Accelerated Creep of Asphalt Concrete at Medium Temperatures

Abstract

In this paper, the accelerated creep under uniaxial tension of hot fine-grained dense asphalt concrete at temperatures of 22–24 °C and stresses from 0.084 MPa to 0.596 MPa is experimentally investigated. The names and brief descriptions of the creep curve and accelerated creep characteristics are given. A mathematical model is proposed that has physical meaning and satisfactorily approximates the asphalt concrete accelerated creep. The model parameters are determined by the Levenberg–Marquardt method. The deformation and time characteristics of the accelerated creep of asphalt concrete are established and analyzed.

1. Introduction

Asphalt concrete is the most widely used road material in the world. It is mainly used for the construction of the upper durable layers of road pavements [1,2,3]. Despite the fact that the cost of bitumen is constantly increasing, asphalt concrete will apparently remain the main road material in the coming decades, as it is today.

It can be said that, since the 1930s, road asphalt concretes have been used for practical applications. Since then, work has been carried out to improve their quality, test methods have been developed and their mechanical behaviors have been studied. Despite all this, in our opinion, the physical and mechanical properties of road asphalt concretes, particularly their mechanical behavior under conditions close to real ones, have not been studied sufficiently.

For example, in Kazakhstan [4], and in many countries around the world, the strength of road asphalt concrete is determined by testing cylindrical samples on a mechanical press until complete failure. The duration of such laboratory tests is only a few seconds, while, in real operating conditions, the service life of asphalt concrete pavements can be 20 years or more [5,6].

It is known that the service life of asphalt concrete pavements and road surfaces depends not only on mechanical loads but also on climatic factors, including freezing and thawing of road pavement structures and subgrade.

The results of our experimental studies show that the mechanical behavior of asphalt concrete under loads at different characteristic moments of time is different. Thus, in work [7], it is shown that an asphalt concrete creep curve consists of three characteristic sections: sections of unsteady, steady and accelerated creep.

The features of an asphalt concrete deformation in regard to unsteady creep, as well as its mathematical description and mechanical interpretation, are contained in papers [7]. A detailed study of the steady creep of asphalt concrete is contained in works [8,9].

The results of the experimental studies presented in this paper show that, at temperatures of 22–24 °C, the duration of the accelerated creep of asphalt concrete, regardless of the stress, is 35% of its long-term strength (the time from the moment the load is applied to complete failure). In other words, the failure of asphalt concrete does not occur “instantaneously” (not quickly) but after a long period of time. In this case, damage first accumulates in the material, and only then do main cracks appear [10].

In the last two decades, two of the most popular methods for assessing the crack resistance of road asphalt concretes have been the so-called “single-edge notched beam test” and “semi-secular bending test”, which are now carried out according to the AASHTO TP 124 standard [11]. According to these methods, asphalt concrete samples in the form of a rectangular beam [12,13] and in the form of a semi-circular disk [14,15] are preliminarily subjected to an artificial initial crack, and then these samples with cracks are tested.

Currently, in order to assess the crack resistance of road asphalt concrete, methods of testing samples in the form of rectangular beams (without pre-applied artificial cracks) for direct tension according to European standards EN 12697-46 [16] are also widely used.

The results of the works [8,9,17] show that a visible to the eye (main) crack in the test sample of an asphalt concrete appears only at the end of the third characteristic section of deformation (at the stage of accelerated creep). Before the main crack appears, the test sample of an asphalt concrete must go through the stages of unsteady-state and steady-state creep, as well as through the first substage of accelerated creep, in which intensive accumulation of damage occurs.

As is known, creep is a fundamental property of viscoelastic materials [18,19,20,21,22,23]. By testing materials for creep, one can obtain a lot of important information about their mechanical behavior. Intensive studies of the creep of road asphalt concretes were carried out in the 1970s–1980s [24,25,26]. In these and many other works of that period, researchers mainly paid attention to determining and describing the long-term strength of asphalt concretes based on the results of their creep testing. At the same time, a relationship between creep and the fatigue strength of an asphalt concrete was established [27]. Subsequently, G.N.Kiryukhin proposed finding the rate of plastic deformation of an asphalt concrete under shear based on the results of creep testing [28]. In the article by B.B.Teltayev et al., reliable correlations were established between the rate of viscoplastic flow, the viscosity of an asphalt concrete and creep stress [8]. The possibilities of using creep data for modeling the mechanical behavior of asphalt concrete under continuous loading at a constant rate and in a thermal stress-restrained specimen test are shown in [16]. Experimental research on and modeling of asphalt concrete creep also feature in the works [29,30,31].

