3.1. Fatigue Life and Distribution
Fatigue life refers to the number of cycles required to induce fatigue failure to a material under cyclic loading.
Table 2 shows the test results for RAC under uniaxial compression fatigue life at different stress levels. In previous studies, the Weibull distribution was usually used to study the fatigue life of concrete [
23,
28,
33,
34,
35]. Thus in this study, the Weibull distribution was also applied to investigate the fatigue life of RAC. When the location parameter or the minimum life parameter was 0, the two-parameter Weibull cumulative distribution function
could be written as follows:
where:
is the failure probability;
N is the fatigue life,
is the characteristic parameter of fatigue life; and
b is the shape parameter of Weibull distribution.
After taking twice natural logarithms of Equation (
1), the following equation could be obtained:
If
,
, and
, then Equation (
2) could be simplified as:
Equation (
3) could be used to test whether a group of fatigue life test data conformed to the two-parameter Weibull distribution. If using graphical regression analysis, the linear relationship between
Y and
X proved good, the hypothesis that fatigue life complies with the Weibull distribution was deemed tenable. According to the fatigue life shown in
Table 2, the corresponding survival rate
p could be calculated:
where:
i is the number of fatigue life scale in ascending order from small to large under the same stress level;
m is the sample size of the fatigue test at a given stress level. Then, the fatigue life at different stress levels was regressed linearly according to Equation (
3). The regression analysis of the relationship of ln(ln(1/
p)) with ln
N under each stress level is shown in
Figure 2. It can be seen from
Figure 2 that at all levels of stress, the correlation coefficient
R was between 0.8789 and 0.9551, and the linear relationship between
X and
Y was significant, indicating that the compressive fatigue life of RAC materials was found to be subject to the Weibull distribution.
Table 3 below lists the Weibull distribution parameters of RAC fatigue life at different stress levels. From shape parameter
b, it can be seen that the influence of fatigue stress level on the dispersion of RAC fatigue life was not significant.
In addition, in the engineering application of concrete, it is generally necessary to establish the
S-
N equation under a specified survival rate according to the reliability requirements. According to Equation (
2), the fatigue life of concrete could be expressed as:
Through Equation (
5), the corresponding fatigue life
N of concrete under a certain stress level
S could be calculated under a given survival rate
p.
To ensure the reliability of the predicted life of concrete at low stress levels, the following single logarithm equation could be used to express the fatigue life:
where A and B are the equation parameters.
The relationship curve of
S-
N, that is, the
S-
N-
p equation, can be obtained by linearly regressing the fatigue life under different failure probabilities in the form of Equation (
5). The single logarithm fatigue equation of RAC under different survival rates
p is shown in
Table 4. In addition, the tested fatigue life and fatigue equation of RAC are plotted in
Figure 3.
In order to compare RAC with natural aggregate concrete of similar compressive strength, a Weibull distribution analysis was performed on the uniaxial compression fatigue life test data of two groups of natural aggregate concrete (NAC-1 and NAC-2) in references [
36,
37,
38] with cubic compressive strengths of 54.2 MPa and 46.8 MPa, respectively. According to these analyses, the
S-N-p equations of NAC-1 and NAC-2 were obtained using the same process as stated above. The fatigue lives of RAC and NAC under a survival rate
under each stress level calculated by Equation (
5) are given in
Table 5. The fatigue equation obtained by the fitting of a single logarithm equation is listed in
Table 4. It can be seen from
Table 4 that the correlation coefficients
R of the fatigue equations of NAC-1 and NAC-2 were greater than 0.99, with a high confidence level.The mix proportion of NAC-1 and NAC-2 are shown in
Table 6 (The fine aggregate was made of natural river sand with a moisture content of 3.8 percent).
The RAC and NAC fatigue equations under the survival rate
p = 0.5 are shown in
Figure 4. It can be seen from
Figure 4 that under the same stress level, the fatigue life of RAC was found to be lower than that of the NAC-1 and NAC-2 in the high cycle fatigue zone, with
, and higher than that of NAC-1 and NAC-2 in the low cycle fatigue zone, with
.
