IRRECOVERABLE AND RECOVERABLE NONLINEAR VISCOELASTIC PROPERTIES OF ASPHALT CONCRETE

Transcription

1 IRRECOVERABLE AND RECOVERABLE NONLINEAR VISCOELASTIC PROPERTIES OF ASPHALT CONCRETE James S. Lai, University of Utah; and Douglas Anderson, Utah State Department of Highways The results are reported of a series of uniaxial ompression reep tests of an asphalt mixture under onstant loadings, multiple-step loadings, and repeated loadings of as many as 1 yles. It has been shown that the nonlinear visoelasti behavior of the asphalt onrete an be represented by a nonlinear generalized Kelvin model that onsists of a nonlinear dashpot onneted in series with a nonlinear Kelvin hain. Thus, the nonlinear reep strains are separated into irreoverable and reoverable strains. It has been shown that the onstitutive equation an be determined by the use of both the reep and the reovery parts of the onstant loading reep test results. The auray in prediting the reep behavior of the asphalt onrete under the multiple-step loadings and the repeated loadings has been shown to be very satisfatory. The importane of the irreoverable strains to the pratial implementation of asphalt pavements subjeted to traffi loading is disussed. A possible way of relating the irreoverable strains to the fatigue life of the material is also disussed. IN the past 1 years, onsiderable interest has been developed in an attempt to haraterize the time and temperature dependene of the mehanial properties of asphalt paving mixtures within the framework of visoelasti theory, espeially the linear visoelasti theory. Most of the test results reported so far on the visoelasti haraterization of asphalt mixtures have been obtained from the onstant stress reep tests, or onstant strain relaxation tests (1-4), and the sinusoidal loading tests (5-8). In most ases, the linear visoelasti behavior of the asphalt onrete was assumed,-and, heneforth, the linear visoelasti material properties in terms of reep ompliane, relaxation modulus, and omplex ompliane and omplex modulus were obtained. For example, the phenomenon of reep an be represented by the following equation under a uniaxial stress state: dt) = f t J(t _ ) arw d a (1) where ( and er are uniaxial strain and stress respetively. J(t) is the uniaxial reep ompliane and is usually determined from onstant stress reep tests. For a given onstant stress, Eq. 1 beomes E'.(t) = J(t)ro or J(t) = dt) O'o (2) In this equation, dt) is the measured reep strain under the onstant stress ro, and the reep ompliane J(t) an be obtained aording to Eq. 2. In priniple, if a material Publiation of this paper sponsored by Committee on Mehanial Properties of Bituminous Paving Mixtures. 73

2 74 is truly linearly visoelasti, Eq. 1 with J(t) determined from Eq. 2 an be used to predit the reep behavior of the material under any kind of uniaxial loading history. However, this is not always true when the material is subjeted to a more omplex loading history. In this report, the time-dependent properties of asphalt onrete are investigated under multiple-step loading histories, inluding several yli loading historip.s, in an effort to verify the appliability of linear visoelasti theory in prediting the reep behavior under multiple-step loadings and also to onstrut a workable onstitutive equation to desribe more losely the time-dependent behavior of asphalt onrete under time- dependent loading histories. In this report, the emphasis is also plaed on distintion between the reoverable and irreoverable reep under loading. This is important from a pratial viewpoint beause the irreoverable reep strain ontributes a large portion of the reep strain of asphalt onrete under external loading, and the irreoverable strain is aumulative under repeated loading, whereas the reoverable reep is not neessarily. MATERIALS AND SPECIMENS A single asphalt mixture was utilized for the investigations. The asphalt ement used in preparing test speimens was from the Amerian Oil Co. with 85 to 1 penetration grade. The asphalt ontent was 9 perent by weight. The gradation of aggregates is shown as follows: Sieve Size 3/a in. No. 4 No. 8 No. 16 No. 5 No. 2 Perentage Passing The test speimens used in this study were made of ompressed asphalt onrete ylinders 2-in. in diameter and 3-in. in length. The asphalt mixture was pressed in a 2-in. diameter mold at 3, psi for 5 min at a temperature of 22 F. The speimens were ured in an oven at 14 F for 72 hours prior to the testing. Also, the randomness of the strain response of eah speimen was minimized by subjeting eah speimen to a 5-psi prestress for 12 min. EXPERIMENTAL APPARATUS PROCEDURES The basi test equipment shown in Figure 1 onsists of a rigid frame A, a loading head B, a 1-to-1 ratio loading lever C, and the deformation measuring devies. A linear variable differential transduer (LVDT) was used to measure the deformation of the speimen. The output from LVDT, whih is diretly related to the total deformation, ould be automatially reorded on a strip-hart reorder. During the test, the speimen was plaed diretly under the loading head, and the L VDT was properly set. A seleted dead weight was then put on the loading lever to produe a onstant load on the speimen. The temperature during the test was kept at 75 ±2 F. RESULTS AND ANALYSIS Nine tests at three stress levels (1, 3, and 5 psi) were performed on prestressed samples. Duration of the loading period varied from 1 to 1, se (i.e., 1, 1, and 1, se). The results of the onstant stress reep tests at three stress levels at three loading periods are shown in Figures 2, 3, and 4. Eah solid line is the average of several repeated test results. In these figures, the reovery strains following eah reep test are also shown. Using Eq. 2, we an obtain the reep ompliane J(t) as follows. One the reep

