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Normalized characterization method for fatigue behavior of cement-treated aggregates based on the yield criterion
Construction and Building Materials 228 (2019) 117099
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Normalized characterization method for fatigue behavior of cement-treated aggregates based on the yield criterion Chaochao Liu, Songtao Lv ⇑, Xinghai Peng, Jianlong Zheng National Engineering Laboratory of Highway Maintenance Technology, Changsha University of Science & Technology, 410004 Hunan, PR China
h i g h l i g h t s The yield surface of cement-treated aggregates (CTA) was established. Proposed a normalized characterization method of fatigue based on the yield criterion. Fatigue behavior of CTA was characterized by the normalized characterization method.
a r t i c l e
i n f o
Article history: Received 15 August 2019 Received in revised form 17 September 2019 Accepted 26 September 2019
Keywords: Normalized characterization Fatigue behavior Cement-treated aggregate Yield criterion
a b s t r a c t The objective of this paper is to propose a normalized characterization method based on the yield criterion to eliminate the impacts of different stress models on the fatigue behaviors characterization of cement-treated aggregates. Based on the theory of octahedral shear stress strength and yield surface mode, the yield surface of cement-treated aggregates was established with four-point bending, compression, and indirect tensile strength value under different loading rates. Then, the fatigue stress and corresponding ultimate strength pffiffiffifficould be characterized by the yield surface, uniformly and uniquely. In the J2 ), the fatigue stress growth route from initial stress to ultimate strength yield surface space (I1, was identified as fatigue stress path. Based on the fatigue stress path, the fatigue stress ratios related to loading rate and stress model were calculated. With these stress ratios, the conventional S-N fatigue equation was modified, on the basis of which a normalized characterization method of fatigue behavior was proposed to characterize the fatigue behavior of cement-treated aggregates. Compared with the traditional characterized method, the normalized characterization method has considered the influence of loading rate and stress model on strength and fatigue test results. The test results demonstrated that different strength and fatigue test results were obtained under distinct stress model and loading rates. All strength values have been expressed indiscriminately in the yield surface. With the normalized characterization method, a unique fatigue curve was gained for cement-treated aggregates under different loading rates and stress models. Therefore, findings of this work realized the normalization characterization of fatigue behavior of cement-treated aggregates. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Semi-rigid pavement has been one of the dominant asphalt pavement structures, taking the advantages of great integrity, the outstanding capacity of dispersing upper load, and strong bearing capacity. Cement-treated aggregate is one of high-quality semirigid base material, which composed of low-portion cement, gradation aggregates, and water. Fatigue cracks of cement-treated aggregates, which resulted in reflection cracks in the surface layer, has been clarified as the main distress of semi-rigid pavement. The fatigue behaviors of cement-treated aggregates dominants the ⇑ Corresponding author. E-mail address: [email protected] (S. Lv). https://doi.org/10.1016/j.conbuildmat.2019.117099 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.
service life and quality of the semi-rigid pavement. Multiple pieces of researches have been conducted by scholars around the world to improve the anti-fatigue capacity of cement-treated aggregates. Also, many methods were proposed to characterize the fatigue behavior and obtain precise anti-fatigue design parameters. Zheng et al. performed a series of laboratory researches to improve the mechanical performance of cement-stabilized macadam by adding additives, such as basalt fibers [1], polypropylene [2,3], and metallic fibers [4]. Wang et al. conducted some tests to reveal the influences of moisture and cement content, aggregate type, compaction, and curing conditions on fatigue performance, and optimize the procedure of preparation of cement-treated aggregates [5–7]. Generally, the mechanical properties of cement-treated aggregates are studied by laboratory or field
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investigations. Fatigue behaviors are usually characterized by the results of compressive, four points bending, tensile, and indirect tensile fatigue test in stress or strain control modes [8–11]. However, it has been discovered that many factors have a considerable impact on the fatigue test results, such as shapes and sizes of sample, loading conditions, test methods, and stress states [12,13]. Under different factors, the fatigue test results present huge distinction, even magnitude difference, which raised a considerable controversy on which test method could obtain an effective evaluation of fatigue performance. Therefore, it is difficult to evaluate the fatigue performance of cement-treated aggregates and obtain accurate design parameters for material and pavement structures. Two circumstances may pay responsibility for the different mechanical performance of cement-treated aggregates occurred under different factors. Firstly, the internal stress and strain of samples varied with test methods and conditions. Existing fatigue models rarely take this influence into account when applied to establish the relationship of fatigue life with stress and strain [14–17]. Another circumstance is that fixed test conditions were set for different test methods. Taking loading rates as an example, when S-N fatigue equation was utilized to characterize fatigue performance, the strength values based on which the stress ratio of fatigue equation calculated are measured under the loading rate of 5 mm/min [18–20]. This loading rate is specified in most countries’ specification to perform different stress states strength test. However, samples subjected loads with the same loading rates under different stress states would generate stress varied under different rates [21]. It implies that the stress ratios calculated by the standard strength value are not comparable for different stress states. With this regard, some efforts have been made to eliminate the influences of different factors on internal stress and strain. Yulij et al. conducted uniaxial and biaxial compression strength test, and identified an appearance of ‘‘critical states” during the accumulation of internal damages which depending on the strain mode. A determination method of the ultimate strength of concrete based on the ‘‘critical states” was suggested [22]. Lv et al. proposed a theory of real stress ratio to illustrate the fatigue behavior of cementtreated aggregates [12,23–25]. The real stress ratio is a corrected stress ratio based on the variation patterns of strength with loading rates. The theory of real stress ratio has revealed the influence of loading rates on the mechanical performance of cement-treated aggregates. With these efforts, the determination standard of ultimate strength was unified. These researches have proposed criterions and methods to determine ultimate strength value at the critical state corresponding to different test conditions. It has been proved by the octahedral shear stress strength theory that when the failure occurred, the stress state of any random point within the object could be characterized uniquely by one point in the stress space [26,27]. Based on the octahedral shear stress strength theory, Desai et al. proposed a kind of response pffiffiffiffi function of yield surface (I1, J 2 ) which could be employed to characterize the mechanical response of geotechnical material and asphalt mixtures under any stress states [28,29]. With the yield surface, different stress caused by different loads under different condition could be characterized uniformly and uniquely with pffiffiffiffi the first stress tensor invariant I1 and shear strength J 2 . pffiffiffiffi Overall, in this paper, a set of yield surface (I1, J 2 ) under different loading conditions were introduced to express different stress and obtain parameters of fatigue equation under corresponding test conditions. Unconfined compressive, indirect tensile, and four-point bending strength tests were performed under different loading rates on cement-treated aggregates. With these test results, the yield surface of cement-treated aggregates
was established based on yield criterion. Different fatigue stress and corresponding ultimate strength were expressed in the yield pffiffiffiffi surface (I1, J 2 ). A unified determination of fatigue stress path was determined based on the stress growth route from fatigue stress to corresponding ultimate strength. Then, a normalized characterization method of fatigue behavior was proposed, which eliminate the interference of the impacts of stress modes and improve the precision of anti-fatigue design parameters of semi-rigid pavement. 2. Methodology In this paper, a characterizing method of fatigue behavior based on yield surface and S-N fatigue equation was proposed for cement-treated aggregates. With this method, different fatigue stress could be expressed in a unified system, and the impacts of stress states on fatigue behavior could be eliminated. The deducing procedures of the method were presented as follows. Fatigue is a cumulative damage process of material under repeated stress at a level less than its ultimate strength, during which the micro-cracks propagated and the bearing surface A reduced continuously. The ultimate load that materials could bear decreases due to the reduction of the actual bearing surface A0 [30]. It could be characterized by a continuous variable, as Eq. (1).
W ¼ A0 =A
ð1Þ 0
where, W is a variable to characterize the fatigue process; A is the residue bearing surface; A is initial bearing surface. The bearing capacity of materials reduces gradually with the propagation of micro-cracks. The fatigue dynamic strength Smax is defined as the ultimate strength of material after a certain cycle of fatigue loading. With the reduction of bearing surface, the internal stress of material caused by the constant fatigue loads increases. The critical stress rNf 1 is specified as the stress level caused by the constant fatigue loads at the loading cycle of N f 1. At the stress peak of this loading cycle, the critical stress rNf 1 approaches the fatigue dynamic strength Smax, expressed as the Eq. (2).
rNf 1 ¼ Smax The critical stress
rNf 1 ¼
ð2Þ
rNf 1 could be computed by Eq. (3).
F tSA ¼ 0 A0Nf 1 ANf 1
ð3Þ
where, t is stress ratio, and other symbols were mentioned above. According to the fatigue equation established by the traditional phenomenological method, Eq. (4) could be derived.
A0Nf 1 A
¼ 1 D Nf 1 ¼
1 Nf
1n
ð4Þ
Then, Eq. (3) could be expressed as Eq. (5).
rNf 1 ¼
tS tS ¼ 1 1 D Nf 1 1=N n
ð5Þ
f
With Eqs. (2) and (5), the relationship of fatigue dynamic strength Smax with fatigue test stress ratio t, strength S and fatigue life Nf of material could be established as Eq. (6).
1 Smax ¼ rNf 1 ¼ t S Nf n
ð6Þ
1 Assume that m ¼ t N f n , then,
Smax ¼ m S
ð7Þ
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The m is defined as the correction factor of fatigue dynamic strength to experimentally determined strength value. The 1 m ¼ t N f n could be transformed to Eq. (8).
