1. Introduction
1.a. Fractography and fracture mechanics in rocks
Plumes represent one of the more important fractographic features commonly occurring on geological tensile fractures. This is useful for studying tectonophysics because the key fracture mechanic parameters, like the tensile stress intensity factor KI (the product of multiplying the fracture stress by the square root of the fracture length) and the palaeostress magnitude, may be obtained from the fractographic features revealed on joint surfaces (Bahat, Reference Bahat1979). Plumes commonly start as straight striae, and their barbs splay towards the two rock boundaries (Fig. 1a). Bahat & Engelder (Reference Bahat and Engelder1984) investigated the fractographies of two joint sets in the Devonian siltstones from the Appalachian Plateau in New York, and distinguished between straight and continuous plumes (maintaining approximately the same morphology throughout its length) of the S-type (Fig. 1a), and the C-type plumes that were neither straight nor continuous. They were either curved and split into various directions (Fig. 1c), or showed rhythmic increase and decrease of plume intensities, that is, enlargement and diminution of depth and widths of the barbs (Fig. 1b). Transitional patterns between the curved and rhythmic (cyclic) styles were also common. Bahat & Engelder (Reference Bahat and Engelder1984) identified in the rhythmic C-type plumes the alternate short and long episodes of cracking, predicted by Secor (Reference Secor1965), suggesting slow fracturing. The latter fracturing was specified to be between the lower ranges of regions I and II of the velocity–stress intensity curve (Fig. 2).
Three plume types propagating from right to left, in the Appalachian Plateau (modified from Bahat & Engelder, Reference Bahat and Engelder1984). (a) Two joints decorated by ‘straight’ plumes meet at a junction (above scale) on a siltstone bed, 21 cm thick, intercalated in shales; both joints are oriented 345°. (b) ‘Rhythmic’ plume displays periodic fan perimeters (solid black lines), designating the fracture fronts where reduced fracture velocities occurred. (c) ‘Curving’ plume shows spreading of barbs to all directions (dashed black lines); both plumes mark joints oriented 335°, cutting adjacent siltstone beds (thickness of lower bed is 18 cm).
Schematic variation of stress intensity factor K with crack velocity V. The sub-critical curve left of KIc is divided into three regions I, II and III. K0 and V0 are the stress corrosion limit and the crack velocity at this limit (Wiederhorn & Bolz, Reference Wiederhorn and Bolz1970). Supercritical growth occurs mostly along a terminal velocity plateau following a rapid increase in crack velocity at KIc (modified from Evans, Reference Evans1974). A velocity overshoot ‘o’ immediately after KIc before the plateau has been predicted by Rabinovitz & Bahat (Reference Rabinovitch and Bahat1979).
Lacazette & Engelder (Reference Lacazette, Engelder, Evans and Wong1992) diverged from the interpretations of Bahat & Engelder (Reference Bahat and Engelder1984) and suggested that the cyclic pattern of the rhythmic C-type plume (Fig. 1c) may arise from dynamic instability of the fracture–fluid–rock system, when KI>KIc (where KIc is the critical stress intensity). Hull (Reference Hull1999, p. 123) describes the increase in roughness beyond the mirror boundary, from the mirror plane, through the mist to the hackle zone, which is associated with increases in crack velocities and stress intensities under unstable conditions. Engelder (Reference Engelder2004) suggests that surface roughness varies in the same manner during stable joint propagation, that is, below KIc. These and additional fractographic descriptions in New York and in other fracture provinces (e.g. Kulander, Barton & Dean, Reference Kulander, Barton and Dean1979; Simón, Arlegui & Pocoví, Reference Simón, Arlegui and Pocoví2006) show how difficult the interpretation of joint fractography may be.
1.b. The mirror boundary
The fracture mechanic boundary between the sub-critical and post-critical regimes is set at KIc (Fig. 2). For a full understanding of the tectonic implications that might be derived by the interpretation of joint fracture surface morphology, there is a fundamental need to distinguish fractographically between these two regimes and identify their particular diagnostic features within the mirror plane and beyond it, respectively. The early history of the joint is recorded on the mirror plane, while the late one is marked on the fringe. Thus, a key criterion in characterizing the mirror plane is identifying the mirror boundary that separates the mirror plane and the fringe. In plutonic rocks, there are occasionally favourite conditions for finding clear boundaries between the mirror and hackle fringes. A definition of the mirror boundary in granite enabled the calculation of the palaeostresses that induced jointing in the rock (Bahat & Rabinovitch, Reference Bahat and Rabinovitch1988). Such clear boundaries also allowed the construction of a semi-quantitative fractographic curve of the fracture velocity (v) versus KI for the analysis of granite fracture conditions during its cooling (Bahat, Bankwitz & Bankwitz, Reference Bahat, Bankwitz and Bankwitz2003). However, the identification of mirror boundaries on joints cutting sedimentary rocks is a more difficult task, as explained below.
