Rate- and strain-dependent brittle deformation of rocks
Abstract
We develop a unifying framework to quantify rate-dependent deformation in the brittle field and establish links between the microscale time-dependent crack growth processes and the macroscopically observed rate dependency. Triaxial deformation experiments have been performed under both constant strain rate and constant stress (creep) conditions on three types of sandstone. The measured relative evolution of P wave speeds as a function of inelastic axial strain is similar for both types of test, despite differences in strain rate of up to 3 orders of magnitude. This similarity indicates that there exists a direct, time-independent link between the microstructural state (as reflected by the variations in P wave speed) and the inelastic axial strain. Comparison of applied stresses between constant strain rate and creep experiments as a function of inelastic strain indicates that creep deformation requires less mechanical work to bring the sample to failure. This energy deficit corresponds to a stress deficit, which can be related to a deficit in energy release rate of the microcracks. We establish empirically that the creep strain rate is given by , where ΔQ is the stress deficit (negative) and σ^{∗} is an activation stress. This empirical exponential relation between creep strain rate and stress deficit is analogous to rate-and-state friction law. We develop a micromechanical approach based on fracture mechanics to determine the evolution of an effective stress intensity factor at crack tips during creep deformation and estimate the activation volume of the stress corrosion reaction responsible for brittle creep.
Key Points
- Deformation by brittle creep requires less energy than by imposed strain rate
- An empirical relation between creep rate and energy deficit is exhibited
- A relation between our empirical law and stress corrosion laws is determined
1 Introduction
Fracturing is the most prevalent deformation mechanism in rocks under upper crustal conditions. Cracks in rocks occur from the grain scale up to the crustal scale. Under all round compressive stress conditions, macroscopic fractures or faults (at the scale of centimeters and above) result from the coalescence of many microscopic, grain-scale cracks [Paterson and Wong, 2005]. Slip along preexisting fault zones and discontinuities generally also requires fracturing at local contacts or asperities. Reactivation of ancient, healed fault zones involves refracturing of cemented, cohesive rock masses. Hence, brittle deformation of intact (i.e., cohesive) rocks is expected to be one of the fundamental underlying processes of fault formation, growth, and sliding.
Deformation in the crust occurs over a wide range of strain rates, from the very slow rates associated with tectonic loading (of the order of 10^{−14} s^{−1} or lower) up to the very fast rates occurring during earthquakes (of the order of 10^{3} s^{−1} or higher). Hence, a complete understanding of fault mechanics requires a detailed knowledge of the time-dependent mechanical behavior of rocks. Such a knowledge should be based on micromechanics but also provide an adequate macroscopic description. The goal of this paper therefore is to establish a simple macroscopic description of the rate dependency of brittle deformation and to link it quantitatively to the corresponding microscale processes. To achieve this, a key challenge is the determination of a relationship between the internal or microstructural state of the rock and macroscopically observable quantities.
As stated above, deformation of rocks in the brittle field proceeds by the progressive growth and coalescence of microcracks. Under nominally dry environmental conditions and rapid loading, crack growth is primarily governed by the applied stresses. Indeed, laboratory experiments show that rock strength is essentially time independent under such conditions [Paterson and Wong, 2005]. At the microscopic scale, time independence means that the crack growth criterion is well modeled by the concept of constant fracture toughness, i.e., a critical stress intensity factor at microcrack tips. At lower strain rates and in the presence of aqueous fluids, i.e., under realistic upper crustal conditions, experiments show that rock fracturing becomes time dependent. This is well demonstrated by the fact that rocks can fail by static fatigue, following a preceding period of creep deformation, under conditions of constant applied stress [e.g., Scholz, 1968; Kranz, 1979, 1980; Kranz et al., 1982; Baud and Meredith, 1997; Heap et al., 2009; Brantut et al., 2013b]. Time-dependent crack growth arises from chemically activated subcritical crack growth processes, such as stress corrosion reactions [see Atkinson, 1984]. Establishing quantitative links between microscopic, grain-scale subcritical cracking and macroscopic, sample-scale to crustal scale brittle creep behavior is a key challenge for our understanding of the time-dependent mechanics of rocks in the Earth's crust.
Experimental rock deformation provides us with several ways to investigate time-dependent brittle deformation. Two main types of experiments can be distinguished: (1) “constant strain rate” experiments in which stress varies as a result of deformation and (2) “creep” experiments in which deformation and deformation rate vary over time as a result of an imposed constant stress.
