent m for each modulus. Also, to ft the normalzed data of Fgure 6, we dvde eqns. (4) by the low stran values G and G and ft the data to equatons of the form, B ( γ ) = A +, ( γ ) + ( γ ) m G 1 γ m ( γ γ ) ( γ ) m B = A + (5) 1+ γ G where the constants A, B and m are determned ndependently for each modulus, as well as m. The low, threshold stran γ =.1 % s taken to be apply to both equatons, so there are only three parameters for each modulus - A, B and m. The data of Fgure 6 were ft by a least squares process to eqns. (5) and Table 1 shows the resultng parameters. We observe that m s approxmately. 6 as noted prevously by Drozdov and Dorfmann [5]. A B m Storage G Loss G Table 1: Curve ft parameters of Kraus model for normalzed storage and loss modul Fgure 7 shows the analytcal curve fts (red sold curves) of eqns. (5) for each modulus, overlad wth the data of Fgure 6. T. J. Rudolph 11 9/1/11

13 1.6 Nornalzed modul G'/G' and G''/G'' o o C C 5 C 45 C Shear stran [%] Fg. 7: Normalzed storage and loss modul wth curve fts. The parameter values of Table 1, together wth eqns. (5) - (7) and (1) (3) then consttute a complete one dmensonal consttutve model for the materal that ncludes temperature, rate (frequency) and stran level effects. We also observe from Fgure 7 that the storage modulus decreases to about one thrd of ts low stran value at strans of about 6%. As wll be seen n the followng sectons, ths s a very mportant factor n ndentaton resstance. Also, from a theoretcal standpont, at hgher stran levels the assumpton of lnear vscoelastcty loses applcablty and the characterzaton of materal wth parameters as above no longer apples. However, the above model for stran ampltude dependence does assume that, at hgher stran levels, the materal modul are scaled up from the low stran level values accordng to Fgure 7. In other words, at hgher stran levels, the materal s assumed to be a lnear vscoelastc materal wth modul determned from the values measured at low stran levels. As a test of how well ths scalng holds for hgher strans, the same materal was tested at a frequency of 1 rad/sec. and at varous stran ampltudes, recordng the stress and stran values at 1 ponts on each cycle of deformaton. Fgure 8 shows the results of ths test at two stran ampltudes; a low and assumed to be lnear stran of.1% (green loop) and a hgher, and notceably nonlnear stran of 3.% (blue loop). At the lower stran level (green) the shape s ellptcal whle at the hgher stran (blue), the Lssajous loop s asymmetrc. T. J. Rudolph 1 9/1/11

14 Stress σ = τ [MPa] Low stran (.1%) Hgh stran (3.%) Stran corrected Stran ε = γ/ [%] Fg. 8: Stress/stran cycles for low and hgh ampltude strans Also shown n Fgure 8 (red) s the stran corrected loop, as determned by the stran correcton modfcaton of the modul accordng to Kraus model. For ths loop, the stran correcton factors appled to the modulus G s. 47 and for G s 1. 85, and whch change the loss tangent tan( δ ) = G / G from. 18 at low stran to. 54 at hgh strans; a factor of 3.. Thus by multplcaton of the low stran modul by these amounts, respectvely, and assumng a stran ampltude of 3.%, the green loop becomes the red loop. In prncple, f the materal were deally lnear vscoelastc, the red and blue loops would concde and they would both be scaled up versons of the low stan (green) path wth the same orentaton of the axes of the ellpse. Fnally then, by Kraus model, or the stran correcton methodology used here, any stran corrected loop wll reman ellptcal,.e., the materal wll be characterzed as lnear vscoelastc, but wth alternate modul. Snce the energy dsspated per unt volume n a harmonc cycle of deformaton of a vscoelastc materal equals the area enclosed n the stress vs. stran graph, t s evdent that ths non-lnear stran model s an approxmaton and may somewhat overestmate the energy dsspaton. 4. Indentaton deformaton and resstance models T. J. Rudolph 13 9/1/11

