# Predictive Models for Correlation of Compaction Characteristics of Weakly Cohesive Soils
## 4 Modeling and Simulation of the Compaction Process
### Numerical Simulation of the Soil Settlement in Compaction Process  
This study investigates the numerical simulation of soil response to static and harmonic inertial dynamic excitation, with a particular focus on the behavior of rollers and the amplitude of vibrations transmitted through poorly cohesive fill soils. By simulating the dynamic interactions between the compactor and the soil, insights are gained into how different soil stiffness levels and damping characteristics influence the overall compaction process and vibration transmissibility.  
The Voigt-Kelvin model is employed for its suitability in modeling the dynamic response of elastic and elastoplastic soils under vibratory loads, capturing the minimal impact of natural viscosity on stress-strain behavior [37; 38; 39; 40; 41]. Through this approach, the study aims to better understand the compactor's effect on various soil conditions and optimize parameters for effective soil compaction. Thereby, a vibratory compactor roller is modeled as a vibrating system based on the Voigt-Kelvin model (Figure 3), with soil stiffness and vibration frequency ranging up to 500 rad/s.  
Main parameters include the viscous damping coefficient c and damping ratio c, which vary to reflect the compaction process. Each pass of the compactor results in intermediate stages of soil layer compaction, where stiffness changes incrementally until the target compaction degree D [%] is achieved. For each level of soil stiffness k\({}_{i}\) reached after a pass, a corresponding damping coefficient c reflects the soil's response to the compactor's dynamic force. The relationship between c and k is given by \(c=2\zeta\sqrt{km}\), with a damping ratio \(\zeta\leq 0.7\), spanning values from 0.16 to 0.7 in the next simulations [13].  
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\({}_{0}\)sin\({}^{\prime}\)t consists in a transfer of energy into the terrain, in order to produce a dynamic response quantifiable by evaluating the soil force response as force Q(t), as seen in Figure 3.  
The dynamic equilibrium equation is presented in Equation (1):  
\[\mathrm{m\ddot{x}}+\mathrm{c\dot{x}}+\mathrm{kx}=\mathrm{m_{0}r\omega^{2}sin \,\omega t}, \tag{1}\]  
Figure 3: Voigt–Kelvin model description.  
where the reaction force of the soil can be expressed as \(\rm{F_{s}=Q(t)=Q_{0}sin(\omega t-\varphi)=cx+kx}\), with  
\[\rm{Q_{0}=\Omega^{2}\frac{m_{0}rk}{m_{0}+m}\sqrt{\frac{1+4\Omega^{2}\zeta^{2}}{ \left(1-\Omega^{2}\right)^{2}+4\Omega^{2}\zeta^{2}}}}, \tag{2}\]  
where \(\rm{p^{2}=\frac{k}{m+m_{0}}}\), \(\Omega=\frac{\omega}{P}\), \(\zeta=\frac{c}{2(m_{0}+m)p}\).  
The transmissibility of dynamic forces determines how effectively the energy generated by a roller is transferred into the soil, ensures uniform compaction, increasing the soil density and stiffness throughout the compacted volume, and is defined as follows:  
\[\rm{T=\sqrt{\frac{1+4\Omega^{2}\zeta^{2}}{\left(1-\Omega^{2}\right)^{2}+4\Omega ^{2}\zeta^{2}}}}. \tag{3}\]