In the latter case, when a rock is held at constant stress (i.e., both the differential stress, denoted Q, and the effective confining pressure, denoted P_{eff}, are maintained constant), it deforms typically in the manner depicted in Figure 1. Three deformation stages have commonly been described, based solely on the observed macroscopic strain-time behavior: an initial decelerating stage, termed primary creep, followed by an apparent constant strain rate stage, termed secondary creep, and finally an accelerating tertiary creep stage, after which the sample fails. It is apparent from Figure 1 that the rock experiences a wide range of strain rates as deformation proceeds. Such experiments hence convey very rich information on the time dependency of the brittle deformation process. However, results from brittle creep experiments are usually used to extract only times to failure and/or secondary creep strain rates, as functions of the applied differential stress, effective pressure, and/or temperature (see Brantut et al. [2013b] for a review). The extraction of a single strain rate from a complete nonlinear strain-time curve evidently obscures the richness and complexity inherent in the creep process. A more comprehensive approach to understand the creep process as a whole, including the primary and tertiary creep stages, remains to be developed.
More generally, if a consistent description of time-dependent brittle deformation is to be determined, it should explain in a unified manner both creep (constant stress) and constant strain rate deformation. In a recent theoretical study [Brantut et al., 2012], we used a micromechanical framework to show how the increasing importance of microcrack interactions with increasing microcrack size could produce the switch from decreasing to increasing crack growth rate during deformation at constant stress (i.e., the transition from primary to tertiary creep). It is the same process that is in fact responsible for the existence of a peak stress in deformation experiments performed at imposed strain rate [Ashby and Sammis, 1990]. The specific theoretical approach of Brantut et al. [2012] hence provides a consistent understanding of the rock fracturing process as a whole and in particular establishes a link between the evolution of strain rate during tests at imposed stress and the evolution of stress during tests at imposed strain rates. However, the model of Brantut et al. [2012] is based on strong assumptions about the microcrack geometry, which are appropriate for modeling low porosity crystalline rocks but are not necessarily appropriate for other rock microstructures.
In this paper, we aim to establish a unified understanding of time-dependent mechanical behavior of rock in the brittle regime, specifically focusing on creep deformation. Our approach is based on the analysis of a new set of data obtained on a range of sandstones under triaxial loading conditions. Section 2 summarizes the sample materials and experimental techniques used in our study. Then, in section 3, we present new experimental results from elastic wave velocity measurements made during both creep and constant strain rate deformation. Because elastic wave velocities are very sensitive to microcrack density and orientation, these results allow us to determine how microcracks develop as a function of the inelastic deformation of the samples and hence provide us with an empirical relation between the internal state of the samples and macroscopically observable quantities such as the inelastic axial strain. In section 4, we analyze the results of the triaxial deformation experiments by comparing, for each increment of change in internal state, the stress and energy required to deform the rock samples under both creep and constant strain rate conditions. This analysis provides us with an empirical relation between stress and strain rate which is valid for the whole creep process (and not just during “secondary” creep). Section 5 is dedicated to the determination of the link between the observed, macroscopic creep law and the underlying microscale subcritical cracking process. To achieve this, we develop a novel, general methodology that allows the determination of empirical micromechanical functions linking the local effective stress intensity factor at microcrack tips to the applied stresses. Finally, we discuss the impact and significance of our results in section 6.
2 Experimental Methods
2.1 Starting Materials and Sample Preparation
Porous sandstones from Darley Dale (northern England) and Bentheim and Wertheim (Germany) were used. Darley Dale and Wertheim sandstones are quartzo-feldspathic; Darley Dale sandstone contains 66% quartz and 21% feldspar, and Wertheim sandstone contains 58% quartz and 33% feldspar. Their average porosity is similar at around 13% [Klein and Reuschlé, 2004; Heap et al., 2009]. Bentheim sandstone is composed mostly of quartz (95%), and its average porosity is 23% [Baud et al., 2006]. Porosities of individual samples are reported in Table 1. Cylindrical samples were prepared by coring sandstone blocks and grinding the ends of the cores to ensure parallelism of the two end surfaces (to ±10 μm accuracy). All the samples were saturated with distilled water for at least 24 h before testing. All saturated samples were then jacketed in an impermeable rubber sleeve before insertion into the deformation apparatus.