Nf ¼ mn
n 1 t
ð8Þ
Suppose that k ¼ mn , then,
Nf ¼ k
n 1 t
ð9Þ
Eq. (9) is the widely used S-N fatigue equation. It could be observed that the functional relationship of stress ratio t and fatigue life Nf is linear when regression analyzed in double logarithmic coordinates. The fatigue properties of materials could be reflected by the parameters of S-N fatigue equation, k, and n. The value of n is adopted to evaluate the sensitivity of fatigue life to the stress level, with the higher value of n representing higher sensitivity. The k reflects the fatigue life of cement-treated aggregates under the stress ratio of 100%, and the larger of k, the higher the position of fatigue curve, which means the better anti-fatigue performance of the material. Currently, the static strength of cement-treated aggregates was measured in the laboratory under fixed loading rates to determine the stress level and calculate the stress ratio for fatigue test. However, it is obvious that the loading rates for a series of fatigue test are varied with the stress level while the loading frequency fixed. The loading rates could be calculated approximately by the Eq. (10) when the sine waveform loading was applied in the fatigue test.
v ¼ 2f r
ð10Þ
where, v is the loading rate; f is the loading frequency, and r is the stress level. Moreover, it has been proved that the loading rates have a significant influence on the mechanic properties of cement-treated aggregates, including strength, modulus, and fatigue [12]. Thought, it is irrational to calculate the stress ratio with the static strength measured under fixed loading rates. It could result in an apparent error in the fatigue curve, even the strength value of cementtreated aggregates measured precisely. In order to correct the stress ratio mentioned above, Lv et al. established the variation patterns of strength with loading rates by conducting strength tests under different loading rates and proposed the theory of real stress ratio [12]. Then, the S-N fatigue equation modified by the real stress ratio t0 , shown as Eq. (11).
Nf ¼ k
n 1 t0
ð11Þ
where t0 is the stress ratio related to loading rates, named real stress ratio. The influence of loading rates on materials mechanic performance has been illustrated while the Eq. (11) was utilized to analyze the fatigue results. The fatigue curves could extend to the strength failure point (1, 1) and the value of k is about 1, which indicates that when the stress level approaches ultimate strength value the fatigue life would be about 1cycle. It is consistent with the actual situation and built a bridge between the fatigue failure and failure strength. It has been proved by the octahedral shear stress strength theory that when the failure occurred, the stress state of any random point within the object could be characterized uniquely by one point in the stress space [28]. All corresponding point of stress space forms a continuous surface which named as yield surface or damage envelope surface. The strength value of material under
Fig. 1. Yield surface model.
any stress states could be determined through its yield surface, as Fig. 1 presents. Combining the theory of yield surface and the S-N fatigue equation modified by real stress ratio, fatigue behaviors of material under any stress states could be characterized. Desai et al. [28] proposed a kind of response function of yield surface which could be employed to characterize the mechanical response of geotechnical material and asphalt mixtures under any stress states. In stress invariant space, the yield surface could be expressed as Eq. (12).
n 2
I1 R a I1PR þ c Pn n J2 u rij ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 ½1 bcosð3hÞ Pn pffiffi where cosð3hÞ ¼ 3 2 3
J3
3
ðJ 2 Þ2
ð12Þ
; 0 6 h 6 p3, h is Cape Roca; J 2 , J3 is the sec-
ond and third deviatoricstress invariants, respectively; Pa is atmospheric pressure; I1 is the first stress tensor invariant; and a, b, c, n, Rare the model parameters. pffiffiffiffi In the space of I1 ; J 2 ; h , the Eq. (12) expressed a closed yield surface, and the model parameter a and b are 0 at peak stress under the unidirectional stress state, so the yield surface will degenerate into a straight line in space [28], and the expression is as Eq. (13): J2 P2n
2 ¼ c I1PR n
! J 2 ¼ cðI1 RÞ2 pffiffiffiffi pffiffiffi ! J 2 ¼ cjI1 Rj
ð13Þ
pffiffiffiffi In (I1, J 2 ) space, the yield surface represented by Eq. (13) is a straight line. The expression on the left part of the upper form is the criterion of the Mises yield condition, and its physical meaning pffiffiffiffi is the shear stress intensity T ¼ J 2 . According to the Desai yield surface model, stress would only occur in one direction. The yield surface in stress invariant space could be easily established with the tensile and compressive strength. In stress invariant space, the coordinates of tensile f c qffiffiffiffiffiffiffi 2 and compressive strength f t failure points are f c0 13 f c and qffiffiffiffiffiffiffi 2 f t0 13 f t , respectively.
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C. Liu et al. / Construction and Building Materials 228 (2019) 117099
In this paper, the Desai yield surface of cement-treated aggregates was established by conducting the strength test on cement-treated aggregates under different stress states, including unconfined compressive, indirect tensile, and four-point bending strength tests. With the yield surface, the ultimate strength, fatigue stress, and stress ratio of fatigue test under different stress states could be obtained in a uniform system. 3. Materials and test design 3.1. Material and mix design 3.1.1. Cement The main components of cement-treated aggregates are cement, graded aggregates, and water. Aggregates are structured by cement hydration products and thus forming strength. The performance of cement has a significant influence on the quality of cement-treated aggregates. The performance indexes of cement, including fineness, setting time, soundness, strength, etc., had been tested, and the test results were displayed in Table 1. 3.1.2. Aggregate and gradation Limestone was adopted as coarse and fine aggregates. The mechanical and physical properties of aggregates were tested according to the Chinese Test Methods of Aggregate for Highway Engineering JTG B42-2005. The test results were displayed in Table 2. The gradation of cement-treated aggregated for this study was determined through the Chinese Construction Specifications for Highway Road Bases (JTJ/T F20-2015) [31], as Fig. 2 showed. 3.1.3. Cement-treated aggregates mix design Previous researches have proved that the cement-treated aggregates produced through the procedure of vibration mixing and compaction are more similar to that of road engineering [32]. Thus, in this research, cement-treated aggregates mixed by vibration mixer, and the vibration compaction test was utilized to determine the optimum moisture content and maximum dry density of cement-treated aggregates. The test was performed according to the standard method of Chinese standard JTG E51-2009. The parameters for the machine was set as Table 3. According to the test results, the optimum moisture and cement content was 4.8% and 4.5%, respectively. And under this mixing proportion, the maximum dry density of mixtures was 2.43 g/cm3. 3.2. Specimens preparation Based on the parameters of design obtained by the works of Section 3.2, the specimens of cement-treated aggregates were manufactured. The samples with the length of 550 mm, the width of 150 mm, and the thickness of 150 mm were utilized for fourpoint bending strength and fatigue tests. The samples with the sizes of U150mm 150 mm were adopted to measure the strength values and fatigue life of cement-treated aggregates under unconfined compressive and indirect tensile stress states. The preparation of sample including mixing, forming in special models,
demolding, and curing at the environment with a temperature of 20 °C and the air humidity over 95%. Each type of tests was conducted with four replicates. 3.3. Experiment design For the purpose of establishing the yield surface of cementtreated aggregates to obtain its failure parameters in a unified model, the strength test was designed and conducted in this paper. The strengths of cement-treated aggregates under general stress states, including unconfined compressive, indirect tensile and four-point bending, were determined under different loading rates. The loading rates are 5 MPa/s, 10 MPa/s, 20 MPa/s, 30 MPa/s, 40 MPa/s, and 50 MPa/s. The strength tests were conducted by Material tests system under the test temperature of 15 °C according to ‘‘test methods of materials stabilized with inorganic binders for highway engineering JTG E51-2009”. A series of fatigue tests were performed on cement-treated aggregates under diverse stress states, including unconfined compressive, indirect tensile and four-point bending. Firstly, the standard strength S of cement-treated aggregates under unconfined compressive, indirect tensile, and four-point bending stress states were measured under the fixed loading rate of 5 mm/minwhich prescribed in JTG E51-2009. Then, based on these standard strength values, the stress levels for the fatigue test were determined as 0.2S, 0.3S, 0.4S, 0.5S, 0.6S, and 0.7S. When the speeds of vehicle are about 60–80 km/s, the loading frequency is nearly 10 Hz. Therefore, in this paper, the fatigue tests were carried out by half-sine-wave loads with loading frequency of 10 Hz. 4. Development of yield surface model for cement-treated aggregates 4.1. Strength properties of cement-treated aggregates under different stress states The strength of cement-treated aggregates under different stress model and loading rates, including the standard strength measured at a fixed loading rate which prescribed in JTG E512009, were measured and illustrated in Fig. 3. The standard strength of cement-treated aggregates under unconfined compressive, indirect tensile and four-point bending stress states were 11.76 MPa, 1.298 MPa, and 1.309 MPa, respectively. It is apparent that the standard strength values distinguish
Table 2 Test results of mechanical and physical properties of aggregates. Items
Test result
Technical requirement
Less than 0.6 mm particle Liquid limit/plastic index Content of flat and elongated particles in coarse aggregate (%) Content of soft stone (%) Crushed stone value (%)
23.5% 3.2 12.1%
liquid limit 28% plastic index 9 20%
0.9 20.3%
3 30%
Table 1 Main technical properties of cement. Items Fineness (the size of the sieve is 80 lm) Setting time (min) Soundness test (mm) Strength of cement mortar (MPa)
2.1 Initial setting (min) Final setting (min) 3 days flexural strength 3 days compressive strength
Test result
Specification
10% 275 408 3 4.7 24.9
180 360 5 2.5 10
C. Liu et al. / Construction and Building Materials 228 (2019) 117099
5
Fig. 2. Gradation curves of cement-treated aggregates.
