Stress-Strain Behavior of Thermoplastic Polyurethane

Transcription

1 Stre-Strain Behaior of Thermoplatic Polyurethane H.J. Qi 1,2, M.C. Boyce 1,* 1 Department of Mechanical Engineering Maachuett Intitute of Technology Cambridge, MA Department of Mechanical Engineering Unierity of Colorado Boulder, CO 839 Submitted in December 23 Reied in July 24 Submitted to Mechanic of Material * Correponding author. Tel: ; fax: addre: [email protected] 1

2 Stre-Strain Behaior of Thermoplatic Polyurethane H.J. Qi 1,2, M.C. Boyce 1 1 Department of Mechanical Engineering, Maachuett Intitute of Technology Cambridge, MA Department of Mechanical Engineering, Unierity of Colorado Boulder, CO 839 Submitted in December 23 Reied in July 24 Abtract The large train nonlinear tre-train behaior of thermoplatic polyurethane (TPU) exhibit trong hyterei, rate dependence and oftening. Thermoplatic polyurethane are copolymer compoed of hard and oft egment. The hard and oft egment phae eparate to form a microtructure of hard and oft domain typically on a length cale of a few ten of nanometer. Studie hae reealed thi domain tructure to eole with deformation; thi eolution i thought to be the primary ource of hyterei and cyclic oftening. In thi paper, experiment and a contitutie model capturing the major feature of the tre-train behaior of TPU, including nonlinear hyperelatic behaior, time dependence, hyterei, and oftening, are preented. The model i baed 2

3 on the morphological oberation of TPU during deformation. A ytematic method to etimate the material parameter for the model i preented. Excellent agreement between experimental reult and model prediction of ariou uniaxial compreion tet confirm the efficacy of the propoed contitutie model. Keyword: Softening; Mullin Effect; Stre-Strain Behaior; Thermoplatic Polyurethane Elatomer; Rubber. 1. Introduction The firt commercial thermoplatic polyurethane (TPU) were etablihed in Germany by Bayer-Fabenfabriken and in the U.S. by B.F. Goodrich in the 195 (Schollenbenger et al., 1958). The Alliance for the Polyurethane Indutry (API) decribe TPU a bridging the gap between rubber and platic, ince TPU offer the mechanical performance characteritic of rubber but can be proceed a thermoplatic. Thi pecial niche of TPU among other polymer and elatomer impart high elaticity combined with high abraion reitance, and reult in a wide array of application ranging from ki boot and footwear to gaket, hoe, and eal. Hydrogen Bond HS SS Figure 1: The alternating tructure of TPU. HS: Hard egment; SS: Soft Segment. 3

4 HD SD HD Iolated HS SD (a) (b) Figure2: Hard domain (HD) and oft domain (SD) of TPU with (a) a low hard egment content (adopted from Petroic and Ferguon, 1991); (b) a high hard egment content (adopted from Ete et al. (1971) and Petroic and Ferguon (1991)). Iolated hard egment (HS) een in (b). Figure 3: Tranmiion Electron Microcope (TEM) image of omium tetroxide tainted TPU (57% oft egment and 43% hard egment). The light region are hard domain and the dark region are oft domain. 4

5 Thermoplatic polyurethane are randomly egmented copolymer (Hepburn, 1982) compoed of hard and oft egment forming a two-phae microtructure (Figure 1). Generally, phae eparation occur in mot TPU due to the intrinic incompatibility between the hard egment and oft egment: the hard egment, compoed of polar material, can form carbonyl to amino hydrogen bond and thu tend to cluter or aggregate into ordered hard domain, wherea the oft egment form amorphou domain. Phae eparation, howeer, i often incomplete, i.e., ome of the hard egment are iolated in the oft domain a illutrated chematically in Figure 2(b) (Petroic and Ferguon, 1991). In many TPU, the hard domain are immered in a rubbery oft egment matrix (Wang and Cooper, 1983; Petroic and Ferguon, 1991). Depending on the hard egment content, the morphology of the hard domain change from one of iolated domain (Figure 2(a), (Petroic and Ferguon, 1991)) to one of interconnected domain (Figure 2(b))(Ete et al., 1971; Petroic and Ferguon, 1991). Uing TEM (Tranmiion Electron Microcope), the interconnected domain tructure wa erified for the TPU 1 ued in thi reearch a hown in Figure 3. The domain ize in Figure 3 i 1~2nm, which i conitent with oberation on ariou other TPU (e.g., Cooper and Tobolky, 1966; Koutky et al., 197; Chen-Tai et al., 1986). For intance, Koutky et al. (197) obered a domain ize of 3nm~1nm for a polyeter-baed polyurethane and 5nm~1nm for a polyether-baed polyurethane; Chen-Tai et al. (1986) obered a hard domain length cale of about 11nm with inter-domain ditance of 13nm for a PBD/TDI/BD-baed polyurethane. The preence of hard domain in egmented polyurethane i ery important to the 1 The proider of the ample would prefer the compoition of the material unreealed. 5

6 mechanical propertie. In egmented polyurethane, hard domain act a phyical crolink, playing a role imilar to chemical crolink in ulcanizate and imparting the material elatomeric behaior. Since hard domain alo occupy ignificant olume and are tiffer than oft domain, they alo function a effectie nano-cale filler and render a material behaior imilar to that of a compoite (Petroic and Ferguon, 1991). At room temperature, oft domain are aboe their gla tranition temperature and impart the material it rubber-like behaior; hard domain are below their glay or melt tranition temperature and are thought to goern the hyterei, permanent deformation, high modulu, and tenile trength. The domain tructure alo impart TPU eratility in mechanical propertie. A wide ariety of property combination can be achieed by arying the molecular weight of the hard and oft egment, their ratio, and chemical type. For intance, TPU with hore hardne of from 6A to 7D are aailable (Payne and Rader, 1993). At preent, thermoplatic polyurethane are an important group of polyurethane product becaue of their adantage in abraion and chemical reitance, excellent mechanical propertie, blood and tiue compatibility, and tructural eratility. Here, the mechanical behaior of a repreentatie TPU i tudied in a erie of compreion tet probing the time-dependent and cyclic loading effect on the large train deformation behaior. A contitutie model for the obered tre-train behaior i then deeloped and compared directly to experimental data. 2 Stre-Strain Behaior The tre-train behaior of TPU demontrate trong hyterei, time dependence, and cyclic oftening. In thi ection, a erie of uniaxial compreion tet are conducted to 6

7 quantitatiely identify thee feature. 2.1 Mechanical Tet Decription Uniaxial compreion tet were conducted uing a computer controlled ero-hydraulic ingle axial tet machine, Intron model 135. The ample material wa a thermoplatic polyurethane with durometer hardne alue of 92A immediately after production and about 94A after 1 year of helf life at room temperature. Sheet of material of about 3mm in thickne were cut into cylinder of about 12mm diameter uing a die cutter. To eliminate potential buckling, the ample height to diameter ratio wa et to be le than 1. In addition, to reduce the contribution of friction due to the interaction with the compreion platen, Teflon heet were placed between the ample and the platen and the initial height/diameter ratio were et to be greater than.5. Specimen were ubjected to contant true train rate loading-unloading cycle and the true tre. true train cure wa documented for each tet. True train wa defined a the logarithm of the compreion ratio determined a the current height oer the initial height, where the current height of the ample wa monitored during teting uing an extenometer. Height meaurement were ued to form a feedback loop with the actuator to define and control the diplacement hitory uch that contant true train rate condition were achieed. True tre wa obtained by multiplying nominal tre by the compreion ratio (Smith, 1974; Petroic and Ferguon, 1991), auming the material incompreible 2. 2 Elatomer are generally incompreible. DSC (Differential Scanning Calorimeter) tet on the ample ued in thi paper howed that the gla tranition temperature wa about -4 C and the melting 7

