Shear Properties of Asphalt Mixtures under Triaxial Compression

: In order to study the inﬂuences of conﬁning pressure and temperature on the shear properties of asphalt mixtures, triaxial tests were conducted at 40 ◦ C, 50 ◦ C, and 60 ◦ C, with the conﬁning pressure ranges from 0 to 1 MPa for the widely used continuous-graded AC (Asphalt Concrete)-13, open-graded OGFC (Open-Graded Friction Course)-13, and gap-graded SMA (Stone Mastic Asphalt)-13 asphalt mixtures in China. A nonlinear regression/prediction model of triaxial strength for asphalt mixtures was proposed. The results show that conﬁning pressure and temperature had a signiﬁcant effect on the shear resistance of asphalt mixtures. With increasing temperature, the shear strength of the asphalt mixture gradually decreased due to the decreasing of cohesion strength; the shear strength of the asphalt mixture increased with the increase of conﬁning pressure. Meanwhile, the cohesion strength increased and the friction angle decreased gradually with the increase of conﬁning pressure. When the conﬁning pressure was close to 0.6 MPa, the Mohr–Coulomb failure envelope bended down, so the linear Mohr–Coulomb criterion is not suitable to describe the failure behavior of asphalt mixtures. Therefore, a power function failure envelope was put forward to characterize the nonlinear shear properties of asphalt mixtures. The nonlinear evolutional laws of shear parameters, which includes cohesion strength and friction angle, were also proposed for asphalt pavement material and structure design. Among these asphalt mixtures, the gap-graded SMA-13 asphalt mixture exhibited better performance on the resistance to shear failure, and it was recommended as the upper layer material to improve the shear performance of asphalt pavement. acquisition, T.H.; Methodology, T.H. S.L.; Project T.H. Resources, S.Q., Writing—review S.L.


Introduction
By the end of 2018, 142,500 km of expressway had been built in China, most of which are composed of the semi-rigid base asphalt pavement structure. The use of semi-rigid base reduces the deflection of the pavement, as well as the bending stress at the bottom of the asphalt layer [1]. Hence, the overall deformation resistance of the pavement structure is greatly enhanced. However, the shear stress in the surface layer is increased inadvertently [2]. Besides this, the overloading phenomenon is widespread in China. Therefore, it is of great importance to enhance the strength, especially the shear strength of asphalt mixtures in the hot areas in China [3]. Otherwise, the rutting damage will appear in the asphalt pavement due to insufficient capacity of shear resistance [4][5][6]. Thus, the performance of asphalt pavement will be greatly reduced, and the service life and service level of pavement will be seriously affected.
In previous studies, the triaxial test and shear formula have been used for the analysis of shear resistance for asphalt mixtures [7][8][9]. In recent years, the Strategic Highway Research Program (SHRP) has proposed three types of tests to evaluate the shear deformation of materials, including triaxial creep and repeated loading tests, simple shear creep and repeated loading tests, and uniaxial creep and repeated loading tests [10,11]. However, these tests have not been widely used in practice in China because of their time consumption. In addition, Ji et al. developed a method for determining the criterion of shear strength based on the large accelerated loading test [12]. Nevertheless, the large accelerated loading test is expensive and complicated to operate. Only a few agencies in China have this equipment, which makes it difficult to be widely used [12]. In order to simplify the test method, Zhang et al. proposed the partial triaxial compression test to evaluate the shear resistance of asphalt mixtures [13]. Although this test method inherits many advantages of triaxial test, its applicability needs further verification because it is a new test method. In order to ensure the asphalt surfaces have adequate shear strength, the Specifications for Design of Highway Asphalt Pavement (JTG D50-2017) in China inspects the integrated penetration strength of the whole asphalt concrete surface layers; the essence is to control the shear strength from the view of asphalt mixtures. Although the uniaxial penetration test was proposed in the specification to simulate the complex stress states of asphalt materials in the pavement structure [14][15][16], it is unreasonable to solve the friction angle and cohesion strength by linear Mohr-Coulomb theory according to the combined tests of uniaxial penetration and unconfined compression, the reason being that the nonlinear shear properties of asphalt mixture were not taken into account [17][18][19].
As mentioned above, on one hand, the shear failure of asphalt pavement materials usually occurs under the effect of overload and heavy load during the hot summer season [20]. On the other hand, the triaxial test has many advantages such as to simulate the complex stress state of the pavement structure, and to solve the friction angle and cohesion strength for asphalt pavement materials and structure design [21][22][23]. However, the use of the linear Mohr-Coulomb theory affects the accuracy of materials and structure design for asphalt pavement. Therefore, the triaxial tests were carried out at various temperatures and confining pressure levels for the widely used continuous-graded AC (Asphalt Concrete)-13, open-graded OGFC (Open Graded Friction Course)-13, and gap-graded SMA (Stone Mastic Asphalt)-13 asphalt mixtures in China [24]. Based on the results, a nonlinear Mohr-Coulomb failure envelope was established, and the nonlinear evolutional laws of cohesion strength and friction angle were also proposed. All of these are important for the design of asphalt pavement materials and structures.