1.c. Objectives of the present study
We set three objectives for the present investigation:
(1) To study three aspects of fracture mechanics that are connected to the formation of joints, starting with the elaboration on the boundaries of mirror planes, which are important in the interpretation of joint fractography.
(2) To extend the studies of the v versus KI curve in relation to fractography from granites to sedimentary rocks, and construct four new semi-quantitative curves of the v versus KI, correlative to plume morphology for two rock types. This would enable evaluation of the conditions of propagation of several joint sets from different localities, aiming at setting the stage for semi-quantitative analyses of their sub-critical jointing conditions.
(3) To demonstrate that the treatment of the fracture mechanics of jointing in sedimentary rocks must be carried out in close conjunction with the variability of the geological parameters.
2. Basic fractographic features on joint surfaces, with a special reference to the mirror plane boundary in sedimentary rocks
2.a. Geometric variations in the context of defining the mirror boundaries
There is a clear distinction between two en échelon segmentation types (Bahat, Reference Bahat1997): the ‘discontinuous breakdown type’, and the ‘continuous breakdown type’. In the ‘discontinuous breakdown type’, the breakdown is ‘discontinuous’ along clear boundaries of the parent joints (Fig. 3a), while in the ‘continuous breakdown type’ the breakdown is ‘continuous’, initiating at various locations on the parent joint, and there is no clear boundary that separates the fringe from the parent joint (Fig. 3b). In the ‘discontinuous breakdown type’, new plumes start to form on the en échelon cracks (e.g. Woodworth, Reference Woodworth1896; Hodgson, Reference Hodgson1961; Bankwitz, Reference Bankwitz1966; Bahat, Reference Bahat1997), suggesting separate fracture events. In the ‘continuous breakdown type’, on the other hand, the plume propagates continuously from the parent joint to the en échelons (Bahat, Reference Bahat1997), implying a probable single fracture event. The literature records ‘transitional’ variations between the ‘continuous breakdown type’ and ‘discontinuous breakdown type’ end-members, which show plumes that cross unclear, irregular or discontinuous (‘zig zag’) boundaries, when continuing to propagate on the en échelon cracks that reside on fringes (e.g. Bankwitz, Reference Bankwitz1965; Roberts, Reference Roberts and Ameen1995). Examples of the ‘continuous breakdown type’ and ‘transitional’ variations are shown by Simón, Arlegui & Pocoví (Reference Simón, Arlegui and Pocoví2006, figs 5b and 3a, b, respectively). In considering these geometric variations in the context of defining the mirror boundaries, it appears that only the ‘discontinuous breakdown type’ provides clear boundaries. They are the ‘shoulders’ that separate the parent joints and fringes in the classic work by Hodgson (Reference Hodgson1961).
Different styles of boundaries separating the parent joints and fringes. (a) A joint cutting Lower Eocene chalk in the Shephela syncline. A 30 cm long plume propagates from left to right in the horizontal, rectangular parent joint. En échelon segments break down discontinuously, in the upper and lower fringes that occur between the parent joint boundaries and the layer boundaries (dashed lines), along which the rock is partly eroded (modified from Bahat, Reference Bahat1997). (b) A joint cutting Middle Eocene chalk, showing a delicate bilateral plume (barely visible) and a continuous breakdown of en échelon segmentation that starts from the centre of the parent joint, where segments propagate downward toward the lower layer boundary. Note en échelon growth in continuation of the plume (at left). Scale bar is 50 cm (modified from Bahat, Reference Bahat1987). (c) A joint from an orthogonal system, cutting chalk near Nazareth, Lower Galil, and arrests at a previous joint at left along a contact ‘a’. A series of coarse, concentric arrest marks that have sharp crests indicate that the joint propagated from its lower right (the origin is hidden under ground) towards its upper left side. The joint surface is also decorated by delicate, radial striae, which become much more intense on cutting the arrest mark crests (at ‘s’ locations). A curved mirror boundary, ‘mb’, separates between the mirror plane at right and the fringe, ‘f’, which is populated by a set of en échelon segments, at left. Note in addition: (1) slight splits of striae to plumes, at some locations (e.g. below the letters AM), and (2) the width of the fringe is determined by the geometric relationship of the orthogonal existing joint at ‘a’, setting a free surface, and the curved mirror boundary. (d) A joint cutting chalk, showing on the mirror plane, ‘m’, the location of fracture origin, ‘o’, concentric, delicate undulations, ‘cu’, and radial, delicate striae, ‘s’. A zone of hackles, ‘h’, resides on the fringe, ‘f’, which is separated from the mirror plane by a mirror boundary, ‘mb’. Note that the fringe forms an angular relationship with the mirror plane (from Bahat, Rabinovitch & Frid, Reference Bahat, Rabinovitch and Frid2005, fig. 2.30a). (e) A profile of the joint shown in (d), maintaining the same inscriptions, and showing the angle ϕ which the fringe forms with the mirror plane.
