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Long-term strength of asphalt concrete and its applications
Construction and Building Materials 244 (2020) 118325
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Long-term strength of asphalt concrete and its applications A.I. Iskakbayev a,b, B.B. Teltayev b,⇑, K.Z. Yestayev b, B.D. Abu b a b
Department of Mechanics, Al-Farabi Kazakh National University, 71 Al-Farabi, Almaty 050040, Kazakhstan Kazakhstan Highway Research Institute, 2A Nurpeissov, Almaty 050061, Kazakhstan
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Long-term strength has been
investigated for an asphalt concrete. Long-term strength of the asphalt
concrete has been described by power function. Activation energy of the asphalt concrete failure depends on its type. Average time between thermal fluctuations depends on temperature and stress.
Thermal fluctuation theory
a r t i c l e
i n f o
Article history: Received 27 May 2019 Received in revised form 13 September 2019 Accepted 30 January 2020
Keywords: Asphalt concrete Uniaxial tension Creep Long-term strength Thermal fluctuation theory Loading with constant rate Thermal stress restrained test
a b s t r a c t The objective of the paper is to show that there is an important mechanical characteristic of an asphalt concrete called as a long-term strength which is determined experimentally in an easy way under scheme of direct creep, and using it one can model the deformation and failure of the asphalt concrete at various types of loading. For this purpose the long-term strength is investigated experimentally for a hot fine-grained asphalt concrete at uniaxial tension under stress from 0.0081 MPa to 3.0 MPa within the range of temperatures from +60 °C to 24 °C. The experiments have been performed in a specifically invented and assembled device which allows determining the mechanical characteristics of materials at tension at various temperatures and types of loading. It is found out that without exclusion for all considered temperatures long-term strength of the asphalt concrete is described by power function with a high accuracy. Exponent characterizing a rate of variation for failure time at variation of stress is increased with the temperature increase: failure time is decreased for 3–7 orders (depending on temperature) at the stress increase for one order. Long-term strength of the asphalt concrete can be also described on the basis of the thermal fluctuation theory. It is found out that activation energy of failure does not depend on stress and temperature, and it is equal to 162.8 kJ/mol; the pre-exponential constant characterizing average time between neighbouring thermal fluctuations depends on stress as well as on temperature; meanwhile the more the temperature the less the effect of stress. Potentials for use are also shown for long-term strength of the asphalt concrete for modeling of the processes of its deformation and failure at loading with a constant rate and in conditions of the thermal stress restrained specimen test. Ó 2020 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. E-mail address: [email protected] (B.B. Teltayev). https://doi.org/10.1016/j.conbuildmat.2020.118325 0950-0618/Ó 2020 Elsevier Ltd. All rights reserved.
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1. Introduction Pavement is subjected to mechanical impact of passing vehicles during the whole period of a highway operation. These impacts are of quasi-cyclic nature and it provided the use of methods of cyclic loading mainly for determination of mechanical characteristics of asphalt concretes [1–11] and other road materials. However, far deeper analysis of test methods show that there is a possibility for the use of characteristics of mechanical behavior of materials obtained under static load for evaluation of their strength and life service at other types of loading including a cyclic one. One of such characteristics is a long-term strength. Long-term strength of a material is referred to as dependence of its failure time on an applied static stress [12–14]. A target stress is applied to a tested sample of a material instantaneously at the initial time moment (t = 0) (it is theoretically, but in practice it is realized for the shortest time), and it is kept constant till the end of the sample failure (t = tf). As it is known the described type of testing in the viscoelasticity theory corresponds to the so-called process of creep [15–17]. The creep test is one of the main in viscoelasticity theory, and its performance is technically more simple compared with other types of testing (for example, test for stress relaxation, deforming with a constant strain rate, deforming with constant loading rate, step loading, cyclic loading, etc.) [18–22]. Long-term strength of solid materials (metals, polymers, alloys, composites, timber, concretes, glass, etc.) has been investigated by S.N. Zhurkov and his co-workers beginning from 50 s of the last century [23,24], and the common thermal fluctuation theory has been proposed. It has been adopted in the above theory that elementary acts of failure process of solid bodies represent by themselves the thermal fluctuation rupture of interatomic bonds. The formula of long-term strength of materials proposed by Zhurkov on the basis of the thermal fluctuation theory has been further modified by Ratner [25–27]. Beginning from 70 s of the last century road researchers tried to describe long-term strength of an asphalt concrete on the basis of the thermal fluctuation theory [28–44]. In the abovementioned works asphalt concrete samples of various composition prepared with the use of bitumens of various grades have been tested at various temperatures (from +60 °C to 45 °C) and at several types of loading: uniaxial tension [36,38,39], transversal bending [30,33,34,37], pure bending [32] and pure shear [28,35]. Most of the authors of these works consider that long-term strength of asphalt concretes can be described on the basis of the thermal fluctuation theory. An interrelation has been determined between characteristics of long-term strength and fatigue strength of an asphalt concrete in the works [45–47]. Particularly, an exponent characterizing slope of fatigue curve is expressed by parameters of long-term strength: activation energy and pre-exponential constant in the formula of S.N. Zhurkov. Kiryukhin G.N. in the work [48] has made an attempt for explanation of fatigue failure of asphalt concretes using the thermal fluctuation theory and determined reliable correlation dependences between activation energy of viscous flow and characteristics of bitumens. And in the work [49] on the basis of the thermal fluctuation theory he has proposed an expression describing shear plastic strain rate of an asphalt concrete and informed that there is a reliable inverse relationship between failure time (long-term strength) and plastic strain rate. He declares that this relationship is provided by common thermal fluctuation mechanism of creep and failure processes. Reliable correlation relationships between viscous plastic flow rate and stress as well as between viscosity and stress at creep of an asphalt concrete has been determined in the paper [50].
As at experimental determining of long-term strength the tests are performed under scheme of creep, the results of these tests contain useful information about deformation of materials. Therefore road researchers apply the tests of creep for evaluation of mechanical behavior of neat and modified asphalt concretes [50–57]. They distinguish static [51,52] and dynamic creep [53–57] of asphalt concretes and it is found out that between test results for these two types of creep there is a positive correlation [52]. Creep strain characterizes well the plastic deformation of an asphalt concrete. Long-term strength of an asphalt concrete can be satisfactorily described by power function at a constant temperature. As early as in 1953H. Hoff described the rate of the steady-state creep of a rod by power function [58]. It has been also proposed for modeling of creep for metals and other viscoelastic materials by L.M. Kachanov [59,14] and Yu.N. Rabotnov [12,13]. At present the power function is often used for describing of characteristics for deformation and failure of an asphalt concrete: fatigue strength [60–62], steady-state creep [50], total dissipation energy at cyclic creep [63] and at modeling of crack growth [64]. The objective of this paper is to show that there is an important mechanical characteristic of an asphalt concrete called as a longterm strength, which is determined experimentally in an easy way, and using it one can model the deformation and failure of the asphalt concrete at various types of loading. This work investigates experimentally long-term strength of a hot fine-grained asphalt concrete prepared with the use of bitumen of grade BND 100/130 at uniaxial tension within the range of temperatures +60 °C and 24 °C. Long-term strength of the asphalt concrete is described by simple power function and on the basis of the thermal fluctuation theory. Several examples are given for the use of long-term strength of the asphalt concrete to solve practical problems connected with their strength at various types of loading. 2. Materials and methods 2.1. Bitumen Bitumen of grade 100–130 which met the requirements of the Kazakhstan standard [65] was used in this study. The bitumen grade on Superpave is PG (Performance Grade) 64–40 [66]. Basic standard indicators of the bitumen are shown in Table 1. Bitumen is produced by the Pavlodar processing plant from the crude oil of Western Siberia (Oil processing plant, Omsk, Russia) by the direct oxidation method. The detailed information about standard characteristics of the bitumen can be found in the paper [50]. 2.2. Asphalt concrete Hot dense asphalt concrete of type B which met the requirements of the Kazakhstan standard [67] was prepared with the
Table 1 Basic standard indicators of the bitumen. Indicator
Measurement unit
Requirements of ST RK 1373
Value
Penetration, 25 °C, 100 gr, 5 s Penetration Index PI Tensility at temperature: 25 °C 0 °C Softening point Fraas point Dynamic viscosity, 60 °C Kinematic viscosity
0.1 mm
101–130
104
– cm
–1.0. . . +1.0
–0.34
90 4.0 43 –22 120 180
140 5.7 46.0 –25.9 175.0 398.0
°C °C Pas mm2/s
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use of aggregate fractions of 5–10 mm (20%), 10–15 mm (13%) and 15–20 mm (10%) from the Novo-Alekseevsk rock pit (Almaty region, Kazakhstan), sand of fraction 0–5 mm (50%) from the plant ‘‘Asphaltconcrete-1” (Almaty city, Kazakhstan) and activated mineral powder (7%) from the Kordai rock pit (Zhambyl region, Kazakhstan). The bitumen content of grade 100–130 was 4.8% by weight of dry mineral material in the asphalt concrete. The content of bitumen was determined under the standard of Kazakhstan [68] and it satisfies the requirements of the standard of Kazakhstan [67] for physical and mechanical characteristics (air void content, water saturation, compression strength at 0 °C, 20 °C and 50 °C, shear resistance at 50 °C etc.). A granulometric composition curve for mineral part of the asphalt concrete is shown in Fig. 1. The detailed information about standard characteristics of the asphalt concrete is contained in the paper [50] published by the authors’ earlier. 2.3. Sample preparation Samples of the hot asphalt concrete in the form of a rectangular prism with dimensions 150x50x50 mm were manufactured as follows. First, the asphalt concrete samples were prepared in the form of a square slab using a Cooper compactor (Cooper, Nottingham, UK, model CRT-RC2S) according to the standard in [69]. Then the samples were cut from the asphalt concrete slabs in the form of a prism. Deviations in sizes of the beams did not exceed two millimeters. 2.4. Creep test Tests on creep were carried out with the asphalt concrete samples in the form of a rectangular prism according to the direct tensile scheme until complete failure was reached. The tests have been performed in a device specifically invented [70] and assembled in Kazakhstan Highway Research Institute which is designed for determining of mechanical characteristics of materials at tension (Figs. 2 and 3). Test temperatures are +60, +48, +36, +24, +12, 0, 12 and 24 °C. Several levels of stress have been selected for each temperature: from 5 to 16. Some parallel samples were tested at each stress: as a rule, 3–5 samples; sometimes the number of parallel samples reached 26. A total 371 samples have been tested for the asphalt concrete. Minimal duration for testing equal to 5.3 s occurred at the temperature of +12 °C and stress of 1.5 MPa. Maximal duration for testing equal to 1 551 000 s has been recorded at
Fig. 1. Granulometric curve of mineral part of the asphalt concrete.