Materials
The mechanical responses of typical pavement structures under heavy tires were analyzed in Reference [2], and the results indicated that the maximum shear stress was generally located at the bottom of upper layer of asphalt pavement. Hence, the triaxial tests were conducted for the widely used upper surface layer materials of asphalt pavement in China, including the continuous-graded AC (Asphalt Concrete)-13, open-graded OGFC (Open Graded Friction Course)-13, and gap-graded SMA (Stone Mastic Asphalt)-13 asphalt mixtures. The gradations were plotted in Figure 1 and the mixture designs are listed in Table 1; the basalt was used as aggregate, the SBS (Styrene Butadiene Styrene) modified bitumen, which is one of the most commonly used modified bitumen in pavement industry, was used as the binder [25,26]. The lignin fiber was used as a component of SMA-13 asphalt mixture. The basic properties of SBS modified bitumen were provided in Table 2 [27].

Specimen Preparation
Current specification of Standard Test Methods of Bitumen and Bituminous Mixtures for Highway Engineering (JTG E20-2011) in China states that the cylindrical specimen with a diameter of 100 mm and a height of 150 mm can be used for triaxial test, the cylindrical specimen can also be tested with other sizes if needed. Meanwhile, the specimen for uniaxial compression test (that is the stress state with confining pressure of 0) of asphalt mixture in the specification is 100 mm in diameter and 100 mm in height. In order to be consistent with the specimen size of the uniaxial compression test, the cylindrical specimen with the diameter of 100 mm and the height of 102 mm was obtained by the gyratory compactor. Then, the upper and lower surfaces of the specimen were polished with a double-sided saw to obtain the specimen with a height of 100 mm.

Specimen Preparation
Current specification of Standard Test Methods of Bitumen and Bituminous Mixtures for Highway Engineering (JTG E20-2011) in China states that the cylindrical specimen with a diameter of 100 mm and a height of 150 mm can be used for triaxial test, the cylindrical specimen can also be tested with other sizes if needed. Meanwhile, the specimen for uniaxial compression test (that is the stress state with confining pressure of 0) of asphalt mixture in the specification is 100 mm in diameter and 100 mm in height. In order to be consistent with the specimen size of the uniaxial compression test, the cylindrical specimen with the diameter of 100 mm and the height of 102 mm was obtained by the gyratory compactor. Then, the upper and lower surfaces of the specimen were polished with a double-sided saw to obtain the specimen with a height of 100 mm.

Triaxial Conditions and Setups
The triaxial test method developed by Zheng and Huang was adopted to conduct the tests. As shown in Figure 2, in this test, the stable confining pressure was applied on the outer cylindrical surface of specimen through the flexible airbag, the axial stress was loaded by a material testing system MTS (Material Testing System) [27,28]. Therefore, the specimen fails at the complex stress state of triaxial compression.