8 TPU ample exhibited a certain amount of permanent et after each loadingunloading cycle. The dimenion (diameter and height) of the ample were meaured after each loading-unloading cycle to enure that the true tre-true train cure alway tarted from the new unloaded pecimen height for each cycle. The meaurement of the dimenion took about 2~3 minute, including re-poitioning the ample on the compreion platen and replacing the Teflon heet wheneer neceary. 2.2 Hyterei Figure 4 how the axial compreion true tre-true train behaior of two freh ample loaded to two different imum train, i.e. ε =. 5 and ε = 1., repectiely. The loading cure how an initially tiff repone, followed by rolloer to a more compliant behaior at a train of about.15, and tiffen again after a train of.7. The unloading path how a large hyterei loop with a reidual train. Additional recoery occurred with time after unloading. Reidual train were meaured r approximately 1 minute after the tet and were found to be ε =. 2 for the ε =. 5 r tet, and ε =. 62 for the ε = 1. tet. temperature wa about 18 C, which confirmed that the ample wa in rubbery tate at room temperature. It i hence reaonable to aume the Poion ratio ν range from.48 to.5. At logarithm compreion train of.5, the error in the tre determination due to the incompreible aumption i 2% (ν=.48) and 1% (ν=.49); at logarithm compreion train of 1., the error i 4% (ν=.48) and 2% (ν=.49). 8

9 2 16 train=.5, 1t cycle train = 1., 1t cycle -True Stre (MPa) True Strain Figure 4: Uniaxial compreion tet on freh ample (N=1) at a train rate ε =.1/, to different imum train ( ε =. 5 and ε = 1., repectiely). N indicate cycle number. 2.3 Time-Dependence Figure 5 how the true tre-true train cure to ε = 1. at three different compreion train rate, i.e. ε 1 =.1/, ε 2 =.5/, and ε 3 =.1/. For the loading portion of the cure, the higher the train rate, the larger the tre. The unloading cure from different train rate tet are nearly identical, uggeting that unloading behaior ha le rate dependence than loading behaior. The reidual train were r r meaured to be ε =. 62 for the ε 1 =.1/ tet, ε =. 46 for the ε 2 =.5/ tet, r and ε =. 43 for the ε 2 =.1/ tet. 9

10 2 16.1/.5/.1/ -True Stre (MPa) True Strain Figure 5: Uniaxial compreion tet at three different train rate. During the proce of loading and unloading, if the tet i upended, time dependence either in the tre repone (tre relaxation when the train i held contant) or in the train repone (creep when the tre i held contant) will be obered for TPU. Stre relaxation tet were conducted during the coure of loadingunloading cycle where the ample wa compreed to a imum train of 1. at a train rate of.1/ with intermittent 6 train holding period at train of.2,.4,.6 and.8 during both loading and unloading a hown in the train hitory plot of Figure 6(a). Figure 6(b) how the correponding true tre-time cure for thi tet. During each hold period, 5% of tre relaxation occurred in the firt 2~5. During loading, the tre wa obered to decreae during the train hold period; whilt during unloading, the tre wa obered to increae during the train holding period (Figure 6(b,c)). Thi behaior i characteritic of the time dependent behaior of more conentional elatomeric material (ee, for example, Lion, 1996; Bergtrom and Boyce, 1998). 1

11 1 2 -True Strain True Stre (MPa) Time () Time () (a) (b) True Stre (MPa) True Strain (c) Figure 6: (a) Applied train hitory for tre relaxation tet; (b) True-tre. time cure for uniaxial compreion tet with a number of intermittently top. (c) True-tre. true train cure for the ame tet. 2.4 Softening Figure 7 how the compreie true tre-true train behaior during the cyclic loadingunloading tet with ε = 1. and ε =.1/. Seeral feature are obered. Firt, in cyclic tet, the tre-train cure in the econd cycle i far more compliant than that 11

12 obered in the firt cycle; thi effect i referred to a oftening behaior. Second, the tre-train behaior tend to tabilize, after a few cycle, with mot oftening occurring during the firt cycle. Third, a the train upon reloading approache the imum train achieed in prior cycle, the tre tend to approach the tre leel of a firt cycle tet at that train. Fourth, the unloading path after a gien train all follow the ame cure, independent of cycle number. Fifth, oftening depend upon train hitory, where larger train produce greater oftening. Lat, the reidual train occur predominantly after the firt cycle, and no ignificant height change are obered after the additional cycle. In the ret of thi paper, the tet whereupon the tre doe not how ignificant decreae from preiou cycle will be referred a the tabilized tet; table cure are typically obered after only 4 cycle. N=1 N=2 N=4 N=1, 2, 4 Figure 7: Cyclic uniaxial compreion tet at train rate number. ε =.1/. N indicate cycle 12

13 ε=.5, N=1 ε=.5, N=4 ε=1., N=1 after the e=.5 tet ε=1., N=4 after the e=.5 tet ε=1., N=1 Figure 8: Uniaxial compreion tet with cyclic training to different imum train. N indicate cycle number. The train rate i ε =.1/. Softening wa further explored by teting to different cyclic train magnitude. Uniaxial compreion tet with ε =. 5 and ε = 1. were conducted on two freh ample, repectiely. The true tre-true train cure hown earlier in Figure 4 are repeated in Figure 8. The ample trained to ε =. 5 wa then ubjected to cyclic loading-unloading with ε =. 5. The true tre-true train cure tabilized after the 4 th cycle. Thi ample wa then trained to ε = 1.. A illutrated in Figure 8, upon training to ε = 1. after preiou excurion to ε =. 5, the true tre-true train cure for ε <. 5 moe along the preiouly tabilized oftened cure; a the train approache.5, the tre approache the preiou imum tre. After ε =. 5, the true tre-true train cure follow the coure hown by the freh ample tet with ε = 1., and the material exhibit the ame behaior a the freh material. The cyclic tet with ε = 1. reult in the tre-train behaior being oftened to the new tabilized cure a defined by ε = 1. cycle tet on an originally freh ample. Thee 13

14 ariation demontrate the trong dependence of the material behaior on the train hitory a well a characteritic feature of cyclic tre-train behaior. 2.5 Equilibrium Path In the tre relaxation tet, the tre relaxe toward an equilibrium tate during the holding period (Figure 6(b)(c) and Figure 9(a)(b)). Figure 9 how the tre relaxation behaior during the 1 t cycle tet and during the 4 th cycle tet after cycling between train of. and 1.. In thee tet, the tre relaxe toward two ditinct equilibrium path after 6 econd for the freh ample and the cycled ample 3. The relaxed alue at any train depend upon the imum train the material ha experienced in it prior loading hitory. During unloading the increae in tre at each hold period i the ame for both the 1 t cycle and the 4 th cycle tet ince both hae been trained to a imum train of 1. before unloading. The tre difference between the tabilized tre from loading. unloading hold period are fairly mall for the 4 th cycle tet. Thee oberation trongly imply that the unloading tre and the loading tre (except the 1 t loading) conerge to the ame equilibrium path, but thi path depend on the imum train experienced on the loading hitory. Preciely determining the equilibrium path for the 1 t cycle tet and the tabilized tet, howeer, i difficult becaue of ambiguity in the concept of long enough time for relaxation. For the firt loading, equilibrium path were determined by finding the point where another 1% tre relaxation would occur at 3 Further tre relaxation can be expected for a longer relaxation time, but the difference between the two equilibrium path i o ignificant that imply accounting for uch a difference a the reult of inufficient relaxation time i unrealitic. 14