Table 3 The parameters of machine set for the vibration compaction test. Frequency (Hz)
Excited force (kN)
Vibration amplitude (mm)
Vibration time (s)
30 ± 2
7.6
1.3 ± 0.05
90 ~ 120
20 18
Strength values (MPa)
16
S c=9.51×v
0.19
2 R =0.91
0.30 2 Sf=0.78× v R =0.89
14
Moreover, when the loading rates increased from 5 MPa/s to 10 MPa/s, the unconfined compressive strength increased by 15%, the indirect tensile strength increased by 20% and the four-point bending strength increased about two times. It denoted that the loading rate has considerable effects on the strength of cementtreated aggregates. Given regards of these two circumstances, the difference of loading rate for different test methods under standard test condition and considerable effects of loading rates on the strength, there is an obvious error in the determination of stress ratio of fatigue test. For precise characterization of fatigue behavior of cementtreated aggregates, it is essential to establish a unified failure strength model for different stress states based on the failure mechanism.
0.08 2 Sit=1.16× v R =0.93
12
4.2. Strength yield surface of cement-treated aggregates
Standard trength values under different test methods
2
0
10
20
30
40
50
60
70
Rates of loading (MPa/s) Fig. 3. Strength properties of CTA under different stress states and loading rates.
with the strength value determined under other loading rates. The strength of cement-treated aggregates under all stress states exhibited a power function growth trend with loading rates, and the fitting results are shown in Fig. 3. Based on the fitting equation and the standard strength values, the fixed loading rate 5 mm/min which is strain control mode could be transformed to the corresponding loading rates of stress control mode, and they were about 3.5 MPa/s, 2 MPa/s and 5.8 MPa/s, shown as the dotted lines in Fig. 3. During the standard strength test, specimens of cementtreated aggregates subjected stress with different loading rates under different stress state even though the strain growth rates were all 5 mm/min. That is because of the difference of modulus under different stress states, which has been proved in many previous pieces researches [12].
According to the yield surface proposed by Desai which introduced in Section 2, the coordinates of yield surface under unconfined compressive, indirect tensile and four-point bending stress states could be expressed with the expression in Table 4. Bringing the strength values of cement-treated aggregates measured previously into the corresponding expression in Table 4, some coordinates of points of the yield surface could be calculated. And the calculation results are displayed in Table 5. pffiffiffiffi In (I1, J 2 ) space, the yield surface of a single stress state is a line, and these lines would intersect at a peak point. Thus, the yield surface of cement-treated aggregates unconfined compressive, indirect tensile, and four-point bending stress states could be represented with three independent lines, as presented in Fig. 4. According to Fig. 4, the solid lines are the yield surface, which is a line under a certain loading rate. It could be observed that the yield surface of cement-treated aggregates diverges outwardly. The point in the yield surface represents a failure point of cement-treated aggregates at a single stress state. Moreover, the failure point at the same stress state could be connected into a straight line which could stretch to the coordinate origin, as the dotted line shown in Fig. 4. Between the yield surface and the coordinate origin, the point of this space means the stress that lower than the failure strength. For example, point A in Fig. 4 means stress at compressive stress state, which could not cause failure to cement-treated aggregates with one loading, based on which the fatigue stress is defined.
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C. Liu et al. / Construction and Building Materials 228 (2019) 117099
Table 4 Expressions of yield surface coordinates under different stress states. Stress state
Strength
r1
r2
r3
I1
Indirect tensile
RT
RT
0
3 RT
2 RT
Unconfined compressive
RC
0
0
RC
RC
Bending
RS
RS
0
0
RS
pffiffiffiffi J2 pffiffiffiffi 39 3 RT pffiffi 3 3 RC pffiffi 3 3 RS
Cape Roca (h) h ¼ arccos p5ffiffiffiffi ¼ 46 52 60° 0°
Table 5 Yield surface coordinates of cement-treated aggregates under different stress states. Stress states
StrengthR (MPa)
I1 (MPa)
pffiffiffiffi J2 (MPa)
Cape Roca (h)
5
Unconfined compressive Indirect tensile Bending
12.91 1.35 1.27
12.91 2.69 1.27
7.45 2.80 0.73
60° 46° 0°
10
Unconfined compressive Indirect tensile Bending
14.78 1.39 1.52
14.78 2.78 1.52
8.53 2.89 0.88
60° 46° 0°
20
Unconfined compressive Indirect tensile Bending
16.75 1.46 1.95
16.75 2.91 1.95
9.67 3.03 1.13
60° 46° 0°
30
Unconfined compressive Indirect tensile Bending
18.13 1.49 2.23
18.13 2.97 2.23
10.47 3.19 1.29
60° 46° 0°
40
Unconfined compressive Indirect tensile Bending
18.69 1.59 2.43
18.69 3.17 2.43
10.79 3.29 1.40
60° 46° 0°
50
Unconfined compressive Indirect tensile Bending
19.51 1.61 2.52
19.51 3.21 2.52
11.26 3.34 1.45
60° 46° 0°
Loading rates
v (MPa/s)
5 MPa/s 1 0MPa/s 2 0MPa/s 3 0MPa/s 4 0MPa/s 5 0MPa/s
stress
11
sion (14).
qffiffiffiffi P J 02 ¼ a A
J 2 (MPa) 9
7 compressive 3Bending A 1
-15
-10
-5
0
ð14Þ
With the increase of loading cycles, the micro-cracks propagated and the bearing surface reduced to A0 . Consequently, the repeated pffiffiffiffi stress increased, approaching to the ultimate strength J2 . The qffiffiffiffi pffiffiffiffi course of repeated stress increase from J 02 to J2 is defined as
Indirect tensile
-20
qffiffiffiffi J 02 due to the certain fatigue loads P is equal to the expres-
5
I1(MPa) Fig. 4. Yield surfaces of cement-treated aggregates under diverseloading rates.
5. Characterizing method of fatigue behavior based on the yield surface 5.1. 5.1Determination of fatigue stress path As mentioned in Section 2, fatigue is a cumulative damage process of material under repeated stress at a level less than its ultimate strength, during which the micro-cracks propagated and pffiffiffiffi the bearing surface A reduced continuously. In (I1, J 2 ) space, at begin of fatigue process, assuming that the specimen is complete and its bearing surface of cement-treated aggregates is A, the shear
the fatigue stress path. The standard strength of cement-treated aggregates under unconfined compressive, indirect tensile and four-point bending stress states were 11.76 MPa, 1.298 MPa, and 1.309 MPa, respectively. Based on these standard strength values, the stress levels for the fatigue test were determined as 0.2S, 0.3S, 0.4S, 0.5S, 0.6S, and 0.7S. The loading frequency is 10 Hz, and then the equivalent stress variation rates could be calculated by the expression (10). And the corresponding strength to stress variation rates, i.e., loading rates, could be obtained with the various patterns of strength with loading rates which established in Section 4.1. Based on these pffiffiffiffi strength values, the corresponding I1, ultimate shear strength J 2 and the shear stress caused by each stress level could be calculated by the expression in Table 4. All calculation results are presented in pffiffiffiffi Table 6 and (I1, J 2 ) space as Fig. 5. As Fig. 5 presents, the ultimate strength points are determined by the stress variation rates. The initial point of each stress levels and stress states corresponds to ultimate strength one by one. The path from the initial point to the ultimate point is fatigue stress path, and the length between the initial point and the ultimate point indicates the anti-fatigue capacities of cement-treated aggregates at a certain loading condition. Taking the unconfined compressive fatigue test at the stress level of 20% of strength as an example, the coordinates of pffiffiffiffi repeated stress in space (I1, J 2 ) is point A in Fig. 5, and the
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C. Liu et al. / Construction and Building Materials 228 (2019) 117099 Table 6 Initial and ultimate point of fatigue test. I1 occurred by the strength corresponding to stress rates (MPa)
pffiffiffiffi J2 occurred by the strength corresponding to stress rates (MPa)
1.36 2.04 2.71 3.39 4.07 4.75
19.77 21.35 22.55 23.53 24.36 25.08
11.41 12.32 13.01 13.57 14.05 14.47
0.52 0.78 1.04 1.29 1.56 1.82
0.54 0.81 1.08 1.35 1.62 1.89
2.65 2.73 2.79 2.85 2.89 2.93
2.75 2.84 2.91 2.96 3.01 3.04
0.27 0.39 0.52 0.65 0.79 0.92
0.15 0.23 0.30 0.38 0.45 0.53
1.28 1.45 1.58 1.69 1.78 1.87
0.74 0.84 0.91 0.97 1.03 1.08
Stress states & the standard strength
Normal stress ratio
Stress level (MPa)
equivalent stress variation ratesv (MPa/s)
I10 occurred under the stress level (MPa)
Unconfined compressive 11.76 MPa
0.2 0.3 0.4 0.5 0.6 0.7
2.35 3.53 4.70 5.88 7.06 8.23
47.04 70.56 94.08 117.60 141.12 164.64
2.35 3.53 4.70 5.88 7.06 8.23
Indirect tensile 1.298 MPa
0.2 0.3 0.4 0.5 0.6 0.7
0.26 0.39 0.52 0.65 0.78 0.91
5.19 7.79 10.38 12.98 15.58 18.17
Four-point bending 1.309 MPa
0.2 0.3 0.4 0.5 0.6 0.7
0.26 0.39 0.52 0.65 0.79 0.92
5.24 7.85 10.47 13.09 15.71 18.33
Fig. 5. Fatigue stress path under different stress states and stress levels.
