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An Introduction to Nonlinear Behavior of Rock
Over the last two decades, studies of nonlinear dynamics in materials, known as nonclassical or anomalous that include rock, damaged materials some ceramics, sintered metals, granular media etc., have increased markedly. These materials exhibit what we term nonequilibrium dynamics at elevated strain amplitudes (>~10^{6}). Specifically, when the material is disturbed by a wave, the modulus decreases. We call this nonlinear fast dynamics. Following this, it takes tens of minutes to hours to return to its equilibrium state. This is called slow dynamics. Further, the apparent mixture of fast and slow dynamics known as conditioning that takes place during nonlinear fast dynamics provides additional complexity not observed in materials whose nonlinearity is due to anharmonicity. The nonequilibrium dynamics is due to mechanically 'soft' inclusions (soft matter) in a 'hard' matrix. For instance, a crack in a solid will induce nonequilibrium dynamics, but a void will not; a sandstone exhibits nonequilibrium dynamics due to distributed soft inclusions, also known as the bond system, but a bar of aluminium does not. Experimental methods and theory have been developed to interrogate nonequilibrium dynamics in solids.
Fundamentally, elastic nonlinearity implies that the stressstrain relation (also known as the equation of state, EOS) is nonlinear. For such a relation, the onedimensional stress (σ)strain (ε) can be described by
where K_{o} is the linear modulus, and β and δ are the first and second order classical nonlinear parameters, normally of order 1^{10} in value. At low dynamic wave amplitudes (strains of less than order 10^{6} under ambient pressure), there is evidence that all (or at least most) solids behave in a manner according to the above equation. At ambient pressure and temperature conditions, for wave amplitudes above approximately 10^{6} strain, the material EOS is thought to be hysteretic. A hysteretic EOS relation as derived by Guyer and McCall known as the PM Space model is,
where α is the hysteretic nonlinear parameter and its sign is a function of the strain derivative ∂u/∂x due to the hysteresis. Eq. (2) is a practical estimate of the dynamics, especially for NDE applications, but does not capture the entirety of nonequilibrium dynamics: the slow dynamics and material conditioning as outlined in Figure 1. As previously noted, slow dynamics means the material takes time to return to its rest state modulus K_{o} relaxing as the logarithm of time. An example of conditioning is as follows: if a rock sample is driven at fixed amplitude for a period of time, the modulus will decrease immediately with the onset of the wave, but then continue to decrease slightly to a new equilibrium value as long as the drive is maintained. Conditioning is a small effect in most materials as can be seen in Figure 1b. It may or may not be correct to think of conditioning as a mix of fast and slow dynamics. In any case, Eq. (2) has been applied broadly to describe the material elastic nonlinearity. The ratedependent effect of conditioning appears to have only a minor influence on estimates of α.
Figure 1  Nonequilibrium dynamics for two types of forcing. The figure illustrates the full nonequilibrium dynamics that includes nonlinear fast dynamics (also known as nonclassical or anomalous nonlinear dynamics), conditioning, and slow dynamics. Figures (a) and (b) show how nonequilibrium dynamics are manifest when a lowamplitude, continuouswave (cw) probewave is input into a sample in the presence of a large amplitude vibration. One sees in (a) the undisturbed probe wave (time A) and the corresponding timeaverage amplitude of the signal in (b) ['cw probe']. At time B, a highamplitude vibration begins and the probewave amplitude changes due to material nonlinearity (see Slow Dynamics Diagnostics section of this paper and Figure 5 for more). From the time the vibration is turned on until it is turned off, nonlinear fast dynamics, including conditioning, take place ('Nonclassical Nonlinear Fast Dynamics  NNFD' in (a) and (b)). As soon as the large amplitude wave is terminated, one sees in (a) and (b) an instantaneous, partial recovery of the amplitude, and then a longer term recovery that is linear with the logarithm of time where slow dynamics is the sole process acting in the system. Figures (ce) show the situation where the sample is disturbed by an impact, such as a tap, in the presence of the probe (time B in (c)). Figure (d) shows a zoom of (c) where one can observe the onset of the tapinduced vibration and its ring down ('NNFD' in (de)). After the vibration energy has dissipated, slow dynamics is the sole process operating in the system, the onset of which is shown in (cd), and the long term behaviour is seen in (e).