15 the ame train. The equilibrium path for the relaxation tet on a preiouly loaded ample i determined by imply finding the midpoint of the point of the ame train on the loading and the unloading path of the tabilized tet (the 4 th cycle tet), auming that the et point under the ame train would conerge at their middle point if gien infinite time. Comparing the cyclic tet with ε =. 5 (Figure 4) to thoe with ε = 1., we conclude that the oftened equilibrium path after cycling at ε =. 5 mut be tiffer than the oftening equilibrium path after cycling at ε = 1.. Therefore, the degree of oftening of the equilibrium path increae with an increae in the prior imum train experienced during the oerall loading hitory N=1 N= N=1 N=4 Equilibrium path for N=1 Equilibrium path for N=4 -True Stre (MPa) True Stre Time () True Strain (a) (b) Figure 9: Uniaxial compreion tet with a number of intermittent train hold period, (a) true-tre. time cure, and (b) true-tre. true-train cure and the equilibrium path for initial and tabilized cycle. N indicate cycle number. 15

16 3 Contitutie Model 3.1 A reiew A contitutie model for the large train deformation of TPU hould addre the three alient feature of the material behaior: 1. Nonlinear large train elatomeric behaior; 2.Time dependence; 3.Softening of the equilibrium path obered during cyclic tet. The experimental data indicate that the tre-train behaior can be decompoed into a time-independent equilibrium path and a time-dependent departure from the equilibrium path, a illutrated in Figure 1. The contitutie model deeloped in Boyce et al (21c) for the tre-train behaior of thermoplatic ulcanizate (TPV) i ued here a a tarting point for the new contitutie model. The equilibrium part of the tretrain behaior act a the backbone of the oerall material tre-train behaior and i taken to originate from the entropy change of long molecular chain in the amorphou oft domain due to orientation of the molecular network with deformation. The ratedependent part i taken to originate from the concomitant internal energy change due to deformation of the hard domain. The icoplatic repone tend to relax the elatic deformation of the hard domain and hence produce the relaxation of the tre-train behaior to the equilibrium behaior with time. A the train rate approache an infiniteimal alue, the hard domain elatic deformation will be fully relaxed and the deiation from the equilibrium path will diminih. The icoplatic behaior come from energy diipation ource; potential ource include platic lip in hard domain, the breakage of hydrogen bond in the hard domain, poible frictional interaction between two hard domain, and the interaction between the oft and hard domain. In thi paper, all energy diipation mechanim are lumped into a ingle icoplatic contitutie 16

17 element. = + Figure 1: Decompoition of material behaior into a rate-independent equilibrium part and a rate dependent part. Within thi framework of decompoition of material behaior into an equilibrium component and a rate-dependent deiation from equilibrium, we attribute the obered train hitory induced oftening behaior to be due to oftening of the equilibrium path. In rubbery material, the oftening of the equilibrium path i referred to a the Mullin effect, o named due to the comprehenie tudy of thi behaior by Mullin on unfilled and filled rubber during the 195 and the 196 (Mullin and Tobin, 1957, 1965; Mullin, 1969). A demontrated in the preiou ection, egmented polyurethane alo demontrate oftening. It i generally belieed the domain tructure of egmented polyurethane i reponible for the oftening, the hyterei, and the correponding energy diipation. 17

18 Replacing phyical crolink in egmented polyurethane by dirupting the hard domain tructure and forming chemical crolink wa hown to reduce the oftening and hyterei; howeer, thi alo reulted in a lo in modulu and tenile trength (Cooper et al. 1976). A number of experimental tudie were conducted to inetigate the relationhip between mechanical propertie and material morphology (Bonart, 1968; Bonart et al., 1969; Bonart and Muller-Riederer, 1981; Ete et al., 1971; Kimura et al. 1974; Sauer et al. 22; Sequela and Prudhomme, 1978; Sung et al., 1979, 198a, 198b; Wilke and Abouzahr, 1981; Yeh et al. 23;). Bonart and coworker (Bonart, 1968; Bonart et al., 1969; Bonart and Muller-Riederer, 1981) ytematically tudied the morphology change during deformation uing X-Ray cattering. They and, more recently, Yeh et. al.(23), found that the tre inhomogeneity in the oft domain could lead to a rotation of the hard domain in order to minimize the oerall energy of deformation. They alo found that at large tretch the hard domain would break down to further accommodate tretch. Ete and coworker (1971) found that during tenile deformation the hard domain were diplaced into a poition where their longer dimenion were predominantly oriented perpendicular to the tretching direction, i.e., a configuration where the hard domain appeared to align perpendicular to the applied tre. To achiee the high degree of hard block orientation, it wa neceary that the hard domain underwent platic deformation, which wa accompanied by the breakage and reformation of hydrogen bond. At ufficiently high train, the hard domain might break into maller unit. At preent, mot elatomer oftening theorie are baed on two concept. The firt theory originate from Blanchard and Parkinon (1952) and Bueche (196, 1961), who 18

19 conidered the increae in tiffne produced by filler to be a reult of rubber-filler attachment proiding additional retriction on the crolinked rubber network. They attributed oftening to reult from the breakdown or looening of ome of thee attachment. Model baed on thi concept include the work of Bueche (196, 1961), Dannenberg (1974), and Rigbi (198), Simo (1987), Goindjee and Simo (1991, 1992), Miehe and Keck (2), and Lion (1996, 1997). The econd theory i due to Mullin, Tobin, Harwood, and Payne (Mullin and Tobin, 1957, 1965; Harwood et al., 1965; Harwood and Payne, 1966a, 1966b; Mullin, 1969), who treated the Mullin effect a the reult of eolution of the microtructure due to a quai-irreerible rearrangement of molecular network during deformation. Model baed on thi concept include Beatty and coworker (Johnon and Beatty, 1993a, 1993b; Beatty and Krihnawamy, 2), Marckmann et al. (22), and Ogden and coworker (Ogden and Roxburgh, 1997, 1999; Dorfman and Ogden, 23; Horgan et al. 23)(See Qi and Boyce (24) for a brief dicuion on thee model.). In filled elatomer, due to the exitence of tiffer filler, uually carbon black, the train (or the tretch) i magnified by an amplification factor due to the greater deformation in the oft domain needed to accommodate the applied train becaue of the ery low train in the tiffer filler domain. Baed on the concept of amplified train, Mullin and Tobin (1957) uggeted that the oftening in rubber ulcanizate wa due to the decreae of olume fraction of the hard domain with train, a a reult of conerion of the hard domain to the oft domain. Micromechanical modeling of rigid particle filled elatomer by Bergtrom and Boyce (1999) reealed that ome of the rubber became trapped among hard particle and could not deform, thu reulting in the effectie 19