Rock Type | Sample | Porosity | P_{eff} | Strain Rate | Peak Differential Stress | Creep Differential Stress | Time to Failure | Notes |
---|---|---|---|---|---|---|---|---|
(%) | (MPa) | (s^{−1}) | (MPa) | (MPa) | (s) | |||
Darley Dale | dd3 | 13.8 | 10 | 10^{−5} | 88.9 | – | – | |
Darley Dale | dd9 | 14.1 | 10 | – | – | 79.3 | 1.9 10^{5} | |
Darley Dale | dd4 | 14.1 | 10 | – | – | 82.7 | 5.6 10^{3} | |
Darley Dale | dd2 | 13.8 | 10 | – | – | 83.3 | 6.2 10^{4} | |
Darley Dale | dd5 | 14.3 | 10 | – | – | 85.9 | 1.4 10^{4} | |
Darley Dale | dds2w | 14.4 | 20 | 10^{−5} | 121.8 | – | – | |
Darley Dale | dds11w | 14.3 | 20 | – | – | 108.4 | 4.5 10^{5} | |
Darley Dale | dds7w | 14.4 | 20 | – | – | 112.0 | 5.3 10^{4} | |
Darley Dale | dds6w | 14.3 | 20 | – | – | 113.7 | 7.0 10^{3} | |
Darley Dale | dds1w | 14.1 | 30 | 10^{−5} | 147.2 | – | – | |
Darley Dale | dds13w | 14.2 | 30 | – | – | 134.9 | 7.1 10^{4} | |
Darley Dale | dds8w | 14.1 | 30 | – | – | 138.8 | 7.1 10^{3} | |
Darley Dale | dds14w | 13.3 | 40 | 10^{−5} | 168.4 | – | – | |
Darley Dale | dds9w | 13.2 | 40 | – | – | 161.8 | 6.7 10^{3} | |
Darley Dale | dds12w | 13.2 | 40 | – | – | 162.4 | 1.2 10^{4} | |
Darley Dale | dds-c2-09 | 12.9 | 40 | 10^{−5} | 163.6 | – | – | V_{P} monitored |
Darley Dale | dds-c2-34 | 12.7 | 40 | – | – | 154.7 | 3.5 10^{4} | V_{P} monitored |
Darley Dale | dds-c2-13 | 13.4 | 40 | – | – | 157.0 | 6.1 10^{3} | V_{P} monitored |
Bentheim | ben4 | 22.6 | 10 | 10^{−5} | 116.1 | – | – | |
Bentheim | ben10t | 22.7 | 10 | 10^{−5} | 119.0 | – | – | |
Bentheim | ben20t | 22.2 | 20 | 10^{−5} | 160.3 | – | – | |
Bentheim | ben30t | 22.2 | 30 | 10^{−5} | 177.9 | – | – | |
Bentheim | ben12 | 22.6 | 10 | – | – | 108.8 | 1.5 10^{5} | |
Bentheim | ben7 | 22.4 | 10 | – | – | 112.9 | 9.7 10^{3} | |
Wertheim | we6 | 12.6 | 10 | 10^{−5} | 115.5 | – | – | |
Wertheim | we4 | 13.1 | 10 | – | – | 95.1 | 7.8 10^{5} | |
Wertheim | we11 | 12.7 | 10 | – | – | 100.7 | 8.7 10^{3} | |
Wertheim | we3 | 12.4 | 10 | – | – | 102.3 | 2.5 10^{4} | |
Wertheim | we2 | 13.0 | 10 | – | – | 106.3 | 4.9 10^{3} |
- ^{a} For experiments performed under constant strain rate conditions, we report both the imposed strain rate and the observed peak differential stress. For experiments performed under creep conditions, we report the imposed differential stress and time to failure.
2.2 Triaxial Deformation Experiments
Deformation tests were performed in conventional triaxial apparatus at the Experimental Geophysics Laboratory of the University of Strasbourg (see full description in Baud et al. [2009]) and at the Rock and Ice Physics Laboratory of University College London (UCL) (see description in Eccles et al. [2005]). Both machines are capable of applying servo-controlled axial load, confining pressure, and pore fluid pressure. The axial shortening of the samples were measured with linear voltage differential transducers (LVDTs) positioned outside the pressure vessel, and the shortening was corrected for the deformation of the loading train. All tests were performed under drained, constant pore pressure conditions. Pore volume changes were monitored by measuring (with an LVDT) the displacement of the actuators applying the servo-controlled pore fluid pressure. We use the change in pore volume to estimate the bulk volumetric strain, assuming constant solid volume. A summary of the samples tested and experimental conditions is presented in Table 1. All experiments were performed in the brittle regime, i.e., all samples failed by localized shear faulting.
Two types of experiment were performed. Samples were deformed either under constant strain rate () or under constant differential stress (creep) conditions. In the latter case, the samples were initially loaded at constant strain rate (10^{−5} s^{−1}) until the target differential stress was reached. Then, the differential stress was held constant (using a servo-control system) throughout the remainder of the experiment as the sample deformed over time (see Brantut et al. [2013b] for more details).
2.3 Wave Velocity Measurements
For some targeted deformation tests on Darley Dale sandstone (both creep and constant strain rate), the evolution of elastic wave velocities with increasing deformation was measured. Here we describe the technique used.