15 Analytcal and computatonal models of rollng resstance due to ndentaton of the belt backng by dlers have been developed and, for desgn purposes, the nfluence of the man parameters such as dler dameter, carry weght, backng thckness and rubber propertes of the backng have been well dentfed wth smple analytcal approaches. Smple models that treat the belt coverng as a vscoelastc layer on a rgd base are that of Jonkers [16], Spaans [17] or Lodewjks [18]. A more rgorous approach, wth the backng modeled as a full two-dmensonal half-space, are those of May, et.al. [19], Hunter [], Morland [1] or Gooder and Loutzenheser []. All these models assume a lnear vscoelastc materal of the backng, and as shown by Lodewjks [18], the twodmensonal models provde slghtly lower values of the ndentaton resstance for equal materal propertes and other system parameters. One can also take a completely computatonal approach to rollng contact problems, as developed by Lynch [4] or Batra, et.al. [5], and as appled to belt cover ndentaton by Wheeler [6], where the cover deforms as a two-dmensonal medum and s modeled by fnte elements. The advantage of a computatonal approach s that a less restrctve deformaton model s possble, such as modelng the entre belt carcass, as may be mportant for cable belts where the deformaton between the steel cables also dsspate energy (cf. Wheeler [7]). Conversely, recourse to computatonal methods at the outset precludes analytcal results that can make parameters explct so that parameter studes mportant for desgn can be tme consumng and expensve. Expermental measurement of ndentaton resstance s possble, but known to be senstve due to the low resstance force of effcent belt constructon n laboratory setups or naccessblty ssues n feld test. The advantages and dsadvantages of the varous deformaton models of the ndentaton process, or the preferred methodology of ths calculaton are not the focus here. We want to prmarly show how the nonlnear consttutve equaton affects the predcted ndentaton results. To do ths t s only mportant to use a method where the stran dependent modul can be ncorporated wthout nvaldatng the assumpton of the model tself. To ths end we focus on the one-dmensonal approaches where a smple one dmensonal consttutve equaton as derved from the test data can be merged drectly nto the ndentaton resstance model, as opposed to the two dmensonal methods where the stran feld produced by a rollng process would be nherently nonhomogeneous and requre unque stran compensated propertes at each pont. Instead we concentrate only on the models of Jonkers [16] and Rudolph and Recks [3], whch s a drect extenson of Lodewjks [18] for a mult-parameter materal. In the case of Jonkers, the equaton for the ndentaton resstance factor f (rato of belt resstance to vertcal load, per unt belt wdth) s, ( δ ) ( π + δ ) cos( δ ) 1 3 ' 4 1+ sn( δ ) 1 3 Wh π tan f = (8) D E where W s the carry load per unt wdth, h s the backng thckness, E s the storage modulus, D s the dler dameter and tan ( δ ) s the loss tangent. The formula (8) s revealng n that, apart from showng the explct dependence of f on the parameters W, 4 3 T. J. Rudolph 14 9/1/11

16 3 4 3 h and D, t shows that f s proportonal to tan( δ ) E 1 = E E. Thus f the storage modulus E decreases and E ncreases, as for materals that exhbt the Payne effect, the rollng resstance wll ncrease drectly wth changes n both modul. The formula of Jonkers n eqn. (8) results from the modelng assumpton that the deformaton process of ndentaton s harmonc,.e., that the materal of the backng contnuously cycles through a compresson phase of ndentaton, followed mmedate by an equal tenson phase. The ndentaton resstance f derves from the energy dsspaton of ths cyclc process and s thus half the area of the ellpse of Fgure 7 wth the stran ampltude determned from equlbrum of the carry load W wth the backng stffness, whch requres a short teratve process to effectvely determne the ndentaton depth of the dler, hence stran, for a gven W. The assumed harmonc deformaton process overestmates the actual case, however, snce for steady belt speeds and unform dstance between dlers, the deformaton s perodc, but not harmonc cycles as assumed by Jonkers. The method of Lodewjks [18] crcumvents the cyclc deformaton process of Jonkers and drectly determnes the contact stress n the transent compresson perod of contact between the dler and backng. The rollng resstance then results from drect calculaton of the moment of that stress about the dler center, hence equvalent force resstng the moton of the belt. Lodewjks developed ths method for a smple three parameter materal ( N = 1 n eqn. (3)) and thus s not well suted for realstc rubber materals where tan ( δ ) s spread over a frequency range as n Fgure 4. Rudolph and Recks [3] generalzed Lodewjks approach by takng the materal model to be an n- parameter Maxwell sold whle retanng the same deformaton assumpton of the Wnkler foundaton. Ther formula for the ndentaton resstance coeffcent can be put nto a form smlar to the Jonkers formula of eqn. (8), and s, f 1/ 3 ( E, τ, ς ) ( E, τ, ς ) 4 / 3 Wh M = (9) D F where the functons M ( E k,ς ) and ( E k,ς ), F are defned by, M = E 8 N 4 3 k 1 3 ( 1+ ς ) ( 1 ς + ς ) + Ek k ( 1+ ς ) + ( 1+ ς ) k ( 1+ k )( k + ς ) e = 1 3 / k and E F 6 N 3 1 ς (1+ ) / k ( + 3ς ς ) + Ek k [( 1 + k )( 1 e ) ( 1 + ς )] = 1 ς. In these defntons, E s the spectral stffness of eqn. (3), k = ( v a) η E s the wave numbers assocated wth each spectral element, v s the belt speed, a + b s the contact length and ς = b a, wth a beng the contact dstance ahead of the dler center and b that behnd. As wth the Jonkers formula (8), the contact length, hence stran, s T. J. Rudolph 15 9/1/11