This paper is devoted to the experimental study of accelerated creep under uniaxial tension in hot fine-grained dense asphalt concrete at temperatures of 22–24 °C and at stresses from 0.084 MPa to 0.596 MPa. The research diagram is shown in Figure 1.

2. Materials and Methods

2.1. Materials

For the creep test, a hot fine-grained dense asphalt concrete of type B was selected which meets the requirements of the Kazakhstan standard ST RK 1225-2019 [32]. A bitumen of the BND 100–130 grade produced by the Pavlodar Petrochemical Plant (Pavlodar, Kazakhstan) by oxidation was adopted as a binder. The bitumen meets the requirements of the Kazakhstan standard ST RK 1373-2013 [33]. The composition of the asphalt concrete is as follows: crushed stone—43% (5–10 mm—20%, 10–15 mm—13%, 15–20 mm—10%), sand—50%, mineral powder—7%, bitumen—4.8%.

2.2. Preparation of Samples

First, the asphalt concrete samples were prepared in the form of rectangular slabs with dimensions of 350 × 350 × 50 mm according to EN 12697-33-2003 [34]. Then, these slabs were cut into rectangular beams with dimensions of 50 × 50 × 150 mm (Figure 2).

2.3. Test

The asphalt concrete samples (beams) were tested for creep under uniaxial tension at temperatures of 22–24 °C until complete failure at different stresses (from 0.084 MPa to 0.596 MPa). During all tests, the temperature and stress were kept constant. The tests were carried out in a special installation with a heat chamber (Figure 3) [8,9]. The diagram of the device is shown in Figure 4. An asphalt concrete sample to be tested (2) is placed in the heat chamber (1) and secured. The temperature regime is maintained inside the heat chamber to ensure a specified constant temperature value in the sample. A rod (4) is secured to the lower end of the sample and a vessel (5) for loading is suspended from its lower end. The loading of the test samples was carried out as follows. A given volume of dry homogeneous sand or homogeneous cast iron shot (pre-calculated load) from the container (6) was loaded into the vessel (5) through the tube (8) over several seconds, excluding the impact phenomenon. The loading rate was regulated by the shutter (7).

A vessel was suspended via a rod to a plate glued to the lower end of the testing sample. A given volume of dry homogeneous sand or homogeneous cast iron shot (pre-calculated load) was loaded into the vessel over several seconds, excluding the impact phenomenon. The elongation of the sample was measured by two clock-type indicators (3) and was recorded by a video camera (10). The test results were then analyzed and processed on a computer.

3. Accelerated Creep

3.1. Creep Curve Characteristics

A typical asphalt concrete creep curve has three characteristic sections (Figure 5): I—section of unsteady-state creep; II—section of steady-state creep; III—section of accelerated creep. The characteristics of the creep curve and their brief descriptions are given in

Table 1. Characteristics of an asphalt concrete creep curve and their brief descriptions.

Characteristic Designation	Characteristic Name	Brief Description
I	section of unsteady-state creep	the first section of creep curve within which strain rate decreases
II	section of steady-state creep	the second section of creep curve within which strain rate is constant
III	section of accelerated creep	the third section of creep curve within which strain rate increases; at the end of this section the test specimen fails
εo	conditionally instantaneous strain	the value of εo depends on magnitude and duration (rate) of full load application
ε1	strain at the end of section I (at the beginning of section II) of creep curve; limiting strain of hardening	maximum strain in section I (minimum strain in section II) of creep curve
ε2	strain at the end of section II (at the beginning of section III) of creep curve	maximum strain in section II (minimum strain in section III) of creep curve
εf	failure strain	maximum strain on creep curve (also in section III) corresponding to failure time tf
Δε23	total strain of accelerated creep	total strain realized in section of accelerated creep
t1	unsteady-state creep duration; limiting time of hardening	at this time point unsteady-state creep ends; from this time point steady-state creep begins
t2	the moment in time corresponding to the end of steady-state creep and the beginning of accelerated creep	at this time point steady-state creep ends; from this time point accelerated creep begins
tf	failure time	t this time point failure of the test sample occurs
Δt23	accelerated creep duration	during this period of time accelerated creep is realized

.