Simón, Arlegui & Pocoví (Reference Simón, Arlegui and Pocoví2006) made the important distinction between ‘rectangular’ and ‘circular’ joints. The ‘rectangular’ joints are decorated by straight, horizontal fringes, which are controlled by the layer boundaries (Fig. 3a). The ‘circular’ joints are not controlled by layer boundaries and have curved fringes (Fig. 3c). As such, the fringe boundaries of the latter joints resemble (rare) curved boundaries of hackle fringes (Fig. 3d, e). Thus, curved ‘discontinuous breakdown type’ boundaries of en échelon fringes can be equated with mirror boundaries.
2.b. Criteria for elucidating the mirror boundaries
We now consider general criteria for elucidating ‘discontinuous breakdown type’ mirror boundaries on joint surfaces. At least one of the following two criteria must be identified on a joint surface for determining a mirror boundary, while the third and fourth criteria can increase the credibility of the definition.
(1) There is an abrupt morphological increase in the size of the cracks across the mirror boundary, from plumes, or delicate radial striae, to rough en échelon cracks on the fringe (Fig. 3c). The transformation of delicate radial striae to hackles on the fringe is another example of a stepwise growth beyond the mirror boundary (Fig. 3d).
There is a fundamental difference between the fractographies exhibited by Figure 3c and d, which represent ‘circular’ joints on the one hand, and Figure 3a, which represents ‘rectangular’ joints on the other. In the ‘circular’ joints the mirrors and fringes were not influenced by boundary effects, which enabled them to reach circular boundaries. Boundaries of ‘circular’ (penny-shaped) joints can be used for calculation of palaeostresses (Bahat & Rabinovitch, Reference Bahat and Rabinovitch1988). On the other hand, in ‘rectangular’ joints, the mirror boundaries are strongly influenced by the layer boundaries (‘boundary effects’), such that they are forced to form parallel to them. This constraint prevents the joint from reaching circular boundaries, which would be attained when the lowest free energy conditions and equilibrium are reached. Such joints cannot be used for palaeostress calculations.
(2) A tilt fringe angle, φ, is formed between the imaginary continuation of the mirror plane and the fringe plane. Maximum f is seen in profile rather than in plan view, that is, looking in the direction that parallels the line of mirror boundary (Fig. 3e).
(3) Apart from size difference between plumes and fringe cracks, there are occasionally changes in the sense of stepping in the transition from the plume to the en échelon segmentation on the fringe that can be readily recognized (Bahat, Reference Bahat1997).
(4) The ratio of mirror plane radius to the radius of the critical flaw should be 15±5 (Bahat & Rabinovitch, Reference Bahat and Rabinovitch1988). Straightforward estimation of this ratio is limited to exposures where the two fractographic parameters are accessible for measurements.
3. Three distinct associations of plume morphologies
As much as detailed studies of individual plumes have proved to be rewarding (e.g. Syme-Gash, Reference Syme-Gash1971), more intriguing are perhaps the investigations of multi-plume associations that occur on different joint sets which crop out at close vicinities in layered rocks, or of the same sets exposed next to each other in granites. Here we make distinctions among three ‘cases’ of plume morphologies on joint surfaces and their fracture mechanic implications:
Case 1. Superposing plume morphologies from ‘early joints’ in layered rocks (those that formed before uplift), on a semi-quantitative v versus KI curve.
Case 2. Superposing plume morphologies from both ‘early joints’, and ‘late joints’ in layered rocks (those that formed by uplift), on a semi-quantitative v versus KI curve.
Case 3. Plotting plume morphologies from joints cutting granites on a semi-quantitative v versus KI curve.
3.a. Case 1
3.a.1. The S- and C-type plumes from the Appalachian Plateau
We consider the particular fractographies that were distinguished on cross fold joint sets (normal and sub-normal to the fold axis) cutting the clastic sediments of the Devonian Catskill Delta in the Appalachian Plateau from New York and Pennsylvania (Bahat & Engelder, Reference Bahat and Engelder1984). These authors found the S-type plumes (Fig. 1a) on the set oriented 345° (NNW-striking) and the C-type plumes (Fig. 1b, c) on the set oriented 335° (NNW-striking). As opposed to scale effects that concern the application of rock strength derived in the laboratory to large rock masses, the application of fracture mechanic parameters such as v versus KI in fractography is not scale-dependent.
