Static and dynamic moduli differ a lot, also in rocks with no dispersion. In particular, the static:dynamic stiffness ratio is very sensitive to the stress path. The ability to model this behavior is important for several applications, for instance in situations where static moduli are needed while only dynamic moduli are accessible by measurements. This paper discusses the impact of friction controlled sliding along grain contacts and crack surfaces. A classical grain pack model which incorporates this effect is found to describe several of the main characteristic features of the static:dynamic relations, suggesting that friction controlled sliding is a dominating effect. Some discrepancies between model and observations can probably be related to the regular structure of the grain pack model versus the irregular microstructure of a real rock.
The term "static moduli" is commonly referring to the set of stiffness parameters defined as a ratio between an applied stress increment and the resulting strain increment, while "dynamic moduli" is referring to the set of stiffness parameters that can be derived from density times an elastic wave velocity squared. Corresponding static and dynamic moduli are equal for a linearly elastic material. For heterogeneous materials like porous rocks, static and dynamic moduli may differ significantly [1,2]. There are several potential causes for this:
a) Strain rate. Deformation of a rock may involve several processes that require some time to complete. The rock's resistance against deformation may therefore depend on the strain rate. The relevant strain rate for a dynamic modulus is primarily given by the frequency of the elastic wave. The strain rate associated with a static modulus depends on the loading conditions. For a standard laboratory test, the "static" strain rate corresponds to an elastic wave with frequency in the lower seismic range .