17 determned by an teratve process of equlbrum, gven the carry load W, wth the stffness of the backng. Formulas (8) and (9) provde two dfferent ways to determne the ndentaton resstance, based on the belt system parameters W, R, h, v and the materal parameters E and τ. Both can be readly used n conjuncton wth eqns. (4) and (5) of Sec. 3 to allow for nonlnear effect of stran ampltude on the stress response of the backng materal. They provde somewhat dfferent results due to nherent assumptons of each, but both are used n the followng secton as a representatve way to show the effects of stran ampltude on the ndentaton resstance calculaton. 5. Indentaton Resstance as Influenced by Stran Level To show the effect of stran levels on the ndentaton resstance, the materal propertes of the tested materal and materal propertes of Sec. 3 are combned wth the two methods of Sec. 4. In the followng calculaton, the varous belt parameters were taken as: R =.75m, h =.635m, v = 5.m/s, T = 5 o C and we let W range from 5 to 4 n/m. Snce stran n the ndentaton zone s largely determned by the carry load, we present results as W. Shown then n Fgure 9 are the calculated values of f vs. W from both formulas (7) and (8), usng both stran adjusted propertes and propertes based on small strans. 7 x 1-3 Indentaton resstance factor f Rudolph and Recks Jonkers Load [N/m] Fg. 9: Indentaton resstance factors for small stran and stran-adjusted propertes. T. J. Rudolph 16 9/1/11

18 As may be seen, there s a dfference n the ndentaton resstance based on the method of calculaton, but both are magnfed smlarly wth respect to the load W, or stran. The departure from the non-stran adjusted values ncreases wth stran levels, or load, and converge at lower strans. For ths partcular materal, there s nearly a factor of two at hgher stran levels,.e., the stran adjusted propertes have a dramatc effect on the ndentaton resstance calculaton, regardless of the methods used to make the calculaton. Furthermore, ths effect would be expected from Jonkers method of eqn. (8) and the effect of stran level on the materal modul as seen n Fgure Summary and Conclusons As has been long known, the storage and loss modul of partcle flled elastomers, such as carbon black flled rubber as s commonly used for conveyor belt backng materal, s dependent on stran levels experenced by the materal. Large conveyor belt desgns and applcatons, due to heavy carry loads, stran the backng well beyond the lnear stran levels. By usng Kraus nonlnear consttutve model wth measured data for a typcal rubber and smple rollng resstance deformaton models, we have shown that the predcted ndentaton resstance s hghly senstve to the materal consttutve parameters, or stran level. For the partcular rubber propertes used n ths study, whch s consdered a typcal belt backng materal, these results would not be unexpected, consderng the drect dependence of the loss tangent tan ( δ ) on the modul and the strong dependence of the ndentaton rollng resstance on the loss tangent. Hence realstc predctve methods for ndentaton predcton should nclude the nonlnear effects of stran, or values of ndentaton resstance, hence power requred to drve the belt, could be serously underpredcted. T. J. Rudolph 17 9/1/11