The C-type plumes are discontinuous, most probably indicating intermittent drastic reduction in velocity, possibly down to arrest (that took place whenever the driving force of pore pressure was exhausted), hence, it periodically vacillates between the stress corrosion limit and region II (Fig. 2), in analogy to the glass experimental results of Kerkhof (Reference Kerkhof1975).
The continuity of the lengthy S-plumes (one of them is about 100 m long: Bahat, Reference Bahat1991, fig. 3.18) and the lack of arrest marks along them indicates that there were no stops or any significant reduction in velocity along its continuous propagation path. This implies (by following the fracture mechanic rule that the fracture stress multiplied by the square root of the fracture length equals a stress intensity constant KI) that KI and the associated fracture velocity are likely to increase with fracture length, possibly up to KIc. We observe, however, that the S-plumes do not end in hackles, hence the propagation of the joint never reached KIc conditions, probably due to the constraint imposed by the overburden. Thus, v and KI are probably around region III. This argument is supported by the following experimental results. The striae induced by Michalske (Reference Michalske1984, fig. 1) developed on a smooth fracture surface of soda-lime silica glass in water when the velocity of fracture propagation reached about 10−2 m/s at around KI=0.7–0.8 MPa m1/2, which was above the mid-range between the stress corrosion limit, KI=0.3 MPa m1/2, and the fracture toughness KIc=0.9±0.1 MPa m1/2, that is, within region III (Fig. 2).
Moreover, the characteristic different appearances of the two plume types in two distinct joint sets cutting siltstone (C-type plumes in set 335° and S-type plumes in set 345°) are in line with our kinetic explanation: they originated under different stress fields (e.g. distinct settings of principal stresses) that induced non-conjugate fractures at different times. Apparently, the inertia of a fast-moving crack (S-type plume) tends to drive it monotonously, while a slow fracture (C-type plume) responds more readily to minute changes in local stresses, and is more likely to deviate from straightness (Fig. 1b, c).
The basic argumentation for suggesting that the S-type plume propagated faster than the C-type plume was outlined by Bahat (Reference Bahat1991, p. 234). Engelder (Reference Engelder2004, fig. 4) showed similar relationships (replacing the S-type plume and the C-type plume, by J1 and J2, respectively).
3.a.2. The construction of the v versus KI curve for the S- and C-type plumes
Bahat, Bankwitz & Bankwitz (Reference Bahat, Bankwitz and Bankwitz2003) introduced a semi-quantitative v versus KI curve correlative to fractographic patterns (Fig. 4). We adapted this approach in constructing semi-quantitative v versus KI curves for the S-type and C-type plumes (Fig. 5). The initiating points of region I occur at KI values between 0.073 MPa m1/2 and 0.14 MPa m1/2. These were calculated according to Atkinson & Meredith (Reference Atkinson, Meredith and Atkinson1987a, fig. 4.7) by extrapolating the lowest two curves of the Tennessee sandstone to a velocity of 10−8 m/s. The Tennessee sandstone is the closest approximating rock to the siltstone and fine-grained sandstone beds in which the two plume types where found in at least four formations by Engelder (Reference Engelder2004). Atkinson & Meredith (Reference Atkinson, Meredith and Atkinson1987a) identified indications of the presence of region II, but have not made definitive observations of region III in rocks, although they are quite common in glasses and ceramics (Wachtman, Reference Wachtman1974). Therefore, a hypothetical region III is added above region II in Figure 5, to facilitate clarifying the ranges of velocities and stress intensities under consideration. The two log v versus log KI curves in region III end in a range of KIc values between 0.45 MPa m1/2 and 0.79 MPa m1/2 (Atkinson & Meredith, Reference Atkinson, Meredith and Atkinson1987b, table 11.3) in Figure 5. We postulate v(KIc)=10−2 m/s, to be somewhat below the v(KIc) of granite (Fig. 4).
A semi-quantitative v versus KI curve for joints in granite from the South Bohemian Batholith in the Czech Republic. The curve is derived from nine criteria, based on laboratory experiments and theoretical considerations, showing the approximate locations of the various fractographic elements (modified after Bahat, Bankwitz & Bankwitz, Reference Bahat, Bankwitz and Bankwitz2003).
A semi-quantitative v versus KI curve for the S-type and C-type plumes representing joints in the Appalachian Plateau; see calculation procedure in the text (Case 1). Plots for the two plume types vary between KIc=0.45 MPa m1/2 and KIc=0.79 MPa m1/2, based on data from Atkinson & Meredith (Reference Atkinson, Meredith and Atkinson1987a).
