Time dependent behavior of rock has long been a subject of interest for the interpretation of geological phenomena. Surprisingly, however, little consideration has been given to the possible consequences of time-dependent rock behavior in the large class of engineering problems which deal with the construction and long-term stability of structures in and on rock. Most likely, the time-dependent properties of rock have not been included in engineering design for three reasons: (!) Time-dependent rock deformations monitored under constant stress conditions in the laboratory are seemingly small. (2) It is difficult to distinguish time-dependent rock deformation in situ from ground motions which are due to external changes, i.e. changes which are unrelated to the material properties. (3) Presently available descriptions of time-dependent rock properties are fragmentary and inadequate for use in design calculations. In addition, field monitoring programs and laboratory experiments which are needed to furnish such descriptions are complex, time consuming and costly.
The time dependent behavior of rock may be divided into time dependent deformation without a loss in load bearing ability on one hand and time-dependent fracture on the other. The time-dependent deformation behavior is generally described in terms of observations which are made under conditions of constant unidirectional stress at different but fixed temperatures and pore water pressures. Accordingly, the measured time dependent strain history is divided into primary (decelerating), secondary ("steady-state") and tertiary (accelerated) stages. The tertiary stage always terminates in fracture and establishes the link with the phenomenon of time-dependent failure. Time-dependent rock deformation can also be divided into recoverable and permanent components. Both are observed when the load acting on a specimen is removed or reduced. The complete deformation history of rock which is necessarily coupled with large irreversible strains upon load removal, is observed at high stress levels and/or at elevated temperatures and pore water pressures. Only the primary state develops at low stress and temperatures.
To model any one of the above phenomena a considerable number of expressions have been proposed [1 - 22]. All of them are based on one of the following approaches. It is assumed that the time-dependent behavior of rock is linear in stress and, therefore, can be represented by linear rheological models consisting of linearly elastic (spring) and viscous (dahspot) elements. Alternatively, time-dependent rock deformation is interpreted by means of physical theories on the atomic, microscopic or macroscopic scale using the concept of thermal activation energy. Finally, a purely empiricial approach may be chosen where curves are fitted to experimental data.
A small but representative sample of existing time-dependent stress-strain models is listed in Equations 1 through 6. All of the equations shown define the variation of the strain parallel to the direction of the numerically greatest stress only. (Mathematical Equation)(Available in full paper) k1 through k6 are experimentally determined constants. Ao denotes the instantaneous rock response which is associated with any stress increment. A 2 is the rate of strain per unit time during secondary creep in constant stress experiments. The remaining terms in Equations (1) to (6) define the primary stage of the strain history under different but fixed loading conditions.