3 75 Figure 1. Basi equipment for unonfined ompressive reep test. Figure 2. Creep and reovery of onstant stress reep tests (t; = 1 se). <Y 5poi I ' "b x - 4. z.o u; / // I/ /..._...,.., ' p i 1.op i 3 Time -Se. 1 se. t 4 5

4 76 Figure 3. Creep and reovery of onstant stress reep tests (t; = 1 se). 8 <Y ' 5. o., '.!: 12 " D PSI / 3 P S l v v-- 5 a 3.p.so 1 f'si 1 se_ '-. 1 1!1 Time - Se Figure 4. Creep and reovery of onstant stress reep tests (t; = 1, se). 1 CY' 5. b I I.: ' t u; 8 2 / --- (/' v v i PSI 3 ps\ 1 PSI ' $1 1. osl 1 se t IZOO Time-Se IGOO Figure 5. Constant stress reep urves plotted in log log sale. ' "' Q u; 1 5, OJ P_-.' 3 p SI -. _ O 13G, psr -,, - i-- l O 16.7

5 urves on the log-strain versus log-time base were replotted, they appeared to be straight lines as shown in Figure 5. Beause the straight lines at three stress levels (Fig. 5) have nearly the same slope (.136), the reep behavior ould be expressed in the following form: where rr is the total reep strain, and Ar is the strain at unit time (t = 1 min). The fat that Ar varies with stress an be seen by ross-plotting the strain values at t = 1 min of Figure 5 and the stress as shown in Figure 6 (open irles) and an be represented by the following relation: 77 (3) Er(t) = Ar(r)t T Ar (r) = a1r + a2r 2 a1 =.159 x 1-3 a2 = x 1-s (4) The reep ompliane J(t), whih is defined as!(t), an be obtained from Eq. 4 as follows: ro In view of Eq. 4 or 5, the asphalt onrete exhibits nonlinear behavior. Although a more popular form of power-law representation, suh as r, an be used as aurately as Eq. 4 to represent the stress dependene of Ar, this type of representation violates the basi invariane requirements as imposed from the ontinuum mehanis standpoint. Furthermore, Eq. 4 an be extended readily to represent the reep behavior under multiple stress states (11), whereas the power-law representation annot. Beause the asphalt onrete was found to be nonlinear from the onstant stress reep test, the seond linearity 1 equirement, the linear superposition priniple, is not appliable any more. However, the modified superposition method (9-13) has been used to desribe the nonlinear reep behavior and has been shown to be appliable in desribing the reep behavior under arbitrary loading from the onstant stress reep test results of many nonlinear visoelasti materials. This modified superposition method is employed here to desribe the reovery behavior. The modified superposition method yields the following form () for the reovery after onstant stress reep: (5) Er(t) = E'[r(t)] - E[r(t - t 1 )] t >ti (6) where E[r(t)J represents the reep strain under onstant stress input, suh as Eq. 4, and t 1 the unloading time. Inserting Eq. 4 into Eq. 6 yields For nr less than 1, as shown in Eq. 4 for the asphalt onrete, Eq. 7 predits that the reovery strain approahes zero. The reovery urves shown in Figures 2, 3, and 4 indiate, however, a large irreoverable strain (permanent set). Apparently Eq. 7 is not apable of desribing the reovery behavior beause either the modified superposition priniple is not appliable for this material or Eq. 4, determined from the onstant stress reep test results, does not represent the atual onstitutive relation of the material. Therefore, an attempt was made using both the reep urves and the reovery urves of Figures 2, 3, and 4 to obtain a better representation of the onstitutive relation of the material. Beause of the large portion of the unreoverable strain of eah reep test and beause the amount of unreoverable strain is dependent on the length of the loading period, (7)

