# Modelling of the inverse creep of road bitumen modified with SBS copolymer/ Keliu bitumo, modifikuoto SBS kopolimeru, atvirkstinio valksnumo modeliavimas/ Ar SBS kopolimeru modificeta celu bitumena inversas sludes modelesana/ SBS kopolumeeriga modifitseeritud teebituumeni vastandroome modelleerimine.

1. Introduction

Since the 1980s polymer modified bitumens have been widely used in road construction as a binder in asphalt mixes. Much research was devoted to demonstrating the beneficial effects of polymer modification on the properties of bitumen and asphalt mixtures (Ho, Zanzotto 2005; Radziszewski 2007; Scholten et al. 2010; Yildirim 2007). Improving the elastic properties of bitumen by the polymer addition is particularly important. It should also be noted that the polymer modified binders, being examples of materials exhibiting viscoelastic properties, are very interesting with respect to rheology. Linear viscoelastic rheological models, which are obtained through the serial or parallel connection of Hookean elastic elements (springs) and Newtonian viscous parts (dashpots), are usually used to describe the phenomena occurring in such materials. Different systems of two-, three-, and four-parameter models have been clearly described in the literature (Barnes et al. 1989; Derski, Ziemba 1968). In the present study, the phenomenon of retarded strain recovery, also called an inverse creep (Bodnar et al. 2006), observed during the tests performed on samples of asphalt binders, has been analyzed. A mathematical description of this phenomenon was made using three viscoelastic models: Burgers, Dual Kelvin+Newton and Dual Kelvin+Maxwell.

The paper presents the theory that a very accurate mathematical description of inverse creep phenomenon of polymer modified bitumen using linear viscoelastic models containing the dual Kelvin element is possible. The objective of the research was modelling the inverse creep using three selected rheological models, on the basis of the results of the experiment carried out on samples of asphalt binders, using a specially developed laboratory methods, evaluation of the modelling results, as well as verification of the formulated theory.

2. Experimental

2.1. Materials

Three asphalt binders were selected for this study: a base bitumen 50/70 penetration grade and two modified binders prepared by blending the base bitumen with the linear styrene-butadiene-styrene (SBS) copolymer, 4% and 8% by weight (marked 50/70 + 4% SBS and 50/70 + 8% SBS, respectively). The base bitumen has been produced from Russian (Ural) crude oil. The specimens of polymer modified binders were mixed at 180[degrees]C (50/70 + 4% SBS) or 190[degrees]C (50/70 + 8% SBS) at a speed of 120 rpm during 2 h. Basic parameters of the tested binders are presented in Table 1.

SBS thermoplastic elastomer, Kraton D-1101CM was used to modify the bitumen properties. According to the manufacturer this is a linear block copolymer based on polystyrene (its content is 31 [+ or -] 1% m), and polybutadiene. It contains anti-oxidant and is supplied as milled powder. The typical properties of Kraton D-1101CM are: elongation at break--880%, tensile strength--33 MPa, 300% modulus--2.9 MPa, specific gravity--940 kg/[m.sup.3].

2.2. Test method

In this study, the modified elastic recovery test was conducted using a standard ductilometer. Test specimens were prepared according to standard EN 13398:2010 "Bitumen and Bituminous Binders--Determination of the Elastic Recovery of Modified Bitumen". However, the test method was extended when compared to the standard procedure described in EN 13398:2010. The test conditions were as follows:

--temperature of the tests: 5, 15, 25, and 35[degrees]C maintained with an accuracy of [+ or -]0.5[degrees]C;

--stretching speed: 50 mm/min;

--max elongation: 200 [+ or -] 0.5 mm; it corresponds to the max value of strain [[epsilon].sub.max] = 6.667 (666.7%);

--the specimens were cut in half after the following periods of time: 0 s (immediately after stretching), 1800 s (only at 15[degrees]C), 3600 s and 7200 s (only at 15[degrees]C);

--strain recovery was measured at the time t: 2, 5, 10, 15, 20, 25, 30, 40, 50, 60, 75, 90, 105, 120, 135, 150, 165 and 180 min after cutting the samples. In order to determine reliable values of strain recovery of the tested binders, the measuring equipment was expanded by adding a specially designed instrument. Stretched samples (after being cut in half) were placed on the 10 mm thick glass with the calibrated scale. Brass pins attached to the glass plate allowed for accurate placement of the specimens in positions corresponding to the elongation of 200 mm.

