# Experimental and Numerical Study in Dynamic Compaction of Weakly-Cohesive Soils
## 2 Theoretical Approaches
### Modeling of Compaction Process with Advanced Kelvin-Voigt Model
implemented a block diagram in the Matlab environment (Figure 5) capable of solving the differential equations motion (ODEs) of the vibratory roller in the compaction process. The working principle during a cycle \"\(i\)\" of the action of the dynamic force of the vibratory roller (corresponding to the contact duration \(t_{ci}\) during a period of action of the load) consists of the following steps of dynamic behaviors of the advanced model:  
* Dynamic behavior in the elastic-plastic deformation regime of the material: From the moment the excitation force exceeds the threshold value of the static friction force, the model highlights both the elastic and plastic deformations that appear in the rheological elements of the composition. This movement is maintained until the instantaneous value of the excitation force begins to decrease and reaches the threshold value of the friction force. * Dynamic behavior in the regime of plastic deformation of the material: upon the further decrease of the excitation force, the ground stores residual deformation, which represents the corresponding initial settlement obtained during the work cycle \"\(i\)\". * Dynamic behavior in the regime of elastic return of the material: When the value of the excitation force falls below the limit value of the consolidation (settlement) force, then the behavior of the material consists of an elastic return (movement that occurs because the value of the force accumulated in the elastic element increases above the value of the threshold of the frictional force in the dissipative element and in this way conditions it to move in the direction of the relaxation of the elastic element). This regime of returning to a state of structural equilibrium, called the normalization regime, evolves until the moment when the value of the excitation force begins to increase and again equals the limit value of the consolidation force. * Dynamic behavior in the material's consolidated plastic regime: After the moment when the excitation force becomes equal to the material's consolidation force, it no longer allows the appearance of deformations in its structure. The duration of this regime is completed when the exciting force of the roll reaches the threshold value of the static friction force specific to the material. When the consolidation limit on the considered cycle \"\(i\)\" is not obtained over the entire range of action in which the exciting force has values lower than the threshold value of static friction, it follows that the material changes its behavior specific to the elastic-plastic regime directly in a behavior specific to the consolidated plastic regime. * Cycle \"\(i+1\)\" is carried out in the same way as previously described, only it must be specified that, at the completion of each complete cycle, the value of the consolidation limit force will increase by the value \(\Delta F\) (different according to the type of material being compacted) until the limit value of the static friction force is reached. At that moment, the consolidation process is completed, and the material continues to take over the external loads (coming from the action of the compactor roller) only with its elastic component. The stiffness of the spring \(k_{p}\) is estimated by determining the permanent plastic deformation of the soil due to a static roller pass in the field [32]:  
\[k_{p}=\frac{F_{s}}{z_{p,L}}, \tag{12}\]  
where \(z_{p,L}\) is the accumulative plastic deformation gained during the loading phase, which equals the plastic deformation of the soil after the roller pass. This means that the in situ stiffness evaluation is greater than the stiffness simulating the elastic soil behavior with the help of the rheologic model. The goal of the compaction process is to increase soil stiffness to the level corresponding to the maximum degree of compaction.