6 78 Figure 6. Creep strain-stress at unit time_ ' a. u Stress-PS[ Figure 7. Nonlinear generalized Kelvin modl!i. Figure 8. Irreoverable strain versus time. 1 " Vi l.o 6 ]., 1.-t i- -, n J "- i i.--- ;,O? SU PE = :: ().3.4 OG I() Time - Min 16.7

7 the reep behavior may be represented by a generalized nonlinear Kelvin model as shown in Figure 7. Here, the nonlinear dashpot may ontribute to the irreoverable strain, and the series of Kelvin models (Kelvin hain) may ontribute to the power-law reep behavior (11). Therefore, tbe total reep strains ( r) of eah test were separated into two parts, the irxeoverable strains (E'p) due to the nonlinear dashpot and the ompletely reoverable strains (.) due to the nonlinear Kelvin hain. The irreoverable strains versus the length of the loading period (1, 1, and 1, se) at eah stress level on a log-log sale were plotted to form three straight lines as shown in Figure 8. Beause these three straight lines have nearly the same slope of.25, the following expression an be obtained for irreoverable strains: Again, by ross-plotting the E'p at t = 1. min and the stress as shown in Figure 6 (losed irles) and using the stress polynomial to fit those points, we an obtain the following equation: 79 (8) E'p(t) = (bio' + bz' 2 ) tnp bi =.844 x 1-3 b2 = x 1-5 np =.25 l (9) The rate of the irreoverable strain (P an be obtained from Eq. 9 as follows: rp(t) = np(bir + b2r 2 ) t"p-l (1) Instead of using Eq. 1, it was found that the "strain hardening theory" that relates the rate of the irreoverable strain to the urrent stress and irreoverable strain yielded a better desription of the irreoverable strain. The rate of the irreoverable strain an be obtained by eliminating the time variable from Eqs. 7 and 1, whih yields or (t) _ {b b 2)1/np (np-ll/np E'p -np io'+ 2' E' (11) (12) (12a) Equations 12 and 12a show that the dashpot is nonlinear and that the "oeffiient of vis - osity" 17 is dependent on the stress and the irreoverable strain. The reoverable strains E' v due to the Kelvin hain were then obtained by subtrating the irreoverable strains E'p from the total strains E'r. Thus, the total reep strains were separated into the reoverable part and the irreoverable part. Again, by using the same tehnique of plotting the E'v versus time in the log-log sale, as shown in Figure 9, and ross-plotting in Figure 6, we an express the reoverable strains in the following form: E'v(t) = (CiO' + C2r 2 ) t"v Ci=.748 x 1-3 (13) C2 = x 1-5 nv =.93