Fig. 2. Device with climatic chamber for determining of mechanical characteristics of materials.
the temperature of 12 °C and stress of 0.369 MPa. The tests have been performed for 3 years and 7 months from 20th April 2015 to 28th November 2018. The tested 371 samples of the asphalt con-
Fig. 3. Refrigerating unit.
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crete have been placed only under loading at various temperatures and stresses for 3 472.8 h, i.e. 145 days. More detailed information about the tested samples of the asphalt concrete is shown in Table 2. 2.5. Loading with a constant rate The device [70] also allows performing the test for samples of materials at loading with a constant rate under the scheme of direct tension to failure. For this purpose the loading has been performed by small cast-iron shot and dry one-size sands carried through access port of a loading vessel. The access port of the vessel has a size (diameter) corresponding to a target rate of loading. 2.6. Thermal stress restrained specimen test The temperature decrease for a tested asphalt concrete sample at this type of test (TSRST) under the standard EN 12697-46 [71] occurs in a thermal chamber with the rate of 10 °C/h till failure. The ends of the tested sample are fixed. Testing of asphalt concrete samples under the scheme have been performed in a special device called TRAVIS and produced in InfraTest Company (Brackenheim, Germany). The whole testing procedure is operated by a computer. Temperature of a tested sample and stress in it are recorded at various time moments.
Fig. 4. Graphs of long-term strength of the asphalt concrete at various temperatures.
3. Results and discussion
increase; at temperature variation for 84 °C from 24 °C to +60 °C the power exponent n is varied in 2.5 times from 7.39 to 2.93, and coefficient A is varied for 8 orders. As values of the coefficients A and n are increased with temperature decrease the graphs of long-term strength should be located upward and have higher slope which is clearly seen in Fig. 4. Dependences of coefficients A and n on temperature are approximated by the following functions with high accuracy:
3.1. Long-term strength
A ¼ 580:37 expð0:2341 T Þ; R2 ¼ 0:9740;
ð2Þ
3.1.1. Power law The graphs of long-term strength are represented for the asphalt concrete at various temperatures in Fig. 4. As it is seen the graphs of long-term strength are described by straight lines in logarithmic coordinates with high accuracy at all the considered temperatures without exception, i.e. dependence of failure time on stress is approximated by power function [12–14,59]:
n ¼ 5:32 expð0:0094 T Þ; R2 ¼ 0:9331:
ð3Þ
tf ¼ A
rn
ð1Þ
where tf is failure time, s; r is stress, MPa; A, n are correlation coefficients. Coefficients A and n of the Eq. (1) are of physical sense. Coefficient A is equal numerically to failure time in seconds at the stress of 1 MPa. And coefficient n characterizes a rate of failure time variation at a stress variation, and shows particularly for what order the failure time is varied at the stress variation for one order. It is seen from Table 3 that there is a reliable correlation relationship between failure time tf and stress r at all considered temperatures. Table 3 and Figs. 5 and 6 show that coefficients A and n depend on temperature: A and n are decreased with the temperature
It is found that there is a reliable correlation relationship between coefficient A and indicator n (Fig. 7): value of preexponential coefficient A of the Eq. (1) (i.e. failure time of the asphalt concrete in seconds at stress equal to 1 MPa) is greatly increased under power dependence with the increase of power exponent n (with the temperature decrease). 3.1.2. Thermal fluctuation approach One of the best known approaches to the explanation of a solid body strength is the thermal fluctuation theory [23,24] according to which failure time of a body s at tensile stress r and an absolute temperature T is determined by equation:
U0 c r ; RT
s ¼ s0 exp
ð4Þ
where U0 is initial activation energy in unstressed condition of a body; c is a structural coefficient characterizing rate of activation energy decrease at failure with stress increase; R is the universal
Table 2 Characteristics of the tested asphalt concrete samples. Temperature, °C
Number of samples
Range of stress, MPa
Average time of failure, s
Total test duration, h
+60 +48 +36 +24 +12 0 12 24 Total:
51 35 30 169 21 21 30 14 371
0.0081–0.075 0.0083–0.1123 0.0189–0.146 0.0551–0.795 0.2–15 0.3–1.2 0.369–3.0 0.785–2.271 –
9–114 79 8.98–85 770.7 27.86–36 501.76 7.9–1 051 652 5.3–129 793.3 209–339 880 79.3–1 551 000 128.7–1 309 591.7 –
32.2 127.4 65.9 1 333.9 125.9 398.4 883.5 505.6 3 472.8
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Table 3 Coefficients of the Eq. (1). Temperature T, °C
60 48 36 24 12 0 12 24
R2
Coefficients A
n
0.0035 0.0040 0.0302 0.8142 42.3923 761.5622 5988.2162 440566.6745
2.9288 3.5538 3.6925 4.3742 4.9626 5.1994 5.1588 7.3839
0.9304 0.9932 0.9646 0.9413 0.9954 0.9936 0.9297 0.7822
Fig. 7. Correlation dependence between coefficients A and n.
Fig. 5. Dependence of coefficient A on temperature.
Fig. 6. Dependence of coefficient n on temperature.