20 fraction of tiff particle to be larger than the phyical fraction, o named occluded olume effect in filled elatomer potulated by earlier worker (Medalia and Krau, 1994). In a tudy of oftening in thermoplatic ulcanizate, where the ulcanizate were the filler, Boyce et al (21a, 21b) howed that the cyclic oftening wa due to the gradual eolution in particle/matrix configuration during preiou loading cycle. The platic deformation of the contiguou thermoplatic phae acted to releae ulcanizate particle creating a peudo-continuou ulcanizate phae and thu a ofter repone during ubequent cycle. Although the material in their tudy wa a ytem of oft filler/hard matrix, in contrat to filled rubber and TPU, which are hard filler/oft matrix, their reearch doe enlighten the eolution of the filler/matrix tructure with deformation. Baed on thee oberation and Mullin and Tobin concept of an eolution in the underlying hard and oft domain microtructure, we (Qi and Boyce, 24) recently propoed a new contitutie model to account for the obered oftening of the equilibrium tre-train behaior of elatomer ia an eolution in the oft and hard domain tructure with tretching. The model wa hown to capture tretch-induced oftening and i thu ued here to repreent the oftening of the equilibrium behaior of the TPU material. In thi paper, we propoe a comprehenie model that capture the obered time dependent nonlinear large train behaior and oftening of the tre-train behaior. The model i contructed following the decompoition of the material behaior a chematically illutrated in Figure 1 and i detailed in the following. 3.2 Contitutie Model Decription The model require three contitutie element, illutrated chematically in Figure 11 for 2

21 a one-dimenional rheological analog to the elatomer deformation model. The icoelatic-platic component conit of a linear elatic pring characterizing the initial elatic contribution due to internal energy change, and a nonlinear icoplatic dahpot capturing the rate and temperature dependent behaior of the material. The equilibrium behaior i modeled with the hyperelatic rubbery pring component capturing the entropy change due to molecular orientation of oft domain and i reponible for the high elaticity (large recoerability) of the oerall deformation. We note that in the actual material, there are contant interchange and interplay between deformation in the oft and hard domain. Here, we aerage out thi interplay by taking the two element to be in parallel in the one-dimenional analog which, in turn, correpond to ubjecting both element (the rubbery pring element and the icoelatic-platic element) to the ame deformation in the general three-dimenional cae. In the following, upercript N denote the ariable acting on the hyperelatic rubbery network pring, whilt upercript V denote the ariable acting on the icoelatic-platic component. Due to the parallel arrangement of thee component, Vicoelatic-platic component (V) Elatic pring Hyperelatic rubbery pring (N) Vico-platic dahpot Figure 11: One-dimenional chematic of the contitutie model. 21

22 where F i the macrocopic deformation gradient; F N = F V = F, (1) N F i the deformation gradient acting on the hyperelatic rubbery network, and V F i the deformation gradient acting on the icoelatic-platic component. The total Cauchy tre i thu gien by where N V = T T, (2) T + N T i the portion of the tre originating from the hyperelatic rubbery behaior; V T i the portion originating from the icoelatic-platic (hard) domain. The rubbery and icoelatic-platic element each require contitutie model a decribed below. 3.3 Hyperelatic Rubbery Network Behaior The tre acting on the hyperelatic rubbery pring, T N, capture the reitance to entropy change in the oft domain due to molecular network orientation (tretching and alignment), and i modeled uing the recent model propoed by Qi and Boyce (24) to account for the obered tretch-induced oftening behaior. Here, the model will be briefly outlined; the detailed decription of the model and dicuion can be found in Qi and Boyce (24). For an iotropic homogeneou elatomer, the Langein chain baed Arruda-Boyce eight-chain model (Arruda and Boyce, 1993) capture the hyperelatic behaior of the material up to large tretch and i ued here to repreent the equilibrium behaior. Taking the rubbery network train to be deiatoric, the Arruda-Boyce eight-chain model gie the Cauchy tre a N µ r N 1 λchain T = L B, (3) 3J λchain N where µ = nkθ, k i Boltzmann contant, Θ i abolute temperature, n i chain r 22

23 denity (number of molecular chain per unit reference olume) of the underlying macromolecular network, and N i the number of rigid link between two crolink (and/or trong phyical entanglement). In thi model, olumetric train 4, which i taken out through F N F will potentially contain a mall N 1 3 N N = J F where J det[ F ] =. B i the iochoric left Cauchy-Green tenor, 1 B = F N F NT, and B = B tr( B)I i the 3 I 1 deiatoric part of B. λ / 3 i the tretch on each chain in the eight-chain chain = network, and I = tr( ) i the firt inariant of B. L i the Langein function defined a 1 B 1 L ( β ) = cothβ. (4) β For thermoplatic polyurethane elatomer, the hard domain play an important role in the mechanical propertie of thermoplatic polyurethane elatomer. In addition to acting a phyical crolink imparting material rubbery tre-train behaior, hard domain occupy a ignificant olume ering a effectie tiff filler in the material (Petroic and Ferguon, 1991). Thi ugget that it i reaonable to model the equilibrium behaior of TPU uing the methodology for elatomer compoite where the elatomer i reinforced with ery tiff particle. Here, the oft domain are treated a the matrix occupying effectie olume fraction, and the hard domain are treated a filler occupying effectie olume fraction h. We emphai that and h hould be regarded a effectie olume fraction and are different from the actual olume fraction 4 In thi comprehenie model, the bulk reitance to olumetric train (i.e. the bulk modulu) will be lumped into the icoelatic-platic component which act in parallel with the rubbery pring element. 23

24 calculated from the material compoition of oft and hard egment. Thi difference will be dicued in more detail following the preentation of the model for the hyperelatic rubbery pring. In filled elatomer, the aerage train (or alternatiely, tretch) in the elatomeric domain i necearily amplified oer that of the macrocopic train ince the tiff filler particle accommodate little of the macrocopic train. For a general three dimenional deformation tate, the firt inariant of the tretch i amplified by (Bergtrom and Boyce, 1999) ( 3) 3 I m, (5) 1 = X I1 + where I 1 i the aerage I m 1 in the matrix, and I 1 i the oerall macrocopic I 1 of the compoite material, and X i an amplification factor dependent on particle olume fraction, ν f, and ditribution. X can take a general form of X = +. Here, we f b f further chooe ( ) ( ) 2 X = to characterize a well-dipered compoite ytem where ν i the oft domain olume fraction. Following Bergtrom and Boyce(2) and Qi and Boyce(24), the Cauchy tre a modified by amplified train i gien by: N X µ N 1 Λ chain T = L B, (6) 3J Λ chain N where 2 Λ i the amplified chain tretch defined a Λ = ( ) + 1 chain 1 2 chain X λ chain and X ( ) 18( 1 ) 2 + = In a TPU, the effectie olume fraction of the oft domain i not equal to the olume fraction of oft egment calculated from the material chemitry. On the one 24

25 hand, phae eparation in TPU i generally incomplete and there exit iolated hard egment, reulting in a maller olume fraction of the hard domain than that obtained from compoition calculation; on the other hand, a portion of the oft domain i occluded by the hard domain, reulting in an increae in the effectie hard domain olume fraction. The latter will dominate the mall to middle deformation, wherea the firt caue the effectie oft domain olume fraction to poibly become larger than the egment compoition calculation at large deformation when all of the oft domain ha been releaed from occluion and iolated hard egment may reole in oft domain. Following Qi and Boyce (24), we take to eole with deformation where initially occluded region of oft domain are gradually releaed with deformation and thi eolution i drien by the local chain tretch, Λ chain, in the oft domain. Upon remoal of applied load, it wa found that the hard domain remained largely in the deformed configuration with ery little recoery (Ete et al. 1971). We thu aume that the configuration change of the hard domain i taken to be permanent. Therefore, we take to remain at it alue attained at the imum chain tretch Λ chain encountered during it loading hitory. Eolution in will be re-actiated a the local chain tretch exceed the preiou imum chain tretch. Therefore, when Λ > chain Λ chain, i modeled to increae with increaing Λ chain, and thu the amplification X decreae with increaing Λ chain. We alo aume arie from an initial alue to a aturation alue a the local chain tretch locking Λ chain reache the locking tretch of the chain, λ chain. The eolution of i taken to obey the following rule: 25