8 8 The onstitutive relation for the asphalt onrete using the generalized Kelvin model is summarized as follows for the onstant stress reep: where (pis given by Eq. 12, and E'v is given by Eq. 13. For the time-dependent stress input, the following equation is utilized: (14) t t (r(t) = f (p(;)d + J (t - ;)"v [C1 + 2C 2'(;}] r(;)d; (15) In Eq. 15, the modified superposition method was again used (in the seond integral) to desribe the reoverable part of the reep strains under time-dependent stress input ( 1' _!.!, _g). RESULTS AND PREDICTION OF CREEP UNDER MULTIPLE-STEP LOADING In order to test and improve the appliation of Eq. 15 to the reep behavior of the asphalt onrete, we performed four multiple-step loading reep tests. The results are shown in Figures 1 through 13. Equations 12, 13, and 15 were utilized to predit the reep behavior under multiplestep loading as follows. The stress inputs of the multiple steps an be expressed in a single algebrai equation using the Heaviside's unit funtion n r(t) = L (O"i - O'i_ 1) H(t - ti) i=o (15a) where H(t - ti) is the Heaviside's unit funtion, whih has the value of 1 when t ti, when t< O, '-1 =, and to=. Inserting Eq. 15a into Eq. 15, after performing the integration, yields the following for the total strain at eah loading step where (16) 2)1/np np E'p ( t ) = [( b1' + b2' t] (17a) (17b) (18a) E"v(t) = (C1' + C2') [t"v_ (t - t1fv] + ( 1 ()' l + 2 ri) ( t - t1l"v (18b) For t,_ 1 < t < t,, r = 1, 2, 3,..., (19a)

9 Figure 9. Reoverable strain versus time. 1 "l 2 Q "...: ;;;.. :i'i o.s u.. a:: x- 5n o :W 5 SI OPE )(- 1 i) Si X: Time - Min. s 1 2 Figure 1. Results and preditions of reep behavior under step loading. e Test data o Nonlinear tt:ery I Time - Min, Figure 11. Results and preditions of reep behavior under step loading '--"-..._----'-----' io Time - Min.

10 82 Figure 12. Results and preditions of reep behavior under step loading. IOr-,...,...,...,-,...,...,...,...,...,...,...,,...,...,...,...r-,...,...,...,...,...,...,...--r,...,...,...,...r-,...,...,...,..,...,...,..._, Test data Nonlinear theory PSI I O'------' _----' _---..._ _----'-----"'----"""" Timt-Min, 18 Figure 13. Results end preditions of reep behavior under step loading. Test data o Nonlinear theory 1Di / "" I ( -- l t ,--' o r----;------t;------t " IY ,, PSI t "._' !._I _I , Time-Min.

11 dt) = (C1" + C2r) [t"v - (t - ti) v] +. + (C1'r-2 + C2"-2) [{t - tr_ 2 )"v - (t - tr_ 1 )"v] + (C1'r-1 + C2r;_) (t - tr_)"v (19b) By adding the predited irreoverable strains (p and the predited reoverable strains ( v the predited total reep strains (r under multiple-step loading an be obtained and are shown as the open irles in Figures 1 through 13 together with the experimental results that are depited by the solid lines. The overall omparisons are quite satisfatory. RESULTS AND PREDICTION OF CREEP UNDER REPEATED LOADING The theory for a loading-unloading type of input was tested by loading speimens for 1 min and allowing them to reover in an unloaded state for 3 min before reloading. Cyles of this type were applied to speimens of 1, 2, 3, 4, and 5 psi, up to eight yles. In addition, a 4-psi, 1-yle repeated test was also onduted. The results are shown in Figures 14 through 19. Again, using Eqs. 15, 19a, and 19b, the reep strains under the repeated loading were alulated as shown in Figures 14 through 18 for the eight-yle tests. The theory predits well for both the loading period and the unloading period of eah yle, though more deviation was observed in the reovery period. These periodi loading tests are partiularly important in the evaluation of asphalt onrete beause pavements are onstantly subjeted to a similar loading pattern. Traffi passing over the pavement reates a loading period that is followed by an unloaded period. The loading time and stress are of primary importane to the total reep strain; thus, heavier, slower moving traffi auses greater "permanent" deformation. Although the total strain (or the shape of the total reep strain output with eah loading yle) is interesting and in many ways useful, the main onern when dealing with this type of material and stress pattern must lie not with the reoverable portion but with the irreoverable strain introdued with eah yle. It an be seen from Figures 2 and 21 that the irreoverable portion auses the aumulation of the total strain. Therefore, when the material is subjeted to a large number of yli loadings, it may seem justified to deal mainly with expressions for the irreoverable strain and to neglet the reoverable strain. In Figure 2, the theoretial and the experimental total strains and the irreoverable strains at the end of eah loading yle were plotted against the number of yles. The differene between total strain and the irreoverable strain, whih equals the reoverable, is small in omparison with the total strain or the irreoverable strain. This figure also points out the range in whih the theory is appliable. The inrease in strain for eah yle remains fairly onstant up to approximately 6 yles. At this point, the rate begins to inrease, and more deformation is observed with eah yle than in the previous yle as shown in Figures 19 and 2. This is also where the theory and the observed strains begin to differ, leading to the onlusion that the equations predit the output within about 2.2 perent strain in this ase, whih is reahed at about 6 yles. A signifiant hange in the shape of the speimen was also observed at about 6 yles. The sample began to take on a barrel shape, inreasing the ross-setional area and most likely weakening a great deal of the internal bonds. In the region lower than 2.2 perent strain, where the inrease in total deformation from one yle to the next remains nearly onstant, the yles are very uniform in shape. Both the loading and reovery periods, after the first few yles, demonstrate onstant strain rates and magnitudes. By letting N be the yle number, t be the duration of the loading period of eah yle, and a be the onstant stress at the load period, Eq. 19a beomes where (p represents the irreoverable strains at the end of the Nth loading yle. The parameters ontribute to the buildup of the irreoverable strain, whih is shown in 83 (2)