gas constant equal to 8.314 J/(Kmol); s0 is a constant corresponding to the period of vibrations of kinetic units equal to 1012–1013 s. The main characteristics of the thermal fluctuation theory of S. N. Zhurkov which determine failure procedure are temperature T and activation barrier:
U ðrÞ ¼ U 0 c r
ð5Þ
As it is seen from this expression the value of activation barrier depends on a stress: it is decreased linearly with the stress increase. The thermal fluctuation theory explains that the more the applied stress the least activation barrier should be overcome to perform failure of a body. The analysis of the results for numerous experimental investigations performed with many materials (metals, polymers, alloys,
etc.) has allowed to S.N. Zhurkov and his co-workers making a conclusion that the failure of a body should be considered as a process where energetic barrier U0 decreased by value cr at the impact of stress r is overcome by means of thermal fluctuations. It has been found that if service life of a body s complies with the Eq. (7) where s0, U0 and c are constant then dependence ‘g s on reciprocal temperature 1T at various stresses should be a spread of straight lines which converge in one point. This characteristic point has been called ‘‘a pole”. It has been found that the pole for many materials is on ordinate axis: T1 = 0 and s0 = 1012–1013 s. The shift of the pole along horizontal axis has been observed for some polymers. This phenomenon has been called ‘‘the pole shift effect”. Many researchers observed the pole shift effect. The most common explanation of the pole shift effect is assumption about variability (dependence on stress and temperature) of coefficients s0, U0 and c [24]. The attempts to describe long-term strength of asphalt concretes on the basis of the thermal fluctuation theory were made earlier in 70–90 of the last century [28–44]. The detailed information about tested asphalt concretes of the works mentioned above is represented in Table 4. The results of these investigations have shown that activation energy of asphalt concrete failure does not depend on a stress, but constant s0 before exponential function in the Eq. (7), on the contrary, depends on a stress. Variation of the pre-exponential member in the equation for the asphalt concrete long-term strength is explained by the fact [31] that great variation of their structure occurs at deformation under load (additional orientations of bonds occur under load especially in bituminous films separating mineral grains) [72,73]. Zolotaryov V.A. and Titar V.S. in their work [30] have divided temperature range from +30 °C to 45 °C into three characteristic zones: viscosity zone, viscoelasticity zone and elastobrittleness zone. However they did not show the specific temperature values corresponding to the limits of the above characteristic zones. They have noted that pre-exponential constant s0 has a fixed value in a half of viscoelasticity zone and in elastobrittleness zone, but it depends on stress in viscosity zone and in a half of viscoelasticity zone, particularly, s0 is decreased with the stress increase according to exponential dependence. It is seen from Table 4 that constant s0 depends on stress (Volkov M.I., Zolotaryov V.A. [28], Kiryukhin G.N. [33], Mozgovoy V.V. [40]) and temperature (Kiryukhin G.N. [34], Noskov V.N., Tarashchinskiy Ye.G. [35]) at high (positive) temperatures and not very low ones (from +60 °C to 15 °C). And constant s0 does not depend either on stress or on temperature at low temperatures (from 25 °C to 45 °C), and it has a fixed value. It follows that, perhaps, the structure of a bitumen in an
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Table 4 The works where description of asphalt concretes long-term strength have been considered on the basis of the thermal fluctuation theory. Authors, works
Type of asphalt concrete, bitumen
Type of test, asphalt concrete sample
Temperature, °C
Stresses, MPa
Equations of long-term strength, conclusions
Zolotaryov V.A., Titar V.S. [30]
Sandy asphalt concrete
Transversal bending, beam: 4 4 16 cm
+30 45 °C
0.5 10
Volkov M.I., Zolotaryov V.A. [28]
Fine-grained asphalt concrete, bitumen grade BND 60/90–5.6%
+60 +18 °C
0.05 0.17
Kiryukhin G.N. [33]
Sandy asphalt concrete, bitumen grade BND 90/130
Pure shear, cylinder: d = 7 cm, h = 12.5 cm Transversal bending, beam: 4 2.5 16 cm Transversal bending, beam: 4 2.5 16 cm Pure bending
Area of viscous flow and half area of viscoelasticity: U ; half area of viscoelasticity and s ¼ C rb exp kT U area of elastic brittleness: s ¼ s0 exp kT U s ¼ C rb exp kT
+50 15 °C
0.16 3.74
Kiryukhin G.N. [34]
Sandy asphalt concrete, bitumen grade BND 90/130
Stabnikov N.V., Kocherova V.I. [32] Noskov V.N., Tarashch-inskiy Ye.G. [35] Aitaliyev Sh.M., Iskakbayev A.I., Teltayev B.B. [37] Aitaliyev Sh.M., Iskakbayev A.I., Issayev D., Teltayev B.B. [36] Gubach L.S., Fisher E. K. [38]
Sandy asphalt concrete with polymer DST 30 (2.5%), bitumen grade BND 60/90 Sandy asphalt concrete of type D, bitumen grade BND 40/60
Nickolskiy Yu.E., Pisklin V.M., Shestakov V.N. [39] Mozgovoy V.V. [40]
Fine-grained asphalt concretes of types A,B,V, bitumens of grades BND 60/90, BND 130/200, BND 200/300
Fine-grained asphalt concrete, bitumen grade BND 60/90 Sandy asphalt concrete, bitumen grade BND 60/90–6%
Sandy asphalt concrete, bitumen grade BND 90/130–7.5%
Sandy asphalt concrete, bitumen grade BND 130/200–7.4%
U0 ; RT
0.16 3.74
1 TÞ s ¼ C rðb0 þb
r
exp
U RT
Polymer asphalt concrete failure can be explained on the basis of the thermal fluctuation theory
Pure shear
+30 +60 °C
0.25 2.0
The thermal fluctuation theory is not acceptable. Formula s ¼ C T m rn is acceptable
Transversal bending, beam: 1.5x4x16 cm Uniaxial tension, beam: 4x4x16 cm
+20 °C
0.07 0.39
s ¼ C expðb rÞ
+20 °C
0.1 0.34
s ¼ C expðb rÞ
Uniaxial tension, cylinder: d = 3 cm, h = 10 cm Uniaxial tension, beam: 6x8x15 cm
5 25 °C
0.4 3.0
s ¼ s0 exp
0 40 °C
0.45 2.35
s ¼ s0 exp
Uniaxial tension, beam: 2x4x16 cm
+40 15 °C
0.04 1,0
s ¼ C rb exp
ð6Þ
U 0 cr kT
U 0 cr kT
U kT
where U0 = 162.8 kJ/mol is an activation energy of the asphalt concrete failure at tension. Having substituted the expressions (6) and (8) into the expression (1) we will have an equation for long-term strength of the asphalt concrete:
tf ¼ 2:916 1029
rð3699:525T þ8:165Þ exp 1
U0 : RT
ð9Þ
As in the expression (4) we have also obtained that the initial activation energy U0 does not depend neither on stress nor on temperature; the stress increase as well as the temperature increase decreases the long-term asphalt concrete strength. Fig. 11 represents the graph showing the accuracy of calculation for the long-term asphalt concrete strength under expression (9). As it is seen the values of the long-term strength calculated under expressions (1) and (9) at various stresses and temperatures comply well.
3.2. Applications
ð7Þ
Having multiplied numerator and denominator of the expres sion 19578:29 1T in the equation (6) with the universal gas constant R = 8.314 J/(K ∙mol) we have:
A ¼ 2:916 1029 exp
+50 15 °C
43:2 RT
1.2 2.5
1 ; R2 ¼ 0: 9851; A ¼ 2:916 1029 exp 19578:29 T 1 2 ; R ¼ 0:9010: T
exp
+20 +20 °C
asphalt concrete is practically not varied during deformation under load at very low temperatures (approximately low than 25 °C). Graphs for dependence of the asphalt concrete fatigue life tf on reciprocal temperature 1T at various stresses constructed based on data of Fig. 1 are represented in Fig. 8. As it is seen all lines converge in one point, i.e. in a pole which has the coordinates: tf0 = 109 s and T1 103 = 2.25, i.e. T = 444.44 K. The obtained coordinates give an opportunity to interpret formally the physical sense of the pole: without dependence on a value of the applied stress (i.e. at any stress) the considered type of an asphalt concrete will be damaged after 109 s at the temperature of 444.44 K (171.3 °C). To describe long-term strength of the asphalt concrete which we tested on the basis of the thermal fluctuation theory we determined the following dependences of coefficient A and indicator n in the expression (1) on reciprocal thermal dynamic temperature (in Kelvin scale) (Figs. 9 and 10)
n ¼ 8:165 þ 3699:525
s ¼ 5:9 1029 r2:0
ð8Þ
Data about long-term strength can be used for determination of an asphalt concrete strength at various conditions of loading. In the work [22] using long-term strength the expression has been obtained according to which it is possible to determine the number of loading cycles to asphalt concrete failure. Further the use of the asphalt concrete long-term strength will be shown at other types of its loading.
A.I. Iskakbayev et al. / Construction and Building Materials 244 (2020) 118325
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Fig. 8. Graphs for dependence of the asphalt concrete long-term strength on reciprocal temperature at various stresses.
Fig. 9. Dependence of coefficient A on reciprocal temperature.
Fig. 11. Graph for evaluation of accuracy of the expression (9).
Fig. 10. Dependence of indicator n on reciprocal temperature.
3.2.1. Loading with a constant rate According to the methods described shortly in Sub-Section 2.5 two series of the asphalt concrete samples have been tested at two different average loading rates: 0.651 MPa/s and 0.005726 MPa/s (Table 5).
Fig. 12 shows the graphs of stress variation in time in two asphalt concrete samples tested at two different loading rates. And Figs. 13 and 14 for these samples represent the graphs for strain variation in time and dependence ‘‘stress–strain”. As it is seen from Fig. 12 the stresses applied to the samples are varied (increased) in time strictly linear. As it has been expected the more loading rate the less failure time. For example, for the considered cases the loading rates differ nearly for two orders which has caused the difference in failure times approximately in 40 times. At linear variation of the stress in the samples the strain is a non-linear one (Fig. 13) which indicates to non-linearity of relationship between stress and strain for the tested type of the asphalt concrete at the test temperature (24 °C). Really, dependences ‘‘stress–strain” have strong non-linearity (Fig. 14). The degree of non-linearity is increased with time, i.e. with the stress increase. The flow effect occurs at small loading rate (r_ =0.005604 MPa/s) practically starting from the stress equal to 0.4 MPa: at the constant stress strain is increased till complete failure.
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Table 5 Results of testing of the asphalt concrete samples at loading with constant rate. Number of series
Loading rate r, MPa
Experiment
Calculation
1
0.650 0.650 0.645 0.647 0.664 0.651
2.4 2.0 2.2 1.9 2.2 2.1
1.9 1.9 1.9 1.9 1.9 1.9
1.56 1.30 1.42 1.23 1.46 1.39
0.005758 0.005604 0.005741 0.005838 0.005690 0.005726
71.2 80.3 74.2 66.8 82.6 75.0
83.0 84.8 83.2 82.1 83.8 83.4
0.41 0.45 0.43 0.39 0.47 0.43
Average 2
Average
Strength rf, MPa
Failure time tf, s
Fig. 12. Graphs for stress variation in the tested samples of the asphalt concrete.