corresponding ultimate point is point B. From point A to point B, the first stress tensor invariant I1increased from 2.352 MPa to pffiffiffiffi 19.77 MPa, and the shear stress intensity J 2 grew from 1.36 MPa to 11.41 MPa. The growth of the first stress tensor invariant I1 and pffiffiffiffi shear stress intensity J 2 is due to the accumulation of internal micro-cracks of cement-treated aggregates under repetition stress. Therefore, the anti-fatigue capacities of cement-treated aggregates could be evaluated by the length of line AB which could be calculated by the coordinates of point A and B. Also, the growth rates of pffiffiffiffi the first stress tensor invariant I1 and shear stress intensity J 2 could be adopted to evaluate the anti-fatigue capacities.
qffiffiffiffi J 02 occurred under the stress level (MPa)
could not reveal the failure strength features during the fatigue process. The prediction value of fatigue life under the stress ratio of 100% is not one cycle. The reason for this circumstance is that the loading rates vary with stress levels and the strength value has been affected by loading rates. Nevertheless, the stress ratio used in traditional S-N fatigue equation is generally obtained with standard strength value, which measured under a fixed loading rate, without correcting based on the impact of loading rates. Moreover, different specimens occur altered stress under different test method, resulting in different test results. Different stress states could be characterized by the first stress pffiffiffiffi tensor invariant I1 and shear stress intensity J 2 based on octahedron theory and yield surface mechanics model, as Section 5.1 expounded. With the yield surface, the difference of internal stress caused by the impacts of load modes, specimen types and loading rates could be unified. And the fatigue stress path could be obtained based on which a new characterizing method for fatigue behavior could be proposed. As the length between initial point A and ultimate point B could be employed to evaluate the anti-fatigue capacities of cementtreated aggregates, the length ratio of line OA to OB could be utilized to substitute the stress ratio of S-N fatigue equation. According to Fig. 5, based on the similar triangle theory, the stress ratio could be expressed by the expression (15).
D¼
OA OA0 C 0 C ¼ ¼ OB OC 0 C 0 B 0
OA 0 where OC 0 is the ratio of initial first stress tensor invariant I1 to ultiqffiffiffiffi 0 mate I1, CC 0 CB is the ratio of initial shear stress intensity J02 to ultipffiffiffiffi mate J2 . Then, the traditional S-N fatigue equation could be modified as Eq. (16).
Nf ¼ k 5.2. A normalized characterization methods of fatigue behavior based on the yield surface
ð15Þ
n 1 D
ð16Þ
5.3. Fatigue behavior characterized by modified S-N equation Usually, the S-N fatigue equation, which reflects the relationship of fatigue life and stress, were utilized to characterize fatigue behaviors of road engineering materials. However, it has been proved in many researches that traditional S-N fatigue equation
qffiffiffiffi In this paper, the ratio of initial shear stress intensity J 02 to pffiffiffiffi ultimate J 2 was taken as the stress ratio of fatigue tests. The
8
C. Liu et al. / Construction and Building Materials 228 (2019) 117099
results of fatigue test, including unconfined compressive, indirect tensile and four-point bending fatigue tests, were analyzed with both traditional S-N fatigue Eq. (9) and modified S-N fatigue Eq. (16), and the results presented as Figs. 6 and 7. It is apparent from Fig. 6 that the fatigue test results under different stress states fitted into typically distinctive and individual fatigue curves when traditional S-N fatigue equation was utilized. The most and irrational circumstance is that the values of parameter k, which reflect the fatigue life of cement-treated aggregates under the stress ratio of 100 percent, are varied dramatically with stress states. The k value under compressive stress is 6193, and it is 80 under bending stress while that of indirect tensile stress is 3.8. Technically, the fatigue life of cement-treated aggregates under the stress ratio of 100 percent is 1 time. Therefore, it is irrational to obtain the anti-fatigue parameter of pavement design by stretch the fatigue curve to point (1,1), which specified in the pavement design method. In Fig. 7, the k values under different stress states are all approximately 1, which means that cement-treated aggregates specimens would be broken with one loading under stress reaching the strength value. Thus, the modified S-N fatigue equation has corrected this error, and the S-N fatigue equation could be developed as Eq. (17).
Nf ¼
n 1 D
ð17Þ
Moreover, the parameter n in fatigue equation is usually applied to evaluate the stress sensibility of fatigue behaviors, the larger the n-value, the more vulnerable of fatigue behavior to the effects of stress. Theoretically, stress sensibility is one of the material attributes, and a certain material has a certain stress sensibility. Based on conventional S-N fatigue equation, the n-values were different for the same cement-treated aggregates under different stress states. It indicates that a certain material had different stress sensibility under different test method, and this raised the considerable controversy on which test method could obtain an effective evaluation for stress sensibility of fatigue behaviors. Comparatively, with the modified S-N fatigue equation, it is found from Fig. 7 that the three fatigue curves of three stress states have very similar n values. The impact of stress states on the analysis of fatigue characteristic has been eliminated, and all fatigue test results could be illustrated with one unified fatigue curve, as shown in Fig. 8.
109
Unconfined compressive Indirect tensile Four-point bending
108
Nf=6193.28•t-7.45
107
R2 =0.93
Nf (time)
6
10
105
Fig. 7. Fatigue curves of cement-treated aggregates fitted by modified S-N fatigue equation.
Nf=80.39•t-7.37 R2 =0.95
104
Fig. 8. Normalized fatigue curve of cement-treated aggregates under different stress states.
Fig. 8 proved that a normalized relationship of fatigue life and stress level could be established for cement-treated aggregates under different stress states. With this normalized model, the kvalue has been proving to be 1, which revealed the relationship of fatigue damage and strength failure. And the fatigue behavior of cement-treated aggregates has been characterized uniquely, without the interference of specimen type, test method, and test conditions. Uniform characterization of fatigue test results obtained by different methods has realized by the normalized characterization methods, which could avoid controversy on test methods and unify the determination of design parameters for anti-fatigue pavement.
103
6. Conclusions
Nf=3.80•t-8.96
102
2
R =0.90 101 0.2
0.3
0.4
0.5
0.6
0.7
0.8
t Fig. 6. Fatigue curves of cement-treated aggregates fitted by traditional S-N fatigue equation.
In this paper, a normalized characterization method of fatigue behavior based on yield surface was proposed to analyze the fatigue test results of cement-treated aggregates under different stress states. Compared with traditional characterization methods of fatigue behavior of cement-treated aggregates, some conclusion could be drawn as follows.
C. Liu et al. / Construction and Building Materials 228 (2019) 117099
(1) The yield surface of cement-treated aggregates was established by the strength test results under different stress states and loading rates, and different ultimate strength could be expressed uniquely by the yield surface. (2) The fatigue stress path was characterized by the yield surface, and consequently a new method to determine the stress ratio of fatigue equation was proposed. (3) The influence of various stress states on the fatigue behaviors of cement-treated aggregates was reduced or even eliminate by the normalized characterization method for fatigue behavior ofcement-treated aggregatesbased on the yield criterion. And the normalization characterization of fatigue behavior of cement-treated aggregates was realized. 7. Data availability Some or all data, models, or code generated or used during the study are available from the corresponding author by request. All data, models, and code generated or used during the study appear in the submitted article. Declaration of Competing Interest 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. Acknowledgements This work was supported by National Natural Science Foundation of China (51578081, 51608085), the Ministry of Transport Construction Projects of Science and Technology (2015318825120), the Projects of Transportation Science and Technology of Hunan (201701). References [1] Y. Zheng, P. Zhang, Y. Cai, Z. Jin, E. Moshtagh, Cracking resistance and mechanical properties of basalt fibers reinforced cement-stabilized macadam, Compos. B Eng. 165 (2019) 312–334, https://doi.org/10.1016/ j.compositesb.2018.11.115. [2] Z. Peng, L. Qingfu, Effect of polypropylene fibre on mechanical and shrinkage properties of cement-stabilised macadam, Int. J. Pavement Eng. 10 (6) (2009) 435–445, https://doi.org/10.1080/10298430802363985. [3] Z. Liu, Experimental research on the engineering characteristics of polyester fiber-reinforced cement-stabilized macadam, J. Mater. Civ. Eng. 27 (10) (2015), https://doi.org/10.1061/(asce)mt.1943-5533.0001251. [4] F.B.-B.H.A. MesbahU, Efficiency of polypropylene and metallic fibres on control of shrinkage and cracking of recycled aggregate mortars, Constr. Build. Mater. 13 (1999) 9. [5] H.-X. Guan, H.-Q. Wang, H. Liu, J.-J. Yan, M. Lin, The effect of intermediate principal stress on compressive strength of different cement content of cement-stabilized macadam and different gradation of AC-13 mixture, Appl. Sci. 8 (10) (2018), https://doi.org/10.3390/app8102000. [6] T.-L. Wang, H.-H. Wang, H.-F. Song, Z.-R. Yue, Z.-H. Guo, Effects of cement content and grain-size composition on engineering properties of high-speedrailway macadam subgrade, Cold Reg. Sci. Technol. 145 (2018) 21–31, https:// doi.org/10.1016/j.coldregions.2017.09.009. [7] J. Wang, Q. Dai, S. Guo, R. Si, Mechanical and durability performance evaluation of crumb rubber-modified epoxy polymer concrete overlays, Constr. Build. Mater. 203 (2019) 469–480, https://doi.org/10.1016/ j.conbuildmat.2019.01.085. [8] J.Z. Songtao Lv, Wenliang Zhong, Characteristies of strength, modulus and fatigue damage for cement stabilized macadam in curing period, China J. Highway Transp. 28 (9) (2015) 8, https://doi.org/10.19721/j.cnki.10017372.2015.09.002. [9] J. Jin, Y. Tan, R. Liu, J. Zheng, J. Zhang, Synergy Effect of Attapulgite, Rubber, and Diatomite on Organic Montmorillonite-Modified Asphalt, J Mater Civil Eng. 31 (2) (2019) 04018388, https://doi.org/10.1061/(ASCE)MT.1943-5533.0002601.
