## 2 Literature Review
### Shear Strain  
As shown in Table 1, shear strain is the most important parameter effecting the damping ratio, according to the presented references [11; 12; 13; 14]. This issue is connected with the so-called threshold strains. In soil dynamics, we can specify three soil reaction zones for loading. Each zone is separated from each other by threshold strains. The first zone, small strain, is characterized by the linear elastic response. In this strain range, the damping ratio has minimum constant values (\(D_{MIN}\)), in contrast to the shear modulus, which takes maximum values (\(G_{MAX}\)). The strain amplitude at which shear modulus decreases to 98% of its maximum value and \(D/D_{MIN}=1.02\) is called the nonlinearity threshold strain (\(\gamma_{t}^{c}\)) [16; 17]. Above \(\gamma_{t}^{c}\), soil behaves nonlinearly but is still elastic. It means that the stress-strain characteristic is curved. However, deformations are recoverable upon unloading [14]. In this zone, increases of damping ratio \(D\) and decreases of shear modulus \(G\) are observed. The degradation threshold strain \(\gamma_{t}^{c}\)[16; 17] is reached when shear modulus decreased to about 80% of the \(G_{MAX}\), and the damping ratio was about 3% higher than the \(D_{MIN}\)[18]. After crossing \(\gamma_{t}^{c}\) deformation becomes irrecoverable.  
It is common practice to replace strain amplitude by normalized shear modulus in equations for damping ratio (Equation (1)) because \(G/G_{MAX}\) values refer to different soil reaction zones: 1-0.98 linear elastic, 0.98-0.80 nonlinear elastic, \(<\)0.8 plastic. Furthermore, it is easier to use from an engineering point of view. In Figure 1, the definition of threshold strains based on \(D\)-\(D_{MIN}\) and normalized \(D/D_{MIN}\) curves is presented.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**Importance to Damping**} & \multirow{2}{*}{**Increasing Parameters**} & \multirow{2}{*}{**Effect on Damping Ratio, According to [13]**} \\  & **Ratio, According to** & & & \\  & **[12]** & **[14]** & & \\ \hline
**Strain amplitude** & *** & *** & Confining pressure (\(\sigma_{0}\)) & Stays constant or decreases with \(\sigma_{0}\) \\ Effective stress & *** & *** & Void ratio (\(e\)) & Decreases with \(e\) \\ Void ratio & *** & * & Geologic age (\(t_{g}\)) & Decreases with \(t_{g}\) \\ Number of loading cycles & *** & ***+ & Cementation (\(\varepsilon\)) & May decrease with \(c\) \\ Degree of saturation & - & * & Overconsolidation ratio & Not affected \\ Overconsolidation ratio & ** & * & Plasticity index (\(PI\)) & Decreases with \(PI\) \\ Frequency of loading & ** & ** & Cyclic strain (\(\gamma_{c}\)) & Increases with \(\gamma_{c}\) \\ Time effect & ** & - & Frequency of loading (\(f\)) & Stays constant or may increase with \(f\) \\ Grain characteristic & * & * & Number of loading cycles & Not significant for moderate \(\gamma_{c}\) and \(N\) \\ Soil structure & * & - & (\(N\)) & Not significant for moderate \(\gamma_{c}\) and \(N\) \\ Soil type and plasticity & - & *** & & \\ \hline \hline \end{tabular}  
* very important, *
* not studied.  
\end{table}
Table 1: Importance of various parameters to damping ratio [12; 14] and the effect of the increase of various parameters on damping ratio [13] of cohesive soils.  
Figure 1: Definition of threshold strains based on \(D\)-\(D_{MIN}\) and normalized \(D/D_{MIN}\) curves.