## 3 Results and Discussion
### Resilient Modulus  
According to the linear relationship between the axial cumulative strain and the axial total strain, a formula can be expressed as:  
\[\varepsilon_{L}=k\varepsilon_{a} \tag{1}\]  
where \(\varepsilon_{L}\) is the axial cumulative strain, \(\varepsilon_{a}\) is the axial total strain, and \(k\) is the ratio of the axial cumulative strain to the axial total strain. Figure 4: Typical stress-strain curve under monotonic loading and cyclic loading. Figure 5: Relationship between axial accumulative strain and axial total strain. Subjected to the monotonic loading, the relationship between the principal stress difference and the recoverable strain can be formulated as:  
\[\sigma_{1}-\sigma_{3}=f(\varepsilon_{a})=g(\varepsilon_{h})=g((1-k)\varepsilon_ {a}) \tag{2}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the principal stresses and \(\varepsilon_{h}\) is the recoverable strain. The resilient modulus of the specimen under the monotonic loading can be defined as:  
\[E_{ur}=\frac{\sigma_{1}-\sigma_{3}}{\varepsilon_{h}}=\frac{\sigma_{1}-\sigma _{3}}{(1-k)\varepsilon_{a}} \tag{3}\]  
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 Figures 6 and 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. Figure 6: Resilient modulus versus axial total strain under different confining pressures. 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:  
\[E_{ur}=ae_{a}^{b} \tag{4}\]  
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:  
Figure 8: Strength and elastic strain corresponding to the resilient modulus. Figure 7: Resilient modulus versus axial total strain under different cement contents. \[\left\{