The strain recovery tests have been conducted on the specimens cut 30, 60 and 120 min after stretching in order to investigate the relationship between the time of relaxation [t.sub.r] (at constant strain--elongation of 200 mm) and strain recovery after the specimens were cut in half. Fig. 1 shows the main stages of the procedure described above.

Some parts of the intended research have not been successful. Specimens of 50/70 penetration grade bitumen, tested at 5[degrees]C broke before they reached the elongation of 200 mm and those examined at 35[degrees]C were too soft to test. Moreover, samples of the binder 50/70 + 8% SBS tested at 35[degrees]C have been broken before the end of the specified time of relaxation [t.sub.r] = 3600 s.

[FIGURE 1 OMITTED]

3. Test results and modelling

In the process of modelling, the Burgers rheological model (Fig. 2a) was used as the basic one. Researchers have shown that it simulates the behaviour of bitumens quite well (Cebon 2000). The following processes occurring in the case of bitumen and asphalt mixtures can be simulated by using the Burgers model: an immediate strain (elastic), creep, strain recovery: immediate (elastic) and retarded, permanent strain and relaxation at constant strain. An example of practical application of the Burgers model to describe the viscoelastic behaviour of asphalt concrete has been presented in the paper (Laurinavicius et al. 2006). The papers (Grabowski et al. 2002; Grabowski, Slowik 2003) demonstrate that the use of simpler viscoelastic models, i.e. two-element--the Maxwell and Kelvin models, as well as three-element--the Jeffreys model do not provide satisfactory results of modelling. The exception is the three-element standard model (consisting of Hooke and Kelvin elements connected in series), which can be used when the test material shows no permanent strain. The differential equation for the Burgers model can be written as follows (Skrzypek 1986):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [sigma]--stress, Pa; [epsilon]--strain; [E.sub.0], [E.sub.1], [[eta].sub.0], [[eta].sub.1]--moduli of elasticity, Pa, and dynamic viscosity, Pas, respectively--according to the diagram shown in Fig. 2a. The total strain in the model at time t is a sum of strain generated in each of the three elements connected in series: Hooke ([[epsilon].sub.H]), Newton ([[epsilon].sub.N]) and Kelvin ([[epsilon].sub.K]):

[epsilon](t) = [[epsilon].sub.H](t) + [[epsilon].sub.N](t) + [[epsilon].sub.K](t). (2)

Since: [[epsilon].sub.H] = [sigma]/[E.sub.0], [[epsilon].sub.N](t) = [sigma]t/[[eta].sub.0] and [[epsilon].sub.K](t) = [sigma]/[E.sub.1][1 - exp([-[E.sub.1]/[[eta].sub.1]])t], then:

[epsilon](t) = [[sigma]/[E.sub.0]] + [[sigma]t/[[eta].sub.0]] + [[sigma]/[E.sub.1]][1 - exp([- [E.sub.1]/[[eta].sub.1]]t)]. (3)

In this paper the author focuses on the problem of mathematical description of the inverse creep of asphalt binders which begins at the moment of unloading (cutting in half) of the samples (t = 0, [sigma] = 0). Strain at time t from the moment of cutting the specimen can be described by the equation:

[epsilon](t) = [[[sigma].sub.0]/[E.sub.0]] + [[[sigma].sub.0]/[E.sub.1]]exp(-t/[[tau].sub.1]) + [[epsilon].sub.[infinity]], (4)

where [[sigma].sub.0]--value of stress just before cutting the sample; [[tau].sub.1]--retardation time ([[tau].sub.1] = [[eta].sub.1]/[E.sub.1]); [[epsilon].sub.[infinity]]--permanent strain (generated in Newtonian dashpot of viscosity [[eta].sub.0]), calculated at t [right arrow] [infinity].

[FIGURE 2 OMITTED]

Since there was no possibility of determining the values of [[sigma].sub.0], the Eq (4) allowed for the calculation of the following parameters of the Burgers model: [[sigma].sub.0]/[E.sub.0], [[sigma].sub.0]/[E.sub.1], [[tau].sub.1] and [[epsilon].sub.[infinity]]. The modelling was carried out employing the least squares of deviation and using the Nonlinear Least Squares Curve Fitter software. The results of strain determination in samples e against time t (19 pairs of results for each approximated curve) were the data used for modelling. The parameters calculated using the Burgers model have been presented in Table 2. The effects of modelling have been shown graphically in Figs 3 and 4. The experiment results have been presented as points and the modelling curves as lines. All the tests of strain recovery were carried out on six specimens. An average value of uncertainty (confidence interval), calculated in accordance with the procedure described in (Slowik 2010), was about 1%--at 95% confidence level.