Triaxial Conditions and Setups
The triaxial test method developed by Zheng and Huang was adopted to conduct the tests. As shown in Figure 2, in this test, the stable confining pressure was applied on the outer cylindrical surface of specimen through the flexible airbag, the axial stress was loaded by a material testing system MTS (Material Testing System) [27,28]. Therefore, the specimen fails at the complex stress state of triaxial compression. As shown in Figure 3, the highest average temperature of China in July was 32.7 °C . Hence, the daily maximum temperature can reach 40 °C or more. According to previous studies [29,30], the temperature of the road surface can be 20 °C higher than the atmospheric temperature in summer. Therefore, three temperatures of 40 °C (104 °F), 50 °C (122 °F ), and 60 °C (140 °F ) were chosen as testing temperatures [31]. The axial loading rate was 1 mm/min to simulate the effect of static load. In this condition, the asphalt mixture is prone to shear failure. Considering that confining pressure significantly affects the triaxial strength properties of the asphalt mixture [32,33], the confining pressure was set to 0, 0.2, 0.4, 0.6, 0.8, and 1.0 MPa, respectively, to analyze the influence of confining pressure. Before the test, the specimen was placed into the temperature control chamber for more than 6 h to ensure that the temperature of the specimen is consistent with the testing temperature. In addition, the lubricant was smeared on the upper and lower surfaces of the specimen to reduce the friction. During the process of conducting the triaxial test, the confining pressure was firstly applied through the airbag and then the axial load was applied by the MTS until the specimen fails.  As shown in Figure 3, the highest average temperature of China in July was 32.7 • C. Hence, the daily maximum temperature can reach 40 • C or more. According to previous studies [29,30], the temperature of the road surface can be 20 • C higher than the atmospheric temperature in summer. Therefore, three temperatures of 40 • C (104 • F), 50 • C (122 • F), and 60 • C (140 • F) were chosen as testing temperatures [31]. The axial loading rate was 1 mm/min to simulate the effect of static load. In this condition, the asphalt mixture is prone to shear failure. Considering that confining pressure significantly affects the triaxial strength properties of the asphalt mixture [32,33], the confining pressure was set to 0, 0.2, 0.4, 0.6, 0.8, and 1.0 MPa, respectively, to analyze the influence of confining pressure. Before the test, the specimen was placed into the temperature control chamber for more than 6 h to ensure that the temperature of the specimen is consistent with the testing temperature. In addition, the lubricant was smeared on the upper and lower surfaces of the specimen to reduce the friction. During the process of conducting the triaxial test, the confining pressure was firstly applied through the airbag and then the axial load was applied by the MTS until the specimen fails.

Triaxial Conditions and Setups
The triaxial test method developed by Zheng and Huang was adopted to conduct the tests. As shown in Figure 2, in this test, the stable confining pressure was applied on the outer cylindrical surface of specimen through the flexible airbag, the axial stress was loaded by a material testing system MTS (Material Testing System) [27,28]. Therefore, the specimen fails at the complex stress state of triaxial compression. As shown in Figure 3, the highest average temperature of China in July was 32.7 °C . Hence, the daily maximum temperature can reach 40 °C or more. According to previous studies [29,30], the temperature of the road surface can be 20 °C higher than the atmospheric temperature in summer. Therefore, three temperatures of 40 °C (104 °F), 50 °C (122 °F ), and 60 °C (140 °F ) were chosen as testing temperatures [31]. The axial loading rate was 1 mm/min to simulate the effect of static load. In this condition, the asphalt mixture is prone to shear failure. Considering that confining pressure significantly affects the triaxial strength properties of the asphalt mixture [32,33], the confining pressure was set to 0, 0.2, 0.4, 0.6, 0.8, and 1.0 MPa, respectively, to analyze the influence of confining pressure. Before the test, the specimen was placed into the temperature control chamber for more than 6 h to ensure that the temperature of the specimen is consistent with the testing temperature. In addition, the lubricant was smeared on the upper and lower surfaces of the specimen to reduce the friction. During the process of conducting the triaxial test, the confining pressure was firstly applied through the airbag and then the axial load was applied by the MTS until the specimen fails.

Triaxial Strength at 40 • C, 50 • C, and 60 • C
The average values of three effective test results of triaxial strength are presented in Table 3. In the triaxial test, the specimens were mainly represented as shear failure, and some diagonal cracks appeared on the surface of the specimen as shown in Figure 4.

Triaxial Strength at 40 °C , 50 °C , and 60 °C
The average values of three effective test results of triaxial strength are presented in Table 3. In the triaxial test, the specimens were mainly represented as shear failure, and some diagonal cracks appeared on the surface of the specimen as shown in Figure 4.   It can be seen from those test results that the triaxial strength is highly influenced by confining pressure and temperature. The triaxial strength increases with the increase of confining pressure, and it increases with the decrease of temperature. All the test results were fitted well with a power function as following:   13. It can be seen from those test results that the triaxial strength is highly influenced by confining pressure and temperature. The triaxial strength increases with the increase of confining pressure, and it increases with the decrease of temperature. All the test results were fitted well with a power function as following: where σ 3 denotes the triaxial compressive strength, MPa, σ 1 is the confining pressure, MPa, f c is the uniaxial compressive strength, MPa. A and n are parameters to be determined by least square method.
The parameters of this model are listed in Table 4.               It can be seen from Table 4 that with the increase of temperature, the parameter A and n decrease, and the σ 3 − σ 1 curves bend down. The increase in amplitude of triaxial strength with confining pressure became larger and larger. Based on the results, a regression/prediction model of triaxial strength considering the effects of temperature and confining pressure for asphalt mixtures is shown in Equation (2).
where σ 3t denotes triaxial compressive strength at different temperatures, σ 1 is the confining pressure, f ct is the uniaxial compressive strength at different temperatures, t is the temperature. The parameters of this triaxial strength model are shown in Table 5.

