#### 3.1. Axial Cumulative Strain

The axial cumulative strain is one of the important indexes to characterize the deformation ability of the lightweight soil. In this study, the axial cumulative strain is defined as the residual strain when the overlying load is removed. As shown in

Figure 4 for sandy lightweight soil, the axial cumulative strain increases with the increasing cyclic loading times, as well as the axial total strain of the specimen. There exists a good linear relationship between the axial cumulative strain and the axial total strain (see

Figure 5), and the ratio of axial cumulative strain to axial total strain remains constant. The same also applies to the tested silty clay lightweight soil specimens. However, the influence of confining pressure, cement content, and soil type on the ratio is very small through inspecting all of the tested specimens in this study (not shown here).

Under lower confining pressures, the cementing structure (with relatively high cement content) of the mixed lightweight soil plays the main role in bearing the load capacity, and the damage of the specimen is very small. In this condition, the EPS particles are still constrained by the cementation force, indicating that the characteristics of larger elastic deformation cannot be brought into full activity. Although the elastic deformation of the lightweight soil would mainly come from the elastic deformation of the EPS particles, the elastic strain is always at a lower level when the stress state is relatively small. Therefore, the difference of the elastic strains under different conditions is not obvious.

#### 3.2. Resilient Modulus

According to the linear relationship between the axial cumulative strain and the axial total strain, a formula can be expressed as:

where

${\epsilon}_{L}$ is the axial cumulative strain,

${\epsilon}_{a}$ is the axial total strain, and

k is the ratio of the axial cumulative strain to the axial total strain.

Subjected to the monotonic loading, the relationship between the principal stress difference and the recoverable strain can be formulated as:

where

${\sigma}_{1}$ and

${\sigma}_{3}$ are the principal stresses and

${\epsilon}_{h}$ is the recoverable strain.

The resilient modulus of the specimen under the monotonic loading can be defined as:

The resilient modulus here can also be defined as the average of the unloading modulus and the reloading modulus [

24]. The unloading modulus is the ratio of the stress at the unloading point to the recoverable strain (i.e., elastic strain), and the reloading modulus is the ratio of the stress at which it is reloaded to the unloading point to the recoverable strain. Although the elastic strain that is mentioned above is very small, the variation will have a significant influence on the resilient modulus of the lightweight soil.

In

Figure 6 and

Figure 7, the resilient modulus decreases with the increasing axial total strain, and it increases with the increasing confining pressure and cement content. Similar to the dynamic modulus [

25], the resilient modulus will gradually approach a same critical value of 40 kPa, even under different confining pressures and cement contents in this study. Due to the increasing axial total strain, the cementation structure of the mixed lightweight soil is gradually damaged and its strength decreases step by step. Simultaneously, the constraint on the deformation of EPS particles is also reduced, and its elastic deformation is playing an increasingly important role afterwards. When the cementation structure is completely destroyed, the mixed lightweight soil tends to be loose, and whose resilient modulus arrives at a same level.

On the one hand, the increasing confining pressure enhances the restraint on the specimen and then on the EPS particles. On the other hand, the increasing confining pressure enhances the destruction of the cementation structure and thus weakens the restraint on the EPS particles. Therefore, the combination of the two actions increases the strength of the mixed lightweight soil increases, weakens the restraint, but increases the elastic deformation of the EPS particles. Obviously, the multiple of strength increase is larger than the multiple of elastic deformation increase, and this can be easily found in

Figure 8.

With increasing cement content, the cementation becomes stronger, as well as the strength of the specimen. Deformation of the EPS particles is constrained, indicating that the elastic deformation would be reduced, which contributes to the increase of resilient modulus of the lightweight soil. Under the tested confining pressures and mixture ratios, the relationship between the resilient modulus and the axial total strain can be established as:

where

a and

b are all the fitting parameters and they are related to the confining pressure and cement content. Subsequently, the fitting relation can be formulated by the following function:

where

a_{1,2},

b_{1,2}, and

c_{1,2} are all the fitting coefficients related to the confining pressure. Then, a general formula can be deduced as:

where

${\sigma}_{3}$ is the confining pressure, α,

β, and

γ are also the fitting coefficients, which can be deduced in

Table 2.

As illustrated in

Figure 9, the resilient modulus of sandy lightweight soil is higher than that of silty clay lightweight soil under confining pressures of 50 kPa and 150 kPa. The same condition also applies to the strength and recoverable elastic strain of the lightweight soil samples (as shown in

Table 3). For the basic physical characteristics, the granular sand has smaller specific surface area and less activity than the silty clay. When cement is added, the granular sand and the cement have a rapid cementation speed with the help of water. The more cemented material is produced, the higher cementation strength would be generated to constrain the deformation of the EPS particles, resulting in smaller elastic deformation and higher resilient modulus. Although the silty clay is flattened with strong activity, the cement’s hydrolysis and the hydration reactions mainly happen around a certain active medium. Therefore, the reaction is slow but needs a long time, and less cementing material is produced within a certain curing time. Moreover, the cemented silty clay has a lower strength to restrict the EPS particles, resulting in a lower strength than the sandy lightweight soil, as well as the resilient modulus. Due to the existence of a layer of bound water film around the fine-grained soil particle, silty clay has better deformation adaptability than sand, indicating that the silty clay lightweight soil has lower strength and resilient modulus than that of the sandy lightweight soil.