Fig. 14. Graphs for dependence ‘‘stress–strain” for the tested samples of the asphalt concrete.
In case of loading with a constant rate r_ a stress r(s) is varied in time according to relationship:
rðtÞ ¼ r_ s:
ð12Þ
Inserting the expression (12) into the integral (10), having performed integrating and represented the obtained solution regarding tf, we have the following expression for failure time:
tf ¼ C r_
n ðnþ1 Þ n : nþ1
ð13Þ
Integral (10) considering the expression (12) can be written as:
Fig. 13. Graphs for strain variation in the tested samples of the asphalt concrete.
In the work [22] the following failure criterion has been obtained for materials of hereditary type:
C n
Z
tf
0
11 tf s n rðsÞds ¼ 1;
ð10Þ
where 1
C ¼ ½A ðn þ 1Þ n :
ð11Þ
The criterion (10) is valid for hereditary materials the long-term strength of which is described by power function of type (1). As it is seen it summarizes damage and contrary to the known Bailey’s integral [74] it considers loading history.
xðtÞ ¼
C r_ n
Z 0
t
ðt sÞn1 1
s ds:
ð14Þ
The expression (14) allows determining of damage x(t) of a material at any time moment t from the beginning of loading with a constant rate x(0) till its complete failure x(tf). Table 5 contains also the results of calculations for failure time of the considered asphalt concrete samples under the expression (13). As it is seen the calculated values of failure time at both (at small and big) loading rates are close to experimental ones. Average difference is 11%. Considering scatter in experimentally determined values of the asphalt concrete long-term strength (Fig. 4) and failure time of the asphalt concrete samples (Table 2) the average difference equal to 11% can be considered as a small one. Graphs of damage accumulation in the asphalt concrete samples in time (with the acting stress increase) are represented in Fig. 15.
A.I. Iskakbayev et al. / Construction and Building Materials 244 (2020) 118325
Fig. 15. Graphs of damage accumulation in the asphalt concrete samples.
9
Fig. 16. Graph of average temperature variation in the asphalt concrete samples in the thermal chamber of the device TRAVIS.
It is seen that at loading with a constant rate the curves of damage accumulation are characterized by a small non-linearity. The rate of non-linearity does not practically depend on a loading rate. As one should expect a rate of damage accumulation depends greatly on a loading rate: the higher the loading rate the bigger the slope of the curve of damage accumulation and less failure time. 3.2.2. Thermal stress restrained specimen test It is well known that one of the main types of an asphalt concrete destruction of highways in the regions with cold winter climate is low temperature cracking [75–78]. Currently there are experimental methods as well as calculation ones for evaluation of low temperature resistance of asphalt concretes and asphalt concrete pavements [69,74–78]. One of the most used experimental methods is the thermal stress restrained specimen test which is shortly described in Section 2.6. Two parallel samples of the asphalt concrete have been tested under this method. Their average failure time was 21 510 s, i.e. 5 h 59 min. As illustrative examples Figs. 16 and 17 represent the graphs showing variation of average temperature and stress in the asphalt concrete samples in time. As it is seen the thermal chamber of the device TRAVIS provides temperature decrease in the asphalt concrete samples with the constant rate of 0.0027 °C/s (9.72 10 °C/h). Initial temperature was equal to 22 °C. Graph of stress variation in time can be approximated by the cubic polynom with a high accuracy:
rðtÞ ¼ 3 1013 t3 þ 1 109 t2 3 105 t þ 0:0538;
Fig. 17. Graph of average stress variation in the asphalt concrete samples in time.
R2 ¼ 0:9905 ð15Þ
It is seen in Fig. 18 that the average critical stress and critical temperature are equal to 3.02 MPa and 34.2 °C respectively. Considering the expression (15) and the expression in Fig. 16 the expression (14) for calculating damage in the asphalt concrete samples will be the following:
xðtÞ ¼
Z 0
t
1 C ðt Þ ðt sÞnðsÞ1 nðsÞ
Fig. 18. Dependence of average stress in the asphalt concrete samples on temperature.
3 103 s3 þ 1 109 s2 3 105 s þ 0:0538 ds ð16Þ C ðsÞ ¼ ½AðsÞ ðnðsÞ þ 1ÞnðsÞ 1
ð17Þ
AðsÞ ¼ 0:000251 exp ½0:2411 ð21:957 0:0027 sÞ;
ð18Þ
nðsÞ ¼ 5:2623 exp ½0:0095 ð21:957 0:0027 sÞ:
ð19Þ
10
A.I. Iskakbayev et al. / Construction and Building Materials 244 (2020) 118325
Potential for use is also shown for characteristics of long-term strength of the asphalt concrete for modeling of the processes of its deformation and failure in conditions of the thermal stress restrained specimen test. 5. Recommendation To continue the research of a long-term strength for asphalt concretes. Meanwhile it is required to determine characteristics of a long term strength for asphalt concretes (A, n, s0, U) with different types and compositions of mineral material, with different grades and content of bituminous binder. Declaration of Competing Interest
Fig. 19. Graph of damage accumulation in the asphalt concrete samples.
Averaged graph of damage accumulation in the asphalt concrete samples in time constructed according to the results of calculations under expression (16) has been shown in Fig. 19 where it is clearly seen that practically for the first 8 000 s (2 h 13 min), i.e. within the interval of the reduced temperature from +22 °C to 0 °C with the rate of 10 °C/h the significant damage is not accumulated in the asphalt concrete samples. Intensive damage accumulation occurs in the samples to their failure with further temperature decrease. 4. Conclusion In this work: 1. It is shown that there is an important mechanical characteristic of an asphalt concrete called as a long-term strength which is determined experimentally in an easy way under scheme of direct creep, and using it one can model the deformation and failure of the asphalt concrete at various types of loading. 2. Long-term strength is investigated experimentally for a hot fine-grained asphalt concrete at uniaxial tension under stress from 0.0081 MPa to 3.0 MPa within the range of temperatures from + 60 °C to 24 °C. 3. The obtained results have shown that without exclusion for all considered temperatures long-term strength of the asphalt concrete is described by power function with a high accuracy. Exponent characterizing a rate of variation (decrease) for failure time at variation (increase) of stress is increased with the temperature increase: failure time is decreased for 3–7 orders (depending on temperature) at the stress increase for one order. 4. Long-term strength of the asphalt concrete can be also described on the basis of the thermal fluctuation theory. It is found out that activation energy of failure does not depend on stress and temperature, and it is equal to 162.8 kJ/mol; the pre-exponential constant characterizing average time between neighbouring thermal fluctuations depends on stress as well as on temperature; meanwhile the more the temperature the less the effect of stress. 5. Using a failure criterion for hereditary materials, proposed by the authors earlier, and characteristics of long-term strength (A, n) expressions have been derived for failure time and damage of a material at any time moment from the start of loading till it full failure in case of load increase with a constant rate. Comparison of the results obtained with the use of the expression for failure time with the results of experiments for the considered asphalt concrete has given a satisfactory coincidence.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A.T. Papagiannakis, E.A. Masad, Pavement design and materials, Wiley & Sons Inc., New Jersey, 2008. [2] MS-4, The asphalt handbook, seventh ed., Asphalt Institute, Lexington, 2007. [3] AASHTO TP 7, Standard test method for determining the permanent deformation and fatigue cracking characteristics of hot mix asphalt (HMA) using the simple shear test (SST) device, 2001. [4] ASTM D 3497, Standard test method for dynamic modulus of asphalt mixtures, 2003. [5] ASTM D 4123, Standard test method for indirect tension test for resilient modulus of bituminous mixtures, 1995. [6] AASHTO TP 62, Standard method of test for determining dynamic modulus of hot-mix asphalt concrete mixtures, 2003. [7] AASHTO T 321, Determining the fatigue life of compacted hot mix asphalt (HMA) subjected to repeated flexural bending, 2008. [8] EN 12697-24, Bituminous mixtures. Test method for hot mix asphalt, Part 24: Resistance to fatigue, 2004. [9] BS EN 12697-26, Bituminous mixtures. Test method for hot mix asphalt, Part 26: Stiffness, 2004. [10] AASHTO T 324, Standard method of test for Hamburg wheel-track testing of compacted asphalt mixtures, 2017. [11] UNI EN 12697-22, Bituminous mixtures. Test method for hot mix asphalt, Part 22: Wheel tracking, 2004. [12] Y.N. Rabotnov, Creep Problems in Structural Members, North-Holland, Amsterdam, 1969. [13] Y.N. Rabotnov, Continuum Mechanics, Nauka, Moscow, 1988. [14] L. Kachanov, Introduction to Continuum Damage Mechanics, Martinus Nijhoff Publishers, Dordrecht, 1986. [15] R.M. Christensen, Theory of Viscoelasticity: An Introduction, Academic Press, New York, 1971. [16] N.N. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction, Springer-Verlag, Berlin, 1989. [17] J.D. Ferry, Viscoelastic Properties of Polymers, third ed., Willey, New York, 1980. [18] A.I. Iskakbayev, B.B. Teltayev, F.K. Andriadi, K. Estayev, E.A. Suppes, A.A. Iskakbayeva, Experimental research of creep, recovery and fracture processes of asphalt concrete under tension, Proceedings of the 24th International Congress on Theoretical and Applied Mechanics, Montreal, Canada, 21-26 August 2016. [19] A.I. Iskakbayev, B.B. Teltayev, C.O. Rossi, Deformation and strength of asphalt concrete under static and step loadings, Proceedings of the AIIT International Congress on Transport Infrastructure and Systems (TIS 2017), Rome, Italy, 2017. [20] B.B. Teltayev, A.I. Iskakbayev, C.O. Rossi, Regularities of creep and long-term strength of hot asphalt concrete under tensile, Functional Pavement Design, Proceedings of the 4th Chinese-European Workshop on Functional Pavement Design, CEW 2016. [21] A.I. Iskakbayev, B.B. Teltayev, C.O. Rossi, G. Yensebayeva, Experimental investigation of an asphalt concrete deformation under cyclic loading, News Nation. Acad. Sci. Rep. Kaz. Ser. Geol. Tech. Sci. 2 (428) (2018) 114–1111. [22] A.I. Iskakbayev, B.B. Teltayev, C.O. Rossi, Modeling of cyclic strength for the asphalt concrete considering damage accumulation, Appl. Sci. 7 (12) (2017) 1– 11. [23] S.N. Zhurkov, Kinetic concept of solid body strength, News Acad. Sci. USSR, Ser. ‘‘Inorg. Materials” 3 (10) (1967) 1767–1777. [24] V.R. Regel, A.I. Slutsker, E. Ye, Tomashevskiy, Kinetic Nature of Solid Body Strength, Nauka, Moscow, 1974. [25] S.B. Ratner, About activation energy of a process of mechanical failure of polymers, Rep. Acad. Sci. USSR 183 (6) (1968) 1297–1300.
