Figure 13 shows that with the increasing MSWI BA content, the compressive failure stress gradually decreases, and the failure strain gradually increases. In cement concrete and geotechnical materials, the curved part at both ends of the stress–strain curve characterizes the compaction stage and the elastic–plastic change stage, while the linear part in the middle represents the elastic deformation stage [
31,
32]. After the incorporation of MSWI BA, the pores in the mixture become more; therefore, with the increase in MSWI BA content, the deformation in the compaction stage gradually becomes larger.
Figure 13a shows that the stress–strain curve shape with different MSWI BA content in the CBM group is similar. However, the stress–strain curve with different MSWI BA content in the CSB group is quite different, as
Figure 13b shows. Comparing
Figure 13a with
Figure 13b shows that the stress–strain curve of CSB-50 is similar to that of the CBM group. Because of this, the large-sized crushed stone in the CBM group can be squeezed into each other to form a skeleton structure, while the CSB group cannot form a skeleton structure well, except for CSB-50.
4.1. Secant Modulus
The secant modulus, also known as secant stiffness, is defined as the slope of the line connecting a point on the stress–strain curve to the origin. In essence, it is the real-time elastic modulus of the material under load. The ratio of real-time stress to failure ultimate stress, that is, the stress level is used as the abscissa. The real-time secant modulus as ordinate. The variation of the secant modulus with stress levels is shown in
Figure 14.
Figure 14 shows that the secant modulus increases rapidly with the increase in stress level, which corresponds to the bending part in the front of the stress–strain curve, that is, the compaction stage. At this stage, the pores and primary cracks are continuously compressed and closed. With the increase in stress level, the growth of secant modulus slows down and a relatively stable growth exists in the medium term, which corresponds to the linear part of the stress–strain curve; that is, the elastic change stage. At this stage, the stress is continuously concentrated, and the recoverable elastic deformation occurs. The secant modulus gradually decreases after reaching the peak, corresponding to the elastic–plastic failure stage of the strain–strain curve. At this stage, the cracks inside the mixture begin to develop, and the growth rate continues to accelerate. When the ultimate stress and ultimate strain are reached, damage occurs.
4.2. Damage Variable
Damage variable
D is often used to describe the response and damage evolution state of materials under load. Generally, the definition method of damage is selected from macroscopic, mesoscopic and microscopic perspectives. The macroscopic perspective is based on the material macroscopic measurable parameters, such as elastic modulus, wave velocity and acoustic emission parameters. The mesoscopic perspective is based on the mesoscopic statistical damage mechanics method. The proportion of the representative volume element of the damaged material is defined as the damage variable, assuming that the failure volume element obeys a certain probability distribution; then, we established the damage evolution model. The microscopic point of view is to define the material microstructure parameters such as the number of micro-defects, length, area and volume [
32].
Lemaitre [
33] proposed a common method to characterize the damage evolution of materials under load by using elastic modulus as the damage variable. Damage variable is defined as the ratio of elastic modulus when damage occurs:
where
Ed is the elastic modulus at the material damage state,
E is the elastic modulus of the material without damage. 0 ≤
D ≤ 1,
D = 0 means that the material has not been destroyed,
D = 1 means that the material has been completely destroyed.
In geotechnical materials, the elastic modulus under the non-destructive state generally takes the initial elastic modulus of the material, which is more conservative for the cement stabilized MSWI BA, and
D < 0 will occur, which deviates from the definition of the damage variable. According to the actual situation of the cement-stabilized MSWI BA elastic modulus during the compression process, Equation (8) is used to determine the damage variable
D:
where
Ep is the peak secant modulus,
Ed is the secant modulus of the elastic–plastic failure stage and
σp is the stress level at the peak secant modulus, 0 ≤
σp ≤ 1.