The UCL triaxial apparatus is equipped with 10 piezoelectric transducers positioned around the sample as shown in Figure 2, connected to a 10 MHz digital recording system. Repeatedly during the tests, a high-voltage (∼100 V), high-frequency (∼1 MHz) pulse was sent sequentially to each transducer, hence producing a mechanical wave with known origin time, while output waveforms were recorded on the nine remaining transducers. Precise P wave arrival times were extracted from the waveform data by using the cross-correlation technique described by Brantut et al. [2011]; the method was improved by resampling the cross-correlation results with interpolated cubic splines, and the resulting precision on the arrival times was 0.05 μs (i.e., around 0.5% relative error on the wave speed). The arrival times were then used to determine the relative P wave speed evolution throughout the tests.
2.4 Inelastic Strain Calculations
For each test, the sample's initial Young's modulus was determined from a fit of the linear part of the differential stress versus axial strain curves. This allowed the nominal elastic strain at any stress to be determined (i.e., the strain that the intact material would have had at the same level of stress) and subtracted from the total recorded axial strain to obtain the inelastic axial strain. In this process, the initial, nonlinear deformation (typically of the order of 0.02%), which correspond to both sample assembly adjustments and closure of preexisting subhorizontal microcracks within the sample [see David et al., 2012], was also subtracted from the total inelastic axial strain.
2.5 Strain Rate Calculations
The recording interval of stress, strain, and pore volume was deliberately varied (within the range 1 s to 1 min) in line with the very wide range in test durations. Also, the calculation of a time derivative such as strain rate from a discretely recorded signal is nontrivial because high frequency noise is amplified when taking differences between successive time steps. We therefore developed a specific method based on Chebyshev interpolation to circumvent this problem. The instantaneous strain rate was estimated as follows. The creep time was mapped to the interval [−1,1], and the strain-time curve was interpolated on N Chebyshev-Lobatto nodes (40≤N≤100, depending on the total number of initial data points). The derivative of strain with respect to the normalized time was then obtained by multiplying the N×N Chebyshev differentiation matrix by the vector formed with the N interpolated points (details of the method and computation can be found in Trefethen [2000]). The strain rate was finally obtained by mapping back the time from the normalized [−1,1] interval to the real creep time in seconds (i.e., from 0 to the time to failure) and reinterpolated on all the recorded time steps. This last stage does not improve the quality of the strain rate estimates but is useful to enable all the variables (stress, strain, etc.) to be correlated within a single time scale.
3 Inelastic Strain as a Proxy for Internal State
In this study we want to compare stresses and strain rates between creep (constant applied stress) and constant applied strain rate tests. Such a comparison is meaningful only if it is performed for samples at the same microstructural state. We thus need to determine a physical quantity that reflects the overall microstructural state of the samples and that we can measure systematically throughout our experiments. In the brittle regime, evolution of the microstructural state during deformation corresponds primarily to the growth of dilatant microcracks between or inside the grains [e.g., Scholz, 2002; Paterson and Wong, 2005]. Because they are traction-free internal surfaces, such microcracks have a strong influence on the elastic wave speeds of rocks [e.g., Sayers and Kachanov, 1995; Fortin et al., 2007]. Hence, by monitoring the evolution of elastic wave speeds during deformation under both creep and constant strain rate conditions, we have an indirect measure of the microstructural state of the samples (i.e., microcrack density and orientation). Here we present the wave speed measurements performed during selected experiments and use them to determine how inelastic axial strain can be used as a convenient proxy for microstructural state.
Figure 3 shows the evolution of the measured P wave speeds, in four different orientations with respect to the axis of compression, as a function of total axial strain, for one constant strain rate experiment (Figure 3a) and two creep experiments (Figures 3b and 3c) performed on Darley Dale sandstone at P_{eff}=40 MPa (samples dd-c2-09, dd-c2-13, and dd-c2-34 in Table 1). The initial P wave speed at P_{eff}=40 MPa before deformation is around 4.3±0.1 km s^{−1}. The evolution of P wave speed with increasing axial deformation follows a well-known pattern [e.g., Ayling et al., 1995], which appears to be common to both types of test. The radial P wave speed (circle symbols) monotonically decreases with increasing axial deformation. Concomitantly, the P wave speeds measured along lower angles to the compression axis, namely, 58° (squares), 39° (diamonds), and 28° (triangles), initially increase at the onset of axial deformation and subsequently decrease after around 0.6% axial strain. The total decrease in P wave speed is larger along the radial direction than along the subaxial directions. This evolution is typical of rocks that crack progressively under compression [Paterson and Wong, 2005, chap. 5]: when the differential stress is applied, the rock initially behaves elastically, with subradial cracks closing (hence the initial increase in subaxial P wave speeds), while subaxial cracks begin to open. Then, when the stress threshold for crack propagation is reached, i.e., after some amount of axial strain, cracks open preferentially in the subaxial direction (hence the large decrease in the radial P wave speed). We hence observe the development of a crack-induced anisotropy.