As shown in

Figure 9a, with a lower confining pressure of 50 kPa and a smaller cement content of 14%, when the axial total strain of the specimen exceeds 8%, the resilient modulus of the sandy lightweight soil is smaller than that of the silty clay lightweight soil. This is rather unusual in the overall condition of these tested samples. When the shear deformation of the specimen becomes larger (i.e., the axial total strain exceeds 8%), the degree of particle breakage increases, and the sandy lightweight specimen starts to loosen, resulting in a decrease in the cohesion, as well as the resilient modulus. Although the same thing happened to the silty clay lightweight soil, the above reduction in cohesion and resilient modulus under the same shear deformation are relatively smaller than the sandy soils. However, such a phenomenon does not exist when the confining pressure and cement content are higher, indicating that the constraint of higher confining pressure excels the loss of cohesion that is caused by shear deformation for sandy lightweight soil, and the higher cement content contributes to the increasing of cementing force in a short term. Within a limited curing time (7 days or 14 days), the resilient modulus of the sandy lightweight soil is usually larger that of silty clay type mixture (see

Table 3).

#### 3.3. Damping Ratio

According to the conventional definition [

26], the damping ratio is related to the energy loss rate in a certain time, and the formula can be expressed as:

where

λ is the damping ratio, ∆

W is the loss energy within a loading-unloading cycle, and

W is the total energy of a complete loading-unloading cycle.

In this study, the damping ratio is calculated by the energy loss rate in a loading-unloading cycle. The typical stress-strain relationship of the tested specimen can be illustrated in

Figure 10. In a complete loading-unloading cycle, the work that is done by extra loads can be defined as: at the beginning OA section, most of the work done by external load is converted into elastic potential energy, indicating that the specimen’s deformation is mainly attributed to the elastic deformation. Subsequently, in the second AB section, plastic deformation is produced in the specimen and it becomes larger and larger. Although, the work done by external load on the specimen is mainly consumed by the plastic deformation and the viscous resistance, there will still be a small part stored as the elastic potential energy. Within the BC section, when the external load is removed, the elastic potential energy that is stored in sections of OA and AB will be gradually released and absolutely consumed by viscous resistance.

According to the general form of stress-strain relationship and energy composition analyses, the EPS beads-mixed lightweight soil belongs to the viscoelastic plastic material. The ∆

W and

W in Equation (7) can be calculated by the following expression:

where

S_{0DA},

S_{AEB},

S_{BFC},

S_{AGIH}, and

S_{0DAEBFC0} are the areas of different regions in

Figure 10. In Equation (8), the former three items are the energy consumed by damping, the last one is the energy consumed by plastic deformation. Afterwards, the damping ratio can be calculated out.

Figure 11 shows the variation of damping ratios versus axial total strains under different confining pressures, cement contents for different soil types. The damping ratio increases slowly with the increasing axial total strain, but its value ranges from 0.07 to 0.08. However, the damping ratios of the lightweight soil obtained by Gao et al. [

15] have a larger variation range from 0.05 to 0.20 when the strain increases from 1% to 10% (see

Figure 11d). This is mainly due to the different test methods and the corresponding formation of tested lightweight samples, with lower cement content and higher volume occupancy of EPS beads. Although the difference of damping ratios are relatively small under different conditions in this study, there are still some characteristics that can be detected. For example, the sandy lightweight soil has lower damping ratio than the silty lightweight soil, and larger confining pressure and higher cement content obtain smaller damping ratios.

There are two kinds of damping in soils, one is the dissipation damping, and the other is the material damping. The former is caused by the diffusing of energy that is accumulated in the soil to the outside world in forms of surface wave and body wave, while the latter is generated from the friction between particles and the viscosity of pore water and air. However, this study is not focusing on the dynamic problem under the action of high frequency. The wave is not considered and the dissipation damping can be assumed to be zero, then the damping can be deduced to be a completely material damping. Actually, the bonding strength of the specimen is relatively strong due to the adopted cement contents, so there is little difference between the dislocation and slip of particles under the confining pressures and the strain levels in this study.

Although little differences in hydration products, pore water, and pore gas have been detected, there are still some other points worth analyzing. The larger the strain is, the greater the damage of the specimen is. The greater the dislocation and slip between particles are, the greater the friction between particles is, and the greater the material damping ratio would be. With increasing confining pressure, the specimen is compressed denser, and then the particles’ dislocation and slip are decreased with smaller friction, as well as the smaller damping ratio. The same situation is still applicable to the cement content. From the previous analyses, more cementing substances are produced in the sandy lightweight soil than in the silty clay lightweight soil for a certain curing time. Therefore, the interaction between grains of sandy lightweight soil is closer and the filling degree of the void is larger. Moreover, the dislocation, slip, and friction between grains are reduced, indicating that the discharge of pore water and pore gas in the lightweight soil is increased, and then a smaller material damping ratio is obtained.