26 λ 1 = A lock chain ( ) Λ 2 chain lock ( λ Λ ) chain chain, (7a) where Λ < Λ Λ, chain chain chain = Λ chain, Λ chain Λ, (7b) chain and A i a parameter that characterize the eolution in with increaing Λ chain. (For a dicuion on thi eolution rule and it effect on the Cauchy tre, ee Qi and Boyce (24).) 3.4 Vicoelatic-platic Element V The tre contribution from the icoelatic-platic component, T, i gien by T V e [ ln V ] V e = h L, (8) V e det F where e L i the fourth-order tenor modulu of elatic contant; V e F i the elatic deformation gradient and V e V i the left tretch tenor of the elatic deformation gradient obtained from the polar decompoition V e V e V e F = V R, where V e R i the rotation tenor. h i the effectie olume fraction of hard domain. It hould be noted that a TPU i not a compoite material, een though a methodology for compoite material aeraging i ued here. Since the icoelatic-platic component capture a ignificant portion of the initial elatic tiffne of the material and lumped energy diipation, the combination of thee behaior i belieed to relate to the hard domain and oft-hard domain interaction. We imply take h =1. 26

27 V V e F = F F V Current configuration F V F V e Initial configuration Relaxed configuration Figure 12: Schematic of decompoition of V F into elatic and ico-platic part. of The elatic deformation gradient i determined from the multiplicatie decompoition V F into elatic and icoplatic contribution (Figure 12) where V V e V F = F F. (9) V F i in a relaxed configuration obtained by elatic unloading. The correponding decompoition of the elocity gradient i L V = V V 1 V e V e 1 V e V V 1 V e 1 F F = F F + F F F F. (1) The elocity gradient of the relaxed configuration i gien by V V V V V L = F 1 F = D + W, (11) where V D and V W are the rate of tretching and the pin, repectiely. We take V W = with no lo in generality a hown in Boyce et al (1988). The ico-platic tretch rate V D i contitutiely precribed to be D V = γ 2τ V V T, (12) where V T i the tre acting on the icoelatic-platic component conected to it 27

28 V V V e relaxed configuration ( T = R T R ); the prime denote the deiator; τ V i the equialent hear tre and V e T V V τ V = T T 2. (13) γ denote the ico-platic hear train rate of the icoplatic component, and i contitutiely precribed to take the form G τ V γ = γ exp 1, (14) kθ where γ i the pre-exponential factor proportional to the attempt frequency, G i the zero tre leel actiation energy, and i the athermal hear trength, which repreent the reitance to the ico-platic hear deformation in TPU. The true tre-true train behaior including relaxation period (Figure 9) how that the amount of tre decreae during loading train hold period i larger than the amount of tre increae during unloading train hold period. Thi effect i ignificant during the 1 t cycle tet. Thi ugget the reitance to icou flow eole with train hitory. Although the mechanim i not clear yet, a poible conjecture i that uch a mechanim change i related to the configuration change of oft and hard domain a eidenced by Ete et al (1971). During the loading coure, the hard domain cluter will irreeribly change their configuration to accommodate local deformation by breaking and reforming the hydrogen bond. We propoe the following eolution rule a a firt tep to capture thi behaior, h 1 =. (15) h 28

29 3.5 Summary of the Contitutie Model The deformation behaior of a TPU i not triial. It i highly nonlinear; it i rate dependent; it i hyteretic; and it often with cyclic loading where the degree of oftening depend on the imum train leel reached in prior cycle. The new contitutie model i ummarized in Table 1 where the material parameter needed to fully capture each feature of the deformation behaior are lited. Due to the ability to ytematically break down the tre-train behaior, a ytematic procedure for determining alue for the material propertie can be deeloped. The three contitutie element in the model each account for different feature of the obered material behaior: i.e. the hyperelatic rubbery oftening pring for equilibrium behaior; the eolution of effectie olume fraction of oft domain for the oftening of equilibrium path; the linear elatic pring accounting for the tiffne contribution of the time dependent behaior; and the icoplatic dahpot accounting for the nonlinear time dependent behaior. It i thu poible to identify the material parameter aociated with the different feature of the material behaior. The Appendix proide a methodology to identify the material parameter. 29

30 Table 1. Summary of contitutie model and material parameter. Equilibrium Stre-Strain Repone T N Hyperelatic Rubbery Spring Element Hyperelatic filler effect Softening µ r, N,,, A X µ N Λ T N chain = L 1 B 3 J Λ N chain ( ) 18 ( 1 ) 2 + X = = Λ chain ( ) exp lock λ chain Λ chain A 1 N T = T + T V Non-equilibrium Rate Dependent Stre-Strain Repone Linear Spring Element Vicoplatic T V Dahpot Time Dependence ν E, γ, G V e T = h L ln V e det F h = 1 γ = γ exp 1 k Θ Element h Softening, = V e [ V ] G τ V 3 i

31 4 Reult The material parameter for the TPU teted in thi tudy were identified uing the procedure outlined in the Appendix and are lited in Table 2. Table 2: Material parameter. Stretch Softening Hyperelatic Rubbery Spring µ N A r (MPa) Elatic-Vicoplatic Component Linear Elatic Spring Vicoplatic Dahpot E ν γ G (MPa) (MPa) ( 2 1 ) (1 2 J ) Uniaxial compreion tet at different contant train rate on freh ample were imulated to erify the propoed contitutie model. Figure 13(a) how the imulation and experimental tre-train cure for the tet at 1 ε =.1 and ε =.1 1. The imulation reult agree ery well with the experimental data and capture the rate dependence of the tre train behaior. Figure 13(b) how the decompoition of the material tre-train behaior into an equilibrium part (ame for both train rate) and a time dependent part, illutrating the methodology of material behaior decompoition. The propoed model doe not fully capture the feature that the unloading cure follow 31

32 the ame path for the tet at different train rate, but the difference i mall. Further improement in the oerall unloading behaior can be achieed by allowing a mall amount of time prior to unloading. Figure 14 how the true tre-true train cure from the imulation when a 3 econd delay wa allotted prior to unloading. -True Stre (MPa) Model,.1/ Model,.1/ Experiment,.1/ Experiment,.1/ -True Stre (MPa) Oerall,.1/ Equilibrium,.1/ Time-dependent,.1/ Oerall,.1/ Equilibrium,.1/ Time-dependent,.1/ True Strain True Strain (a) (b) Figure 13: (a) True Stre-true train cure for uniaxial compreion tet; (b) Decompoition of the tre-train behaior into an equilibrium part and a time dependent part. ε t() Figure 14: True Stre-true train cure for uniaxial compreion tet with ε =.1/ and 3 econd delay before unloading. The inet how the true train loading hitory. 32

33 Simulation on cyclic loading-unloading tet were alo conducted. Recall that the TPU ample howed reidual train (permanent et) after unloading. In the experiment, the dimenion of the ample (diameter and height) were meaured between cycle, and were ued a the new dimenion for the ample o that the true tre-true train cure alway tarted from the new unloaded pecimen height for each cycle. Such a proce of haing the true tre-true train cure begin at the origin by meauring the dimenion before each tet correpond to imply hifting the true tre-true train cure leftward r by the amount of reidual train ε. In imulation, the true tre-true train cure were firt obtained baed on the initial dimenion of the ample and were then hifted leftward by the reidual train meaured after 2 minute idling time between the two cyclic imulation, o choen correponding to the 2 minute period between cycle ued to meaure the dimenion change and repoition the ample on the platen for the next cycle of loading. Figure 15(a) how the loading hitory for the tet with ε = 1. and ε =.1 1. Due to the hift of the cure, the econd cycle imulation wa loaded to the train r 1 + ε, where r ε i the reidual train meaured from the firt cycle imulation. Simulation on cyclic loading-unloading tet were conducted at a train rate of 1 ε =.1 (Figure 15(b)) and ε =.1 1 (Figure 15(c)). The model capture the loading and unloading path at both train rate. The reidual train after 2min idling time in the imulation wa.5, which wa ery cloe to the reidual train of.4~.6 obered in the experiment. Figure 15(d) and (e) how the decompoition of the material tre-train behaior into an equilibrium part and a time dependent part for each cycle of the tet. From Figure 15(d) and (e), the equilibrium part during loading and unloading for the econd cycle follow the ame path, indicating that additional oftening doe not 33