Combined with the definition of the damage variable and
Figure 14, it can be seen that the CBM group is damaged at about 0.9 stress level, while the damage stress level of CSB group is quite different; CSB-100 is damaged at 0.6 stress level; therefore, the CBM group have more resistant to load than the CSB group. The CSB group is not suitable for the construction of pavement bases for heavy-traffic roads, so as to avoid damage caused by heavy traffic, the use of the CSB group should be avoided.
4.3. Constitutive Model
Scholars are more focused on the constitutive relationship of the material under load [
34]. The uniaxial compressive stress–strain constitutive model of cement concrete proposed by Guo is segmented form, in which the ascending section is a polynomial function, and the descending section is a rational fractional function [
35]. Yan proposed an improved Duncan-Chang constitutive model to simulate the stress–strain of cement-stabilized macadam during an unconfined compressive process [
36]. Zhang established the damage constitutive model of cement-stabilized cinder macadam during the unconfined compressive process through regression analysis [
37].
Through a large number of simulation tests on the existing models, finding that the existing models cannot well simulate the stress–strain process of the unconfined compression test of cement-stabilized MSWI BA. According to the characteristics of the compressive stress–strain curve of the cement-stabilized MSWI BA mixture, a segmented constitutive model is proposed to describe the compressive process of the cement-stabilized MSWI BA mixture. The constitutive model is divided into three stages, namely, the compaction stage, the elastic change stage and the elastic–plastic failure stage. The compaction stage and the elastic change stage are simulated by polynomial function with reference to the Guo model. In the elastic–plastic failure stage, the damage constitutive model is established based on the statistical damage theory.
According to the strain equivalence hypothesis proposed by Lemaitre, the damage constitutive relationship in the elastic–plastic failure stage can be obtained [
38]:
where
σ is the nominal stress,
Ep is the peak secant modulus, namely the reference elastic modulus,
ε is the strain and
D is the damage variable.
The development of internal damage to the mixture is complex and random under load, which is difficult to be described by a single characteristic variable. The mixture can be discretized into micro-unit; the damage obeys random distribution and has statistical regular. According to the principle of Weibull distribution, the damage probability density function is expressed as follows [
39]:
where
h,
F0 are Weibull distribution parameters,
F is the random distribution variable of micro-unit strength in the mixture.
The
D in the elastic–plastic failure stage can be defined as the ratio of the number of damaged micro-unit to the total number of micro-units in the mixture from the microscopic point of view:
where
Nd is the number of damaged micro-unit in the mixture and
Nt is the total number of micro-units.
According to Equation (10), the number of damaged micro-unit is:
Then, the damage variable of the mixture in the elastic–plastic failure stage is:
The damage constitutive model of the elastic–plastic change stage of the mixture can be obtained by Equation (9):
According to the characteristics of the compressive stress–strain curve, the damage constitutive model in the elastic–plastic failure stage should satisfy the following boundary conditions:
where
εm is the ultimate compressive failure strain of the mixture,
σm is the failure ultimate stress and
εp is the strain at the peak secant modulus.
Substituting the above boundary conditions into Equations (13) and (14) can obtain Equations (16) and (17) [
40]:
The segmented constitutive model of stress–strain curve can be expressed by Equation (18):
where
εs is the strain at the elastic change stage,
εp is the strain at the peak secant modulus,
εm is the ultimate failure strain and
a,
b,
c,
k,
d,
F0,
h are related model parameters.
The stress–strain curves of CSB-75 and CSB-100 have no obvious linear part. Assuming that the constitutive model is only composed of the compaction stage and elastic–plastic change stage. The relevant model parameters of each group of mixtures are calculated as
Table 8 shows.
The comparison between the test curve and the model curve of each group is shown in
Figure 15.
Figure 15 shows that the test curve and the model curve are basically coincident, indicating that the established segmented constitutive model can well simulate the stress–strain relationship of the mixture with different MSWI BA content, which can provide a reference for the study of deformation evolution and mechanical behavior analysis of cement-stabilized MSWI BA pavement base under load.