Since it was found that the results of modelling carried out by using the Burgers model are not fully satisfactory, both qualitatively (shape of the model curves compared to the experiment results) and quantitatively (values of the coefficient of determination [R.sup.2]), more complicated models developed by serial connection of two Kelvin elements with Newtonian dashpot (Dual Kelvin+Newton--Fig. 2b) or with the Maxwell element (Dual Kelvin+Maxwell--Fig. 2c), described also by (Cebon 2000) as a general viscoelastic material, were used. For these models, the Eq (4) takes the following forms, respectively:

[epsilon](t) = [[[sigma].sub.0]/[E.sub.1]]exp(-t/[[tau].sub.1]) + [[[sigma].sub.0]/[E.sub.2]]exp(-t/[[tau].sub.2]) + [[epsilon].sub.[infinity]] (5)

or

[epsilon](t) = [[[sigma].sub.0]/[E.sub.0]] + [[[sigma].sub.0]/[E.sub.1]]exp(-t/[[tau].sub.1]) + [[[sigma].sub.0]/[E.sub.2]]exp(-t/[[tau].sub.2]) + [[epsilon].sub.[infinity]], (6)

where [[tau].sub.1] = [[eta].sub.1]/[E.sub.1]; [[tau].sub.2] = [[eta].sub.2]/[E.sub.2]; [E.sub.0], [E.sub.1], [E.sub.2], [[eta].sub.1], [[eta].sub.2]--moduli of elasticity, Pa, and dynamic viscosity, Pas, respectively--according to the diagrams shown in Figs 2b and 2c.

[FIGURE 3 OMITTED]

Calculated values of the Dual Kelvin+Newton model parameters are presented in Table 3. The effects of modelling have been shown graphically in Figs 5 and 6. The corresponding results achieved by using the Dual Kelvin+Maxwell model have been presented in Table 4, as well as in Figs 7 and 8.

4. Discussion

Simulation of the inverse creep of asphalt binders was possible due to modification and a significant extension of the standard method for determination of elastic recovery, described in EN 13398:2010. In particular, the measurement of the length of the samples cut in half using a calibrated glass plate yielded a very positive result contributing to the improvement of readings of the results. It also allowed us to obtain measurement reliability and uncertainty at an acceptable level. There are two important elements of the study, i.e. a significant lengthening of the observation time up to 180 min in the case of strain recovery (according to EN 13398:2010 the measurement is finished after 30 min since cutting the samples) and an attempt to analyze the impact of the time of relaxation on the inverse creep curves of the tested asphalt binders. This phenomenon was studied at length in the case of the tests conducted at a temperature of 15[degrees]C (time of relaxation [t.sub.r] was equal to 0, 1800, 3600 and 7200 s). In other cases (tests carried out at 5, 25 and 35[degrees]C), only the behaviour of samples cut immediately after stretching ([t.sub.r] = 0 s) and samples whose constant level of strain has been maintained for a period [t.sub.r] = 3600 s were compared.

[FIGURE 4 OMITTED]