34 occur during the econd loading cycle TrueStrain Time () (a) (b) -True Stre (MPa) Oerall, N=1 Equilibrium, N=1 Time-dependent, N=1 Oerall, N=2 Equilibrium, N=2 Time-dependent, N= True Strain (c) (d) 34

35 -True Stre (MPa) Oerall, N=1 Equilibrium, N=1 Time-dependent, N=1 Oerall, N=2 Equilibrium, N=2 Time-dependent, N= True Strain (e) Figure 15: Numerical imulation on cyclic loading tet: (a) loading hitory; Stre-train behaior for (b) ε =.1/ ; and (c) ε =.1/ ; Decompoition of the tre-train behaior into an equilibrium part and a time dependent part for (d) ε =.1/ ; and (e) ε =.1/. N indicate cycle number. Figure 16(a) how the cyclic loading to different imum train at ε =.1/ where the ample wa ubjected to three loading-unloading cycle: the firt cycle wa loaded to ε =. 5, and the econd and the third cycle were loaded to ε = 1.. The correponding experimental reult are alo preented in the figure. The numerical imulation adequately capture the oftening effect during the cyclic tet. It i noted that the experimental reult ued to obtain material parameter did not include the tet with loading to ε =. 5, but the model predict the oftening repone correponding to.5 train. Figure 16(b) how the eolution of the effectie olume fraction of oft domain during thi deformation coure. The eole with train from the original =. 4 and reache =. 59 at ε =. 5. Thi alue for i retained until the train exceed.5 upon reloading, whereupon tart increaing again. 35

36 -True Stre (MPa) Model, N=1 Model, N=2 Model, N=3 Experiment, N=1 Experiment, N=2 Experiment, N=3 ν 3 rd Loading 2 nd Unloading 2 nd Loading 1 t Unloading 1 t Loading 2 nd Loading True Strain (a) (b) Figure 16: Cyclic loading to different imum train, i.e. ε =. 5 firt, then ε = 1. with train rate of ε =.1/. (a) true tre-true train cure; (b) eolution of effectie olume fraction of oft domain. For both imulation and experiment, the cure are the firt loading to ε =. 5 then unloading (N=1), followed by reloading to ε = 1. then unloading (N=2), finally reloading to ε = 1. then unloading (N=3). In experiment, the ubequent reloading occurred after the tre-train behaior tabilized. Figure 17 how the numerical imulation of the relaxation tet at ε =.1/. For the firt cycle (Figure 17(a)), the numerical imulation capture the decreae/increae of the tre during each hold period during loading/unloading, except for the top at ε = 8% during unloading due to relatie low tre drop at the tranition from loading to unloading in the imulation. Figure 17(c) and (d) how the one-dimenional decompoition of the material tre-train behaior into an equilibrium part and a time dependent part for each cycle of the tet. 36

37 2 16 Experiment, N=1 Model, N=1 2 Experiment, N=4 Model, N=4 -True Stre (MPa) True Stre (MPa) True Strain True Strain (a) (b) Oerall, N=1 Equlibrium, N=1 Time dependent, N=1 15 Oerall, N=2 Equlibrium, N=2 Time dependent, N=2 -True Stre (MPa) 1 5 -True Stre (MPa) True Strain True Strain (c) (d) Figure 17: Numerical imulation of relaxation tet: (a) the 1 t cycle; (b) the 2 nd cycle cure; Decompoition the material tre-train behaior into an equilibrium part and a time dependent part: (c) the 1 t cycle; (d) the 2 nd cycle cure. N indicate cycle number. 5 Concluion The large train tre-train behaior of thermoplatic polyurethane wa inetigated in thi paper. It wa hown by uniaxial compreion tet that TPU exhibited ery complicated tre-train behaior, which ha trong rate dependence, hyterei, and 37

38 oftening, where the oftening i eident upon reloading. A et of experiment ha been preented which acted to iolate thee ariou dependencie and feature of the tretrain behaior. In eeking a contitutie that can be applied to general 3-dimenional engineering analyi, uch a complicated behaior impede any attempt at a imple phenomenological cure-fit model and dictate the need for a phyically-baed contitutie model. A contitutie model accounting for the rate dependent hyterei behaior of polyurethane material and the oftening behaior i then preented. The contitutie model decompoe the material behaior into a rate-independent equilibrium part and a rate-dependent icoelatic-platic part. For the oftening of the equilibrium path, the model adopt the concept of amplified train, and take the train amplification factor to eole with loading hitory due to microtructural reorganization of the oft and hard domain which act to increae the olume fraction of the effectie oft domain. Comparion of numerical imulation of uniaxial compreion tet with experimental data erify the propoed contitutie model. The model adequately capture the oerall nonlinear behaior, the oftening (Mullin effect), and the time dependent behaior of the TPU. The underlying material tructure of TPU undergoe ignificant change during large deformation. A a firt approach, thee change are emulated in the propoed contitutie model by imply taking the olume fraction of the oft domain to eole with deformation. Howeer, tructural change will alo alter other tructure-dependent procee uch a the time-dependent icou repone and will reult in aniotropy of the tructure and behaior. The underlying phyical proce for thee eolution rule i not 38

39 clear. In the future, micromechanical modeling, together with adanced experiment whereby tructural eolution i monitored with deformation, hould be ued to explore thee important apect. Appendix: Parameter Identification for the Contitutie Model In order to identify the material parameter in the propoed contitutie model, two type of cyclic tet are neceary: Cyclic relaxation tet (tenion or compreion) with intermittent holding period at ditinct train during both loading and unloading coure to identify equilibrium tre-train repone of the material; cyclic tet (tenion or compreion) at different train rate to identify rate-dependent behaior of the material. Here, the procedure of parameter identification i exemplified uing the cyclic uniaxial compreion tet and relaxation tet decribed aboe. A.1 Material Parameter Identification for Hyperelatic Rubbery Softening Spring The material parameter aociated with the hyperelatic rubbery component of the contitutie model can be determined uing the two equilibrium path (the 1 t cycle tet and the 4 th cycle tet after cyclic loading to a train of 1.) preented in Figure A1. The initial Young moduli for thee two cure were meaured from the initial lope of the cure, ( ) E r 24MPa and E () 1 r 14MPa, where E ( ) denote the initial Young r modulu for the 1 t cycle equilibrium path, and () 1 E denote the initial Young modulu r for the tabilized equilibrium path after a imum cyclic train of 1.. In the following, a upercript denote the ariable for the 1 t cycle equilibrium path, and a upercript 1 denote the ariable for the tabilized equilibrium path. The ratio i E ( ) ( 1) r E r = 1.7, and 39