As stated in [7], in the unsteady-state creep section, the process of asphalt concrete strengthening occurs as a result of the strain rate decreasing over time. In the steady-state creep section, the deformation process occurs at a constant rate. It is assumed that, in this section, the number of broken bonds in the structural elements of asphalt concrete (destruction) is equal to the number of restored bonds per unit of time. In the accelerated creep section, irreversible accumulation of damage begins in places of stress concentration (pores, microcracks); this results in a macroscopic crack forming by merging some of the specified defects. Then, the macroscopic crack spreads (grows) and, at the end of the accelerated creep section, complete failure of the material occurs (division of the asphalt concrete sample into parts) [10].

It is known that one of the main types of destruction of road asphalt concrete pavement is fatigue cracking (Figure 6a) [35]. In regions with cold winter climates, low-temperature cracking of asphalt concrete pavements occurs (Figure 6b). Therefore, we believe that, in the study of the strength and durability of asphalt concrete, it is important to model the accelerated creep and determine the parameters characterizing accumulation of damage and destruction at this stage.

3.2. Mathematical Model

For a mathematical description of the accelerated creep of asphalt concrete, the following expression is proposed:

$${\epsilon }_{III}\left(t\right)=\dot{{\epsilon }_{II}}·{\displaystyle \frac{\left(n+1\right)·t_f}{\left(n+1-q\right)}}·\left[1-{\left(1-{\displaystyle \frac{t}{t_f}}\right)}^{\left({\displaystyle \frac{n+1-q}{n+1}}\right)}\right],$$

(1)

where n + 1 − q > 0;

• $\dot{{\epsilon }_{II}}$—steady-state creep rate;
• t_f—failure time;
• n—parameter characterizing damage rate;
• q—parameter characterizing accelerated creep rate;
• t—time: 0 ≤ t ≤ t_f; 0 ≤ t/t_f ≤ 1.

Model (1) is not just a mathematical expression selected to approximate the obtained experimental data. It has a clear physical meaning. Thus, at t = 0 we can ascertain the following: ε_III(0) = 0. This is correct since, at the initial moment of time, the accelerated creep deformation is zero. At t = t_f, we obtain the following from it:

$${\epsilon }_{III}\left(t_f\right)={\epsilon }_f=\dot{{\epsilon }_{II}}·{\displaystyle \frac{\left(n+1\right)·t_f}{\left(n+1-q\right)}},$$

(2)

where ε_f—failure strain, %.

Taking into account expression (2), model (1) can be rewritten as follows:

$${\epsilon }_{III}\left(t\right)={\epsilon }_f·\left[1-{\left(1-{\displaystyle \frac{t}{t_f}}\right)}^{\left({\displaystyle \frac{n+1-q}{n+1}}\right)}\right].$$

(3)

As can be seen from expressions (1) and (2), the accelerated creep strain, including the failure strain ε_f, is determined by the steady-state creep rate ($\dot{{\epsilon }_{II}}$), the failure time t_f and the parameters n and q.

3.3. Levenberg–Marquardt Method

In order to determine with high accuracy the values of parameters n and q in model (1), when approximating the experimentally obtained accelerated creep strains from the asphalt concrete samples, the Levenberg–Marquardt method was used and the calculation process was implemented in MathCAD 14 [36].

We will represent the problem of determining the parameters and model (1) as a least squares problem in the following form:

$$F\left(x\right)={{\displaystyle \sum }}_{i=1}^N{f}_i\left(x\right){{\displaystyle \sum }}_{i=1}^N{\left[{\epsilon }_{III}\left(t_i\right)-{\epsilon }_{IIIe}\left(t_i\right)\right]}^2\to min,$$

(4)

In the Levenberg–Marquardt method, the direction of searching for a local minimumof a function $F\left(x\right)$ is defined as the solution to the system of equations [37]:

$$\left[J_{𝓀}^T\left(x\right)J_{𝓀}\left(x\right)+{\lambda }_{𝓀}I\right]p_{𝓀}=-J_{𝓀}\left(x\right)f_{𝓀}\left(x\right)$$

(5)

where $J_{𝓀}$—Jacobian matrix;

• I—identity matrix;
• λ—the Marquardt parameter;
• p—search direction;
• $𝓀$—step number.

After finding the search direction $p_{𝓀}$ in step $𝓀$, the next point is determined by the following formula:

$$x_{𝓀+1}=x_{𝓀}+p_{𝓀}$$

(6)

The process of searching for new solutions to system (5) stops as soon as condition (4) is met.

The initial values of the parameters n₀ and q₀ were determined by selection, which involved visually assessing the closeness of the approximating model graph to the experimentally obtained values of the strain.