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Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Long-term strength of asphalt concrete and its applications A.I. Iskakbayev a,b, B.B. Teltayev b,⇑, K.Z. Yestayev b, B.D. Abu b a b
Department of Mechanics, Al-Farabi Kazakh National University, 71 Al-Farabi, Almaty 050040, Kazakhstan Kazakhstan Highway Research Institute, 2A Nurpeissov, Almaty 050061, Kazakhstan
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Long-term strength has been
investigated for an asphalt concrete. Long-term strength of the asphalt
concrete has been described by power function. Activation energy of the asphalt concrete failure depends on its type. Average time between thermal fluctuations depends on temperature and stress.
Thermal fluctuation theory
a r t i c l e
i n f o
Article history: Received 27 May 2019 Received in revised form 13 September 2019 Accepted 30 January 2020
Keywords: Asphalt concrete Uniaxial tension Creep Long-term strength Thermal fluctuation theory Loading with constant rate Thermal stress restrained test
a b s t r a c t The objective of the paper is to show that there is an important mechanical characteristic of an asphalt concrete called as a long-term strength which is determined experimentally in an easy way under scheme of direct creep, and using it one can model the deformation and failure of the asphalt concrete at various types of loading. For this purpose the long-term strength is investigated experimentally for a hot fine-grained asphalt concrete at uniaxial tension under stress from 0.0081 MPa to 3.0 MPa within the range of temperatures from +60 °C to 24 °C. The experiments have been performed in a specifically invented and assembled device which allows determining the mechanical characteristics of materials at tension at various temperatures and types of loading. It is found out that without exclusion for all considered temperatures long-term strength of the asphalt concrete is described by power function with a high accuracy. Exponent characterizing a rate of variation for failure time at variation of stress is increased with the temperature increase: failure time is decreased for 3–7 orders (depending on temperature) at the stress increase for one order. Long-term strength of the asphalt concrete can be also described on the basis of the thermal fluctuation theory. It is found out that activation energy of failure does not depend on stress and temperature, and it is equal to 162.8 kJ/mol; the pre-exponential constant characterizing average time between neighbouring thermal fluctuations depends on stress as well as on temperature; meanwhile the more the temperature the less the effect of stress. Potentials for use are also shown for long-term strength of the asphalt concrete for modeling of the processes of its deformation and failure at loading with a constant rate and in conditions of the thermal stress restrained specimen test. Ó 2020 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. E-mail address: [email protected] (B.B. Teltayev). https://doi.org/10.1016/j.conbuildmat.2020.118325 0950-0618/Ó 2020 Elsevier Ltd. All rights reserved.
2
A.I. Iskakbayev et al. / Construction and Building Materials 244 (2020) 118325
1. Introduction Pavement is subjected to mechanical impact of passing vehicles during the whole period of a highway operation. These impacts are of quasi-cyclic nature and it provided the use of methods of cyclic loading mainly for determination of mechanical characteristics of asphalt concretes [1–11] and other road materials. However, far deeper analysis of test methods show that there is a possibility for the use of characteristics of mechanical behavior of materials obtained under static load for evaluation of their strength and life service at other types of loading including a cyclic one. One of such characteristics is a long-term strength. Long-term strength of a material is referred to as dependence of its failure time on an applied static stress [12–14]. A target stress is applied to a tested sample of a material instantaneously at the initial time moment (t = 0) (it is theoretically, but in practice it is realized for the shortest time), and it is kept constant till the end of the sample failure (t = tf). As it is known the described type of testing in the viscoelasticity theory corresponds to the so-called process of creep [15–17]. The creep test is one of the main in viscoelasticity theory, and its performance is technically more simple compared with other types of testing (for example, test for stress relaxation, deforming with a constant strain rate, deforming with constant loading rate, step loading, cyclic loading, etc.) [18–22]. Long-term strength of solid materials (metals, polymers, alloys, composites, timber, concretes, glass, etc.) has been investigated by S.N. Zhurkov and his co-workers beginning from 50 s of the last century [23,24], and the common thermal fluctuation theory has been proposed. It has been adopted in the above theory that elementary acts of failure process of solid bodies represent by themselves the thermal fluctuation rupture of interatomic bonds. The formula of long-term strength of materials proposed by Zhurkov on the basis of the thermal fluctuation theory has been further modified by Ratner [25–27]. Beginning from 70 s of the last century road researchers tried to describe long-term strength of an asphalt concrete on the basis of the thermal fluctuation theory [28–44]. In the abovementioned works asphalt concrete samples of various composition prepared with the use of bitumens of various grades have been tested at various temperatures (from +60 °C to 45 °C) and at several types of loading: uniaxial tension [36,38,39], transversal bending [30,33,34,37], pure bending [32] and pure shear [28,35]. Most of the authors of these works consider that long-term strength of asphalt concretes can be described on the basis of the thermal fluctuation theory. An interrelation has been determined between characteristics of long-term strength and fatigue strength of an asphalt concrete in the works [45–47]. Particularly, an exponent characterizing slope of fatigue curve is expressed by parameters of long-term strength: activation energy and pre-exponential constant in the formula of S.N. Zhurkov. Kiryukhin G.N. in the work [48] has made an attempt for explanation of fatigue failure of asphalt concretes using the thermal fluctuation theory and determined reliable correlation dependences between activation energy of viscous flow and characteristics of bitumens. And in the work [49] on the basis of the thermal fluctuation theory he has proposed an expression describing shear plastic strain rate of an asphalt concrete and informed that there is a reliable inverse relationship between failure time (long-term strength) and plastic strain rate. He declares that this relationship is provided by common thermal fluctuation mechanism of creep and failure processes. Reliable correlation relationships between viscous plastic flow rate and stress as well as between viscosity and stress at creep of an asphalt concrete has been determined in the paper [50].