40 from eqn.(6), E E ( ) r () 1 r ( ) () 1 () 1 2 [ ( 1 ) + 18( 1 ) ] () 1 [ ( 1 ) 18( 1 ) ] 2 + = = (A1) X X () 1 () 1 where () 1 i effectie olume fraction of oft domain at ε = 1., and remain contant during the 4 th cycle tet. The chemical compoition of the TPU ued in the current tudy i 57% oft egment and 43% hard egment. Therefore, baed on the argument in the dicuion of the effectie olume fraction, it i reaonable to aume that =. 4 and =. 8. From eqn. (A1), 1 ().75, ( ) X = 9.58 and () X 1 = Equilibrium path, N=1 Equilibrium path, N=4 -True Stre (MPa) True Strain Figure A1: Equilibrium path from the 1 t relaxation tet and the 4 th relaxation tet. N indicate cycle number. Since in the 4 th cycle tet, both and X remain contant, it i conenient to ue the tabilized equilibrium path to determine parameter µ r and N in the Arruda-Boyce model. We found µ r = 1. 4MPa and N = 6.. The locking chain tretch hence i λ = N = locking chain 4

41 In a uniaxial compreion tet, the compreion ratio in the axial direction i λ 1 and i related with compreion train ε by λ ε 1 = e. Auming material i incompreible, the lateral tretch ratio are λ 2 = λ3 = 1/ λ1. Then the macrocopic equialent tretch i λ chain = λ + λ + λ = e 2ε + 2e 3 ε 2e 3 ε (A2) For ε = 1., λ The amplified chain tretch ratio i thu = chain eqn.(7), we obtained A Figure A2 how the cure fitting uing the etimated parameter Λ chain µ r = 1. 4MPa, N = 6., A = 1. 4, =. 4, =. 8.. From 2 -True Stre (MPa) Equilibrium path, N=1 Equilibrium path, N=4 Arruda-Boyce model, N=1 Arruda-Boyce model, N= True Strain Figure A2: Material parameter identification for the rubbery hyperelatic pring. N indicate cycle number. 41

42 -True Stre (MPa) N=1, train=.5 N=1, train=1., after train=.5 N=2, train= True Strain Figure A3: The tre-train behaior of equilibrium behaior during cyclic loading to a imum train of.5 for firt cycle, then reloading to 1. for two cycle. N indicate cycle number. The obtained parameter for the equilibrium path were ued to imulate the tet where the ample wa ubjected to three load-unloading cycle: the firt one to a imum train of.5, wherea the lat two to a imum train of 1.. Figure A3 how the numerical imulation. Clearly, the loading to the imum train of.5 how le oftening in the tre-train behaior than that with the imum train of 1.. A.2 Material Parameter identification for icoelatic-platic component From the one dimenion implification of eqn.(2), the axial tre acting on the icoelatic-platic component V T i determined by T V N = T T. (A3) The elatic modulu E for the elatic pring in the icoelatic-platic component can be determined ince the initial Young modulu of the material i the ummation of the contribution from hyperelatic rubbery pring and the elatic pring. The initial oerall 42

43 Young modulu wa meaured from the true tre-true train cure in Figure 7, E 55MPa. Hence, ν he = E Er 31MPa, where ν h = 1. Therefore E 51MPa. Poion ratio wa choen a ν =. 48 to enure mall material compreibility. For the material parameter aociated with the icoplatic dahpot element,, G, and γ can be determined uing the loading cure. From eqn.(a3), for the tet at difference train rate were contructed, a hown in Figure A4(a). V T ε plot The equialent hear train γ and hear tre τ are related to train and tre in uniaxial compreion tet by 1 V γ = 3ε, τ = T (A4) 3 The equialent ico-platic hear train i obtained by ubtracting the elatic hear deformation from the total equialent hear train γ = γ τ / G, where G / 3 (A5) E The equialent ico-platic hear tre and hear train cure at different train rate hereby were contructed for the loading path (Figure A4(b)). Eqn.(14) decribe the icoplatic flow rate and can be rewritten a τ = c ln γ + b, (A6) D b = D lnγ, D c =, ( ) G D =. (A7) k Θ Eqn. (A6), together with Figure A4(b), proide inight into the eolution of during deformation. Indeed, a detailed eolution rule for can be identified by contructing τ γ cure at a number of γ from Figure A4(b), and then inetigating the ariation of 43

44 the lope and interception of the cure with repect of γ. Here, for the ake of breity, =. we aume eole with effectie olume fraction of hard domain, i.e. ( ) A hown in the reult, uch implification generally can gie good prediction of the tre-train behaior of the material. h h 1 -Non-equilibrium Stre (MPa) /.5/.1/ τ (MPa) True Strain γ (a) (b) Figure A4: (a) rate. V T ε plot at different train rate; (b) τ -γ plot at different train From Figure A4(b), the equialent hear tre at each equialent hear train rate approximate to a contant alue at large equialent hear train. For γ 1 =.173/ (from ε =.1/ ), τ 1 = 2.9MPa ; for γ 2 =.866 / (from ε =.5/ ), τ 2 = 3.8MPa ; for γ =.173/ 3 (from ε =.1/ ), τ 3 = 4.3 MPa. Uing leat quare fit gie c =.6, b = From eqn (A7), we obtained 8.3 γ = e D, (A8a) 44

45 D =. (A8b).6 Uing a kinked model, Argon (1973) and Argon and Beono (1977) predicted for a wide range of glay polymer,. 77G =. (A9) 1 Although the reitance to flow i a more complicated mechanim in the TPU, we ue thi expreion a a guideline. From eqn(a9), ~ 2. 52MPa. = 2. 52MPa wa thu ued a a tarting point for identifying material parameter. From eqn. (A8), we obtained = 1 =. 2 2 γ 1.94 and G J Reference Argon, A.S., A theory for the low temperature platic deformation of glay polymer. Phil. Mag., 28, Argon, A.S., Beono, M.I., Platic deformation in polyimide, with new implication on the theory of platic deformation of glay polymer. Phil. Mag., 35, Arruda, E.M., Boyce, M.C., A Three-dimenional contitutie model for the large tretch behaior of elatomer. J. Mech. Phy. Solid, 41, Beatty, M.F., Krihnawamy, S., 2. A theory of tre-oftening in incompreible iotropic material. J. Mech. Phy. Solid, 48, Bergtrom, J., Boyce, M.C., Mechanical behaior of particle filled elatomer. Rubber Chem. Tech., 72,

46 Bergtrom, J.S., Boyce, M.C., 2. Large train time-dependent behaior of filled elatomer. Mech. Mater., 32, Blanchard, A.F., Parkinon, D., Breakage of carbon-rubber network by applied tre. Ind. Eng. Chem., 44, 799. Bonart, R., X-ray inetigation concerning the phyical tructure of cro-linking in urethane elatomer. J. Macromol. Sci., Phy, B2, 115. Bonart, R., Morbitzer, L., Hentze, G., X-ray inetigation concerning the phyical tructure of cro-linking in urethane elatomer, II. J. Macromol. Sci. Phy., B3, Bonart, R., Muller-Riederer, G., Modellortellungen zur molekulorientierung in gedehnten egmentierten polyurethan elatomeren. Colloid Polym. Sci., 259, Boyce, M.C., Socrate, S., Yeh, O.C., Kear, K., Shaw, K., 21a. Micromechanim of Deformation and Recoery in Thermoplatic Vulcanizate. J. Mech. Phy. Solid, 49, Boyce, M.C., Yeh, O.C., Socrate, S., Kear, K., Shaw, K., 21b. Micromechanim of the Cyclic Softening in Thermoplatic Vulcanizate. J. Mech. Phy. Solid, 49, Boyce, M.C., Kear, K., Socrate, S., Shaw, K., 21c. Deformation of thermoplatic ulcanizate. J. Mech. Phy. Solid, 49, Boyce, M.C., Park, D.M., Argon, A.S., Large inelatic deformation of glay polymer, Part I: rate dependent contitutie model. Mech. Mater., 7, Bueche, F., 196. Molecular bai for the Mullin effect. J. Appl. Polym. Sci., 4, 17-46