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Normalized characterization method for fatigue behavior of cement-treated aggregates based on the yield criterion Chaochao Liu, Songtao Lv ⇑, Xinghai Peng, Jianlong Zheng National Engineering Laboratory of Highway Maintenance Technology, Changsha University of Science & Technology, 410004 Hunan, PR China
h i g h l i g h t s The yield surface of cement-treated aggregates (CTA) was established. Proposed a normalized characterization method of fatigue based on the yield criterion. Fatigue behavior of CTA was characterized by the normalized characterization method.
a r t i c l e
i n f o
Article history: Received 15 August 2019 Received in revised form 17 September 2019 Accepted 26 September 2019
Keywords: Normalized characterization Fatigue behavior Cement-treated aggregate Yield criterion
a b s t r a c t The objective of this paper is to propose a normalized characterization method based on the yield criterion to eliminate the impacts of different stress models on the fatigue behaviors characterization of cement-treated aggregates. Based on the theory of octahedral shear stress strength and yield surface mode, the yield surface of cement-treated aggregates was established with four-point bending, compression, and indirect tensile strength value under different loading rates. Then, the fatigue stress and corresponding ultimate strength pffiffiffifficould be characterized by the yield surface, uniformly and uniquely. In the J2 ), the fatigue stress growth route from initial stress to ultimate strength yield surface space (I1, was identified as fatigue stress path. Based on the fatigue stress path, the fatigue stress ratios related to loading rate and stress model were calculated. With these stress ratios, the conventional S-N fatigue equation was modified, on the basis of which a normalized characterization method of fatigue behavior was proposed to characterize the fatigue behavior of cement-treated aggregates. Compared with the traditional characterized method, the normalized characterization method has considered the influence of loading rate and stress model on strength and fatigue test results. The test results demonstrated that different strength and fatigue test results were obtained under distinct stress model and loading rates. All strength values have been expressed indiscriminately in the yield surface. With the normalized characterization method, a unique fatigue curve was gained for cement-treated aggregates under different loading rates and stress models. Therefore, findings of this work realized the normalization characterization of fatigue behavior of cement-treated aggregates. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Semi-rigid pavement has been one of the dominant asphalt pavement structures, taking the advantages of great integrity, the outstanding capacity of dispersing upper load, and strong bearing capacity. Cement-treated aggregate is one of high-quality semirigid base material, which composed of low-portion cement, gradation aggregates, and water. Fatigue cracks of cement-treated aggregates, which resulted in reflection cracks in the surface layer, has been clarified as the main distress of semi-rigid pavement. The fatigue behaviors of cement-treated aggregates dominants the ⇑ Corresponding author. E-mail address: [email protected] (S. Lv). https://doi.org/10.1016/j.conbuildmat.2019.117099 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.
service life and quality of the semi-rigid pavement. Multiple pieces of researches have been conducted by scholars around the world to improve the anti-fatigue capacity of cement-treated aggregates. Also, many methods were proposed to characterize the fatigue behavior and obtain precise anti-fatigue design parameters. Zheng et al. performed a series of laboratory researches to improve the mechanical performance of cement-stabilized macadam by adding additives, such as basalt fibers [1], polypropylene [2,3], and metallic fibers [4]. Wang et al. conducted some tests to reveal the influences of moisture and cement content, aggregate type, compaction, and curing conditions on fatigue performance, and optimize the procedure of preparation of cement-treated aggregates [5–7]. Generally, the mechanical properties of cement-treated aggregates are studied by laboratory or field
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investigations. Fatigue behaviors are usually characterized by the results of compressive, four points bending, tensile, and indirect tensile fatigue test in stress or strain control modes [8–11]. However, it has been discovered that many factors have a considerable impact on the fatigue test results, such as shapes and sizes of sample, loading conditions, test methods, and stress states [12,13]. Under different factors, the fatigue test results present huge distinction, even magnitude difference, which raised a considerable controversy on which test method could obtain an effective evaluation of fatigue performance. Therefore, it is difficult to evaluate the fatigue performance of cement-treated aggregates and obtain accurate design parameters for material and pavement structures. Two circumstances may pay responsibility for the different mechanical performance of cement-treated aggregates occurred under different factors. Firstly, the internal stress and strain of samples varied with test methods and conditions. Existing fatigue models rarely take this influence into account when applied to establish the relationship of fatigue life with stress and strain [14–17]. Another circumstance is that fixed test conditions were set for different test methods. Taking loading rates as an example, when S-N fatigue equation was utilized to characterize fatigue performance, the strength values based on which the stress ratio of fatigue equation calculated are measured under the loading rate of 5 mm/min [18–20]. This loading rate is specified in most countries’ specification to perform different stress states strength test. However, samples subjected loads with the same loading rates under different stress states would generate stress varied under different rates [21]. It implies that the stress ratios calculated by the standard strength value are not comparable for different stress states. With this regard, some efforts have been made to eliminate the influences of different factors on internal stress and strain. Yulij et al. conducted uniaxial and biaxial compression strength test, and identified an appearance of ‘‘critical states” during the accumulation of internal damages which depending on the strain mode. A determination method of the ultimate strength of concrete based on the ‘‘critical states” was suggested [22]. Lv et al. proposed a theory of real stress ratio to illustrate the fatigue behavior of cementtreated aggregates [12,23–25]. The real stress ratio is a corrected stress ratio based on the variation patterns of strength with loading rates. The theory of real stress ratio has revealed the influence of loading rates on the mechanical performance of cement-treated aggregates. With these efforts, the determination standard of ultimate strength was unified. These researches have proposed criterions and methods to determine ultimate strength value at the critical state corresponding to different test conditions. It has been proved by the octahedral shear stress strength theory that when the failure occurred, the stress state of any random point within the object could be characterized uniquely by one point in the stress space [26,27]. Based on the octahedral shear stress strength theory, Desai et al. proposed a kind of response pffiffiffiffi function of yield surface (I1, J 2 ) which could be employed to characterize the mechanical response of geotechnical material and asphalt mixtures under any stress states [28,29]. With the yield surface, different stress caused by different loads under different condition could be characterized uniformly and uniquely with pffiffiffiffi the first stress tensor invariant I1 and shear strength J 2 . pffiffiffiffi Overall, in this paper, a set of yield surface (I1, J 2 ) under different loading conditions were introduced to express different stress and obtain parameters of fatigue equation under corresponding test conditions. Unconfined compressive, indirect tensile, and four-point bending strength tests were performed under different loading rates on cement-treated aggregates. With these test results, the yield surface of cement-treated aggregates
was established based on yield criterion. Different fatigue stress and corresponding ultimate strength were expressed in the yield pffiffiffiffi surface (I1, J 2 ). A unified determination of fatigue stress path was determined based on the stress growth route from fatigue stress to corresponding ultimate strength. Then, a normalized characterization method of fatigue behavior was proposed, which eliminate the interference of the impacts of stress modes and improve the precision of anti-fatigue design parameters of semi-rigid pavement. 2. Methodology In this paper, a characterizing method of fatigue behavior based on yield surface and S-N fatigue equation was proposed for cement-treated aggregates. With this method, different fatigue stress could be expressed in a unified system, and the impacts of stress states on fatigue behavior could be eliminated. The deducing procedures of the method were presented as follows. Fatigue is a cumulative damage process of material under repeated stress at a level less than its ultimate strength, during which the micro-cracks propagated and the bearing surface A reduced continuously. The ultimate load that materials could bear decreases due to the reduction of the actual bearing surface A0 [30]. It could be characterized by a continuous variable, as Eq. (1).
W ¼ A0 =A
ð1Þ 0
where, W is a variable to characterize the fatigue process; A is the residue bearing surface; A is initial bearing surface. The bearing capacity of materials reduces gradually with the propagation of micro-cracks. The fatigue dynamic strength Smax is defined as the ultimate strength of material after a certain cycle of fatigue loading. With the reduction of bearing surface, the internal stress of material caused by the constant fatigue loads increases. The critical stress rNf 1 is specified as the stress level caused by the constant fatigue loads at the loading cycle of N f 1. At the stress peak of this loading cycle, the critical stress rNf 1 approaches the fatigue dynamic strength Smax, expressed as the Eq. (2).
rNf 1 ¼ Smax The critical stress
rNf 1 ¼
ð2Þ
rNf 1 could be computed by Eq. (3).
F tSA ¼ 0 A0Nf 1 ANf 1
ð3Þ
where, t is stress ratio, and other symbols were mentioned above. According to the fatigue equation established by the traditional phenomenological method, Eq. (4) could be derived.
A0Nf 1 A
¼ 1 D Nf 1 ¼
1 Nf
1n
ð4Þ
Then, Eq. (3) could be expressed as Eq. (5).
rNf 1 ¼
tS tS ¼ 1 1 D Nf 1 1=N n
ð5Þ
f
With Eqs. (2) and (5), the relationship of fatigue dynamic strength Smax with fatigue test stress ratio t, strength S and fatigue life Nf of material could be established as Eq. (6).
1 Smax ¼ rNf 1 ¼ t S Nf n
ð6Þ
1 Assume that m ¼ t N f n , then,
Smax ¼ m S
ð7Þ
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The m is defined as the correction factor of fatigue dynamic strength to experimentally determined strength value. The 1 m ¼ t N f n could be transformed to Eq. (8).