Specimens of asphalt binders used for testing were selected in such a way as to obtain large differences in their viscoelastic properties. Thus, a sample of the bitumen 50/70 modified with the addition of 8% of SBS copolymer can be regarded as an example of highly modified bitumen which under certain conditions shows a complete strain recovery (no permanent strain). A sample of the bitumen 50/70 with the addition of 4% of SBS copolymer can be considered an example of medium-modified bitumen, which shows a permanent strain. A sample of the 50/70 base bitumen has been used in the study as a reference--an example of unmodified bitumen which shows unfavourable elastic properties and a considerable degree of permanent deformation. The procedure for modelling of the observed inverse creep started with the use of the Burgers model (Fig. 2a)--a very popular and often described in the literature. This model has been chosen due to the fact that the viscous and elastic elements were connected in such a way that the model can be used to describe all the phenomena observed during testing the asphalt binders, i.e. immediate strain, creep, strain recovery - both immediate and retarded, permanent strain, and also relaxation at constant strain. Upon analyzing the results of modelling carried out by using the Burgers model it can be noted that despite the relatively high values of the coefficient of determination [R.sup.2] (from 0.952 to 0.999), the modelling curves differ significantly from the results of the experiment. The most significant incompatibilities have been found in the case of the immediate strain recovery and permanent strain values. For a number of specimens, it was observed that the values of immediate strain recovery, calculated using the Burgers model are larger than the corresponding results achieved 2 min after cutting the samples. A similar problem occurred in the case of permanent deformation, where the calculated values [[epsilon].sub.[infinity]] (when t [right arrow] [infinity]) were higher than the results obtained 180 min after cutting the samples. The above-mentioned incompatibilities, which can be clearly seen in Figs 3a, 3b and 4b, are hard to accept. The discussion presented above leads to the conclusion that the Burgers model is not useful for quantitative description of the inverse creep of the analyzed asphalt binders. It was therefore decided to use other, more complicated viscoelastic models. Applying the generalized Kelvin or Maxwell models has been taken into consideration. However, in the analyzed case, they are not useful due to the limitations of their construction (the first one shows no permanent deformation, and the second one does not show the complete strain recovery). Thus, two models constructed through modification of the Burgers one, has been chosen. The first one has been made by replacing the free Hookean spring (modulus of elasticity [E.sub.0]) in the Burgers model with a Kelvin element (of the parameters [E.sub.2] and [[eta].sub.2]). Thus, a model consisting of two Kelvin elements and a Newtonian dashpot connected in series has been developed. This model does not describe the immediate strain recovery. However, due to the use of the dual Kelvin element (two retardation times), it effectively describes the large curvature observed when analyzing the strain recovery against time relationship, resulting in high-speed inverse creep in the initial phase of the test--just after cutting the samples. As an effect, high values of the coefficient of determination [R.sup.2] (from 0.992 to 1.000) have been achieved. The third model was obtained by serial connection of previously described one with the additional Hookean spring (modulus of elasticity [E.sub.0]). The result is a model consisting of two Kelvin and one Maxwell elements connected in series (Fig. 2c). The results achieved by using this model show the best compatibility with the results of the experiment, as evidenced by the highest value of the coefficient of determination [R.sup.2] (from 0.997 to 1.000). Furthermore, a description of the relaxation effect of reducing the value of immediate strain recovery, as well as increase of the permanent strain value in the case of polymer modified asphalt binders can be regarded as correct.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

It can be said that those two models containing dual Kelvin element yielded acceptable results. The main difference is due to their behaviour just after cutting the samples. The model which contains a free Hookean spring (Fig. 2c) describes the phenomenon of immediate strain recovery. On the other hand, there is no such possibility in the case of the model without that element (Fig. 2b). Due to the test method used, it was not possible to accurately measure the length of the samples right after unloading. The first reliable results have been obtained 2 min after cutting the samples. Therefore, it is difficult to say which of the two models simulate, in a manner closer to reality, the course of the observed inverse creep of samples previously subjected to tension.

The author has observed a different nature of the strain changes with regard to time in the latest studies of copolymer SBS modified binders, carried out using Bending Beam Rheometer (BBR) at the following temperatures: -40[degrees]C, -32[degrees]C, -24[degrees]C and -16[degrees]C. On the basis of those test results, a very high value of the immediate deflection recovery of the tested beams (measured already 0.5 s after unloading the beam) was discovered. It can be assumed that the use of the Dual Kelvin+Maxwell model in this case should be the most appropriate. On the other hand, the mathematical description proposed by using the Dual Kelvin+Newton model, could differ significantly from the experiment results.

[FIGURE 7 OMITTED]

One of the practical implications resulting from the application of mathematical methods for describing the inverse creep of asphalt binders specimens (the Dual Kelvin+Maxwell model in particular) discussed in the present paper is a possibility of calculating the values of two important properties, which would be practically impossible to determine following the laboratory test method used in the study. These are: immediate strain recovery ([[sigma].sub.0]/[E.sub.0]) and permanent strain [[epsilon].sub.[infinity]]. Based on these parameters, we can precisely determine the coefficient of retardation [[alpha].sub.s] (Judycki 1989), which shall take the following, modified form (symbols as in the previous formulas):

[[alpha].sub.s] = (1 - [[[sigma].sub.0]/[E.sub.0]([[epsilon].sub.max] - [[epsilon].sub.[infinity]])]) 100, %. (7)

The coefficient of retardation [[alpha].sub.s] is a measure of strain recovery rate after cutting the samples, and may be a useful parameter to assess the effectiveness of bitumen modification by polymer addition. The author intends to develop this problem further in the publications to follow.