Nonlinear Mohr-Coulomb Failure Criterion
Based on the triaxial results, the Mohr-Coulomb circles can be drawn, and all the tangent points can be obtained by two Mohr-Coulomb circles tangent to each other. Considering that the Mohr-Coulomb criterion is one of the most popular strength theories in highway engineering, the triaxial compressive test results are plotted in the σ − τ space to investigate the shear failure properties of asphalt mixtures, and the results are shown in Figures 8-10 (take the results of 60 • C as example). The data is fitted with linear function and power function respectively, and the nonlinear function is shown in Equation (3).
where σ represents the normal stress in the shear failure plane, τ is the shear stress in the shear failure plane, a, b and c are constants. In fact, c is cohesion strength when σ = 0. By curve fitting of the test data, all constants obtained are listed in Table 6.           As can be seen from Figures 8-10 that the linear Mohr-Coulomb criterion is not suitable to describe the shear failure behavior of asphalt mixtures, the failure envelopes of asphalt mixtures can be interpolated by a power function. With the increase of temperature, the parameters a, b, and c decrease. The failure envelopes of these asphalt mixtures bend down when the confining pressure is close to 0.6 MPa. It is clear that the failure strength of asphalt mixtures is highly dependent on the confining pressure and temperature. Moreover, this power functional Mohr-Coulomb failure envelope under the loading speed of 1 mm/min can be validated by the nonlinear failure envelope in the I 1 − √ J 2 space proposed by Wang et al. [15] (where I 1 represents the first invariant of stress tensor, J 2 denotes the second invariant of stress deviator). It is also reasonable to predict the failure envelope of asphalt mixtures in σ − τ space considering the influence of temperature and confining pressure as following: where σ represents the normal stress in the shear failure plane; τ t is the shear stress in the shear failure plane at different temperatures; t is the temperature; m 0 , n 0 , m 1 , n 1 , m 2 , and n 2 are constants. All the parameters of this model are listed in Table 7. Consider linear case, i.e., b = 1 and a = tan ϕ, where ϕ is defined as friction angle, linear Mohr-Coulomb failure criterion is given in Equation (5) [27,28]. Furthermore, the strength parameters that are cohesion strength and friction angle obtained by linear Mohr-Coulomb theory are shown in Table 8.  Table 8 and Figure 11 show that according to the linear Mohr-Coulomb theory, the cohesion strength of asphalt mixtures decreases with the increase of temperature; the friction angle slightly decreases with temperature. Both cohesion strength and friction angle of asphalt mixture do not change with confining pressure. In the Mohr-Coulomb criterion, there are two important constants included, i.e., the cohesion strength c and friction angle ϕ. The expression of friction angle ϕ for asphalt mixtures can be obtained as follows.
Appl. Sci. 2019, 9, x; doi: FOR PEER REVIEW 9 of 17 where  represents the normal stress in the shear failure plane; t  is the shear stress in the shear failure plane at different temperatures; t is the temperature; m0, n0, m1, n1, m2, and n2 are constants. All the parameters of this model are listed in Table 7.  , where  is defined as friction angle, linear Mohr-Coulomb failure criterion is given in Equation (5) [27,28]. Furthermore, the strength parameters that are cohesion strength and friction angle obtained by linear Mohr-Coulomb theory are shown in Table 8.   Table 8 and Figure 11 show that according to the linear Mohr-Coulomb theory, the cohesion strength of asphalt mixtures decreases with the increase of temperature; the friction angle slightly decreases with temperature. Both cohesion strength and friction angle of asphalt mixture do not change with confining pressure. In the Mohr-Coulomb criterion, there are two important constants included, i.e., the cohesion strength c and friction angle  . The expression of friction angle  for asphalt mixtures can be obtained as follows.
Change the partial differential to ordinary differential, and then Change the partial differential to ordinary differential, and then ϕ = arctan(abσ b−1 ) In addition, the tangent equation passing through the point (σ m , τ m ) can be derived from Equation (3) Let σ = 0, then Equation (8) gives Finally, denote τ σ=0 as C, and substitute Equation (3) into Equation (9), then one can obtain the expression for cohesion strength as where C is cohesion strength under different confining pressure, a, b, c are parameters in Equation (3). As a result, the nonlinear evolutional laws of Mohr-Coulomb strength parameters cohesion strength C and friction angle ϕ are obtained and shown in Table 9. In Figures 12-14, the dotted lines from top to bottom represent the friction angle and cohesion strength of SMA-13, AC-13, and OGFC-13 obtained by linear Mohr-Coulomb theory, respectively.
It can be seen from Figures 12-14 that the linear Mohr-Coulomb theory cannot reflect the nonlinear variation of the cohesion strength and friction angle with testing conditions. Besides this, the linear Mohr-Coulomb theory in current asphalt pavement design produces a relatively large deviation in the calculation of cohesion strength and friction angle, which makes the design of pavement materials not accurate. Therefore, it is more appropriate to use the nonlinear failure criterion in the design of the asphalt pavement materials and structures.
As can be seen from Table 9, the increase of temperature degrades the triaxial strength of asphalt mixtures. Compared with the continuous-graded AC-13 asphalt mixture and the open-graded OGFC-13 asphalt mixture, the gap-graded SMA-13 asphalt mixture exhibits better performance on the resistance to shear damage under complex stress conditions. It is mainly due to the framework dense structure of SMA-13 asphalt mixture was formed by coarse aggregates interlocked with each other and fine aggregates filled with voids. In addition, there are some fibers in the SMA-13 asphalt mixture which improves the shear strength [34,35]. Considering the overloading phenomenon that generally exists in China, the SMA-13 asphalt mixture is recommended as the upper layer material to improve the shear performance of the asphalt pavement structure. As can be seen from Table 9, the increase of temperature degrades the triaxial strength of asphalt mixtures. Compared with the continuous-graded AC-13 asphalt mixture and the open-graded OGFC-13 asphalt mixture, the gap-graded SMA-13 asphalt mixture exhibits better performance on the resistance to As can be seen from Table 9, the increase of temperature degrades the triaxial strength of asphalt mixtures. Compared with the continuous-graded AC-13 asphalt mixture and the open-graded OGFC-13 asphalt mixture, the gap-graded SMA-13 asphalt mixture exhibits better performance on the resistance to As can be seen from Table 9, the increase of temperature degrades the triaxial strength of asphalt mixtures. Compared with the continuous-graded AC-13 asphalt mixture and the open-graded OGFC-13 asphalt mixture, the gap-graded SMA-13 asphalt mixture exhibits better performance on the resistance to