12 Figure 14. Results and preditions of reep behavior under yli loading (1-psi stress amplitude). s , PSI Test data Nonllm!Or tluwry 2 "' Time - Min, Figure 15. Results and preditions of reep behavior under yli loading (2-psi stress amplitude). Test do!o Nonlinear theory "n n n PSI ".' u; 4 ' '-----'--_.... -'--' Time- Min Figure 16. Results and preditions of reep behavior under yli loading (3-psi stress amplitude). 1,..._ , , , , , Test data o Nnlineor theory... "' 6 3 PSI. :;, " 4 B Time - Min.

13 Figure 17. Results and preditions of reep behavior under yli loading (4-psi stress amplitude). 1 - Test data o Nonlinear theory ' f-: fo'f---i--+---t---,-\-----,--t----,:-----,l-----,-=:::l-----,i:r----,+----,---,---,-1 2 r.::onn Time- Min, Figure 18. Results and preditions of reep behavior under yli loading (5-psi stress amplitude). IOr-,,,...,,,,,_,..,,,T"",,,...,,,,,-,-,,,,,--,,,...,,,,,...,...,,---,, 8 ' ' 9 u; 4 o...,,,,,,,,_,,,,,_.,,,_.,,,,_,,,,,...,,,_.,---,---,,_,,,,, Time - Min. Figure 19. Results and preditions of reep behavior under yli loading (4-psi stress amplitude, 1 yles). 6 4 N I x, : 4 - PSI 2 4 G I - - Test data Nonlinear t htor't B J-'.,. 1.-"""\ '"" r - pq, 1 ' I 4 9 I ) - ( -,_< - "} v ( ( ( l ov O o ov { < I > Time-Min

14 Figure 2. Irreoverable and theoretial strains versus yle number. (4 psi Cyli Load) Vi eo 1 12 Figure 21. Irreoverable and reoverable portions of total strain in a repeated loading test. Reoverable Portion LJ Irreoverable Portion z ;, a:... Vl D D TIME