Comparison of Figures 3a, 3b, and 3c reveals not only that the qualitative evolution of the P wave speeds is similar for both creep and constant strain rate deformation conditions but also that the quantitative changes in P wave speeds (aside from some variability in the initial values) are remarkably similar. Despite several orders of magnitude difference in times to failure (and therefore in strain rate), the rate of decrease in P wave speed per unit of axial strain appears to be similar for all three tests. This observation suggests that axial strain is a potential proxy for the microstructural state of the samples. However, a significant part of the total axial strain corresponds to the elastic deformation of the specimen, and hence differences in applied differential stress from one experiment to another will generate differences in axial strain that are not due solely to crack growth. This introduces a bias that can be avoided by processing the data to remove the elastic strain, which leads us to use the inelastic axial strain as our proxy for microstructural state.
Figure 4 shows a synoptic plot of the evolution of the calculated normalized axial and radial P wave speed as a function of inelastic axial strain for the same three experiments. The wave speed variations are normalized by the initial wave speed prior to deformation at the appropriate confining pressure. Remarkably, despite the wide range of strain rates (from 10^{−5} s^{−1} down to 6.9 10^{−8} s^{−1}) and the different experimental conditions (imposed constant strain rate or applied stress), the relative change of P wave speed is quantitatively very similar for all the tests. This similarity in the evolution of P wave speeds with inelastic strain between the three tests implies that the microstructural state of the samples can be considered to be a simple function of the amount of inelastic axial strain.
We note that while we consider this interpretation to be valid over the range of strain rates and stresses investigated here, it will not necessarily be valid for other arbitrary stresses, strains, and strain rates. In general, the constitutive behavior of cracked materials is path dependent, i.e., the stress versus strain response depends on the material's deformation history. This is all the more true since the main micromechanism of deformation is cracking, which is irreversible under the ambient temperature conditions of our tests. Despite these limitations, for the range of stress conditions investigated here, the wave velocity data show that the inelastic strain is an appropriate proxy for the microstructural state of the samples.
4 Energy Deficit and Creep Strain Rates
In the previous section we established that the inelastic axial strain is a relevant proxy for the microstructural state of the samples. In this section we make use of this tool to investigate how the creep strain rate evolution is related to the applied creep stress and to the stress applied (for the same microstructural state) in constant strain rate experiments.
4.1 Comparison of Creep and Constant Strain Rate Data
Figure 5 shows the differential stress (denoted Q) as a function of the inelastic axial strain for one creep experiment (in black) and one constant strain rate experiment (in grey) performed on Darley Dale sandstone at 30 MPa effective pressure. As expected, the behavior is the same during the loading stage, since the imposed condition (constant strain rate of 10^{−5} s^{−1}) is the same. In the creep experiment, once the creep stress is reached, the stress is maintained constant. By contrast, the stress keeps evolving during the constant strain rate experiment. In the latter case, the differential stress first increases until it reaches a peak (around 147 MPa) at 0.005 inelastic axial strain; it then decreases gradually. No macroscopic dynamic stress drop occurred, since the machine stiffness is high enough to prevent runaway deformation in the strain-weakening regime. After 0.01 inelastic strain, the stress applied to the sample under creep conditions could no longer be maintained as the deformation rate increased dramatically (the sample experienced tertiary creep, see Figure 1).
A striking observation is that the sample deformed under constant stress fails macroscopically (i.e., the deformation rate started to increase in an uncontrollable manner) when the two curves (constant strain rate and constant stress) intersect each other. For the whole range of inelastic strain experienced by the samples in both tests, we can calculate the difference ΔQ between the differential stress applied during creep (which is constant) and the differential stress applied during deformation at constant strain rate. Creep starts at ΔQ=0 and then becomes negative as the creep stress remains below the stress applied during constant strain rate deformation, and we observe in Figure 5 that creep deformation becomes unstable when the stress difference returns to ΔQ=0.
Figure 6 shows the evolution of both the stress difference ΔQ and the creep strain rate (on a log scale) as a function of the inelastic axial creep strain. There is a remarkable similarity between the two curves. Several fundamental observations can be made. First, the creep strain rate is never constant, in qualitative agreement with the recent micromechanical model of Brantut et al. [2012]. Second, the minimum creep strain rate is achieved when the stress difference is largest (most negative), i.e., when the differential stress is at the peak during the constant strain rate test.
We note that the transition from the decelerating branch to the accelerating branch in Figure 7 is equivalent to the peak stress in a constant strain rate experiment and is therefore also likely to be associated with the onset of localization of deformation onto a fault plane.
4.2 Effect of Confining Pressure
The procedure described above was repeated for a set of experiments performed on Darley Dale sandstone under effective pressures ranging from 10 to 40 MPa. The results are compiled in Figure 8, which again shows the normalized creep strain rate as a function of the stress difference.