4. Results and Discussion

4.1. Approximation of Accelerated Creep

Figure 7 shows examples of approximation of the asphalt concrete accelerated creep at different stresses using the proposed model (1). As can be seen from this figure, and as other similar results have shown, the model (1) approximates the experimentally obtained values of strain in the accelerated creep section with high accuracy.

The values of the parameters of the accelerated creep model (1) found by the Levenberg–Marquardt method for the asphalt concrete samples, the graphs of which are shown in Figure 7, are given in

Table 2. Values of the parameters of the accelerated creep model.

Stress σ, MPa	Steady-State Creep Rate έII, %/C	Failure Time tf, c	Parameters
			n	q	$\frac{\mathit{n} + 1 - \mathit{q}}{\mathit{n} + 1}$
0.0840	0.6367 × 10−4	1800	1.0729	1.3500	0.3488
0.1566	8.2298 × 10−4	180	1.3147	0.7798	0.6631
0.1852	5.8545 × 10−4	180	0.8388	0.9654	0.4750
0.2323	30.9657 × 10−4	40	0.5854	0.7843	0.5053
0.3053	92.2767 × 10−4	20	0.1099	0.3895	0.6491
0.5570	305.5211 × 10−4	11	0.9998	1.0758	0.4621

. It is seen that the values of the parameters n and q are in the range from 0.50 to 1.31 and from 0.39 to 1.35, respectively. The scattering of values of the combined parameter (n + 1 − q)/(n + 1) is relatively small (from 0.35 to 0.66), and its average value is 0.5172.

4.2. Deformation Characteristics

In the accelerated creep section, two strains are key: (1) strain at the beginning of the accelerated creep section (at the end of the steady-state creep section) ε₂; (2) strain at the end of the accelerated creep section (failure strain) ε_f. It should be expected that, at a constant temperature, they depend on the stress. In Figure 8 and Figure 9, it is seen that the dependences are well described by a power function. In this case, it can be approximately assumed that the influence of small stresses (up to 0.2 MPa) is significant; the values of these strains at stresses from 0.2 MPa to 0.6 MPa are practically constant and are approximately equal to 1% and 2%, respectively. Strans ε₂ and ε_f decrease according to a power dependence from 1.4% to 1% and from 3.9% to 2%, respectively, with an increase in stress from 0.0084 MPa to 0.2 MPa.

Of course, an important characteristic is also the total strain (Δε₂₃) accumulated in the accelerated creep section which can be determined by the formula:

ε₂₃ = ε_f − ε₂.

(7)

The dependence of the strain Δε₂₃ on the stress is shown in Figure 10. As expected, the approximation of this characteristic strain is similar to the previous strains ε₂ and ε_f: In general, the dependence is described by a power function; the influence of stress occurs at its small values (up to 0.2 MPa); the strain Δε₂₃ within the stress range from 0.2 MPa to 0.6 MPa is practically constant.

Figure 11 shows the values of the relative accelerated creep strain Δε₂₃/ε_f at different stresses. The values of Δε₂₃/ε_f are in the range of 38.6% to 64.8%. The average value is 51%. In other words, the value of the total strain accumulated in the accelerated creep section is, on average, equal to half the failure strain.

4.3. Time Characteristics

The following time characteristics of accelerated creep are important from a practical point of view: (1) the start time of the accelerated creep section (the end of the steady-state creep section) t₂ is the time counted from the moment of the end of loading of a test specimen until the start of the accelerated creep; (2) the end time of the accelerated creep section (long-term strength) tf is the time counted from the moment of the end of loading of a specimen until its complete failure; (3) the duration of the accelerated creep section Δt₂₃ which is determined by the formula:

Δt₂₃ = t_f − t₂.

(8)

The dependences of the time characteristics t₂ and tf on stress are shown in Figure 12 and Figure 13, where it is evident that they are described with high accuracy by a power function. A strong influence of stress occurs at its small values (up to 0.3 MPa). With an increase in stress from 0.084 MPa to 0.3 MPa (by 3.6 times), the characteristic times t₂ and tf decrease by three orders of magnitude, i.e., by thousands of times; with a further change in stress from 0.3 MPa to 0.6 MPa, the times t₂ and tf decrease by five times.

It should be noted that the values of the exponents in the formulas in Figure 12 and Figure 13, equal to 3.11 and 3.01, respectively, are very close. The slight difference in their values can be explained by the scatter of experimental results. Then, for both dependencies, these exponents can be taken to be the same and equal to their arithmetic mean values: (3.11 + 3.01)/2 = 3.06.