47 114. Bueche, F., Mullin effect and rubber-filler interaction. J. Appl. Polym. Sci. 5, Chen-Tai, C.H.Y., Thoma, E.L., MacKnight, W.J., Schneider, N.S., Structure and morphology of egmented polyurethane. 3. Electron microcopy and mall angle X- ray cattering tudie of amorphou random egmented polyurethane. Polym., 27, 659. Cooper, S.L., Tobolky, A.V., Propertie of linear elatomeric polyurethane. J. Polym. Sci., 1, Cooper, S.L., Huh, D.S., Morri, W.J., Encyclopedia of Polymer Science and Technolgy, Supplementary Volume, Wiley, New York, 521. Dannenberg, E.M., The effect of urface chemical interaction on the propertie of filler-reinforced rubber. Rubber Chem. Tech., 48, Dorfmann, A., Ogden, R.W., 23. A peudo-elatic model for loading, partial unloading and reloading of particle-reinforced rubber. Int. J. Solid Struct., 4, Ete, G.M., Seymour, R.W., Cooper, S.L., Infrared tudie of egmented polyurethane elatomer. II, Macromol., 4, Goindjee, S., Simo, J., A micro-mechanically baed continuum damage model for carbon black-filled rubber incorporating Mullin effect. J. Mech. Phy. Solid, 39, Goindjee, S., Simo, J., Tranition from micro-mechanic to computationally efficient phenomenology: carbon black filled rubber incorporating Mullin effect. J. Mech. Phy. Solid, 4,

48 Harwood, J.A.C., Mullin, L., Payne, A.R., Stre oftening in natural rubber ulcanizate, Part II. J. Appl. Polymer Sci., 9, Harwood, J.A.C., Payne, A.R., 1966a. Stre oftening in natural rubber ulcanizate, Part III. J. Appl. Polymer Sci., 1, Harwood, J.A.C., Payne, A.R., 1966b. Stre oftening in natural rubber ulcanizate, Part IV. J. Appl. Polymer Sci., 1, Hepburn, C., Polyurethane Elatomer, Applied Science, London. Horgan, C.O., Ogden, R.W., Saccomandi, G., 23. A theory of tre oftening of elatomer baed on finite chain extenility. To be appeared in Proc. R. Soc. Lond. A. Johnon, M.A., Beatty, M.F., 1993a. The mullin effect in uniaxial extenion and it influence on the tranere ibration of a rubber tring. Comtinuum Mech. Thermodyn., 5, Johnon, M.A., Beatty, M.F., 1993b. A contitutie equation for the Mullin effect in tre controlled uniaxial extenion experiment. Comtinuum Mech. Thermodyn., 5, Kimura, I., Ihihara, H., Ono, H., Yahihara, N., Nomura, S., Kawai, H., Morphology and deformation mechanim of egmented poly(urethaneurea) in relation to pherulitic crytalline texture. Macromol., 7, Koutky, J.A., Hien, N.V., Cooper, S.L., 197. Some reult on electron microcope inetigation of polyether-urethane and polyeter-urethane block copolymer. Polymer Lett., 8, Lion, A., A contitutie model for carbon black filled rubber, experimental inetigation and mathematical repreentation. Continuum Mech. Thermodyn., 6, 48

49 Lion, A., A phyically baed method to repreent the thermo-mechanical behaior of elatomer. Acta Mechanica, 123, Marckmann, G.,Verron, E., Gornet, L., Chagnon, G., Charrier, P., Fort, P., 22. A theory of network alteration for the Mullin effect. J. Mech. Phy. Solid, 5, Medalia, A.I., Krau, G., Science and Technology of Rubber, Academic Pre Inc., New York. Miehe, C., Keck, J., 2. Superimpoed finite elatic-icoelatic-platoelatic repone with damage in filled rubbery polymer: Experiment, modeling and algorithmic implementation. J. Mech. Phy. Solid, 48, Mullin, L., Tobin, N.R., Theoretical model for the elatic behaior of fillerreinforced ulcanized rubber. Rubber Chem. Technol., 3, Mullin, L., Tobin, N.R., Stre oftening in natural rubber ulcanizate, Part I. J. Appl. Polymer Sci., 9, Mullin, L., Softening of rubber by deformation. Rubber Chem. Tech., 42, Ogden, R.W, Roxburgh, D.G., 1999a. A peudo-elatic model for the Mullin effect in filled rubber, Proc. R. Soc. Lond. A, 455, Ogden, R.W, Roxburgh, D.G., 1999b. An energy-baed model of the Mullin effect, in Proceeding of the Firt European Conference on Contitutie Model for Rubber, Belkema, Rotterdam. Payne, M.T., Rader, C.P., Thermoplatic elatomer: a riing tart, in: Elatomer 49

50 Technology Handbook (ed. By Cheremiinoff, N.P.), CRC pre. Petroic, Z., Ferguon, J., Polyurethane Elatomer. Prog. Polym. Sci., 16, Qi, J.H., Boyce, M.C., 24. Contitutie Model for Stretch-Induced Softening of the Stre-Stretch Behaior of Elatomeric Material. J. Mech. Phy. Solid., aailable online a Article in Pre. Rigbi, Z., 198. Reinforcement of rubber by carbon black. Ad. Poly. Sci., 36, Sauer, B.B., Mclean, R.S., Brill, D.J., Londono, D.J., 22. Morphology and orientation during the deformation of egmented elatomer tudied with mall-angle X-ray cattering and atomic force microcopy. J. Ploym. Sci., 4, Sequela R., Prudhomme, J., Effect of cating olent on mechanical and tructural propertie of polydiene-hydrogenated polytyrene-polyioprene-polytyrene and polytyrene-polybutadiene-polytyrene block copolymer. Macromol., 11, Schollenbenger, C.S., Scott, H., Moore, G.R., Polyurethane VC, a irtually crolinked elatomer. Rubber World, 137, 549. Simo, J.C., On a fully three-dimenional finite-train icoelatic damage model: formulation and computational apect. Compu. Method Appl. Mech. Eng., 6, Smith, T.L., Tenile trength of polyurethane and other elatomeric block copolymer. J. Ploym. Sci., 12, Sung, C.S.P., Smith, T.W., Hu, C.B., Sung, N.H., Hyterei behaior in polyether poly(urethaneurea) baed on 2,4-toluene diiocyanate, ethylenediamine, and 5

51 poly(tetramethylene oxide). Macromol., 12, Sung, C.S.P., Hu, C.B., Wu, C.S., 198a. Propertie of egmented poly(urethaneurea) baed on 2,4-toluene diiocyanate I. Macromol., 13, Sung, C.S.P., Smith, T.W., Sung, N.H., 198b. Propertie of egmented poly(urethaneurea) baed on 2,4-toluene diiocyanate, II. Macromol., 13, Wang, C.B., Cooper, S.L., Morphology and propertie of egmented polyether polyurethaneurea. Macromol. 16, Wilke, G.L., Abouzahr, S., SAXS tudie of egmented polyether poly(urethaneurea) elatomer. Macromol., 14, Yeh, F., Hiao, B.S., Sauer, B.B., Michel, S., Sieler, H.W., 23. In itu tudie of tructure deelopment during deformation of a egment poly(urethane-urea) elatomer. Macromol., 36,