The empirical exponential relation between strain rate and stress difference (equation 2) holds for all the effective pressures investigated. Notably, the activation stress σ^{∗} does not seem to depend upon effective pressure in any systematic manner: σ^{∗} is around 1.0 MPa at P_{eff}=10 MPa (see above), 1.2 MPa at P_{eff}=20 MPa, 2.2 MPa at P_{eff}=30 MPa, and 1.6 MPa at P_{eff}=40 MPa. These values are within the range determined above for all the sandstones tested at P_{eff}=10 MPa.
4.3 Energy Deficit
In the previous sections we established a macroscopic, empirical description of time-dependent brittle deformation of the tested sandstones samples: we discovered that the strain rate was an exponential function of the stress deficit and that the confining pressure was apparently not quantitatively influencing this relation. Here we use a thermodynamic approach to understand the origin of our empirical description and what it implies in terms of the energetics of brittle deformation.
The comparison between equations 9 and 11 indicates that the activation stress σ^{∗} should reflect the parameter b^{′}/RT, which gives the sensitivity of crack growth rate to variations in crack tip energy release rate. Because the stress intensity factor K is a linear function of the applied stresses, the same observation can be made (b^{′}/RT being replaced by b/RT) if we compare equations 2 and 10. These analogies also shows that the activation stress is a combination of the subcritical cracking parameters and a term (e.g., H^{′}) linking the internal state evolution (e.g., crack length) to the macroscopically averaged quantity (here, inelastic strain).
Despite the generality of our approach, the practical use of a relation like 9 is limited by our poor knowledge, in practice, of the correct general meaning of H. Hence, the precise relation between ΔW^{p′} and, say, the actual energy release rate of the microcracks within the rock sample is poorly constrained.
5 Extraction of Microscale Processes
In the previous section we established an empirical exponential relation between strain rate and stress deficit during creep deformation and showed that such a relation should be linked to small-scale subcritical crack growth laws. In this section we develop a more specific framework, based on fracture mechanics, which will allow us to determine precisely a link between the macroscopic stress dependency parameter of creep strain rate (the activation stress σ^{∗}) and microscale subcritical crack growth parameters.
5.1 Empirical Micromechanical Functions
In conventional micromechanical approaches, one makes precise assumptions about the microcrack geometry with respect to the applied stresses, and explicit formulations of the functions α(l) and β(l) can then be established. This is the case, for instance, in the “wing crack” model of Ashby and Sammis [1990] for triaxial conditions (further developed for generalized stress states by Deshpande and Evans [2008]; Bhat et al. [2011]). However, the assumptions about the crack size and geometry are generally very restrictive; even then, some simplifications have to be made in order to access analytical formulations of K_{I} in terms of crack length and stresses. Such “ab initio” micromechanical models are complex and need the introduction of numerous, sometimes poorly constrained, microscopic parameters.
Here we adopt a different approach, based directly on experimental data. Instead of looking for a simplified, tractable microcrack geometry, we will attempt to determine in the most direct way the functions α(l) and β(l) by inverting experimental data.
5.2 Determination of K_{I} Evolution
In order to determine the evolution of an effective K_{I} during the experiments, we need to invert for the evolution of α and β at all stages (i.e., as a function of an effective microcrack length l) throughout deformation. Here again we make use of the observation that inelastic axial strain is a reasonable proxy for the microstructural state of the samples, where microstructural state is now identified relative to an effective crack length l.
We now consider that, during the deformation tests performed at a constant strain rate of 10^{−5} s^{−1}, the imposed deformation is so fast that the microcracks grow at speeds much higher than the typical subcritical crack growth rates, and therefore the crack growth criterion is essentially determined by the time-independent fracture toughness. Equivalently, we can also consider that an imposed deformation rate induces a constant crack growth rate and therefore that K_{I} remains constant during such tests. In all cases, we assume that the effective stress intensity factor is close to the fracture toughness of the material (K_{IC}) for the samples deformed at constant (“fast,” 10^{−5} s^{−1}) strain rate. K_{IC} is a material constant, equal to 1 MPa m^{1/2} for pure quartz in wet conditions [Atkinson, 1984]. Using this constant value for K_{I} in equation 12, we see that the functions and can be obtained as functions of the inelastic axial strain by inverting the evolution of differential stress Q at various effective pressures P_{eff} during constant strain rate tests. The inversion is performed by a least squares method on the data obtained at P_{eff}=10, 20, and 30 MPa. The fits are shown for Darley Dale sandstone in Figure 9 (left), and the resulting empirical micromechanical functionsand are shown in Figure 9 (right).
