Small-Strain Stiffness, Shear-Wave Velocity, and Soil Compressibility

The small-strain shear modulus depends on stress in uncemented soils. In effect, the shear-wave velocity, which is often used to calculate shear stiffness, follows a power equation with the mean effective stress in the polarization plane $V_s=\alpha {\left({\sigma }_m^{\prime }/1\hspace{0.17em}\text{kPa}\right)}^{\beta }$, where the $\alpha$ factor is the velocity at 1 kPa, and the $\beta$ exponent captures the velocity sensitivity to the state of stress. The small-strain shear stiffness, or velocity, is a constant-fabric measurement at a given state of stress. However, parameters $\alpha$ and $\beta$ are determined by fitting the power equation to velocity measurements conducted at different effective stress levels, so changes in both contact stiffness and soil fabric are inherently involved. Therefore, the $\alpha$ and $\beta$ parameters should be linked to soil compressibility $C_C$. Compiled experimental results show that the $\alpha$ factor decreases and the $\beta$ exponent increases as soil compressibility $C_C$ increases, and there is a robust inverse relationship between $\alpha$ and $\beta$ for all sediments: $\beta \approx 0.73-0.27\log \left[\alpha /\left(\text{m}/\text{s}\right)\right]$. Velocity data for a jointed rock mass show similar trends, including a power-type stress-dependent velocity and inverse correlation between $\alpha$ and $\beta$; however, the $\alpha \text{-}\beta$ trend for jointed rocks plots above the trend for soils.