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
}= α(σ_{m}/1
kPa)^{β}, where the α factor
is the velocity at 1 kPa, and the β 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 α and
β 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 α and β parameters
should be linked to soil compressibility Ϲ_{Ϲ}.
Compiled experimental results show that the α factor
decreases and the β exponent increases as soil compressibility Ϲ_{Ϲ} increases, and there is a robust inverse
relationship between α and
β for all sediments: β ≈0.73
– 0.27 log [α/(m/s)]. Velocity data for a jointed rock
mass show similar trends, including a power-type stress-dependent velocity and inverse
correlation between α and
β; however, the α-β trend for jointed rocks plots above the
trend for soils.