Asphalt is a representative viscoelastic material that has different mechanical performances under diverse temperatures, loading frequencies, and stress states [
18]. In order to accurately analyze the difference in asphalt tensile and compressive resilient modulus under different loading frequencies, modulus tests were carried out at different loading frequencies.
4.1. Contrastive Analysis of Modulus Test Results
The average value of the moduli from the last five cycles was defined as the dynamic resilient modulus of the asphalt. Dynamic moduli obtained by specified test methods are plotted in
Figure 4. The comparison of six kinds of dynamic loading state under different frequencies is shown in
Figure 5. The fitting curve of different dynamic resilient moduli under different loading frequencies is shown in
Figure 6. Fitting parameters of different dynamic resilient moduli under different loading frequencies is shown in
Table 5.
The dynamic resilient modulus from the four test methods increased with loading frequency. As the loading frequency increased, the increase rate of each modulus slowed down. The modulus value rose with the increase of frequency under different stress states. The following phenomena were found under the same frequency: unconfined compressive modulus > indirect tensile modulus > direct tensile modulus > four-point bending flexural modulus (
Figure 4).
The dynamic tension and compression-resilient modulus from the indirect tensile test and four-point bending test increased with increasing loading frequency, and the fastest increase was within 0.1–1 Hz.
Compression resilient modulus divided by tension resilient modulus from the four-point bending test was similar under the five various loading frequencies, and the average value was approximately 1.20. Compression resilient modulus divided by tension resilient modulus from the indirect tensile test was similar for all loading frequencies, and the average value was approximately 1.18. Compression resilient modulus divided by tension resilient modulus from the direct tension experiment and unconfined compression experiment were similar for all different loading frequencies, and the average value was approximately 1.28. The compression resilient modulus and tension resilient modulus showed a similar relationship. The tension and compression moduli of the asphalt were notably differentiated.
The dynamic compression resilient modulus from the four-point bending test and the indirect tensile experiment was similar to the dynamic compression resilient modulus from the unconfined compression test (
Figure 5). The dynamic tension resilient modulus from the four-point bending test and the indirect tensile experiment was similar to the dynamic tension resilient modulus from the direct test. Modulus experimental results had high susceptibility to sample form, size, and forced state. Notably, the modulus revealed variability and discreteness under various forced states. Nevertheless, even at identical forced state and loading frequency, the modulus of specimens can be relatively discrete. Asphalt modulus increased exponentially with loading frequency under various stress states (
Figure 6). The stress state affected parameters
a,
b in the modulus equation, and the size of coefficients
a and
b varied widely. Coefficient
a represents the declivity of the fatigue curve, and the parameter ambition represents the sequence of unconfined compressive > indirect tensile > direct tensile > flexural tensile. The power function modulus equations of modulus varied greatly inside diverse stress states.
There was a modulus deviation in the different test results. It could be attributed to the difference in the stress states under different test conditions. A large number of studies have shown that the modulus of asphalt is different when using different testing methods that have diverse stress conditions. It is easy to see that the compression and tension resilient moduli obtained from the four tests were similar. To reduce or even eliminate the effect of different forced status, loading frequency, specimen form, and specimen size, it is necessary to establish a standardized model for the stiffness characteristics of asphalt inside diverse stress states, on account of modulus and loading frequency variation.
4.2. Standardization Analysis
Data standardization includes two aspects: similar trends of data processing and dimensionless processing. Making the trend of data processing the same mainly solves the problem of different data properties, as the dimensionless processing of data makes data comparable.
Standardization analysis involves mapping the data to a range from 0 to 1 to remove the unit limit of the data and convert it into a dimensionless pure value for comparison with the indicators of diverse units and magnitudes. The data standardization process does not change the physical meaning of the data, but makes it comparable under the same scale. After standardization, the modulus of each asphalt mixture for each different loading state was converted into dimensionless parameter values that could be comprehensively analyzed.
Data pre-processing is an effective method to solve the problems related to the original data, which paves the way for further processing of the original data [
19]. Standardization is one of the familiar data pre-processing methods for establishing classification and regression models for most researchers [
20,
21]. In min–max normalization, features are normalized in the range [0, 1] using the following equation:
min
A represents the minimum values of feature
A. max
A represents the maximum values of feature
A. The original value of data
A is expressed in
v. The normalized processing value of data
A is expressed as
v′. The maximum eigenvalue and minimum eigenvalue are mapped to 1 and 0, respectively. A standardization model for the tensile and compressive characteristics of asphalt inside diverse stress states and a standardization model for the compressive, tensile, indirect tensile, and flexural characteristics of asphalt under diverse stress states was established based on the variation of modulus and loading frequency. The loading frequency corresponding to the 60 km/h speed of the vehicle was about 10 Hz, so 0.1 and 50 Hz were chosen as the limiting conditions in the test, considering the actual state of vehicle speed. The standardization of tensile ratio and compressive ratio for the dynamic loading state under different frequencies are shown in
Figure 7 and
Figure 8. The standardization of modulus ratio versus frequency ratio achieved by the specified test methods is shown in
Figure 9.
The fitting results obtained from
Figure 7,
Figure 8 and
Figure 9 inside diverse stress states showed a strong linear correlation in the modulus and frequency ratios, and the fitting correlation coefficient was extremely high. Comparisons with conventional dynamic modulus test results and modulus ratio also continuously increased with frequency ratio [
6]. The difference in fitting results under diverse stress states, on account of the new approach of stiffness analysis, was extremely lessened, and it was difficult to indicate the difference in stiffness experiment results from one stress state to another. Any of the four moduli can be directly substituted into the standardization model for the flexural, compressive, tensile, and indirect tensile characteristics of asphalt mixture under diverse stress states to obtain the modulus at other frequencies.
On account of the new method of stiffness analysis, in the standardized modulus equation, the discreteness in modulus equation coefficients inside diverse stress states was greatly reduced. The values of coefficients
a and
b were very close to each other (
Table 6), and the values of coefficient varied with the experimental conditions and stress status [
21,
22].
Consequences achieved from the standardization analysis modulus test revealed that modulus ratio and frequency ratio in diverse stress states were synchronously fit by the same frame of reference. Consequently, for modulus features, the standardization analysis approach is solves the conundrum that commonly plagues conventional modulus test results, and is a probable option for accurately and consistently characterizing the modulus features of asphalt inside diverse stress conditions. The united form of the standardized stiffness equations for various frequencies was obtained. The goal of standardizing modulus characteristics inside diverse stress states was accomplished, and provided a theoretical model and technical foundation for the scientific transformation of material modulus to structural modulus [
23].
There is a hypothetical relationship, as follows:
where
E is the dynamic modulus from the test;
f is the loading frequencies; and
a and
b are fitting parameters. The fitting process on the logarithmic frequency scale is shown in
Figure 7,
Figure 8 and
Figure 9.