15 Eq. 2. Furthermore, if the maximum strain <max (as approximated by the pmax) is hosen to be the failure riterion of a material, Eq. 2, after rewriting into the following form, implies that 87 (21) The longer the loading time during eah yle and the larger the stress amplitude are, the fewer the number of yles the material an withstand. Although Eq. 21 is derived from the visoelasti haraterization of asphalt onrete, some similar onlusions, suh as the effet of load duration, effet of ultimate strain (tensile strain), and magnitude of stress amplitude on the fatigues life of asphalt onrete, have been reahed in other fatigue studies (14,.!_. CONCLUSION It has been shown that the nonlinear visoelasti behavior of an asphalt onrete an be represented by a nonlinear generalized Kelvin model that is made of a nonlinear dashpot onneted in series with a nonlinear Kelvin hain. The nonlinear dashpot aounts for the time-dependent irreoverable strain (visous flow), and the nonlinear Kelvin hain aounts for the power-law time-dependent reoverable strain. It has been shown that the onstitutive equation an be determined relatively simply by utilizing both the reep and reovery parts of the onstant stress reep test results. The auray in prediting the reep behavior of the asphalt onrete under multiple-step loading and repeated loadings using the proposed onstitutive equation is very satisfatory. It has also been shown that an equation relating the number of yles to failure to the applied stress amplitude and the duration of eah yle similar to the existing fatigue theories an be derived from the irreoverable reep strains. It is hoped that this study will lead to a better understanding of the time-dependent behavior of asphalt onrete. In this report, only a single asphalt onrete mixture was utilized for the investigations, whih were onduted under only one temperature (75 ±F). Though it is antiipated that varying the mixtures and the testing temperatures would definitely affet the reep behavior of asphalt onrete, some of the preliminary experimental results indiate that the differene of the reep behavior of different asphalt onrete mixtures at different temperatures (above the glass transition temperature) is more quantitative than qualitative. REFERENCES 1. Seor, K. E., and Monismith, C. L. Analysis of Triaxial Test Data on Asphalt Conrete Using Visoelasti Priniples. HRB Pro., Vol. 4, 1961, pp Monismith, C. L., and Seor, K. E. Visoelasti Behavior of Asphalt Conrete Pavements. Pro. Internat. Conf. on the Strutural Design of Asphalt Pavements, Univ. of Mihigan, Pagen, C. A. An Analysis of the Thermorheologial Response of Bituminous Conrete. Ohio State Univ., PhD thesis, Fitzgerald, J. E., and Lai, J. S. Initial Evaluation of the Effet of Syntheti Rubber Additives on the Thermorheologial Properties of Asphalt Mixtures. Highway Researh Reord 313, 197, pp Pagazian, H. S. The Response of Linear Visoelasti Materials in the Frequeny Domain With Emphasis on Asphalt Conrete. Pro. Internat. Conf. on the Strutural Design of Asphalt Pavement, Univ. of Mihigan, Pagen, C. A. Rheologial Response of Bituminous Conrete. Highway Researh Reord 67, 1965, pp Kallas, B. F., and Riley, S. C. Mehanial Properties of Asphalt Pavement Materials. Pro. Seond Internat. Conf. on the Strutural Design of Asphalt Pavements, Univ. of Mihigan, 1967.

16 88 8. Swami, S. A., Goetz, W. H., and Harr, M. E. Time and Load hldependent Properties of Bituminous Mixtures. Highway Researh Reord 313, 197, pp Findley, W. N., and Lai, J. S. A Modified Superposition Priniple Applied to Creep of Nonlinear Visoelasti Material. Trans. So. of Rheology, Vol. 11, No. 2, 1967, pp Lai, J. S., and Findley, W. N. Stress Tielaxation of Nonlinear Visoelasti Material Under Uniaxial Strain. Trans. So. of Rheology, Vol. 12, No. 2, 1968, pp Findley, W. N., Lai, J. S., and Onaian, K. Creep and Stress Relaxation of Non Linear Visoelasti Materials. To be published. 12. Pipkin, A. C., and Rogers, T. G. A Nonlinear hltegral Representation for Visoelasti Behavior. Jour. of Mehanis and Physis of Solids, Vol. 16, 1968, pp Stafford, R.. On Mathematial Forms for the Material Funtions in Nonlinear Visoelastiity. Jour. Mehanis and Physis of Solids, 1969, pp Deaon, J. A. Materials Charaterization-Experimental Behavior. HRB Spe. Rept. 126, 1971, pp Finn, F. N. Fators hlvolved in the Design of Asphalt Pavement Surfaes. NCHRP Rept. 39, 1967, 112 pp.