# Experimental and Numerical Study in Dynamic Compaction of Weakly-Cohesive Soils
## 2 Theoretical Approaches
### Modeling of Compaction Process with Advanced Kelvin-Voigt Model  
Therefore, the Kelvin-Voigt Model can describe the predominantly elastic behavior of the terrain, characteristic of vibration compaction works. However, if one also desires to simulate the viscous dissipative phenomenon, then a new rheological model with viscous-elastic and dissipative behavior is needed. The Coulomb friction theory is widely used to describe frictional behavior in soils. Weakly cohesive soils can exhibit friction that does not conform only to [PERSON]'s simple model due to their specific characteristics. Other friction concepts relevant to weakly cohesive soils apart from Coulomb are non-Coulomb friction (for silts or loans), non-linear friction because of capillary forces that influence the friction between particles, viscous friction (in saturated soils), adhesion and Van der Waals forces that act in addition to normal friction (in clays or other fine-grained soils), time-dependent friction (in the case of some weakly cohesive soils, especially those with high clay content), etc. [17]. The proposed model considers only the Coulomb friction between soil particles. Thus, an additional elastic spring with stiffness \(\mathrm{k_{p}}\) and a Coulomb friction element \(\mathrm{F_{fr}}\) (with the dynamic force kept constant relative to velocity [31]) were introduced, connected in series with the generalized Kelvin-Voigt model (Figure 3). The spring deforms proportionally during the loading phase (compression) but remains locked otherwise (during unloading and loss of contact) and simulates the plastic deformations during the loading phase. Hence, a model is proposed to describe the viscous-elastic and dissipative behavior, with emphasis on the slow settlement of weakly cohesive soils subjected to the cyclic loads transmitted by the vibratory roller. With this model, it is possible to simulate the complex rheological behavior of the interaction between the vibratory roller and the terrain under various excitation laws, where the descriptive parameters can vary over a wide range. Accordingly, the force transmitted to the terrain can be expressed by the following equation:  
\[\mathrm{F_{s}}=\mathrm{F_{ve}}+\mathrm{F_{p}}, \tag{9}\]  
Figure 3: Rheological model for the simulation of roller–terrain interaction in the vibratory compaction process. where \(F_{\rm{ve}}\) represents the viscous-elastic force (\(F_{\rm{ve}}=\rm{kx}+c\dot{x}\)) of the model, and \(F_{\rm{p}}\) is the plastic force with the expression  
\[F_{\rm{p}}=\left\{ \begin{matrix}k_{\rm{p}}x_{\rm{p}}+F_{\rm{fr}}{\rm{sign}}v\; \;{\rm{for}}\;\;F_{\rm{e}}>F_{\rm{p}},\\ 0\;\;{\rm{for}}\;\;F_{\rm{e}}\leq F_{\rm{p}}\end{matrix} \right., \tag{10}\]  
where \(k_{\rm{p}}\) represents the plastic stiffness, \(x_{\rm{p}}\) is the plastic deformation of the terrain, \(F_{\rm{fr}}\) is the friction force, and sign is the signum function. The characteristic equation for modeling the friction force \(F_{\rm{fr}}\) (see Figure 4) as a mathematical formulation for Matlab implementation is  
\[F_{\rm{fr}}=\left[F_{\rm{c}}+(F_{\rm{brk}}-F_{\rm{c}})e^{-c_{\rm{v}}|v|} \right]{\rm{sign}}(v)+f_{\rm{v}}v, \tag{11}\]  
where \(F_{\rm{brk}}\) represents the breakaway friction force, \(F_{\rm{c}}\) is the Coulomb friction force, \(F_{\rm{v}}\) is the viscous friction force, \(f_{\rm{v}}\) is viscous friction coefficient, \(v=v_{\rm{R}}-v_{\rm{C}}\) is the relative speed between the two connection points of the model (R, respectively C, on the diagram in Figure 5), and \(c_{\rm{v}}\) is a coefficient depending on the velocity of load application. The behavior of the advanced Kelvin-Voigt model is nonlinear due to the presence of the element simulating friction. The mathematical solution can be obtained either through specific numerical methods to approximate the result or by implementing the model in specialized software to carry out the computational calculations. Therefore, the authors  
Figure 4: Model for friction force \(F_{\rm{f}}\). Figure 5: Block diagram for friction force modeling in Matlab.