Under the action of fatigue loading, the failure process of the concrete material essentially matched the process of crack evolution and development in the ITZ and matrix. According to the International Union of Laboratories and Experts in Construction Materials, Systems and Structures (RILEM) report, the fatigue failure of concrete materials is attributed to two mechanisms: the degradation of the bond between the matrix and coarse aggregate, and the development of cracks in the matrix. These two failure mechanisms either act alone or coexist in time [
39]. Hsu [
40] considered that for low cycle and high amplitude fatigues, the matrix failure was dominant, and the continuous “through-cracks” that had formed due to the fatigue crack extending into the mortar had led to the final failure. For high cycle and low amplitude fatigues, the debonding of the interface between matrix and aggregate had led to the failure of the material, and the slow and gradual development of bonding cracks between mortar and coarse aggregate had resulted in the material fatigue failure. In addition, Zheng [
41] further subdivided the fatigue component into three zones by introducing the influence factors of matrix and ITZs into the fatigue performance. For the low cycle fatigue zone corresponding to
, the dominant mechanism of fatigue damage was matrix cracking, and the matrix property was deemed the main factor affecting fatigue life, which was named the dominant matrix cracking zone. In the high fatigue zone cycle corresponding to
, the bond cracking initiated, propagated to the interfacial zone, and extended to the matrix was deemed the dominating reason for fatigue failure. Thus, the properties of the interfacial zones were the main factors affecting fatigue life. These zones are called the dominant bond cracking zones. Correspondingly, in the region of
, the fatigue damage to concrete material mainly developed due to matrix cracking within the interfacial zones. The properties of the matrix and interfacial zone were both found to significantly impact on the fatigue performance of concrete, which region is called the transition zone. Accordingly, the fatigue zoning of concrete in
Figure 4 was achieved as follows: (i) matrix cracking control zone, (ii) transition zone, and (iii) interface cracking control zone.
In the high fatigue cycle region (
), the fatigue life of RAC was lower than that of the NAC-1 and NAC-2, which mainly related to the nature of the multiple ITZs of RAC. Nanoindentation tests were conducted on the ITZ of RAC at the age of 90 days. The selected indented areas on the sample are shown in
Figure 5a–c, containing ITZ
, ITZ
and ITZ
of RAC, respectively. After the indentation test, the obtained modulus lattice data were processed using the function of Contour-Color fill in the Origin software. Thus, the contour maps of indentation modulus of different ITZs were obtained, as shown in
Figure 5d–f. It can be seen from
Figure 5 that there are three kinds of ITZs (ITZ
, ITZ
and ITZ
) in RAC.In any ITZ, the indentation modulus is lower than that of the new and old mortars, which indicates that the three kinds of ITZs are weak connecting components of RAC. Due to the existence of multiple weak ITZs, RAC is more prone to produce bond cracks than NAC in the crack control zone of the interface zone, which has a more detrimental effect on fatigue life.
In the low fatigue cycle region (
), the fatigue life of RAC was found to be higher than that of NAC-1 and NAC-2. The water to binder ratio is an important factor determining the porosity of concrete matrix, and the existence of pores has an adverse effect on the concrete matrix performance [
42]. It can be seen from
Table 1 and
Table 2 that the effective water to binder ratio of RAC is 0.4, while that of NAC-1 and NAC-2 is much greater than 0.4 (the sand used for RAC is dried sand with a water content of about 0, while the mass water content of sand used for NAC is 3.8%). Thus, the effective water to binder ratio of RAC is lower than that of NAC-1 and NAC-2. This suggests that the RAC matrix has a lower porosity than NAC-1 and NAC-2, and its matrix property is better, having a stronger ability to resist crack development. Meanwhile in the crack control zone of matrix, the matrix property is the main factor affecting the fatigue life of concrete. This may be the main reason that the fatigue life of RAC is higher than that of NAC-1 and NAC-2 in the low cycle fatigue region (
). Moreover, it is the main factor that the cube compressive strength of RAC is higher than that of NAC-1 and NAC-2.
In the matrix cracking control zone (i), the fatigue life of RAC is higher than that of NAC-1 and NAC-2. However, in the interface cracking control zone (iii), the fatigue life of RAC is lower than that of NAC-1 and NAC-2, and this gap increases gradually with the decreasing stress level. In another way, it suggests that in the high cycle fatigue region (), the existence of multiple ITZs has a greater adverse effect on the fatigue life of RAC than NAC.