The sub-critical part of the log v versus log KI curve can be estimated as follows. A well-known, semi-empirical law states that the velocity increases with KI as (e.g. eq. 1.144 in Bahat, Rabinovitch & Frid, Reference Bahat, Rabinovitch and Frid2005):
where a and n (called the sub-critical crack growth indices) are constants. We use for KI=0.073 MPa m1/2 the value of n=14 and for KI=0.14 MPa m1/2 the value of n=26, according to Atkinson & Meredith (Reference Atkinson, Meredith and Atkinson1987b, table 11.6) for the Tennessee sandstone in the corresponding different conditions. Constant a is not required for drawing the v and KI curve in region I, since we know the initial point of region I, and the n values determine the slopes of the curves.
Both C and S plumes propagated in sub-critical ranges (Table 1), such that the ranges of KI were limited by the two curves shown in Figure 5. The periodic smooth fracture of the C plume that follows the rhythmic increase in barb intensity (Fig. 1c) represents conditions between the lower limits of region I, possibly at about 10−6 m/s fracture velocity, in the smooth zone, and about 10−4 m/s in the zone that shows the radial short barbs. The latter value is somewhat above 4×10−5 m/s, the approximate upper value of arrest marks in plate glass (Kerkhof, Reference Kerkhof1975), and is assumed to correspond to the lower ranges of plume velocities. We suggest that region II for the S and C plumes varies between the latter two values, 10−4 m/s and 4×10−5 m/s, S closer to the upper value and C nearer to the lower one. Savalli & Engelder (Reference Savalli and Engelder2005, fig. 13B) suggested that region II for these rocks falls close to 10−2 m/s. The S plume mostly propagated at the high velocities of striae (Michalske, Reference Michalske1984), between around 2×10−4 m/s and close to 10−2 m/s (Tables 1, 2).
Plume morphologies on joint surfaces with their fracture mechanic implications
Source: *Bahat & Engelder, Reference Bahat and Engelder1984; **Bahat, Reference Bahat1987; ***Bahat, Bankwitz & Bankwitz, Reference Bahat, Bankwitz and Bankwitz2003.
Conditions of fracture velocities for assumed ranges of stress intensities* for joints cutting siltstone and chalk
*See text for sources and criteria leading to stress intensity assumptions.
See description of fracture provinces in Table 1.
3.b. Case 2
3.b.1. The ‘Lower Eocene plume’ and ‘Middle Eocene plume’ in the Beer Sheva syncline
The Beer Sheva syncline is an asymmetrical fault-fold basin within the Syrian Arc (Krenkel, Reference Krenkel1924), a sigmoid fold system that stretches from Syria in the north, through Israel to Egypt in the south. This study concerns the Mor Formation and the Horsha Formation, from the Lower and Middle Eocene, respectively, each of them about 100 m thick. The Mor Formation consists of chalk layers (40 to 90 cm thick) alternating with beds of chert nodules, up to 10 cm thick. The cross-fold single layer joints oriented 328° form the dominant set, arrested at the boundaries of the chalk layers with chert beds. Chert does not occur in the overlying Horsha Formation, which consists only of chalk layers of various thicknesses. The single layer joints of the Horsha Formation vary considerably in orientation (Bahat, Reference Bahat1987).
Practically all the joints that reveal their fractographies in the Beer Sheva fracture province are decorated by plumes. Case 2 relates to two plume types in this syncline (Bahat, Reference Bahat1987, Reference Bahat1999). The plumes that decorate single-layer burial joints, which cut the Lower Eocene chalks, are coarse and closely associated with abundant rough arrest marks (e.g. shown on the book cover of Bahat, Reference Bahat1991). On the other hand, the plumes that mark single-layer uplift joints in the Middle Eocene chalks are delicate and are mostly not associated with arrest marks (Fig. 3b).
These distinct fractographies most likely recorded propagation at different fracture velocities: slower in the Lower Eocene chalks, and faster in the Middle Eocene chalks. The Lower Eocene single layer joints developed in the burial stage by extension (horizontal ó3 positive). The coarse plumes and rough arrest marks appeared when fracturing propagated, while being permanently covered by additional sediments under increasing overburden stresses. On the other hand, previous studies have shown that the Middle Eocene single layer joints had been formed by uplift, under growing tensile conditions when upper sediments were being removed by erosion (Bahat, Reference Bahat1999). Hence, fracturing initiated near to the surface, and as elevation progressed, compensation by erosion gradually exposed the rock from levels of greater depths to this maximal tension (Bahat, Reference Bahat1991, p. 296), that is, these joints grew in response to a stress gradient in which the horizontal least principal stress (ó3 negative) migrated downward as the erosion progressed.
Correspondingly, it is likely that fracture velocity was greater when propagating close to the ground surface under tensile conditions, compared to the velocity under conditions of increasing compression with depth. Although faster propagating than the Lower Eocene joints, no hackles have been observed on any Middle Eocene joint, implying conditions of KI below KIc.
Thus, whereas the coarse plumes from the Lower Eocene, which are commonly associated with arrest marks, correspond to a certain fracture velocity range, signifying the Lower Eocene plumes, the delicate plumes from the Middle Eocene that are not associated with arrest marks correspond to another range, signifying the Middle Eocene plumes (Fig. 6).