Nf ¼ mn
n 1 t
ð8Þ
Suppose that k ¼ mn , then,
Nf ¼ k
n 1 t
ð9Þ
Eq. (9) is the widely used S-N fatigue equation. It could be observed that the functional relationship of stress ratio t and fatigue life Nf is linear when regression analyzed in double logarithmic coordinates. The fatigue properties of materials could be reflected by the parameters of S-N fatigue equation, k, and n. The value of n is adopted to evaluate the sensitivity of fatigue life to the stress level, with the higher value of n representing higher sensitivity. The k reflects the fatigue life of cement-treated aggregates under the stress ratio of 100%, and the larger of k, the higher the position of fatigue curve, which means the better anti-fatigue performance of the material. Currently, the static strength of cement-treated aggregates was measured in the laboratory under fixed loading rates to determine the stress level and calculate the stress ratio for fatigue test. However, it is obvious that the loading rates for a series of fatigue test are varied with the stress level while the loading frequency fixed. The loading rates could be calculated approximately by the Eq. (10) when the sine waveform loading was applied in the fatigue test.
v ¼ 2f r
ð10Þ
where, v is the loading rate; f is the loading frequency, and r is the stress level. Moreover, it has been proved that the loading rates have a significant influence on the mechanic properties of cement-treated aggregates, including strength, modulus, and fatigue [12]. Thought, it is irrational to calculate the stress ratio with the static strength measured under fixed loading rates. It could result in an apparent error in the fatigue curve, even the strength value of cementtreated aggregates measured precisely. In order to correct the stress ratio mentioned above, Lv et al. established the variation patterns of strength with loading rates by conducting strength tests under different loading rates and proposed the theory of real stress ratio [12]. Then, the S-N fatigue equation modified by the real stress ratio t0 , shown as Eq. (11).
Nf ¼ k
n 1 t0
ð11Þ
where t0 is the stress ratio related to loading rates, named real stress ratio. The influence of loading rates on materials mechanic performance has been illustrated while the Eq. (11) was utilized to analyze the fatigue results. The fatigue curves could extend to the strength failure point (1, 1) and the value of k is about 1, which indicates that when the stress level approaches ultimate strength value the fatigue life would be about 1cycle. It is consistent with the actual situation and built a bridge between the fatigue failure and failure strength. It has been proved by the octahedral shear stress strength theory that when the failure occurred, the stress state of any random point within the object could be characterized uniquely by one point in the stress space [28]. All corresponding point of stress space forms a continuous surface which named as yield surface or damage envelope surface. The strength value of material under
Fig. 1. Yield surface model.
any stress states could be determined through its yield surface, as Fig. 1 presents. Combining the theory of yield surface and the S-N fatigue equation modified by real stress ratio, fatigue behaviors of material under any stress states could be characterized. Desai et al. [28] proposed a kind of response function of yield surface which could be employed to characterize the mechanical response of geotechnical material and asphalt mixtures under any stress states. In stress invariant space, the yield surface could be expressed as Eq. (12).
n 2
I1 R a I1PR þ c Pn n J2 u rij ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 ½1 bcosð3hÞ Pn pffiffi where cosð3hÞ ¼ 3 2 3
J3
3
ðJ 2 Þ2
ð12Þ
; 0 6 h 6 p3, h is Cape Roca; J 2 , J3 is the sec-
ond and third deviatoricstress invariants, respectively; Pa is atmospheric pressure; I1 is the first stress tensor invariant; and a, b, c, n, Rare the model parameters. pffiffiffiffi In the space of I1 ; J 2 ; h , the Eq. (12) expressed a closed yield surface, and the model parameter a and b are 0 at peak stress under the unidirectional stress state, so the yield surface will degenerate into a straight line in space [28], and the expression is as Eq. (13): J2 P2n
2 ¼ c I1PR n
! J 2 ¼ cðI1 RÞ2 pffiffiffiffi pffiffiffi ! J 2 ¼ cjI1 Rj
ð13Þ
pffiffiffiffi In (I1, J 2 ) space, the yield surface represented by Eq. (13) is a straight line. The expression on the left part of the upper form is the criterion of the Mises yield condition, and its physical meaning pffiffiffiffi is the shear stress intensity T ¼ J 2 . According to the Desai yield surface model, stress would only occur in one direction. The yield surface in stress invariant space could be easily established with the tensile and compressive strength. In stress invariant space, the coordinates of tensile f c qffiffiffiffiffiffiffi 2 and compressive strength f t failure points are f c0 13 f c and qffiffiffiffiffiffiffi 2 f t0 13 f t , respectively.
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In this paper, the Desai yield surface of cement-treated aggregates was established by conducting the strength test on cement-treated aggregates under different stress states, including unconfined compressive, indirect tensile, and four-point bending strength tests. With the yield surface, the ultimate strength, fatigue stress, and stress ratio of fatigue test under different stress states could be obtained in a uniform system. 3. Materials and test design 3.1. Material and mix design 3.1.1. Cement The main components of cement-treated aggregates are cement, graded aggregates, and water. Aggregates are structured by cement hydration products and thus forming strength. The performance of cement has a significant influence on the quality of cement-treated aggregates. The performance indexes of cement, including fineness, setting time, soundness, strength, etc., had been tested, and the test results were displayed in Table 1. 3.1.2. Aggregate and gradation Limestone was adopted as coarse and fine aggregates. The mechanical and physical properties of aggregates were tested according to the Chinese Test Methods of Aggregate for Highway Engineering JTG B42-2005. The test results were displayed in Table 2. The gradation of cement-treated aggregated for this study was determined through the Chinese Construction Specifications for Highway Road Bases (JTJ/T F20-2015) [31], as Fig. 2 showed. 3.1.3. Cement-treated aggregates mix design Previous researches have proved that the cement-treated aggregates produced through the procedure of vibration mixing and compaction are more similar to that of road engineering [32]. Thus, in this research, cement-treated aggregates mixed by vibration mixer, and the vibration compaction test was utilized to determine the optimum moisture content and maximum dry density of cement-treated aggregates. The test was performed according to the standard method of Chinese standard JTG E51-2009. The parameters for the machine was set as Table 3. According to the test results, the optimum moisture and cement content was 4.8% and 4.5%, respectively. And under this mixing proportion, the maximum dry density of mixtures was 2.43 g/cm3. 3.2. Specimens preparation Based on the parameters of design obtained by the works of Section 3.2, the specimens of cement-treated aggregates were manufactured. The samples with the length of 550 mm, the width of 150 mm, and the thickness of 150 mm were utilized for fourpoint bending strength and fatigue tests. The samples with the sizes of U150mm 150 mm were adopted to measure the strength values and fatigue life of cement-treated aggregates under unconfined compressive and indirect tensile stress states. The preparation of sample including mixing, forming in special models,
demolding, and curing at the environment with a temperature of 20 °C and the air humidity over 95%. Each type of tests was conducted with four replicates. 3.3. Experiment design For the purpose of establishing the yield surface of cementtreated aggregates to obtain its failure parameters in a unified model, the strength test was designed and conducted in this paper. The strengths of cement-treated aggregates under general stress states, including unconfined compressive, indirect tensile and four-point bending, were determined under different loading rates. The loading rates are 5 MPa/s, 10 MPa/s, 20 MPa/s, 30 MPa/s, 40 MPa/s, and 50 MPa/s. The strength tests were conducted by Material tests system under the test temperature of 15 °C according to ‘‘test methods of materials stabilized with inorganic binders for highway engineering JTG E51-2009”. A series of fatigue tests were performed on cement-treated aggregates under diverse stress states, including unconfined compressive, indirect tensile and four-point bending. Firstly, the standard strength S of cement-treated aggregates under unconfined compressive, indirect tensile, and four-point bending stress states were measured under the fixed loading rate of 5 mm/minwhich prescribed in JTG E51-2009. Then, based on these standard strength values, the stress levels for the fatigue test were determined as 0.2S, 0.3S, 0.4S, 0.5S, 0.6S, and 0.7S. When the speeds of vehicle are about 60–80 km/s, the loading frequency is nearly 10 Hz. Therefore, in this paper, the fatigue tests were carried out by half-sine-wave loads with loading frequency of 10 Hz. 4. Development of yield surface model for cement-treated aggregates 4.1. Strength properties of cement-treated aggregates under different stress states The strength of cement-treated aggregates under different stress model and loading rates, including the standard strength measured at a fixed loading rate which prescribed in JTG E512009, were measured and illustrated in Fig. 3. The standard strength of cement-treated aggregates under unconfined compressive, indirect tensile and four-point bending stress states were 11.76 MPa, 1.298 MPa, and 1.309 MPa, respectively. It is apparent that the standard strength values distinguish
Table 2 Test results of mechanical and physical properties of aggregates. Items
Test result
Technical requirement
Less than 0.6 mm particle Liquid limit/plastic index Content of flat and elongated particles in coarse aggregate (%) Content of soft stone (%) Crushed stone value (%)
23.5% 3.2 12.1%
liquid limit 28% plastic index 9 20%
0.9 20.3%
3 30%
Table 1 Main technical properties of cement. Items Fineness (the size of the sieve is 80 lm) Setting time (min) Soundness test (mm) Strength of cement mortar (MPa)
2.1 Initial setting (min) Final setting (min) 3 days flexural strength 3 days compressive strength
Test result
Specification
10% 275 408 3 4.7 24.9
180 360 5 2.5 10
C. Liu et al. / Construction and Building Materials 228 (2019) 117099
5
Fig. 2. Gradation curves of cement-treated aggregates.