Conclusions and Recommendations
Triaxial compressive tests were conducted with confining pressure levels ranges from 0 to 1 MPa at 40 • C, 50 • C, and 60 • C. The results could be applied to investigate the shear failure behavior of asphalt mixtures. The following conclusions can be drawn: (1) The failure behavior of asphalt mixtures SMA-13, AC-13, and OGFC-13 are affected significantly by the confining pressure and temperature. It was observed that the increase of temperature degrades the triaxial strength of asphalt mixtures mainly due to the decrease of cohesion strength; when the confining pressure increases, the shear failure strength of asphalt mixture increases with the increase of cohesion strength, while the friction angle of asphalt mixtures decreases.
(2) The linear Mohr-Coulomb criterion is not suitable to describe the failure behavior of asphalt mixtures since its shear failure envelope is highly affected by the testing conditions. Therefore, a nonlinear regression/prediction model of shear strength for asphalt mixtures considering the influence of temperature and confining pressure was proposed. The nonlinear evolutional laws of cohesion strength and friction angle are also proposed. It is reasonable to use the nonlinear failure criterion in the design of asphalt pavement materials and structures.
(3) Compared with the continuous-graded AC-13 asphalt mixture and the open-graded OGFC-13 asphalt mixture, the gap-graded SMA-13 asphalt mixture exhibits better performance on the resistance to shear damage under complex stress condition. Considering the overloading phenomenon that generally exists in China, the SMA-13 asphalt mixture is recommended as the upper layer material to improve the shear performance of asphalt pavement structure.
(4) Triaxial compressive tests should be done at different loading rates in the future. And further study is required for asphalt mixtures with varying bitumen-aggregate ratio, or varying amount of asphalt binder and so on in the lab to improve the shear resistance of asphalt mixtures at high-temperature conditions.