Investigation of the relationship between dynamic and static deformation moduli of rocks

Abstract

The determination of deformation parameters of rock material is an essential part of any design in rock mechanics. The goal of this paper is to show, that there is a relationship between static and dynamic modulus of elasticity (E), modulus of rigidity (G) and bulk modulus (K). For this purpose, different data on igneous, sedimentary and metamorphic rocks, all of which are widely used as construction materials, were collected and analyzed from literature. New linear and nonlinear relationships have been proposed and results confirmed a strong correlation between static and dynamic moduli of rock species. According to rock types, for igneous rocks, the best correlation between static and dynamic modulus of elasticity (E) were nonlinear logarithmic and power ones; for sedimentary rocks were linear and for metamorphic rocks were nonlinear logarithmic and power correlation. Moreover, with respect to different published linear correlations between static modulus of elasticity (E stat ) and dynamic modulus of elasticity (E dyn ), an interesting correlation for rock material constants was established. It was found that the static modulus of elasticity depends on the dynamic modulus only with one parameter formula.

Wave Propagation in Rocks – Investigating the Effect of Rheology

“…A consequence of the second law of thermodynamics is that dynamic values must be larger than static ones. This prediction has been confirmed recently for many various types of rock [12][13][14]. For the Hooke model with the force equilibrium assumption (i.e., neglecting the material time derivative of velocity in the Cauchy momentum equation), analytical calculations are also available for simple yet relevant -symmetric enough -geometries and boundary conditions.…”

“…Namely, when measuring Young's modulus (or, in 3D, the two elasticity coefficients) of a solid, the speed of uniaxial loading, or the frequency of sound in a wave-based measurement, may influence the outcome and a sufficient interpretation may come in terms of a PTZ model. Indeed, in rock mechanics, dynamic elastic moduli are known to be larger than their static counterparts [13][14][15], in accord with the thermodynamics-originated inequality in Eq. (32b) (or its 3D version).…”

Four Spacetime Dimensional Simulation of Rheological Waves in Solids and the Merits of Thermodynamics

“…Accordingly, the wave propagation speeds follow. This, on one side, illustrates how the PTZ model can interpret that the dynamic elasticity coefficients of rocks are larger than their static counterpart [ 1 , 2 , 3 , 4 ]. On the other side, the nontrivial—frequency dependent, therefore, dispersive—wave propagation indicates that the numerical solution of PTZ wave propagation problems should contain the minimal possible amount of dispersion error, in order to give account of the dispersive property of the continuum model itself.…”

“…This viscoelastic/rheological reaction may not be simply explained by a viscosity-related additional stress (the Kelvin–Voigt model of rheology), but the time derivative of stress may also be needed in the description, with the simplest such model being the so-called standard or Poynting–Thomson–Zener (PTZ) one [see its details below]. Namely, the PTZ model is the simplest model that enables describing both creep (declining increase of strain during constant stress) and relaxation (declining decrease of stress during constant strain), as well as the simplest one, via which it is possible to interpret that the dynamic elasticity coefficients of rocks are different from, and larger than, their static counterpart [ 1 , 2 , 3 , 4 ]. Related to the latter aspect, high-frequency waves have a larger propagation speed in PTZ media than low-frequency ones [ 4 ], which makes this model relevant for, e.g., seismic phenomena and acoustic rock mechanical measurement methods.…”

Elastic and Frictional Properties of Fault Zones in Reservoir-Scale Hydro-Mechanical Models—A Sensitivity Study

“…However, mechanical properties like Young's modulus and unconfined compressive strength (UCS) derived from lab tests are usually significantly larger (1.5×-10×) than the corresponding values of a larger rock mass [24,37]. This scale-dependence can be accounted for by using empirical correlations, but this inevitably adds uncertainties to the material properties assigned to the fault zone in the numerical model [38,41,42]. If no cores are available, an option is to calculate dynamic Young's modulus from well logs (p-and s-wave velocities, density).…”