As at experimental determining of long-term strength the tests are performed under scheme of creep, the results of these tests contain useful information about deformation of materials. Therefore road researchers apply the tests of creep for evaluation of mechanical behavior of neat and modified asphalt concretes [50–57]. They distinguish static [51,52] and dynamic creep [53–57] of asphalt concretes and it is found out that between test results for these two types of creep there is a positive correlation [52]. Creep strain characterizes well the plastic deformation of an asphalt concrete. Long-term strength of an asphalt concrete can be satisfactorily described by power function at a constant temperature. As early as in 1953H. Hoff described the rate of the steady-state creep of a rod by power function [58]. It has been also proposed for modeling of creep for metals and other viscoelastic materials by L.M. Kachanov [59,14] and Yu.N. Rabotnov [12,13]. At present the power function is often used for describing of characteristics for deformation and failure of an asphalt concrete: fatigue strength [60–62], steady-state creep [50], total dissipation energy at cyclic creep [63] and at modeling of crack growth [64]. The objective of this paper is to show that there is an important mechanical characteristic of an asphalt concrete called as a longterm strength, which is determined experimentally in an easy way, and using it one can model the deformation and failure of the asphalt concrete at various types of loading. This work investigates experimentally long-term strength of a hot fine-grained asphalt concrete prepared with the use of bitumen of grade BND 100/130 at uniaxial tension within the range of temperatures +60 °C and 24 °C. Long-term strength of the asphalt concrete is described by simple power function and on the basis of the thermal fluctuation theory. Several examples are given for the use of long-term strength of the asphalt concrete to solve practical problems connected with their strength at various types of loading. 2. Materials and methods 2.1. Bitumen Bitumen of grade 100–130 which met the requirements of the Kazakhstan standard [65] was used in this study. The bitumen grade on Superpave is PG (Performance Grade) 64–40 [66]. Basic standard indicators of the bitumen are shown in Table 1. Bitumen is produced by the Pavlodar processing plant from the crude oil of Western Siberia (Oil processing plant, Omsk, Russia) by the direct oxidation method. The detailed information about standard characteristics of the bitumen can be found in the paper [50]. 2.2. Asphalt concrete Hot dense asphalt concrete of type B which met the requirements of the Kazakhstan standard [67] was prepared with the
Table 1 Basic standard indicators of the bitumen. Indicator
Measurement unit
Requirements of ST RK 1373
Value
Penetration, 25 °C, 100 gr, 5 s Penetration Index PI Tensility at temperature: 25 °C 0 °C Softening point Fraas point Dynamic viscosity, 60 °C Kinematic viscosity
0.1 mm
101–130
104
– cm
–1.0. . . +1.0
–0.34
90 4.0 43 –22 120 180
140 5.7 46.0 –25.9 175.0 398.0
°C °C Pas mm2/s
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use of aggregate fractions of 5–10 mm (20%), 10–15 mm (13%) and 15–20 mm (10%) from the Novo-Alekseevsk rock pit (Almaty region, Kazakhstan), sand of fraction 0–5 mm (50%) from the plant ‘‘Asphaltconcrete-1” (Almaty city, Kazakhstan) and activated mineral powder (7%) from the Kordai rock pit (Zhambyl region, Kazakhstan). The bitumen content of grade 100–130 was 4.8% by weight of dry mineral material in the asphalt concrete. The content of bitumen was determined under the standard of Kazakhstan [68] and it satisfies the requirements of the standard of Kazakhstan [67] for physical and mechanical characteristics (air void content, water saturation, compression strength at 0 °C, 20 °C and 50 °C, shear resistance at 50 °C etc.). A granulometric composition curve for mineral part of the asphalt concrete is shown in Fig. 1. The detailed information about standard characteristics of the asphalt concrete is contained in the paper [50] published by the authors’ earlier. 2.3. Sample preparation Samples of the hot asphalt concrete in the form of a rectangular prism with dimensions 150x50x50 mm were manufactured as follows. First, the asphalt concrete samples were prepared in the form of a square slab using a Cooper compactor (Cooper, Nottingham, UK, model CRT-RC2S) according to the standard in [69]. Then the samples were cut from the asphalt concrete slabs in the form of a prism. Deviations in sizes of the beams did not exceed two millimeters. 2.4. Creep test Tests on creep were carried out with the asphalt concrete samples in the form of a rectangular prism according to the direct tensile scheme until complete failure was reached. The tests have been performed in a device specifically invented [70] and assembled in Kazakhstan Highway Research Institute which is designed for determining of mechanical characteristics of materials at tension (Figs. 2 and 3). Test temperatures are +60, +48, +36, +24, +12, 0, 12 and 24 °C. Several levels of stress have been selected for each temperature: from 5 to 16. Some parallel samples were tested at each stress: as a rule, 3–5 samples; sometimes the number of parallel samples reached 26. A total 371 samples have been tested for the asphalt concrete. Minimal duration for testing equal to 5.3 s occurred at the temperature of +12 °C and stress of 1.5 MPa. Maximal duration for testing equal to 1 551 000 s has been recorded at
Fig. 1. Granulometric curve of mineral part of the asphalt concrete.
Fig. 2. Device with climatic chamber for determining of mechanical characteristics of materials.
the temperature of 12 °C and stress of 0.369 MPa. The tests have been performed for 3 years and 7 months from 20th April 2015 to 28th November 2018. The tested 371 samples of the asphalt con-
Fig. 3. Refrigerating unit.
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crete have been placed only under loading at various temperatures and stresses for 3 472.8 h, i.e. 145 days. More detailed information about the tested samples of the asphalt concrete is shown in Table 2. 2.5. Loading with a constant rate The device [70] also allows performing the test for samples of materials at loading with a constant rate under the scheme of direct tension to failure. For this purpose the loading has been performed by small cast-iron shot and dry one-size sands carried through access port of a loading vessel. The access port of the vessel has a size (diameter) corresponding to a target rate of loading. 2.6. Thermal stress restrained specimen test The temperature decrease for a tested asphalt concrete sample at this type of test (TSRST) under the standard EN 12697-46 [71] occurs in a thermal chamber with the rate of 10 °C/h till failure. The ends of the tested sample are fixed. Testing of asphalt concrete samples under the scheme have been performed in a special device called TRAVIS and produced in InfraTest Company (Brackenheim, Germany). The whole testing procedure is operated by a computer. Temperature of a tested sample and stress in it are recorded at various time moments.
Fig. 4. Graphs of long-term strength of the asphalt concrete at various temperatures.
3. Results and discussion
increase; at temperature variation for 84 °C from 24 °C to +60 °C the power exponent n is varied in 2.5 times from 7.39 to 2.93, and coefficient A is varied for 8 orders. As values of the coefficients A and n are increased with temperature decrease the graphs of long-term strength should be located upward and have higher slope which is clearly seen in Fig. 4. Dependences of coefficients A and n on temperature are approximated by the following functions with high accuracy:
3.1. Long-term strength
A ¼ 580:37 expð0:2341 T Þ; R2 ¼ 0:9740;
ð2Þ
3.1.1. Power law The graphs of long-term strength are represented for the asphalt concrete at various temperatures in Fig. 4. As it is seen the graphs of long-term strength are described by straight lines in logarithmic coordinates with high accuracy at all the considered temperatures without exception, i.e. dependence of failure time on stress is approximated by power function [12–14,59]:
n ¼ 5:32 expð0:0094 T Þ; R2 ¼ 0:9331:
ð3Þ
tf ¼ A
rn
ð1Þ
where tf is failure time, s; r is stress, MPa; A, n are correlation coefficients. Coefficients A and n of the Eq. (1) are of physical sense. Coefficient A is equal numerically to failure time in seconds at the stress of 1 MPa. And coefficient n characterizes a rate of failure time variation at a stress variation, and shows particularly for what order the failure time is varied at the stress variation for one order. It is seen from Table 3 that there is a reliable correlation relationship between failure time tf and stress r at all considered temperatures. Table 3 and Figs. 5 and 6 show that coefficients A and n depend on temperature: A and n are decreased with the temperature
It is found that there is a reliable correlation relationship between coefficient A and indicator n (Fig. 7): value of preexponential coefficient A of the Eq. (1) (i.e. failure time of the asphalt concrete in seconds at stress equal to 1 MPa) is greatly increased under power dependence with the increase of power exponent n (with the temperature decrease). 3.1.2. Thermal fluctuation approach One of the best known approaches to the explanation of a solid body strength is the thermal fluctuation theory [23,24] according to which failure time of a body s at tensile stress r and an absolute temperature T is determined by equation:
U0 c r ; RT
s ¼ s0 exp
ð4Þ
where U0 is initial activation energy in unstressed condition of a body; c is a structural coefficient characterizing rate of activation energy decrease at failure with stress increase; R is the universal
Table 2 Characteristics of the tested asphalt concrete samples. Temperature, °C
Number of samples
Range of stress, MPa
Average time of failure, s
Total test duration, h
+60 +48 +36 +24 +12 0 12 24 Total:
51 35 30 169 21 21 30 14 371
0.0081–0.075 0.0083–0.1123 0.0189–0.146 0.0551–0.795 0.2–15 0.3–1.2 0.369–3.0 0.785–2.271 –
9–114 79 8.98–85 770.7 27.86–36 501.76 7.9–1 051 652 5.3–129 793.3 209–339 880 79.3–1 551 000 128.7–1 309 591.7 –
32.2 127.4 65.9 1 333.9 125.9 398.4 883.5 505.6 3 472.8
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Table 3 Coefficients of the Eq. (1). Temperature T, °C
60 48 36 24 12 0 12 24
R2
Coefficients A
n
0.0035 0.0040 0.0302 0.8142 42.3923 761.5622 5988.2162 440566.6745
2.9288 3.5538 3.6925 4.3742 4.9626 5.1994 5.1588 7.3839
0.9304 0.9932 0.9646 0.9413 0.9954 0.9936 0.9297 0.7822
Fig. 7. Correlation dependence between coefficients A and n.
Fig. 5. Dependence of coefficient A on temperature.