[FIGURE 8 OMITTED]

5. Conclusions

The Burgers model which is regarded as a very useful model describing rheological phenomena occurring in the case of bitumen and asphalt mixes does not give fully satisfactory results of modelling of the inverse creep of the investigated asphalt binders.

Much better compatibility of the modelling results in comparison with the results of the experiment ([R.sup.2] > 0.99) was obtained by using models consisting of the dual Kelvin element connected in series with Newton or Maxwell elements. Applying the Dual Kelvin+Maxwell allows for the calculation of two important values, i.e. immediate strain recovery and permanent strain of the tested binders, which are difficult to measure by the laboratory method.

The results of modelling carried out in accordance with the procedure presented in the paper can be used in practice to determine the coefficient of retardation in order to assess the efficiency of bitumen modification with polymers.

doi: 10.3846/bjrbe.2012.10

Acknowledgments

The research was supported by Poznan University of Technology, the Project 12-274/11 DS.

The author wishes to thank Mrs. Katarzyna Szczechowska for her laboratory assistance during the testing program.

Received 18 April 2011; accepted 4 July 2011

References

Barnes, H. A.; Hutton, J. F.; Walters, K. 1989. An Introduction to Rheology. Amsterdam: Elsevier Science Publishers B. V. 199 p. ISBN 0444874690.

Bodnar, A.; Chrzanowski, M.; Latus, P. 2006. Reologia konstrukcji pretowych. Cracow: Cracow University of Technology. 242 p. ISBN 8372424098 (in Polish).

Cebon, D. 2000. Handbook of Vehicle-Road Interaction. Lisse: Swets & Zeitlinger Publishers B. V. 601 p. ISBN 9026515545.

Derski, W.; Ziemba, S. 1968. Analiza modeli reologicznych. Warsaw: Polish Scientific Publishers PWN. 160 p. (in Polish).

Grabowski, W.; Kuczma, M.; Slowik, M. 2002. Mathematical Modelling of Rheological Properties of Polymer Modified Bitumens, Foundations of Civil and Environmental Engineering 2: 27-42.

Grabowski, W.; Slowik, M. 2003. Assessment of Selected Rheological Properties of Polymer Modified Bitumens Applied in Poland, International Journal of Pavements 2(3): 14-23.

Ho, S.; Zanzotto, L. 2005. The Low Temperature Properties of Conventional and Modified Asphalt Binders Evaluated by the Failure Energy and Secant Modulus from Direct Tension Tests, Materials and Structures 38: 137-143. http://dx.doi. org/10.1007/BF02480586

Judycki, J. 1989. Elasticity of Road Bitumens Modified with Elastomers, Archives of Civil Engineering 35(3-4): 373-385 (in Polish).

Laurinavicius, A.; Oginskas, R.; Zilioniene, D. 2006. Research and Evaluation of Lithuanian Asphalt Concrete Road Pavements Reinforced by Geosynthetics, The Baltic Journal of Road and Bridge Engineering 1(1): 21-28.

Radziszewski, P. 2007. Modified Asphalt Mixtures Resistance to Permanent Deformations, Journal of Civil Engineering and Management 13(4): 307-315.

Scholten, E. J.; Vonk, W.; Korenstra, J. 2010. Towards Green Pavements with Novel Class of SBS Polymers for Enhanced Effectiveness in Bitumen and Pavement Performance, International Journal of Pavement Research and Technology 3(4): 216-222.

Skrzypek, J. 1986. Plastycznosc i peizanie. Warsaw: Polish Scientific Publishers PWN. 360 p. ISBN 8301062207 (in Polish).

Slowik, M. 2010. Analiza niepewnosci pomiarowych przy ocenie wlasciwosci asfaltow drogowych, Archives of Institute of Civil Engineering 7: 129-144 (in Polish).