Table 3 The parameters of machine set for the vibration compaction test. Frequency (Hz)
Excited force (kN)
Vibration amplitude (mm)
Vibration time (s)
30 ± 2
7.6
1.3 ± 0.05
90 ~ 120
20 18
Strength values (MPa)
16
S c=9.51×v
0.19
2 R =0.91
0.30 2 Sf=0.78× v R =0.89
14
Moreover, when the loading rates increased from 5 MPa/s to 10 MPa/s, the unconfined compressive strength increased by 15%, the indirect tensile strength increased by 20% and the four-point bending strength increased about two times. It denoted that the loading rate has considerable effects on the strength of cementtreated aggregates. Given regards of these two circumstances, the difference of loading rate for different test methods under standard test condition and considerable effects of loading rates on the strength, there is an obvious error in the determination of stress ratio of fatigue test. For precise characterization of fatigue behavior of cementtreated aggregates, it is essential to establish a unified failure strength model for different stress states based on the failure mechanism.
0.08 2 Sit=1.16× v R =0.93
12
4.2. Strength yield surface of cement-treated aggregates
Standard trength values under different test methods
2
0
10
20
30
40
50
60
70
Rates of loading (MPa/s) Fig. 3. Strength properties of CTA under different stress states and loading rates.
with the strength value determined under other loading rates. The strength of cement-treated aggregates under all stress states exhibited a power function growth trend with loading rates, and the fitting results are shown in Fig. 3. Based on the fitting equation and the standard strength values, the fixed loading rate 5 mm/min which is strain control mode could be transformed to the corresponding loading rates of stress control mode, and they were about 3.5 MPa/s, 2 MPa/s and 5.8 MPa/s, shown as the dotted lines in Fig. 3. During the standard strength test, specimens of cementtreated aggregates subjected stress with different loading rates under different stress state even though the strain growth rates were all 5 mm/min. That is because of the difference of modulus under different stress states, which has been proved in many previous pieces researches [12].
According to the yield surface proposed by Desai which introduced in Section 2, the coordinates of yield surface under unconfined compressive, indirect tensile and four-point bending stress states could be expressed with the expression in Table 4. Bringing the strength values of cement-treated aggregates measured previously into the corresponding expression in Table 4, some coordinates of points of the yield surface could be calculated. And the calculation results are displayed in Table 5. pffiffiffiffi In (I1, J 2 ) space, the yield surface of a single stress state is a line, and these lines would intersect at a peak point. Thus, the yield surface of cement-treated aggregates unconfined compressive, indirect tensile, and four-point bending stress states could be represented with three independent lines, as presented in Fig. 4. According to Fig. 4, the solid lines are the yield surface, which is a line under a certain loading rate. It could be observed that the yield surface of cement-treated aggregates diverges outwardly. The point in the yield surface represents a failure point of cement-treated aggregates at a single stress state. Moreover, the failure point at the same stress state could be connected into a straight line which could stretch to the coordinate origin, as the dotted line shown in Fig. 4. Between the yield surface and the coordinate origin, the point of this space means the stress that lower than the failure strength. For example, point A in Fig. 4 means stress at compressive stress state, which could not cause failure to cement-treated aggregates with one loading, based on which the fatigue stress is defined.
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C. Liu et al. / Construction and Building Materials 228 (2019) 117099
Table 4 Expressions of yield surface coordinates under different stress states. Stress state
Strength
r1
r2
r3
I1
Indirect tensile
RT
RT
0
3 RT
2 RT
Unconfined compressive
RC
0
0
RC
RC
Bending
RS
RS
0
0
RS
pffiffiffiffi J2 pffiffiffiffi 39 3 RT pffiffi 3 3 RC pffiffi 3 3 RS
Cape Roca (h) h ¼ arccos p5ffiffiffiffi ¼ 46 52 60° 0°
Table 5 Yield surface coordinates of cement-treated aggregates under different stress states. Stress states
StrengthR (MPa)
I1 (MPa)
pffiffiffiffi J2 (MPa)
Cape Roca (h)
5
Unconfined compressive Indirect tensile Bending
12.91 1.35 1.27
12.91 2.69 1.27
7.45 2.80 0.73
60° 46° 0°
10
Unconfined compressive Indirect tensile Bending
14.78 1.39 1.52
14.78 2.78 1.52
8.53 2.89 0.88
60° 46° 0°
20
Unconfined compressive Indirect tensile Bending
16.75 1.46 1.95
16.75 2.91 1.95
9.67 3.03 1.13
60° 46° 0°
30
Unconfined compressive Indirect tensile Bending
18.13 1.49 2.23
18.13 2.97 2.23
10.47 3.19 1.29
60° 46° 0°
40
Unconfined compressive Indirect tensile Bending
18.69 1.59 2.43
18.69 3.17 2.43
10.79 3.29 1.40
60° 46° 0°
50
Unconfined compressive Indirect tensile Bending
19.51 1.61 2.52
19.51 3.21 2.52
11.26 3.34 1.45
60° 46° 0°
Loading rates
v (MPa/s)
5 MPa/s 1 0MPa/s 2 0MPa/s 3 0MPa/s 4 0MPa/s 5 0MPa/s
stress
11
sion (14).
qffiffiffiffi P J 02 ¼ a A
J 2 (MPa) 9
7 compressive 3Bending A 1
-15
-10
-5
0
ð14Þ
With the increase of loading cycles, the micro-cracks propagated and the bearing surface reduced to A0 . Consequently, the repeated pffiffiffiffi stress increased, approaching to the ultimate strength J2 . The qffiffiffiffi pffiffiffiffi course of repeated stress increase from J 02 to J2 is defined as
Indirect tensile
-20
qffiffiffiffi J 02 due to the certain fatigue loads P is equal to the expres-
5
I1(MPa) Fig. 4. Yield surfaces of cement-treated aggregates under diverseloading rates.
5. Characterizing method of fatigue behavior based on the yield surface 5.1. 5.1Determination of fatigue stress path As mentioned in Section 2, fatigue is a cumulative damage process of material under repeated stress at a level less than its ultimate strength, during which the micro-cracks propagated and pffiffiffiffi the bearing surface A reduced continuously. In (I1, J 2 ) space, at begin of fatigue process, assuming that the specimen is complete and its bearing surface of cement-treated aggregates is A, the shear
the fatigue stress path. The standard strength of cement-treated aggregates under unconfined compressive, indirect tensile and four-point bending stress states were 11.76 MPa, 1.298 MPa, and 1.309 MPa, respectively. Based on these standard strength values, the stress levels for the fatigue test were determined as 0.2S, 0.3S, 0.4S, 0.5S, 0.6S, and 0.7S. The loading frequency is 10 Hz, and then the equivalent stress variation rates could be calculated by the expression (10). And the corresponding strength to stress variation rates, i.e., loading rates, could be obtained with the various patterns of strength with loading rates which established in Section 4.1. Based on these pffiffiffiffi strength values, the corresponding I1, ultimate shear strength J 2 and the shear stress caused by each stress level could be calculated by the expression in Table 4. All calculation results are presented in pffiffiffiffi Table 6 and (I1, J 2 ) space as Fig. 5. As Fig. 5 presents, the ultimate strength points are determined by the stress variation rates. The initial point of each stress levels and stress states corresponds to ultimate strength one by one. The path from the initial point to the ultimate point is fatigue stress path, and the length between the initial point and the ultimate point indicates the anti-fatigue capacities of cement-treated aggregates at a certain loading condition. Taking the unconfined compressive fatigue test at the stress level of 20% of strength as an example, the coordinates of pffiffiffiffi repeated stress in space (I1, J 2 ) is point A in Fig. 5, and the
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C. Liu et al. / Construction and Building Materials 228 (2019) 117099 Table 6 Initial and ultimate point of fatigue test. I1 occurred by the strength corresponding to stress rates (MPa)
pffiffiffiffi J2 occurred by the strength corresponding to stress rates (MPa)
1.36 2.04 2.71 3.39 4.07 4.75
19.77 21.35 22.55 23.53 24.36 25.08
11.41 12.32 13.01 13.57 14.05 14.47
0.52 0.78 1.04 1.29 1.56 1.82
0.54 0.81 1.08 1.35 1.62 1.89
2.65 2.73 2.79 2.85 2.89 2.93
2.75 2.84 2.91 2.96 3.01 3.04
0.27 0.39 0.52 0.65 0.79 0.92
0.15 0.23 0.30 0.38 0.45 0.53
1.28 1.45 1.58 1.69 1.78 1.87
0.74 0.84 0.91 0.97 1.03 1.08
Stress states & the standard strength
Normal stress ratio
Stress level (MPa)
equivalent stress variation ratesv (MPa/s)
I10 occurred under the stress level (MPa)
Unconfined compressive 11.76 MPa
0.2 0.3 0.4 0.5 0.6 0.7
2.35 3.53 4.70 5.88 7.06 8.23
47.04 70.56 94.08 117.60 141.12 164.64
2.35 3.53 4.70 5.88 7.06 8.23
Indirect tensile 1.298 MPa
0.2 0.3 0.4 0.5 0.6 0.7
0.26 0.39 0.52 0.65 0.78 0.91
5.19 7.79 10.38 12.98 15.58 18.17
Four-point bending 1.309 MPa
0.2 0.3 0.4 0.5 0.6 0.7
0.26 0.39 0.52 0.65 0.79 0.92
5.24 7.85 10.47 13.09 15.71 18.33
Fig. 5. Fatigue stress path under different stress states and stress levels.
corresponding ultimate point is point B. From point A to point B, the first stress tensor invariant I1increased from 2.352 MPa to pffiffiffiffi 19.77 MPa, and the shear stress intensity J 2 grew from 1.36 MPa to 11.41 MPa. The growth of the first stress tensor invariant I1 and pffiffiffiffi shear stress intensity J 2 is due to the accumulation of internal micro-cracks of cement-treated aggregates under repetition stress. Therefore, the anti-fatigue capacities of cement-treated aggregates could be evaluated by the length of line AB which could be calculated by the coordinates of point A and B. Also, the growth rates of pffiffiffiffi the first stress tensor invariant I1 and shear stress intensity J 2 could be adopted to evaluate the anti-fatigue capacities.