Fig. 6. Dependence of coefficient n on temperature.
gas constant equal to 8.314 J/(Kmol); s0 is a constant corresponding to the period of vibrations of kinetic units equal to 1012–1013 s. The main characteristics of the thermal fluctuation theory of S. N. Zhurkov which determine failure procedure are temperature T and activation barrier:
U ðrÞ ¼ U 0 c r
ð5Þ
As it is seen from this expression the value of activation barrier depends on a stress: it is decreased linearly with the stress increase. The thermal fluctuation theory explains that the more the applied stress the least activation barrier should be overcome to perform failure of a body. The analysis of the results for numerous experimental investigations performed with many materials (metals, polymers, alloys,
etc.) has allowed to S.N. Zhurkov and his co-workers making a conclusion that the failure of a body should be considered as a process where energetic barrier U0 decreased by value cr at the impact of stress r is overcome by means of thermal fluctuations. It has been found that if service life of a body s complies with the Eq. (7) where s0, U0 and c are constant then dependence ‘g s on reciprocal temperature 1T at various stresses should be a spread of straight lines which converge in one point. This characteristic point has been called ‘‘a pole”. It has been found that the pole for many materials is on ordinate axis: T1 = 0 and s0 = 1012–1013 s. The shift of the pole along horizontal axis has been observed for some polymers. This phenomenon has been called ‘‘the pole shift effect”. Many researchers observed the pole shift effect. The most common explanation of the pole shift effect is assumption about variability (dependence on stress and temperature) of coefficients s0, U0 and c [24]. The attempts to describe long-term strength of asphalt concretes on the basis of the thermal fluctuation theory were made earlier in 70–90 of the last century [28–44]. The detailed information about tested asphalt concretes of the works mentioned above is represented in Table 4. The results of these investigations have shown that activation energy of asphalt concrete failure does not depend on a stress, but constant s0 before exponential function in the Eq. (7), on the contrary, depends on a stress. Variation of the pre-exponential member in the equation for the asphalt concrete long-term strength is explained by the fact [31] that great variation of their structure occurs at deformation under load (additional orientations of bonds occur under load especially in bituminous films separating mineral grains) [72,73]. Zolotaryov V.A. and Titar V.S. in their work [30] have divided temperature range from +30 °C to 45 °C into three characteristic zones: viscosity zone, viscoelasticity zone and elastobrittleness zone. However they did not show the specific temperature values corresponding to the limits of the above characteristic zones. They have noted that pre-exponential constant s0 has a fixed value in a half of viscoelasticity zone and in elastobrittleness zone, but it depends on stress in viscosity zone and in a half of viscoelasticity zone, particularly, s0 is decreased with the stress increase according to exponential dependence. It is seen from Table 4 that constant s0 depends on stress (Volkov M.I., Zolotaryov V.A. [28], Kiryukhin G.N. [33], Mozgovoy V.V. [40]) and temperature (Kiryukhin G.N. [34], Noskov V.N., Tarashchinskiy Ye.G. [35]) at high (positive) temperatures and not very low ones (from +60 °C to 15 °C). And constant s0 does not depend either on stress or on temperature at low temperatures (from 25 °C to 45 °C), and it has a fixed value. It follows that, perhaps, the structure of a bitumen in an
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Table 4 The works where description of asphalt concretes long-term strength have been considered on the basis of the thermal fluctuation theory. Authors, works
Type of asphalt concrete, bitumen
Type of test, asphalt concrete sample
Temperature, °C
Stresses, MPa
Equations of long-term strength, conclusions
Zolotaryov V.A., Titar V.S. [30]
Sandy asphalt concrete
Transversal bending, beam: 4 4 16 cm
+30 45 °C
0.5 10
Volkov M.I., Zolotaryov V.A. [28]
Fine-grained asphalt concrete, bitumen grade BND 60/90–5.6%
+60 +18 °C
0.05 0.17
Kiryukhin G.N. [33]
Sandy asphalt concrete, bitumen grade BND 90/130
Pure shear, cylinder: d = 7 cm, h = 12.5 cm Transversal bending, beam: 4 2.5 16 cm Transversal bending, beam: 4 2.5 16 cm Pure bending
Area of viscous flow and half area of viscoelasticity: U ; half area of viscoelasticity and s ¼ C rb exp kT U area of elastic brittleness: s ¼ s0 exp kT U s ¼ C rb exp kT
+50 15 °C
0.16 3.74
Kiryukhin G.N. [34]
Sandy asphalt concrete, bitumen grade BND 90/130
Stabnikov N.V., Kocherova V.I. [32] Noskov V.N., Tarashch-inskiy Ye.G. [35] Aitaliyev Sh.M., Iskakbayev A.I., Teltayev B.B. [37] Aitaliyev Sh.M., Iskakbayev A.I., Issayev D., Teltayev B.B. [36] Gubach L.S., Fisher E. K. [38]
Sandy asphalt concrete with polymer DST 30 (2.5%), bitumen grade BND 60/90 Sandy asphalt concrete of type D, bitumen grade BND 40/60
Nickolskiy Yu.E., Pisklin V.M., Shestakov V.N. [39] Mozgovoy V.V. [40]
Fine-grained asphalt concretes of types A,B,V, bitumens of grades BND 60/90, BND 130/200, BND 200/300
Fine-grained asphalt concrete, bitumen grade BND 60/90 Sandy asphalt concrete, bitumen grade BND 60/90–6%
Sandy asphalt concrete, bitumen grade BND 90/130–7.5%
Sandy asphalt concrete, bitumen grade BND 130/200–7.4%
U0 ; RT
0.16 3.74
1 TÞ s ¼ C rðb0 þb
r
exp
U RT
Polymer asphalt concrete failure can be explained on the basis of the thermal fluctuation theory
Pure shear
+30 +60 °C
0.25 2.0
The thermal fluctuation theory is not acceptable. Formula s ¼ C T m rn is acceptable
Transversal bending, beam: 1.5x4x16 cm Uniaxial tension, beam: 4x4x16 cm
+20 °C
0.07 0.39
s ¼ C expðb rÞ
+20 °C
0.1 0.34
s ¼ C expðb rÞ
Uniaxial tension, cylinder: d = 3 cm, h = 10 cm Uniaxial tension, beam: 6x8x15 cm
5 25 °C
0.4 3.0
s ¼ s0 exp
0 40 °C
0.45 2.35
s ¼ s0 exp
Uniaxial tension, beam: 2x4x16 cm
+40 15 °C
0.04 1,0
s ¼ C rb exp
ð6Þ
U 0 cr kT
U 0 cr kT
U kT
where U0 = 162.8 kJ/mol is an activation energy of the asphalt concrete failure at tension. Having substituted the expressions (6) and (8) into the expression (1) we will have an equation for long-term strength of the asphalt concrete:
tf ¼ 2:916 1029
rð3699:525T þ8:165Þ exp 1
U0 : RT
ð9Þ
As in the expression (4) we have also obtained that the initial activation energy U0 does not depend neither on stress nor on temperature; the stress increase as well as the temperature increase decreases the long-term asphalt concrete strength. Fig. 11 represents the graph showing the accuracy of calculation for the long-term asphalt concrete strength under expression (9). As it is seen the values of the long-term strength calculated under expressions (1) and (9) at various stresses and temperatures comply well.
3.2. Applications
ð7Þ
Having multiplied numerator and denominator of the expres sion 19578:29 1T in the equation (6) with the universal gas constant R = 8.314 J/(K ∙mol) we have:
A ¼ 2:916 1029 exp
+50 15 °C
43:2 RT
1.2 2.5
1 ; R2 ¼ 0: 9851; A ¼ 2:916 1029 exp 19578:29 T 1 2 ; R ¼ 0:9010: T
exp
+20 +20 °C
asphalt concrete is practically not varied during deformation under load at very low temperatures (approximately low than 25 °C). Graphs for dependence of the asphalt concrete fatigue life tf on reciprocal temperature 1T at various stresses constructed based on data of Fig. 1 are represented in Fig. 8. As it is seen all lines converge in one point, i.e. in a pole which has the coordinates: tf0 = 109 s and T1 103 = 2.25, i.e. T = 444.44 K. The obtained coordinates give an opportunity to interpret formally the physical sense of the pole: without dependence on a value of the applied stress (i.e. at any stress) the considered type of an asphalt concrete will be damaged after 109 s at the temperature of 444.44 K (171.3 °C). To describe long-term strength of the asphalt concrete which we tested on the basis of the thermal fluctuation theory we determined the following dependences of coefficient A and indicator n in the expression (1) on reciprocal thermal dynamic temperature (in Kelvin scale) (Figs. 9 and 10)
n ¼ 8:165 þ 3699:525
s ¼ 5:9 1029 r2:0
ð8Þ
Data about long-term strength can be used for determination of an asphalt concrete strength at various conditions of loading. In the work [22] using long-term strength the expression has been obtained according to which it is possible to determine the number of loading cycles to asphalt concrete failure. Further the use of the asphalt concrete long-term strength will be shown at other types of its loading.
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Fig. 8. Graphs for dependence of the asphalt concrete long-term strength on reciprocal temperature at various stresses.
