# Resilient Modulus Characterization of Compacted Cohesive Subgrade Soil
## 1 Introduction
In the small-strain, elastic zone the shear modulus is calculated using Equation (1):  
\[G=\rho\cdot V_{S}^{2}, \tag{1}\]  
in which \(\rho\) stands for soil density and \(V_{s}\) stands for the shear wave speed. The shear wave speed is estimated using, e.g., the bender element test (BE). The bender elements are piezoelectric cantilever strips, which are placed on top of the soil specimen's bottom side. The electric signal produces compressional (P) and shear (S) waves. The wave produced by the bender element propagates through the soil sample and induces a voltage in the second bender element. The wave propagation data recorded by the emitter and receiver of the bender element as a function of time leads to estimating the shear and compression wave velocities [28]. The repeated loading conditions in which the plastic strain occurs takes place in intermediate and large strain zones. The resilient modulus \(M_{r}\) is based on elastic theory, although subgrade materials, themselves, are not elastic. If the load is small enough, after a large number of repetitions, the soil can behave in an elastic manner; however, the deformation is nearly fully, but not fully, recoverable [29; 30; 31; 32]. The phenomenon of plastic deformation decreases during cyclic loading and is connected with the shakedown concept, while the state after numerous cycles, in which no permanent deformation occurs, is called the \"resilient state\". This specific elastic state is characterized by resilient modulus \(M_{r}\). The influence of such factors as confining pressure, deviator stress, moisture, saturation degree on resilient modulus value was reported by many studies [33; 34; 35; 36; 37; 38]. The conditions under which the soil subgrade works are characterized by long-term repeating loads. The \(M_{r}\) value is calculated using Equation (2):  
\[M_{r}=\frac{\sigma_{max}}{\Delta\varepsilon_{r}}, \tag{2}\]  
in which \(\Delta\sigma\) stands for the deviator stress and \(\Delta\varepsilon_{r}\) stands for the recoverable strain. The resilient modulus value can be obtained by repeat loading triaxial tests of the tested soil. Numerous methods and numerical models have been proposed in order to obtain the \(M_{r}\) value [39; 40]. One of them is the \(k\)-\(\theta\) model, called the \"Uzan-Witczak model\", which describes the resilient modulus, characteristic for varying confining pressure [41]. This model is applicable for various types of soil. Its coefficients (\(k_{1}\), \(k_{2}\), and \(k_{3}\)) for a certain type of soil remain the same, in regard to the stress state. The Uzan-Witczak model equation is presented in the Equation (3):  
\[M_{r}=k_{1}p_{a}\bigg{(}\frac{\theta}{p_{a}}\bigg{)}^{k_{2}}\bigg{(}\frac{q_{ max}}{p_{a}}\bigg{)}^{k_{3}}, \tag{3}\]  
in which \(\theta\) stands for the bulk stress \(\theta=\sigma_{x}+\sigma_{y}+\sigma_{z}=3\sigma_{c}+\Delta q\), where \(\sigma_{c}\) is confining pressure, \(\Delta q\) is stress magnitude, \(p_{a}\) stands for the atmospheric pressure (normalizing factor), \(q_{max}\) is the stress deviator equal to \(q_{max}=\sigma_{1}-\sigma_{3};k_{1}\), \(k_{2}\), and \(k_{3}\) are regression constants that are a function of the soil properties [42]. The resilient modulus value of granular materials is dependent from several parameters, among which the most important are the stress level, confining pressure and moisture content, the \(M_{r}\) value decreases along with the increase in water content [43]. The \(M_{r}\) value in certain, constant physical soil conditions, changes with strain level. A method for describing this phenomena is degradation curve which presents a change of the modulus at a given strain level to the maximal material modulus [44]. The difference between the elastic moduli in small-strain and plastic zones is presented in Figure 1.  
In this article, the resilient modulus \(M_{r}\) characteristic for cohesive soils was presented and a new analytical model for resilient modulus calculation is proposed.