3.2.1. Dynamic Stress–Strain Backbone Curves
According to the dynamic triaxial test data, the dynamic stress–strain curve for the improved soil forms an elliptical hysteresis circle during one cycle of cyclic stress loading and unloading, and the average values for the stress and strain at the apex of the hysteresis circle of 3~8 levels during the dynamic loading process are selected to obtain the trend for the dynamic strain and corresponding dynamic stress. The dynamic stress–strain curves for PVA-amended expansive soils with different circumferential pressures are plotted in
Figure 5, and the dynamic stress–strain curves for soils amended with different frequencies are plotted in
Figure 6.
As shown in
Figure 5, the dynamic stress–strain curves increase with increasing circumferential pressure. From the curves measured at different frequencies, the deformation under the same dynamic stress is minimized for σ
3 = 300 kPa. The main reason for this difference is that the high-confining pressure specimen has greater compactness and less porosity than the low confining pressure specimen during the consolidation process. In addition, the contact between the PVA in the improved soil and the soil particles is more compact, so the dynamic strain under the same stress is relatively small.
Figure 6 shows that when the confining pressure and dynamic stress are the same, the higher the frequency, the lower the dynamic strain. Analysis shows that in a cycle period, the higher the frequency, the shorter the dynamic stress that acts on the specimen [
27]. At this time, the deformation of the specimen occurs too late to fully develop, and the test soil exhibits a certain degree of viscosity, resulting in stiffness; that is, the higher the frequency, the lower the dynamic strain under the action of the same dynamic stress.
By comparing the curves shown in
Figure 5 and
Figure 6, the slope of the stress–strain curve is greater in the early stage of dynamic strain development (ε
d ≤ 0.4), the dynamic strain of the soil responds to stress faster, and the specimen is stabilized in terms of force and deformation because elastic deformation accounts for a larger proportion of the deformation. As the dynamic strain continues to increase (ε
d > 0.4), the slope of the curve decreases, and the links between the soil particles in the specimen are destroyed, resulting in a gradual increase in plastic deformation as a proportion of the deformation of the soil sample [
28].
3.2.2. Changing Law for the Dynamic Elastic Modulus
When the soil is subjected to dynamic loading, nonlinear dynamic strains and dynamic stresses are generated, as shown in
Figure 5 and
Figure 6, and the change rule does not follow Hooke’s law. Therefore, the dynamic elastic modulus is needed to describe the nonlinear characteristics of the soil under dynamic loading. The average values of the dynamic stress and dynamic strain at the apex of the 3~8 level hysteresis loops during dynamic loading were selected to calculate the average dynamic elastic modulus, and according to the average dynamic elastic modulus and average dynamic strain, the E
d-ε
d curves under different confining pressures were plotted, as shown in
Figure 7.
As shown in
Figure 7, the trend for the dynamic elastic modulus development is similar for different confining pressures, and when the frequency is the same, the dynamic elastic modulus increases with increasing confining pressure. This is because, the greater the confining pressure, the smaller the pore ratio between the soil bodies, the greater the soil compactness, the greater the stress needed to produce relative sliding between the soil particles, and the greater the dynamic elastic modulus. The modulus of dynamic elasticity decreases with increasing dynamic strain. In the early stage of development (ε
d ≤ 0.4%), the modulus of dynamic elasticity decreases obviously with increasing dynamic strain and then gradually tends to stabilize. The difference in the modulus of dynamic elasticity between different confining pressures is also reduced.
The E
d-ε
d curves for different frequencies are plotted for certain enclosure pressures, as shown in
Figure 8.
Figure 8 shows that the dynamic elastic modulus increases with increasing frequency at the same confining pressure. This is because when cyclic loads are applied, the higher the frequency, the shorter the vibration period, the less time the load has to act on the soil, and the lower the amount of kinetic energy that is transferred [
27]. When the frequency of load action is higher, the deformation and rebound process for the soil due to the external force is untimely, which leads to a decrease in the rate of increase in the dynamic strain in the soil. Therefore, the dynamic elastic modulus is greater than that of soil with sufficient rebound. At the early stage of deformation of the modified soil (ε
d ≤ 0.4%), the dynamic elastic modulus decreases rapidly, and with a gradual increase in dynamic strain (ε
d > 0.4%), the decrease in the dynamic elastic modulus slows.
The relationship between the dynamic elastic modulus E
d and the dynamic strain ε
d of the soil can be expressed by fitting relationship (3) [
27,
28]:
where
is the initial dynamic elastic modulus and
is the fitting parameter
The fitted curve between the dynamic modulus of elasticity and dynamic strain for PVA-amended expansive soil at a confining pressure of σ
3 = 100 kPa and frequency
f = 1 Hz is shown in
Figure 9, where the specific parameters for the fitted equation are given by
At this time, the curve fitting coefficient is 0.98. According to the fitting results shown in
Figure 9, Equation (3) can provide a more complete expression for the change rule between the dynamic elastic modulus E
d and the dynamic strain ε
d of PVA-amended expansive soil.
The fitted parameters at other frequencies and envelope pressures are shown in
Table 6.