The function α, which corresponds to the effect of differential stress (i.e., a measure of the average shear stress) on the effective stress intensity factor, is always positive: as expected, larger differential stress tend to increase K_{I} at crack tips. During deformation, α remains almost constant, only increasing slightly at inelastic strain larger than 1%. The function β, which quantifies the effect of effective pressure on the effective stress intensity factor, is always negative: this is also expected, since larger confining pressures tend to close cracks and decrease K_{I}. β decreases continuously with increasing inelastic deformation. This is also consistent with basic fracture mechanics concepts: large cracks tend to be easier to close (i.e., K_{I} is more sensitive to closing stresses).
The relation between the normalized creep strain rate and K_{I}−K_{IC} is shown in Figures 11 and 12 for the available data on Darley Dale sandstone and Bentheim sandstone, respectively. In these Figures, the strain rate data beyond the minimum strain rate (i.e., beyond the peak stress in the constant strain rate experiments) have been omitted to emphasize the portion of the experiments during which the strain is not yet localized along a fault plane (and hence is more representative of an effective stress intensity factor and crack size).
5.3 Interpretation
6 Discussion and Implications
6.1 A General Method of Analysis: Empirical Micromechanical Functions
In section 5.2, we established a general methodology that allows the determination of an effective K_{I} throughout a deformation test, using only macroscopic observations. The application of this method requires that there exists an appropriate macroscopically measurable proxy for crack length (in our case, inelastic axial strain) in order to make meaningful comparisons between tests conducted under different conditions. Then, the only assumption required is that fast deformation occurs with K_{I}≈K_{IC} at microcrack tips. If at least two stress-strain curves are available for two different effective pressures, the empirical micromechanical functions can be determined using relation 12.
This method of analysis, which relies on LEFM, allows the determination of an empirical micromechanical model with a minimal number of assumptions and only one parameter, K_{IC}, which is a well-documented material property [see, for instance, Atkinson and Meredith, 1987]. Hence, we can bypass the mathematical complexities and the large number of assumptions or loosely constrained parameters needed for the development of ab initio micromechanical models, such as those of Costin [1985] or Ashby and Sammis [1990]. As a point of comparison, however, we can compute the functions α and β given by the model of Ashby and Sammis [1990] and compare them with our experimentally derived values. The exact expressions for α and β can be found in Appendix B. Unfortunately, this model cannot be fitted to porous rocks such as Darley Dale sandstone [see Baud et al., 2013]; hence, we can only draw qualitative comparisons by choosing a rock type for which the model is applicable. For that purpose, we choose parameter values relevant to Westerly granite, which can be found in Brantut et al. [2012] (see Appendix B). Figure 13 shows the evolution of and as functions of the crack length l. Both functions evolve in a manner remarkably similar to those inverted from our experimental data (Figure 9, right). Furthermore, the numerical values derived from the model and from the experiments are within the same range, even though the parameter values in the model do not correspond to the same rock type. This strong similarity between the results from processing our experimental data and ab initio approaches, such as the wing crack model of Ashby and Sammis [1990], reinforces our approach. This qualitative comparison exercise should be complemented with a more systematic quantitative analysis. However, such an analysis is beyond the scope of this study but could be performed using experimental data for Westerly granite or any other appropriate rock type.
Our overall methodology is unfortunately only approximate, because it implicitly assumes that the loading path dependence of the constitutive behavior can be neglected. Hence, it can be used only to compare tests performed under similar stress conditions. This is legitimate in the context of brittle creep, because the initial loading of the sample is always the same, and the constant applied stress in the creep tests remains close to the applied stress in the constant strain rate tests (for the same inelastic axial strain). Further developments are needed to explore how path dependence affects the results obtained from experiments performed under significantly different conditions. In other words, using our functions α and β determined from triaxial constant strain rate tests to predict the behavior under generalized loading configurations (e.g., true triaxiality) might result in serious bias.
6.2 A “Rate-and-State” Deformation Law for Brittle Rocks
Nevertheless, the semiempirical formulation 2 can be seen as a useful description of the rate dependency of deformation in the brittle regime. This relation is also directly comparable to that of Lockner [1998], which was determined for Westerly granite based on the most complete data set existing for a single rock type. Lockner [1998] included an explicit, empirical description for the nonlinearity of the stress-strain behavior at constant strain rate (i.e., our reference curves at 10^{−5} s^{−1}). In our analysis, we bypass this issue by introducing the concept of stress deficit. Our methodology allows for a very simple treatment of the time dependency, which can be applied routinely to typical triaxial experiments. The determination of the activation stress only requires two experiments: one at fast, constant strain rate and one under constant stress conditions (at the same effective pressure). Then, once σ^{∗} is known, equation 2 can be used to make reasonable predictions of times to failure and creep strain rates. Note that the only major problem with using this method for making predictions is that sample variability can significantly offset the strain rates and hence the time to failure, as shown for instance in Figure 6. The problem of sample variability, or, equivalently, the variability in initial microstructural state [see Brantut et al., 2013b], remains the main issue for precise predictions in natural materials such as rocks.