19 References 1. Fletcher, W.P. and A.N. Gent: Non-lnearty n the dynamc propertes of vulcanzed rubber compounds, Trans. Inst. Rubber Ind. 9, (1953), Payne A.R.: A note on the exstence of a yeld pont on the dynamc modulus of loaded vulcanzates, J. of Appl. Polymer Sc. 3 (196), 17-???? 3. Payne, A.R.: The Dynamc Propertes of Carbon Black-Loaded Natural Rubber Vulcanzates, Part I, J. Appl. Polymer Sc. 6 (19), (196), Ulmer, J.D.: Stran dependence of dynamc mechancal propertes of carbon blackflled rubber compounds, Rubber Chemstry and Technology 69 (1996), Drozdov, A. D., and A. Dorfmann: The Payne Effect for Partcle-Renforced Elastomers, Polymer Engneerng and Scence 4 (), Veweg, S., R. Unger, K. Schröter, E. Donth and G. Henrch: Frequency and temperature dependence of the small-stran behavor of carbon-black flled vulcanzates, Polymer Networks Blends 5, (1995) Veweg, S., R. Unger, G. Henrch and E. Donth: Comparson of dynamc shear propertes of styrene-butadene vulcanzates flled wth carbon black or polymerc fllers, J. Appl. Polymer Sc. 73, (1998), Wang, M-J; The role of fller networkng n dynamc propertes of flled rubber, Annual Mtg. of the Rubber Dvson, Amercan Chemcal Socety, May 5-8, (1998), Indanopols, Indana, Paper No Wenman, A.S., and K.R. Rajagopal: Mechancal Response of Polymers: An Introducton; Cambrdge Unversty Press, Cambrdge, UK, (). 1. Ferry, J.D.: Vscoelastc Propertes of Polymers; 3 rd Edton, Wley, New York, (198). 11. Wllams, M., R. Landel and J. Ferry: The temperature dependence of relaxaton mechansms n amorphous polymers and other glass-formng lquds; Journal of the Amercan Chemcal Socety, 77 (1955), Harrell, R., PolyScCon LLC, Temperature, Frequency and Stran Dependent Vscoelastc Property Characterzaton of Cured Elastomer Samples XXXX and XXXX, Oct., (6). 13. Rudolph, T.J.: Appled Rubber Belt Cover Loss Predcton from Indentaton; 8 SME Annual Meetng and Exhbt, Salt Lake Cty, UT, Feb. 4-7, (8). 14. Kraus, G.: J. Appl. Polym. Sc., Appl. Polym. Symp., 39 (1984), Lon, A., and C. Kardelky: The Payne effect n fnte vscoelastcty: Consttutve modelng based on fractonal dervatves and ntrnsc tme scales, Int. J. Plastcty, (4), Jonkers, C.O.: The ndentaton rollng resstance of belt conveyors - A theoretcal approach, Fördern und Heben, 3, 4 (198), Spaans, C.: The calculaton of the man resstance of belt conveyors; Bulk Solds Handlng, 11, 4 (1991), Lodewjks, G.: The rollng resstance of conveyor belts; Bulk Solds Handlng, 15, 1 (1995) May, W.D., E.L. Morrs and D. Atack: Rollng frcton of a hard cylnder over a vscoelastc materal; Journal of Appled Physcs, 3, 11 (1959), T. J. Rudolph 18 9/1/11

20 . Hunter, S.C.: The rollng contact of a rgd cylnder wth a vscoelastc half space; Journal of Appled Mechancs, 8 (1961), Morland, L.W.: A plane problem of rollng contact n lnear vscoelastcty theory; Journal of Appled Mechancs, 9 (196), Gooder, J.N. and C.B. Loutzenheser: Movng contact problems of a rgd profle on a vscoelastc base; Dvson of Engneerng Mechancs, Stanford Unversty, Techncal Report No. 148, (1964). 3. Rudolph, T.J. and A.V.Recks: Vscoelastc ndentaton and resstance to moton of conveyor belts usng a generalzed Maxwell model of the backng materal; J. Rubber Chemstry and Technology, June (6). 4. Lynch F.: A Fnte Element Method of Vscoelastc Stress Analyss wth Applcaton to Rollng Contact Problems. Internatonal Journal for Numercal Methods n Engneerng, 1 (1969), Batra R., M. Levnson and E. Betz: Rubber Covered Rolls - The Thermo-vscoelastc Problem. A fnte Element Soluton; Internatonal Journal for Numercal Methods n Engneerng, 1 (1976), Wheeler, C.: Analyss of the Man Resstances of Belt Conveyors; PhD Thess, The Unversty of Newcastle, Australa, (3). T. J. Rudolph 19 9/1/11