A semi-quantitative v versus KI curve for joints in chalks from the Beer Sheva syncline; see calculation procedure in the text (Case 2). The frames LEP (for the coarse plume from the Lower Eocene chalks) and MEP (for the delicate plume from the Middle Eocene chalks), mark the fracture velocity limits of the two joints, below and above region II, respectively. Range of terminal velocities, between A and B for dry chalk and between C and D for wet chalk. KIo is the stress intensity at the stress corrosion limit.
3.b.2. The construction of the v versus KI curve for the ‘Lower Eocene plume’ and the ‘Middle Eocene plume’ in the Beer Sheva syncline
The construction of the logarithmic v versus KI curve for joints in chalks from the Beer Sheva syncline (Fig. 6) is based on data from various outcrops. The value KIc~0.17 MPa m1/2 was borrowed from the data on chalk by Dibb, Hughes & Poole (Reference Dibb, Hughes and Poole1983). The range of the initiating points of region I used in Figure 6 was between KI (v=10−8)=0.03 MPa m1/2 and 0.065 MPa m1/2. It was calculated according to Atkinson & Meredith (Reference Atkinson, Meredith and Atkinson1987a, fig. 4.14) by extrapolating the lowest curves of the calcite rocks to low velocities. This yielded the range of KI/KIc to be from 0.16 to 0.38, respectively. Thus, KI (v=10−8) varied from 0.16×KIc=0.16×0.17~0.03, up to 0.38×0.17=0.065 MPa m1/2.
The two curves were created by joining the respective points of KI (v=10−8) to the KIc ones by following the procedure outlined for case 1 in constructing the sub-critical part of the log v versus log KI. According to Atkinson & Meredith (Reference Atkinson, Meredith and Atkinson1987a, table 11.6), n for wet marble is ~9. Chalks may have different stiffnesses, depending on their particle and water constitution as well as their lithological histories (Mimran, Reference Mimran1977, Reference Mimran, Schneidermann and Harris1985). The chalks under consideration here are rather indurated (Bahat, Reference Bahat1987), that is, relatively stiff. Since no additional constants from carbonate rocks are to be found, we used the closest available one, from wet marble. The crack velocity at KIc, vC is postulated to be between 10−4 m/sec and 10−2 m/s, crossing KIc=0.17 MPa m1/2 at two locations.
Both plumes propagated in sub-critical ranges, limited by the two curves shown in Figure 6. The ‘Lower Eocene plume’ is estimated to have propagated between about 10−6 m/s and 4×10−5 m/s, close to the range of rib markings in plate glass and somewhat below it (Kerkhof, Reference Kerkhof1975), mainly below region II. The ‘Middle Eocene plume’ is estimated to have propagated between about 5×10−3 m/s, a little below the fracture velocity of striae in soda-lime glass under water (Michalske, Reference Michalske1984) and about 10−4 m/s, mainly above region II. This velocity range was lower than for the S plumes that showed a high degree of continuity (Fig. 1a; Bahat, Reference Bahat1991, fig. 3.18). The curves of the ‘Lower Eocene plume’ and the ‘Middle Eocene plume’ are constructed such that region II has the 10−4 m/s and 4×10−5 m/s values, respectively (Fig. 6), as in Figure 5, although it could change somewhat, because this region is controlled by the rate of reactant (water solution) transport to the crack tip (Wiederhorn, Reference Wiederhorn1967), conditions unknown to us in these two rocks.
The range of terminal velocities between A and B for dry chalk is adapted from T. Levi (unpub. M.Sc. thesis, Ben Gurion Univ. Negev, 2003) and Bahat, Rabinovitch & Frid (Reference Bahat, Rabinovitch and Frid2005, p. 466), and taken to be between 710 m/s and 340 m/s. However, the range of terminal velocities between C and D for wet chalk is assumed to be more realistic, between 200 m/s and 100 m/s. Crack velocities are generally proportional to sound velocities in the medium. The latter are proportional to (G/r)1/2, where G is the shear modulus and r is the density. According to G. Hayati (unpub. Ph.D. thesis, Israel Inst. Technology, 1975, fig. 4.34), the elastic moduli dependence on wetness for chalks having densities around 1400 kg/m3 (similar to the density of the present chalk), is G(dry)/G(wet)~9. Assuming a density increase of ~1.3 when wet (assuming minimal change in volume as water fills in the pores, when porosity is about 0.3), the decrease of velocity is about (9×1.3)1/2~3.4, yielding 710/3.4 and 340/3.4 as the velocity limits in wet chalks, that is, the maximum and minimum terminal velocities, C and D, respectively, in Figure 6.