The validity of equation 2 for strength predictions under different imposed constant strain rates could potentially be checked against triaxial data obtained over a range of strain rates. If applied to estimate the strength (or, more precisely, the stress deficit) of a rock under deformation at constant strain rates slower than the reference strain rate (10^{−5} s^{−1}), equation 2, according to our estimates for σ^{∗}, would predict a decrease in strength of a few megapascals per decade of strain rate. Again, because of natural sample variability, such small variations in strength would be rather difficult to observe consistently in a data set obtained at different imposed constant strain rates. A more reliable methodology would be to perform strain rate steps during tests, similarly to the velocity steps typically performed during friction experiments. Such an approach will be tested and assessed in future work.
7 Conclusions
We performed triaxial experiments on three types of sandstone and compared the results obtained under constant strain rate (10^{−5} s^{−1}) and creep (constant stress) conditions. The evolution of P wave speeds as a function of inelastic axial strain is the same in both types of test, despite differences in strain rates of up to 3 orders of magnitude. This similarity indicates that inelastic axial strain is an appropriate proxy for the microstructural state of the samples, under both creep and constant strain rate conditions.
The comparison of differential stress versus inelastic strain curves for both types of test reveals that creep deformation requires less mechanical work to bring the sample to failure. This energy deficit corresponds to a stress deficit, which can be related to a deficit in energy release rate for the cracks, or, equivalently, to a deficit in crack tip stress intensity factors. The evolution of strain during creep tests is well described, empirically, by an exponential of the stress deficit. This description introduces an activation stress σ^{∗} that can be easily quantified by measuring the slope of the logarithm of strain rate versus stress deficit curves. An interesting particular consequence of these observations is that the creep strain rate is in fact never constant. The so-called secondary creep stage is simply an inflection period, during which strain rate remains close to or at its minimum for an extended period of time.
Based on linear fracture mechanics, and the assumption that crack growth is the dominant deformation mechanism, we developed a method to estimate the evolution of an effective stress intensity factor at crack tips during creep deformation. The estimated evolution mirrors the evolution of the stress deficit. Further assuming that the relation between crack size and inelastic strain rate is approximately linear before peak stress is reached, we determined a relation between crack growth rate v and stress intensity factor K_{I}. By analyzing the slope of those K_{I}−v curves using the stress corrosion law of Freiman [1984], we determined an activation volume for the stress corrosion reaction responsible for brittle creep in our samples. This activation volume is of the same order of magnitude as the molar volume of quartz and is close to the activation volumes determined independently from typical subcritical crack growth tests [Freiman, 1984].
The exponential relation between strain rate and stress deficit determined for brittle deformation of intact rocks has the same form as rate-and-state friction laws which relate slip rate to applied shear stress on an interface. The stress dependency of slip rate in rate-and-state friction laws arises from thermally activated processes [e.g., Nakatani, 2001; Rice et al., 2001] which have an activation volume very similar to the one determined in our analysis for stress corrosion cracking. This similarity opens the way for further studies aimed at establishing the precise the relation between fracture of intact rocks and friction along rock interfaces.
Acknowledgments
N.B. is grateful to Harsha Bhat for useful discussions and especially clarifying some thermodynamical aspects of brittle deformation. We thank reviewer Hiroyuki Noda and Associate Editor Reid Cooper for their useful comments. The authors also thank Thierry Reuschlé for his contribution in setting up the triaxial equipment in Strasbourg. Part of this research was funded by UK NERC grant NE/G016909/1 to P.G.M.
Appendix A: Volumetric Strain Evolution
Here we compare the evolution of inelastic volumetric strain as a function of inelastic axial strain for both constant strain rate and creep experiments. The inelastic volumetric strain was calculated from the porosity change of the samples (measured with a pore volumometer), corrected from the drained modulus estimated in the elastic part of the differential stress versus volumetric strain behavior. Figure A1 shows the inelastic volumetric strain as a function of inelastic axial strain for two experiments performed on Darley Dale sandstone at 30 MPa effective pressure, under constant strain rate (grey) and creep (black) conditions. The evolution is remarkably similar between the two experimental conditions. For the case of creep, the volumetric strain oscillates near the reference curve of the constant strain rate experiment. This oscillation can be attributed to thermal effects (the experiment was conducted over several days). These thermal effects affect most of our creep data and cannot be easily corrected. For the purpose of this study, we consider therefore that the inelastic volumetric strain does not significantly differ from one test to another when the effective pressure conditions are the same.