Description:

_Journal of Structural Geology 32 (2010) 1485e1487_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _A note on the scaling relations for opening mode fractures in rock_ _Christopher H. Scholz_ _Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY, USA_ _article info_ _Article history: Received 30 April 2010 Received in revised form 29 August 2010 Accepted 18 September 2010 Available online 29 September 2010_ _Keywords: Fractures Joints Dikes Veins Scaling relations_ _abstract_ _It is well established that shear cracks in rock (faults) obey linear displacement-length scaling and thus have scale invariant driving stresses. Several recent papers have claimed that for opening mode cracks in rock (joints, veins, and dikes) displacement obeys square root scaling with fracture length. This is a fundamentally different mode of behavior, because, unlike shear cracks, opening mode cracks would then be unstable under constant stress boundary conditions. Here the same data are reexamined and it is found, to the contrary, that for opening mode cracks in rock, fracture toughness Kc scales with L and hence displacement scales linearly with L. The conflicting view resulted from data misinterpretation. This resolves the discrepancy between the behavior of shear and opening mode cracks in rock._ _? 2010 Elsevier Ltd. All rights reserved._ _1. Introduction_ _The basic scaling law for cracks relates the displacement of the crack walls d to crack length L through the stress drop (driving stress) Ds with a linear relation. In the case of an elastic crack this is:_ _dmax ? Ds2 1 E n2 L : (1)_ _If cracks grow under constant stress loading, we would expect Ds to be scale independent and dmax to scale linearly with L. This is indeed the scaling relation found for shear cracks, i.e., faults and earthquakes (for a recent review, see Scholz, 2007). Shear cracks in rock are stable under constant stress loading because fracture energy Gc increases linearly with L, hence there is no Griffith-type instability (Cowie and Scholz, 1992). In contrast, for opening mode (tensile) cracks, i.e., joints, veins, and dikes, Olson (2003) and Schultz et al. (2008a) have argued that dmax scales as L._ _In the linear elastic fracture mechanics (LEFM) formulation (e.g., Lawn and Wilshaw, 1975), the criterion for crack propagation is:_ _Kc ? Ds L 2p : (2)_ _where Kc is the fracture toughness, which is often assumed in LEFM to be a material constant. Combining (1) and (2) gives:_ _dmax ? Kc?1q n2 p L E p 8 : (3)_ _so that the L scaling can be interpreted as meaning that Kc is scale invariant. In such a case, Eq. (2) shows that the equilibrium driving stress of the crack must fall as the crack grows. Such cracks would be unstable under constant stress loading and therefore must be assumed to grow under constant displacement boundary conditions (e.g., Segall, 1984)._ _These results suggest that tensile and shear cracks behave in fundamentally different ways. Why this should be is very puzzling. The differences between shear and tensile cracks are (a) they have different geometrical terms in their crack-tip stress fields, and (b) the former supports residual friction between its walls whereas the latter does not. Neither of these appear to offer an explanation for this fundamental difference in behavior: the friction of faults, for example, is differenced out in the Ds term. This demands a reexamination of the scaling of joints, dikes, and veins proposed by Olson (2003) and Schultz et al. (2008a)._ _2. Reanalysis of the data_ _The observations relevant to this problem are shown in Fig. 1 (modified from Schultz et al., 2008a). There dmax is plotted vs. L for various data sets of opening mode fractures. Each data set is fit with a function dmax ? C L, where:_ _Culpepper Florence Lake Ethiopia Moros Ledeve Ship Rock_ _C, M1 2 6.2 ? 10?4 6.8 ? 10?4 8.8 ? 10?2 b 2.5 ? 10?3_ _1 ? 10?2 e E,a GPa_ _20 40 73 20 20 19_ _L, Log-med, m_ _0.7 0.8 2000 0.3 5 2900_ _Kc, MPa m?2_ _7.6 16.8 3682 31 124 850_ _a Elastic moduli average low pressure values from Birch (1966). b From Schultz et al. (2008b)._ _Fig. 1. Compilation of data on joints, veins and dikes, from Schultz et al., 2008a. Fits to L scaling are in heavy lines. Large cross is Ship Rock master dike._ _C ? KcE?p1 ? p????_??n8??2?:_ _In this interpretation the value of C varies by more than three orders of magnitude and tends to progressively increase with the length range of each dataset. This great variation in C cannot be due to variation of the elastic constants, which do not vary by more than about a factor of two or three between different rock types. It must reflect large variations of Kc between data sets._ _Olson (2003) recognized this problem and citing Pollard (1987), noted that the km scale Ship Rock dikes were associated with 10 m wide joint clusters that constitute brittle process zones. Because fracture energy Gc represents the sum of the surface energy expended in creating all the cracks in the process zone, this can be expected to result in a much greater fracture energy Gc and hence Kc for those dikes than for the smaller scale veins and joints. He did not follow this line of inquiry further. Pollard and Segall (1987), discussing the same dike family, derived an expression for the process zone which shows that the process zone size (and hence Gc) scales linearly with L. Available data support this. At the laboratory (cm) scale, Mode I fracture propagation in rock is accompanied by the development of a mm scale process zone consisting of a volumetric region of microcracking surrounding the fracture tip (Peng, 1975; Swanson, 1987). The size of the process zone increases with fracture length, resulting in a corresponding increase of Kc and Gc with L (Labuz et al., 1985, 1987; Peck et al., 1985a,b). Segall and Pollard (1983) noted that the terminations of joints of length 1e10 m in the Sierra Nevada consist of cm scale arrays of sub-parallel cracks, suggesting process zones intermediate in scale to those just described. Engvik et al. (2005, 2009) identified mineralized haloes that surround dikes with their process zones and showed that they scale linearly with dike displacement._ _These observations suggest that Kc scales with L, which contradicts the interpretation shown in Fig. 1. To evaluate this, an average Kc was calculated for each data set in Fig. 1, using Eq. (4), the published C value in Olson (2003) and Schultz et al. (2008a) and appropriate elastic constants for the host rocks. This procedure was not followed for the Ship Rock data set because that consists of echelon segments of a single long dike, which, for reasons given later, are not appropriate for this analysis. In the Ship Rock case Kc was calculated for the entire dike using Eq. (3), with L ? 2900 m and dmax ? 3.9 m (large cross in Fig. 1; dmax is the corrected value from Delaney and Pollard (1981)) and an average value of E for the shale host rock of 19 GPa (Birch, 1966). The results are given vs. the logarithmic mean L for each data set in Table 1 and Fig. 2._ _Kc_ _Fig. 2 makes ? 26.8L?54, R2 clear that ? 0.87. It Kc scales with L. The fit to the data yields thus appears that KcfL. Inserting this into Eq. (3) indicates that dmaxfL, which implies that the correct interpretation of the data in Fig. 1 is scaling along the dotted lines of constant dmaxL rather than the solid square root lines. From that we see that most of the data lie in the range dmax L ? 10?2e10?3._ _These results are consistent with the earlier findings of Vermilye and Scholz (1995), who reported linear scaling for veins and dikes over about seven orders of magnitude in length scale, with dmaxL ratios in the range 10?2e10?3. They also found that the dmaxL ratios of segmented veins were systematically much smaller than those of isolated single segment veins. They attributed this difference to the effect of elastic stress interactions between segments, following the analysis of Pollard et al. (1982)._ _Olson (2003) reinterpreted the Vermilye and Scholz data for two localities: Culpepper Quarry and Florence Lake. He lumped together the single and multiple segment veins in both cases and fit each to a single curve with exponent w 1?2. Examination of his Fig. 5 shows that the single segment and multiple segment data in each case occupy different populations, with the multiple segment veins having distinctly smaller aspect ratios (these two populations of different aspect ratio are also clear in his Fig. 4, although there he did not distinguish in the figure between single and multiple segment veins). Because the longer veins are the multiple segment ones, this conspires to skew the fit to have a lower exponent. Vermilye and Scholz emphasized that the multiple segment veins are not self-similar to the single segment ones (see their Fig. 9). They are thus qualitatively different objects and one cannot define a scaling law that includes both._ Ключевые слова: dike, scholz, mode crack, vein dike, fracture energy, set, solid earth, elastic constant, shear, data, engvik, birch, data set, shear crack, fundamentally, ds, problem, dowding, ship rock, conclusion, mode, segment, resistance, mode fracture, swanson, physical, geological, loading, elastic, joint vein, shah, america, fracture, geophysical, pollard, solid, length, host rock, interaction, dmaxfl, note, rock, segment vein, earth, international, linear scaling, propagation, wilshaw, structural, olson schultz, behavior, linear, kc, crack, geological society, ratio, karma, scaling relation, large cross, geology, segall, single, constant stress, process, journal geophysical, process zone, range, single segment, population, fracture length, vermilye, reason, aperture, stress, criterion, scale linearly, displacement, press, opening mode, joint, delaney, schultz, exponent, vein, mechanics, constant, wa, relation, fracture propagation, doe, result, ethiopian dike, formation, function, eq, attributed, zone, cooke, law, assumed, tensile, ship, scale, linearly, stockhert, barton, paper, ol, journal structural, growth, interpreted, dmax, vermilye scholz, smaller, peng, dmaxl ratio, analysis, order, increase, calculated, consistent, case, lin, dataset, unstable, olson, structural geology, geophysical solid, gc, argued, multiple segment, elsevier, society, multiple, interpretation, greater, gordon, opening, fault, journal, crack rock, host, rock mechanics, difference, scaling, magnitude, energy