3.c. Case 3
Case 3 relates to a study of ten distinct fractographies on ten adjacent joints from the same set at the Borsov granite quarry in the South Bohemian Batholith (also termed South Bohemian Pluton) from the Czech Republic (Bahat, Bankwitz & Bankwitz, Reference Bahat, Bankwitz and Bankwitz2003; Bankwitz et al. Reference Bankwitz, Bankwitz, Thomas, Wemmer, Kämpf, Cosgrove and Engelder2004), which belongs to the internal zone of the Variscan belt of Europe. The country rock consists predominantly of kyanite–sillimanite-bearing gneisses and schists of Late Proterozoic to Early Palaeozoic age metamorphism at 320–330 Ma (Petrakakis, Reference Petrakakis1997; Gerdes et al. Reference Gerdes, Friedl, Parrish and Finger2003: Tropper et al. Reference Tropper, Deibl, Finger and Kaindl2006).
Compared to the uniformity of the fractographic features on each joint set from the Beer Sheva sedimentary syncline, and to a large extent, also on the two sets from the Appalachian Plateau, there is a great fractographic variability in the Borsov granite quarry. Whereas in the two sedimentary provinces the plume features prevail, with a minor appearance of arrest marks, all confined to the sub-critical side of the v versus KI diagram (Fig. 2), the fractography from the Borsov granite quarry consists essentially of almost all the known, conventional brittle fracture elements (Bahat, Bankwitz & Bankwitz, Reference Bahat, Bankwitz and Bankwitz2003). A semi-quantitative curve of v versus KI was constructed for the joints at Borsov, showing that these joints were distributed between very low and very high values of v and KI (along most of the curve shown in Fig. 4; see also Table 1). This wide spread of values was interpreted as an indication that the joints had been formed by high pore pressures in the cooling granites, which locally varied considerably at the locations of individual joints.
4. Discussion
4.a. The absence of hackles in the investigated sedimentary rock layers
Cases 1 and 2 here received treatments as close to quantitative as possible, noting some of the existing limitations of the method (Bahat, Bankwitz & Bankwitz, Reference Bahat, Bankwitz and Bankwitz2003). In these relatively thin layers (up to about 50 cm in the siltstones of the Appalachian Plateau, and about 90 cm thickness in the chalks of the Beer Sheva syncline), plane stress conditions gradually increase from the middle of the rock layers towards their boundaries (Bahat, Reference Bahat1991, pp. 26, 246). Under these conditions, KIc values reach their maximum (Broek, Reference Broek1982), compared to low KIc values in thick layers where plane strain conditions prevail. Accordingly, the difficulty of attaining fracture toughness (KIc) conditions in thin sedimentary layers is why hackle formation in these rocks is likely to be minimal. In granites, on the other hand, hackles are more likely to occur, where plane strain conditions are more common.
4.b. The multiple geological influences on plume morphologies that must be considered in conjunction with the fracture mechanic analysis
We explained above our reasons for suggesting why in case 1 the coarser plumes were faster than the smoother ones, and in case 2 the plumes with the more delicate morphologies were faster than the plumes with the coarser morphologies. The implication is that when correlating KI and v conditions to plume characteristics in different fracture provinces, various geological parameters have to be taken into account, because these two fracture mechanic properties represent the sum of mechanical parameters that stem from the overall, local geological conditions, some of which influence the jointing process. We consider below several such parameters, including: (1) layer thickness, (2) joint genetics, (3) pore pressure, (4) supporting fractographic observations, (5) lithology and (6) interdependent relationships.
(1) It has been remarked that plumes like the S type are correlative with the reduction of layer thickness (Roberts, Reference Roberts1961; Syme-Gash, Reference Syme-Gash1971). This relationship can be linked to the ‘fracture slanting’ model, stemming from studies by Hertzberg (Reference Hertzberg1976), Broek (Reference Broek1982) and others. This is the reason why there is an increase in the occurrence of coarse S-type plumes in thinner layers. Apparently, slant cracks in thinner plates propagate more rapidly than those in thicker ones (Bank-Sills & Schur, Reference Bank-Sills and Schur1989).
(2) As mentioned above, while single layer jointing of the Lower Eocene chalks was of the burial genetic type, jointing of the Middle Eocene chalks took place during uplift(s). A somewhat similar correlation is seen in the Appalachian Plateau. Arrest marks of the C type occurred on slowly propagating joints (Fig. 1c) from the burial stage under increasing overburden stresses, while the rock has been constantly covered by additional sediments. On the other hand, arrest marks were absent from the S-type joints that formed during the syntectonic stage (Bahat, Rabinovitch & Frid, Reference Bahat, Rabinovitch and Frid2005, p. 205), when tectonic forces partly overcame the retarding influence of the overburden on fracture propagation.