Fig. 9. Dependence of coefficient A on reciprocal temperature.
Fig. 11. Graph for evaluation of accuracy of the expression (9).
Fig. 10. Dependence of indicator n on reciprocal temperature.
3.2.1. Loading with a constant rate According to the methods described shortly in Sub-Section 2.5 two series of the asphalt concrete samples have been tested at two different average loading rates: 0.651 MPa/s and 0.005726 MPa/s (Table 5).
Fig. 12 shows the graphs of stress variation in time in two asphalt concrete samples tested at two different loading rates. And Figs. 13 and 14 for these samples represent the graphs for strain variation in time and dependence ‘‘stress–strain”. As it is seen from Fig. 12 the stresses applied to the samples are varied (increased) in time strictly linear. As it has been expected the more loading rate the less failure time. For example, for the considered cases the loading rates differ nearly for two orders which has caused the difference in failure times approximately in 40 times. At linear variation of the stress in the samples the strain is a non-linear one (Fig. 13) which indicates to non-linearity of relationship between stress and strain for the tested type of the asphalt concrete at the test temperature (24 °C). Really, dependences ‘‘stress–strain” have strong non-linearity (Fig. 14). The degree of non-linearity is increased with time, i.e. with the stress increase. The flow effect occurs at small loading rate (r_ =0.005604 MPa/s) practically starting from the stress equal to 0.4 MPa: at the constant stress strain is increased till complete failure.
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Table 5 Results of testing of the asphalt concrete samples at loading with constant rate. Number of series
Loading rate r, MPa
Experiment
Calculation
1
0.650 0.650 0.645 0.647 0.664 0.651
2.4 2.0 2.2 1.9 2.2 2.1
1.9 1.9 1.9 1.9 1.9 1.9
1.56 1.30 1.42 1.23 1.46 1.39
0.005758 0.005604 0.005741 0.005838 0.005690 0.005726
71.2 80.3 74.2 66.8 82.6 75.0
83.0 84.8 83.2 82.1 83.8 83.4
0.41 0.45 0.43 0.39 0.47 0.43
Average 2
Average
Strength rf, MPa
Failure time tf, s
Fig. 12. Graphs for stress variation in the tested samples of the asphalt concrete.
Fig. 14. Graphs for dependence ‘‘stress–strain” for the tested samples of the asphalt concrete.
In case of loading with a constant rate r_ a stress r(s) is varied in time according to relationship:
rðtÞ ¼ r_ s:
ð12Þ
Inserting the expression (12) into the integral (10), having performed integrating and represented the obtained solution regarding tf, we have the following expression for failure time:
tf ¼ C r_
n ðnþ1 Þ n : nþ1
ð13Þ
Integral (10) considering the expression (12) can be written as:
Fig. 13. Graphs for strain variation in the tested samples of the asphalt concrete.
In the work [22] the following failure criterion has been obtained for materials of hereditary type:
C n
Z
tf
0
11 tf s n rðsÞds ¼ 1;
ð10Þ
where 1
C ¼ ½A ðn þ 1Þ n :
ð11Þ
The criterion (10) is valid for hereditary materials the long-term strength of which is described by power function of type (1). As it is seen it summarizes damage and contrary to the known Bailey’s integral [74] it considers loading history.
xðtÞ ¼
C r_ n
Z 0
t
ðt sÞn1 1
s ds:
ð14Þ
The expression (14) allows determining of damage x(t) of a material at any time moment t from the beginning of loading with a constant rate x(0) till its complete failure x(tf). Table 5 contains also the results of calculations for failure time of the considered asphalt concrete samples under the expression (13). As it is seen the calculated values of failure time at both (at small and big) loading rates are close to experimental ones. Average difference is 11%. Considering scatter in experimentally determined values of the asphalt concrete long-term strength (Fig. 4) and failure time of the asphalt concrete samples (Table 2) the average difference equal to 11% can be considered as a small one. Graphs of damage accumulation in the asphalt concrete samples in time (with the acting stress increase) are represented in Fig. 15.
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Fig. 15. Graphs of damage accumulation in the asphalt concrete samples.
9
Fig. 16. Graph of average temperature variation in the asphalt concrete samples in the thermal chamber of the device TRAVIS.
It is seen that at loading with a constant rate the curves of damage accumulation are characterized by a small non-linearity. The rate of non-linearity does not practically depend on a loading rate. As one should expect a rate of damage accumulation depends greatly on a loading rate: the higher the loading rate the bigger the slope of the curve of damage accumulation and less failure time. 3.2.2. Thermal stress restrained specimen test It is well known that one of the main types of an asphalt concrete destruction of highways in the regions with cold winter climate is low temperature cracking [75–78]. Currently there are experimental methods as well as calculation ones for evaluation of low temperature resistance of asphalt concretes and asphalt concrete pavements [69,74–78]. One of the most used experimental methods is the thermal stress restrained specimen test which is shortly described in Section 2.6. Two parallel samples of the asphalt concrete have been tested under this method. Their average failure time was 21 510 s, i.e. 5 h 59 min. As illustrative examples Figs. 16 and 17 represent the graphs showing variation of average temperature and stress in the asphalt concrete samples in time. As it is seen the thermal chamber of the device TRAVIS provides temperature decrease in the asphalt concrete samples with the constant rate of 0.0027 °C/s (9.72 10 °C/h). Initial temperature was equal to 22 °C. Graph of stress variation in time can be approximated by the cubic polynom with a high accuracy:
rðtÞ ¼ 3 1013 t3 þ 1 109 t2 3 105 t þ 0:0538;
Fig. 17. Graph of average stress variation in the asphalt concrete samples in time.
R2 ¼ 0:9905 ð15Þ
It is seen in Fig. 18 that the average critical stress and critical temperature are equal to 3.02 MPa and 34.2 °C respectively. Considering the expression (15) and the expression in Fig. 16 the expression (14) for calculating damage in the asphalt concrete samples will be the following:
xðtÞ ¼
Z 0
t
1 C ðt Þ ðt sÞnðsÞ1 nðsÞ
Fig. 18. Dependence of average stress in the asphalt concrete samples on temperature.
3 103 s3 þ 1 109 s2 3 105 s þ 0:0538 ds ð16Þ C ðsÞ ¼ ½AðsÞ ðnðsÞ þ 1ÞnðsÞ 1
ð17Þ
AðsÞ ¼ 0:000251 exp ½0:2411 ð21:957 0:0027 sÞ;
ð18Þ
nðsÞ ¼ 5:2623 exp ½0:0095 ð21:957 0:0027 sÞ:
ð19Þ
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Potential for use is also shown for characteristics of long-term strength of the asphalt concrete for modeling of the processes of its deformation and failure in conditions of the thermal stress restrained specimen test. 5. Recommendation To continue the research of a long-term strength for asphalt concretes. Meanwhile it is required to determine characteristics of a long term strength for asphalt concretes (A, n, s0, U) with different types and compositions of mineral material, with different grades and content of bituminous binder. Declaration of Competing Interest
Fig. 19. Graph of damage accumulation in the asphalt concrete samples.
Averaged graph of damage accumulation in the asphalt concrete samples in time constructed according to the results of calculations under expression (16) has been shown in Fig. 19 where it is clearly seen that practically for the first 8 000 s (2 h 13 min), i.e. within the interval of the reduced temperature from +22 °C to 0 °C with the rate of 10 °C/h the significant damage is not accumulated in the asphalt concrete samples. Intensive damage accumulation occurs in the samples to their failure with further temperature decrease. 4. Conclusion In this work: 1. It is shown that there is an important mechanical characteristic of an asphalt concrete called as a long-term strength which is determined experimentally in an easy way under scheme of direct creep, and using it one can model the deformation and failure of the asphalt concrete at various types of loading. 2. Long-term strength is investigated experimentally for a hot fine-grained asphalt concrete at uniaxial tension under stress from 0.0081 MPa to 3.0 MPa within the range of temperatures from + 60 °C to 24 °C. 3. The obtained results have shown that without exclusion for all considered temperatures long-term strength of the asphalt concrete is described by power function with a high accuracy. Exponent characterizing a rate of variation (decrease) for failure time at variation (increase) of stress is increased with the temperature increase: failure time is decreased for 3–7 orders (depending on temperature) at the stress increase for one order. 4. Long-term strength of the asphalt concrete can be also described on the basis of the thermal fluctuation theory. It is found out that activation energy of failure does not depend on stress and temperature, and it is equal to 162.8 kJ/mol; the pre-exponential constant characterizing average time between neighbouring thermal fluctuations depends on stress as well as on temperature; meanwhile the more the temperature the less the effect of stress. 5. Using a failure criterion for hereditary materials, proposed by the authors earlier, and characteristics of long-term strength (A, n) expressions have been derived for failure time and damage of a material at any time moment from the start of loading till it full failure in case of load increase with a constant rate. Comparison of the results obtained with the use of the expression for failure time with the results of experiments for the considered asphalt concrete has given a satisfactory coincidence.
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