qffiffiffiffi J 02 occurred under the stress level (MPa)
could not reveal the failure strength features during the fatigue process. The prediction value of fatigue life under the stress ratio of 100% is not one cycle. The reason for this circumstance is that the loading rates vary with stress levels and the strength value has been affected by loading rates. Nevertheless, the stress ratio used in traditional S-N fatigue equation is generally obtained with standard strength value, which measured under a fixed loading rate, without correcting based on the impact of loading rates. Moreover, different specimens occur altered stress under different test method, resulting in different test results. Different stress states could be characterized by the first stress pffiffiffiffi tensor invariant I1 and shear stress intensity J 2 based on octahedron theory and yield surface mechanics model, as Section 5.1 expounded. With the yield surface, the difference of internal stress caused by the impacts of load modes, specimen types and loading rates could be unified. And the fatigue stress path could be obtained based on which a new characterizing method for fatigue behavior could be proposed. As the length between initial point A and ultimate point B could be employed to evaluate the anti-fatigue capacities of cementtreated aggregates, the length ratio of line OA to OB could be utilized to substitute the stress ratio of S-N fatigue equation. According to Fig. 5, based on the similar triangle theory, the stress ratio could be expressed by the expression (15).
D¼
OA OA0 C 0 C ¼ ¼ OB OC 0 C 0 B 0
OA 0 where OC 0 is the ratio of initial first stress tensor invariant I1 to ultiqffiffiffiffi 0 mate I1, CC 0 CB is the ratio of initial shear stress intensity J02 to ultipffiffiffiffi mate J2 . Then, the traditional S-N fatigue equation could be modified as Eq. (16).
Nf ¼ k 5.2. A normalized characterization methods of fatigue behavior based on the yield surface
ð15Þ
n 1 D
ð16Þ
5.3. Fatigue behavior characterized by modified S-N equation Usually, the S-N fatigue equation, which reflects the relationship of fatigue life and stress, were utilized to characterize fatigue behaviors of road engineering materials. However, it has been proved in many researches that traditional S-N fatigue equation
qffiffiffiffi In this paper, the ratio of initial shear stress intensity J 02 to pffiffiffiffi ultimate J 2 was taken as the stress ratio of fatigue tests. The
8
C. Liu et al. / Construction and Building Materials 228 (2019) 117099
results of fatigue test, including unconfined compressive, indirect tensile and four-point bending fatigue tests, were analyzed with both traditional S-N fatigue Eq. (9) and modified S-N fatigue Eq. (16), and the results presented as Figs. 6 and 7. It is apparent from Fig. 6 that the fatigue test results under different stress states fitted into typically distinctive and individual fatigue curves when traditional S-N fatigue equation was utilized. The most and irrational circumstance is that the values of parameter k, which reflect the fatigue life of cement-treated aggregates under the stress ratio of 100 percent, are varied dramatically with stress states. The k value under compressive stress is 6193, and it is 80 under bending stress while that of indirect tensile stress is 3.8. Technically, the fatigue life of cement-treated aggregates under the stress ratio of 100 percent is 1 time. Therefore, it is irrational to obtain the anti-fatigue parameter of pavement design by stretch the fatigue curve to point (1,1), which specified in the pavement design method. In Fig. 7, the k values under different stress states are all approximately 1, which means that cement-treated aggregates specimens would be broken with one loading under stress reaching the strength value. Thus, the modified S-N fatigue equation has corrected this error, and the S-N fatigue equation could be developed as Eq. (17).
Nf ¼
n 1 D
ð17Þ
Moreover, the parameter n in fatigue equation is usually applied to evaluate the stress sensibility of fatigue behaviors, the larger the n-value, the more vulnerable of fatigue behavior to the effects of stress. Theoretically, stress sensibility is one of the material attributes, and a certain material has a certain stress sensibility. Based on conventional S-N fatigue equation, the n-values were different for the same cement-treated aggregates under different stress states. It indicates that a certain material had different stress sensibility under different test method, and this raised the considerable controversy on which test method could obtain an effective evaluation for stress sensibility of fatigue behaviors. Comparatively, with the modified S-N fatigue equation, it is found from Fig. 7 that the three fatigue curves of three stress states have very similar n values. The impact of stress states on the analysis of fatigue characteristic has been eliminated, and all fatigue test results could be illustrated with one unified fatigue curve, as shown in Fig. 8.
109
Unconfined compressive Indirect tensile Four-point bending
108
Nf=6193.28•t-7.45
107
R2 =0.93
Nf (time)
6
10
105
Fig. 7. Fatigue curves of cement-treated aggregates fitted by modified S-N fatigue equation.
Nf=80.39•t-7.37 R2 =0.95
104
Fig. 8. Normalized fatigue curve of cement-treated aggregates under different stress states.
Fig. 8 proved that a normalized relationship of fatigue life and stress level could be established for cement-treated aggregates under different stress states. With this normalized model, the kvalue has been proving to be 1, which revealed the relationship of fatigue damage and strength failure. And the fatigue behavior of cement-treated aggregates has been characterized uniquely, without the interference of specimen type, test method, and test conditions. Uniform characterization of fatigue test results obtained by different methods has realized by the normalized characterization methods, which could avoid controversy on test methods and unify the determination of design parameters for anti-fatigue pavement.
103
6. Conclusions
Nf=3.80•t-8.96
102
2
R =0.90 101 0.2
0.3
0.4
0.5
0.6
0.7
0.8
t Fig. 6. Fatigue curves of cement-treated aggregates fitted by traditional S-N fatigue equation.
In this paper, a normalized characterization method of fatigue behavior based on yield surface was proposed to analyze the fatigue test results of cement-treated aggregates under different stress states. Compared with traditional characterization methods of fatigue behavior of cement-treated aggregates, some conclusion could be drawn as follows.
C. Liu et al. / Construction and Building Materials 228 (2019) 117099
(1) The yield surface of cement-treated aggregates was established by the strength test results under different stress states and loading rates, and different ultimate strength could be expressed uniquely by the yield surface. (2) The fatigue stress path was characterized by the yield surface, and consequently a new method to determine the stress ratio of fatigue equation was proposed. (3) The influence of various stress states on the fatigue behaviors of cement-treated aggregates was reduced or even eliminate by the normalized characterization method for fatigue behavior ofcement-treated aggregatesbased on the yield criterion. And the normalization characterization of fatigue behavior of cement-treated aggregates was realized. 7. Data availability Some or all data, models, or code generated or used during the study are available from the corresponding author by request. All data, models, and code generated or used during the study appear in the submitted article. Declaration of Competing Interest 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. Acknowledgements This work was supported by National Natural Science Foundation of China (51578081, 51608085), the Ministry of Transport Construction Projects of Science and Technology (2015318825120), the Projects of Transportation Science and Technology of Hunan (201701). References [1] Y. Zheng, P. Zhang, Y. Cai, Z. Jin, E. Moshtagh, Cracking resistance and mechanical properties of basalt fibers reinforced cement-stabilized macadam, Compos. B Eng. 165 (2019) 312–334, https://doi.org/10.1016/ j.compositesb.2018.11.115. [2] Z. Peng, L. Qingfu, Effect of polypropylene fibre on mechanical and shrinkage properties of cement-stabilised macadam, Int. J. Pavement Eng. 10 (6) (2009) 435–445, https://doi.org/10.1080/10298430802363985. [3] Z. Liu, Experimental research on the engineering characteristics of polyester fiber-reinforced cement-stabilized macadam, J. Mater. Civ. Eng. 27 (10) (2015), https://doi.org/10.1061/(asce)mt.1943-5533.0001251. [4] F.B.-B.H.A. MesbahU, Efficiency of polypropylene and metallic fibres on control of shrinkage and cracking of recycled aggregate mortars, Constr. Build. Mater. 13 (1999) 9. [5] H.-X. Guan, H.-Q. Wang, H. Liu, J.-J. Yan, M. Lin, The effect of intermediate principal stress on compressive strength of different cement content of cement-stabilized macadam and different gradation of AC-13 mixture, Appl. Sci. 8 (10) (2018), https://doi.org/10.3390/app8102000. [6] T.-L. Wang, H.-H. Wang, H.-F. Song, Z.-R. Yue, Z.-H. Guo, Effects of cement content and grain-size composition on engineering properties of high-speedrailway macadam subgrade, Cold Reg. Sci. Technol. 145 (2018) 21–31, https:// doi.org/10.1016/j.coldregions.2017.09.009. [7] J. Wang, Q. Dai, S. Guo, R. Si, Mechanical and durability performance evaluation of crumb rubber-modified epoxy polymer concrete overlays, Constr. Build. Mater. 203 (2019) 469–480, https://doi.org/10.1016/ j.conbuildmat.2019.01.085. [8] J.Z. Songtao Lv, Wenliang Zhong, Characteristies of strength, modulus and fatigue damage for cement stabilized macadam in curing period, China J. Highway Transp. 28 (9) (2015) 8, https://doi.org/10.19721/j.cnki.10017372.2015.09.002. [9] J. Jin, Y. Tan, R. Liu, J. Zheng, J. Zhang, Synergy Effect of Attapulgite, Rubber, and Diatomite on Organic Montmorillonite-Modified Asphalt, J Mater Civil Eng. 31 (2) (2019) 04018388, https://doi.org/10.1061/(ASCE)MT.1943-5533.0002601.
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