The initial dynamic elastic modulus E0 is the dynamic elastic modulus of the soil when subjected to initial dynamic loading and is usually known as the initial shear modulus or initial compression modulus. This dynamic elastic modulus is an important mechanical parameter reflecting the stiffness of the soil and can be used to assess the stability and load-bearing capacity of the improved soil.
Figure 10 shows the initial dynamic elastic modulus versus the frequency of the enclosing pressure.
Figure 10a shows that the initial dynamic elastic modulus increases with increasing enclosing pressure.
Figure 10b demonstrates that the initial dynamic elastic modulus increases with increasing frequency relative to the enclosing pressure, but the increase trend is slow. Therefore, the circumferential pressure has a large effect on the initial dynamic elastic modulus, while the frequency also has an effect on the initial dynamic elastic modulus, but the effect is small. This is because the increase in the circumferential pressure will lead to the specimen being subjected to a greater lateral constraint force, and the deformation of the specimen under dynamic loading will be smaller, so the initial dynamic elastic modulus will increase.
The effect of the loading frequency on the initial dynamic elastic modulus of the specimen manifests during the loading process, and the initial dynamic elastic modulus is generated during the initial loading of the specimen. Thus, the effect of the frequency on the initial dynamic elastic modulus can be analyzed only within a very small strain interval. When the specimen is subjected to high-frequency loading, at the beginning of the initial dynamic loading, the specimen will experience stress and strain. At this time, the stress–strain curve is similar to a straight line, so the specimen viscosity is relatively small. The initial dynamic elastic modulus increases with increasing frequency, but the effect of frequency on the initial dynamic elastic modulus is relatively small compared to the effect of confining pressure.
3.2.3. Change Rule for the Damping Ratio
Figure 11 shows the curves for the damping ratio
λ versus the dynamic strain
εd calculated by elliptic hysteresis curve fitting for different enclosing pressures.
The damping ratio λ of the soil reflects the proportion of energy dissipated due to the internal damping effect of the soil under dynamic loading and characterizes the ability of the soil to absorb energy and resist vibration. It can be observed from
Figure 11 that the damping ratio
λ of the soil increases with the development of the dynamic strain, and the growth rate is low at the beginning of the development of the dynamic strain (ε
d ≤ 0.4%). When the dynamic strain continues to increase (ε
d > 0.4%), the growth rate of the damping ratio
λ first increases and then decreases and tends to stabilize with increasing dynamic strain. Under the same frequency conditions, the damping ratio gradually decreases with increasing confining pressure, and the development trend of the damping ratio
λ under different confining pressures is more or less the same.
In the early stage of dynamic strain development (εd ≤ 0.4%), the proportion of elastic deformation in the deformation of the soil is greater, and the energy consumption for stress transfer is less. With the development of dynamic strain (εd > 0.4%), relative sliding of the soil particles occurs, the proportion of plastic deformation gradually increases, the number of pores inside the specimen increases, and the work performed by the internal friction resistance increases, resulting in a drastic increase in the damping ratio. When the dynamic strain continues to increase, the internal structure of the specimen changes. Under the applied force, the pore space produced before is partially compacted, and the soil particles are rearranged to form a temporary stable state when the growth rate of the damping ratio begins to decrease. The damping ratio thus stabilizes with the development of dynamic strain.
At a certain frequency, an increase in the circumferential pressure leads to a denser arrangement of internal particles in the specimen, and PVA can be better dissolved into the internal pores of the soil specimen. At this time, the pore space in the sample is reduced, and the contact area of the soil particles inside the sample is increased, which improves the efficiency of dynamic load transfer. The relative motion of soil particles inside the specimen under high-confining pressure is reduced, so the percentage of plastic strain is not as good as that for the specimen under lower confining pressure, so the damping ratio decreases with increasing confining pressure.
The
f-ε
d curves for different frequencies are plotted in
Figure 12. From
Figure 12, it can be observed that the development trends for the
λ-ε
d relationship curves between the damping ratio and dynamic strain at different frequencies and confining pressures are basically the same, and the damping ratio grows less in the early stage of the development of the dynamic strain (ε
d ≤ 0.4%); when the dynamic strain continues to develop (ε
d > 0.4%), the rate of increase in the damping ratio first increases and then decreases. In addition, compared with
Figure 12, when the frequency is the only variable, the damping ratio of the soil decreases with increasing frequency, and the difference in the damping ratio between neighboring circumferential pressures is greater than that between neighboring frequencies, which indicates that the frequency has a small effect on the damping ratio.
The main development trends for the λ-εd relationship curves under different frequencies and different peripheral pressures are basically the same because the development process for soil sample deformation is basically the same, and at the early stage of deformation of the soil samples (εd ≤ 0.4%), the proportion of elastic deformation is greater. As the deformation of the soil samples continues to develop (εd > 0.4%), the proportion of plastic deformation gradually increases, and at this time, the growth rate of the soil damping ratio increases. This indicates that different confining pressures and frequencies affect the magnitude of the damping ratio of the soil but have less influence on the development process of soil deformation.
As the frequency increases, the damping ratio of the soil decreases. This is because, at a certain critical damping coefficient for the soil sample, the magnitude of the damping ratio depends on the amount of energy lost during one cycle. When the control frequency is the only variable, an increase in frequency results in a reduction in the time available for the actuator rod to act on the specimen and less energy being transferred, so the damping ratio decreases.