# Experimental and Numerical Study in Dynamic Compaction of Weakly-Cohesive Soils
## 2 Theoretical Approaches  
### Modeling of Compaction Process with Kelvin-Voigt Model  
The relevant literature (e.g., [25; 26; 27]) demonstrates the behavior of vibratory compaction soils can be interpreted based on rheologic models with lumped or non-linear parameters. The Kelvin-Voigt model is commonly used to simulate the behavior of viscoelastic materials, such as soil during vibration compaction. The model consists of a spring (representing the elastic component) and a dashpot (representing the viscous component) arranged in parallel to capture the elastic response and the time-dependent deformation under load due to viscosity (Figure 2). With the help of the Kelvin-Voigt model, we can predict how soil undergoes permanent settlement due to the viscous component. This aspect puts into evidence a simulation of how materials like soil or asphalt respond under mechanical pressure during compaction [5; 6; 28; 29; 30]. By adjusting its constitutive parameters (soil rigidity k and damping c), the model can be adapted to simulate different types of soils, and each reacts differently to compaction forces' action. Thus, the Kelvin-Voigt model can simulate the strain response of the soil in time considering the following:  
* The initial loading phase in which, as compaction begins, stress is applied to the material and the elastic and viscous responses contribute to the total deformation;
* The phase in which the deformation varies in time due to the change of the viscous term, depending on the state of the material, leading to different deformation speeds;
* The equilibrium phase that occurs after some time if the compaction stress remains constant, the rate of deformation decreases, and the material approaches a deformation in equilibrium, where the elastic response is predominant;
* The discharge phase (if applicable) upon completion of compaction when the material has an elastic recovery, tending to return to its original shape. If we consider that terrain will be modeled by viscoelastic elements with identical characteristics for a certain state and technological regime, then we make it possible to approach the simulation of compaction with the help of a linear viscoelastic rheological system. The effect of the kinematic excitation x(t) = A\({}_{0}\)sinot consists of a transfer of energy into the terrain to produce a dynamic response quantifiable by evaluating the soil force response as force Q(t), as seen in Figure 2.  
Figure 2: Kelvin-Voigt model description. The dynamic equilibrium equation is presented in Equation (1)  
\[\mathrm{m\ddot{x}}+\mathrm{c\dot{x}}+\mathrm{kx}=\mathrm{m_{0}r\omega^{2}sin\, \omega t}, \tag{1}\]  
where the reaction force of the soil can be expressed as \(\mathrm{F_{s}}=\mathrm{Q(t)}=\mathrm{Q_{0}sin(\omega t}-\varphi)\) = \(\mathrm{c\dot{x}}+\mathrm{kx}\) with  
\[\mathrm{Q_{0}}=\Omega^{2}\frac{\mathrm{m_{0}rk}}{\mathrm{m_{0}+m}}\sqrt{ \frac{1+4\Omega^{2}\xi^{2}}{\left(1-\Omega^{2}\right)^{2}+4\Omega^{2}\xi^{2}}}, \tag{2}\]  
where \(\mathrm{p^{2}}=\frac{\mathrm{k}}{\mathrm{m+m_{0}}}\), \(\Omega=\frac{\omega}{\mathrm{p}}\), \(\zeta=\frac{\mathrm{c}}{2\left(\mathrm{m_{0}+m}\right)\mathrm{p}}\). The dynamic transmissibility \(\mathrm{T}\) of motion is defined as follows:  
\[\mathrm{T}=\sqrt{\frac{1+4\Omega^{2}\xi^{2}}{\left(1-\Omega^{2}\right)^{2}+4 \Omega^{2}\xi^{2}}}, \tag{3}\]  
and the amplitude of the motion is  
\[\mathrm{A}=\frac{\mathrm{m_{0}r}}{\mathrm{m_{0}+m}}\cdot\frac{\Omega^{2} \mathrm{A_{st}}}{\sqrt{\left(1-\Omega^{2}\right)^{2}+4\Omega^{2}\xi^{2}}}, \tag{4}\]  
which means that \(\mathrm{A}=\mathrm{A}\) (\(\Omega\), \(\dot{Q}\)) is a function expressed in the relative pulsation \(\Omega\) and the damping ratio \(\zeta\). Thus, the factors affecting the transmissibility in the compaction process are represented by the force transmitted to the ground (static moment of the vibrator, mass of the roller, pulsation) and, respectively, stiffness and damping properties of the terrain layer.