Yildirim, Y. 2007. Polymer Modified Asphalt Binders, Construction and Building Materials 21(1): 66-72. http://dx.doi. org/10.1016/j.conbuildmat.2005.07.007

Mieczyslaw Slowik

Poznan University of Technology, Institute of Civil Engineering, 5, Piotrowo St., PL 61-138 Poznan, Poland

E-mail: mieczyslaw.slowik@put.poznan.pl
```Table 1. Basic properties of the investigated asphalt binders

Properties                               Bitumen

50/70            50/70 + 4% SBS

Penetration at          68.4 [+ or -] 1.4     54.4 [+ or -] 1.1
25[degrees]C, dmm

Softening Point         44.9 [+ or -] 1.0     53.0 [+ or -] 1.1
(R&B), [degrees]C

Fraass Breaking         -7.5 [+ or -] 1.8     -9.3 [+ or -] 2.3
Point, [degrees]C

Viscoelasticity
Interval (R&B -         52.4 [+ or -] 2.0     62.3 [+ or -] 2.6
Fraass
BP),[degrees]C

Properties                  Bitumen

50/70 + 8% SBS

Penetration at          43.4 [+ or -] 1.2
25[degrees]C, dmm

Softening Point         97.6 [+ or -] 1.6
(R&B), [degrees]C

Fraass Breaking        -16.0 [+ or -] 2.5
Point, [degrees]C

Viscoelasticity
Interval (R&B -        113.6 [+ or -] 3.0
Fraass
BP),[degrees]C

Table 2. Results of modelling of the inverse creep of the
tested asphalt binders--the Burgers model parameters

SBS                                  [[sigma]    [[sigma]
copolymer         T,       [t.sub.r],   .sub.0]/    .sub.0]/
content, %    [degrees]C       s        [E.sub.0]   [E.sub.1]

0                  15           0          0.201       0.398
0                  15          1800        0.117       0.415
0                  15          3600        0.052       1.050
0                  15          7200       -0.008       0.543
0                  25           0          0.148       0.754
0                  25          3600        0.063       0.480
4                  5            0          2.414       2.748
4                  5           3600        0.864       3.237
4                  15           0          3.103       2.142
4                  15          1800        0.943       3.325
4                  15          3600        0.590       3.335
4                  15          7200        0.494       3.282
4                  25           0          3.023       2.723
4                  25          3600        1.407       3.905
4                  35           0          3.842       2.317
4                  35          3600        2.195       3.126
8                  5            0          3.089       2.398
8                  5           3600        1.327       3.890
8                  15           0          4.165       2.426
8                  15          1800        3.074       3.436
8                  15          3600        2.617       3.698
8                  15          7200        2.424       3.816
8                  25           0          5.629       1.037
8                  25          3600        5.731       0.851
8                  35           0          6.498       0.168

SBS                         [[epsilon]
copolymer     [[tau].sub.1],      .sub.      [R.sup.2]
content, %          s          [infinity]]

0                   3565           6.068        0.972
0                   6459           6.134        0.988
0                  37905           5.565        0.997
0                  18080           6.132        0.996
0                   6372           5.764        0.988
0                   4148           6.124        0.989
4                   1205           1.504        0.974
4                   2097           2.565        0.988
4                   1020           1.422        0.952
4                   1736           2.399        0.982
4                   2361           2.742        0.989
4                   2861           2.890        0.989
4                   1208           0.920        0.966
4                   1490           1.355        0.980
4                   614            0.508        0.966
4                   746            1.346        0.970
8                   1202           1.179        0.964
8                   1882           1.450        0.984
8                   573            0.076        0.977
8                   651            0.156        0.978
8                   882            0.352        0.975
8                   987            0.427        0.975
8                   139            0.001        0.999
8                   552            0.084        0.962
8                   301            0.001        0.990

Table 3. Results of modelling of the inverse creep of the tested
asphalt binders--parameters of the Dual Kelvin+Newton model

SBS                                 [[sigma]
copolymer        T,       [t.sub.r],   .sub.0]/    [[tau].sub.1],
content, %   [degrees]C       s        [E.sub.0]         s

0                15           0          0.409          6600
0                15          1800        0.437          7742
0                15          3600        1.251         46563
0                15          7200        0.435         12317
0                25           0          1.026         13830
0                25          3600        0.579         12206
4                5            0          2.131          1831
4                5           3600        2.742          3131
4                15           0          1.464          1966
4                15          1800        2.513          3130
4                15          3600        2.709          4307
4                15          7200        2.887          5601
4                25           0          2.094          1939
4                25          3600        2.900          2567
4                35           0          1.521          1047
4                35          3600        1.864          1470
8                5            0          1.830          1953
8                5           3600        3.137          3089
8                15           0          1.659          891
8                15          1800        2.280          1052
8                15          3600        2.500          1503
8                15          7200        2.646          1680
8                25           0          0.484          210
8                25          3600        0.535          981