Using Formula (8), the duration of the accelerated creep section can be calculated:

Δt₂₃ = 7.15·σ^3.06 − 4.63·σ^3.06 = 2.52·σ^3.06.

(9)

The relative duration of the accelerated creep section:

$${\displaystyle \frac{\mathsf{\Delta }{t}_{23}}{t_f}}={\displaystyle \frac{2.52·{\sigma }^{3.06}}{7.15·{\sigma }^{3.06}}}=0.35.$$

(10)

Thus, it has been established that the relative duration of the accelerated creep of asphalt concrete at temperatures of 22–24 °C does not depend on stress and is equal to 35% of the long-term strength.

A joint consideration of the experimental results obtained in Section 4.2 and Section 4.3 allows us to formulate the following important finding: the deformation characteristics of the accelerated creep of asphalt concrete change only at low (up to 0.2 MPa) stresses, while at medium and high stresses (from 0.2 MPa to 0.6 MPa) they remain practically constant. The latter can be explained by the fact that, with an increase in stress, the effect of increasing the deformation rate and the effect of reducing the duration of the accelerated creep period are completely mutually compensated; in other words, with an increase in stress, due to a reduction in the time of action of the load, the strain does not have time to develop. This finding is of great practical importance, as, since the processes of damage accumulation and destruction of asphalt concrete pavements are significantly influenced by the values of the load from the axles (wheels) of vehicles, in models of road pavements, including tensile strain in asphalt concrete layers, the values of the deformation characteristics of asphalt concretes can be taken as the same for medium and heavy vehicles.

5. Conclusions

The results of this experimental study on accelerated creep under uniaxial tension in hot fine-grained dense asphalt concrete at temperatures of 22–24 °C and stresses from 0.084 MPa to 0.596 MPa allow us to formulate the following conclusions:

• A mathematical model is proposed that has physical meaning and satisfactorily approximates the accelerated creep of asphalt concrete. The model includes strain and time of failure, rate of steady-state creep and parameters characterizing rates of damage and deformation at the stage of accelerated creep. In order to determine with high accuracy the parameters n and q of the model when approximating the accelerated creep strains, the Levenberg–Marquardt method is used and is implemented in MatCAD 14. A brief description of the Levenberg–Marquardt method is given.
• The initial accelerated creep strain, failure strain and accelerated creep strain decrease with increasing stress. Their stress dependencies are described by power functions. It has been established that only small stresses (up to 0.2 MPa) affect the deformation characteristics of the accelerated creep of asphalt concrete, while medium and large stresses (from 0.2 MPa to 0.6 MPa) have practically no effect. The relative accelerated creep strain (the ratio of the accelerated creep strain to the failure strain) at different stresses is 38.6–64.8%, with an average value of 51%. In other words, the total strain accumulated at the accelerated creep stage is, on average, equal to half the failure strain.
• There are reliable correlation dependences at the start and end points of the accelerated creep section, as well as throughout the duration of the accelerated creep section on the stress, described with high accuracy by power functions. A strong influence of stress occurs at its low values (up to 0.3 MPa). The relative duration of accelerated creep (the ratio of the duration of accelerated creep to the long-term strength) does not depend on stress and is equal to 0.35, i.e., the duration of accelerated creep is equal to one third of the failure time.
• The phenomenon of maintaining the deformation characteristics of the accelerated creep of asphalt concrete at medium and high stresses (from 0.2 MPa to 0.6 MPa) can be explained by the fact that, with an increase in stress, the effect of increasing the deformation rate and the effect of reducing the duration of the accelerated creep period are completely mutually compensated; in other words, with an increase in stress, due to a reduction in the time of action of the load, the strain does not have time to develop. This finding is of great practical importance, as, since the processes of damage accumulation and destruction of asphalt concrete pavements are significantly influenced by the values of the load from the axles (wheels) of vehicles, in models of road pavements, including tensile strain in asphalt concrete layers, the values of the deformation characteristics of asphalt concretes can be taken as being the same for medium and heavy vehicles.
• The limitations of the current research are that the accelerated creep of asphalt concrete was investigated only at temperatures of 22–24 °C and that the effect of stress on the n and q parameters of the model was not assessed. It is recommended that similar experimental studies at other temperatures are carried out in the future to establish the temperature dependences of the characteristics of the accelerated creep of asphalt concrete and, based on the results of these studies, determine the effect of stress on the n and q parameters of the model.