8                35           0          0.109          441

SBS       [[sigma]                     [[epsilon]
copolymer    .sub.0]/    [[tau].sub.2],      .sub.      [R.sup.2]
content, %   [E.sub.1]         s          [infinity]]

0              0.262          174            5.989        0.997
0              0.129           91            6.100        0.992
0              0.054           59            5.362        0.998
0             -0,021          656            6.252        0.997
0              0.213          350            5.411        0.998
0              0.179          807            5.900        0.998
4              3.098           97            1.428        0.998
4              1.476          258            2.397        0.998
4              3.863           84            1.333        0.998
4              1.920          293            2.188        0.999
4              1.477          487            2.421        0.999
4              1.334          565            2.407        1.000
4              3.737           86            0.827        0.996
4              2.547          221            1.172        0.998
4              4.680           66            0.464        0.998
4              3.524          124            1.261        0.998
8              3.735           81            1.096        0.997
8              2.227          240            1.222        0.997
8              4.964           61            0.043        0.999
8              4.280           90            0.102        0.999
8              3.885          117            0.262        0.998
8              3.673          128            0.320        0.998
8              6.183           33            0.000        1.000
8              6.063           40            0.069        1.000
8              6.558           23            0.000        1.000

Table 4. Results of modelling of the inverse creep of the tested
asphalt binders--parameters of the Dual Kelvin+Maxwell model

SBS                                  [[sigma]    [[sigma]
copolymer         T,       [t.sub.r],   .sub.0]/    .sub.0]/
content, %    [degrees]C       s        [E.sub.0]   [E.sub.1]

0                  15           0          0.102       0.441
0                  15          1800        0.087       1.264
0                  15          3600        0.019       1.263
0                  15          7200        0.002       0.434
0                  25           0          0.076       1.492
0                  25          3600        0.025       0.713
4                  5            0          1.711       1.663
4                  5           3600        0.502       2.321
4                  15           0          2.188       1.181
4                  15          1800        0.406       2.256
4                  15          3600        0.254       2.507
4                  15          7200        0.140       2.844
4                  25           0          2.431       1.467
4                  25          3600        0.701       2.494
4                  35           0          3.305       0.718
4                  35          3600        1.407       1.257
8                  5            0          2.428       1.368
8                  5           3600        0.800       2.638
8                  15           0          3.474       0.951
8                  15          1800        2.010       1.621
8                  15          3600        1.655       1.853
8                  15          7200        1.608       1.878
8                  25           0          5.034       0.375
8                  25          3600        5.300       0.361
8                  35           0          6.413       0.089

SBS                         [[sigma]
copolymer     [[tau].sub.1],   .sub.0]/    [[tau].sub.2],
content, %          s          [E.sub.2]         s

0                   8973          0.191          368
0                  44966          0.112          983
0                  47077          0.034           76
0                  12316         -0.022          624
0                  25259          0.196          713
0                  18898          0.194         1074
4                   2685          1.938          302
4                   4522          1.618          645
4                   2994          2.033          225
4                   3908          1.907          480
4                   5584          1.650          749
4                   6349          1.384          708
4                   4481          2.170          399
4                   3310          2.393          416
4                   3133          2.274          281
4                   2479          2.806          280
8                   3563          1.904          321
8                   4729          2.244          588
8                   1549          2.228          218
8                   1475          2.961          208
8                   2167          2.953          274
8                   2681          2.956          337
8                   229           1.257           68
8                   1523          0.946          156
8                   496           0.165           95

SBS        [[epsilon]
copolymer        .sub.      [R.sup.2]
content, %    [infinity]]

0                 5.932        0.998
0                 5.203        0.997
0                 5.350        0.997
0                 6.253        0.997
0                 4.903        0.999
0                 5.734        0.998
4                 1.355        0.999
4                 2.226        1.000
4                 1.265        0.999
4                 2.097        0.999
4                 2.256        0.999
4                 2.298        1.000
4                 0.600        0.999
4                 1.079        0.999
4                 0.370        0.999
4                 1.196        0.998
8                 0.966        0.999
8                 0.985        1.000
8                 0.013        0.999
8                 0.074        0.999
8                 0.205        1.000
8                 0.226        1.000
8                 0.000        1.000
8                 0.059        0.998
8                 0.000        0.998
```
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