Thus, there is a correlation between the fracture conditions characteristic to the various genetic jointing stages, the joint velocities and the corresponding induced fractographies. In the two investigated sedimentary fracture provinces, jointing in the burial stage was slow because it was constrained by heavy overburden, which resulted in fractographies enriched in arrest marks. During the syntectonic and uplift stages, overburden pressures were neutralized by counter-stresses that induced faster fracturing as recorded by plumes, mostly without arrest marks.
(3) The discovery of the C plumes on joint surfaces (Fig. 1c) suggested to Bahat & Engelder (Reference Bahat and Engelder1984) that the mechanism proposed by Secor (Reference Secor1965) for fracture propagation at depth is applicable in interpreting certain jointing processes. Secor's model fits quite well the formation mechanism of burial joints which are primarily driven by pore pressure, but it is doubtful whether this mechanism can also explain the growth of uplift joints, which result from remote tension, quite likely caused by bending (Price, Reference Price1974; Bahat & Rabinovitch, Reference Bahat and Rabinovitch1988). The role of pore pressure in forming syntectonic joints depends on the overall stress conditions imposed by opposing remote tectonic stresses and local pore pressures, and may change from case to case. It appears that the formation of the S-type plumes, which do not reveal periodicity analogous to that exhibited by the rhythmic C-type plume, was less affected by pore fluid pressures than the latter plume type, and perhaps not affected at all.
(4) Supporting evidence for the interpretation suggested above comes from several experimental observations, which have shown that arrest marks precede striae (and plumes) as fracture propagation proceeds (e.g. Bahat, Reference Bahat1991, pp. 136, 235). Therefore, the presence or absence of arrest marks is a diagnostic criterion in establishing the velocity ranges of plume propagation.
(5) Different lithologies would supply different amounts and pressures of pore driving forces for jointing.
(6) It appears that there are some interdependent relationships between the influences of the layer thickness, joint genetics and pore pressure on the plume morphology. The increase of overburden pressure during the burial stage imposes a constraint on the jointing velocity. This is overcome by periodic bursts of pore pressure, as often recorded on rhythmic plumes (Fig. 1c).
Thus, given an increase in layer thickness, a lithology that would supply proper amounts of fluids and strong overburden pressures would tend to create rhythmic short C-type plumes and arrest marks. On the other hand, decreasing overburden pressures and reduced effectiveness of pore pressures would favour the formation of lengthy, coarse S-type plumes and absence of arrest marks. When all other conditions are the same, thicker layers would be marked by more delicate plumes, whereas thinner layers would be decorated by coarser plumes.
4.c. Isotropic and anisotropic conditions
There is a need to distinguish between two fracture cases: (a) an idealistic, isotropic one (remote from boundary conditions), and (b) a fracture influenced by boundary conditions, rendering it anisotropic. Plumes will not form under conditions of no resistance to pure tensile fracture, often seen on a dynamically fractured glass under idealistic, isotropic conditions. However, dynamic fracture in glass, or glass ceramic (which is dominated by the glassy phase), when propagating between neighbouring boundaries, induces a series of curving striae that form a plume (Bahat et al. Reference Bahat, Frid, Rabinovitch and Palchik2002). In fact, these striae are even split (microscopically) into secondary ones, imitating plumes (Goldbaum et al. Reference Goldbaum, Frid, Bahat and Rabinovitch2003), a process characteristic of plumes often seen on joint surfaces. In addition, on the joint surface of a given rock, coarseness of plumes generally changes inversely with fracture velocity. Delicate plumes are created when some resistance is translated to local mode III shear, which retards the propagation of joints, due to the ‘mode III crack closure effect’ (Tshegg, Reference Tschegg1983). When the latter parameter becomes more pronounced, coarse plumes are developed.
5. Limitations and open questions
There are limitations in our comprehension of the observations. Particularly, the calculations lead us to estimate (rather than determine) our geological results, due to certain restrictions, as itemized below.
(1) We investigate fracture processes only at their post-mortem stage, measure strain results via their fractographic record, and translate it to KI and v parameters. By doing this, we probably introduce errors into our calculations.
(2) At the base of our calculations we made the following assumptions: (a) The shape of the v versus KI diagram has a universal shape, at least for the sub-critical regime. (b) We have used results of materials (rocks) similar to the ones treated because of lack of experimental results for the actual ones. This may be important, especially for the exponent value n. (c) Velocity values at KIc are hard to estimate experimentally, and we have therefore used a wide range of values.
Consequently, the final estimates for the velocities and KI values of the S and C plumes, as well as for the Lower Eocene plume and Middle Eocene plume cases cannot be considered to be precise. However, the approximate fracture mechanic conditions do clearly emerge.
(3) Although, generally considered to indicate slow fracture velocities (e.g. Kulander, Barton & Dean, Reference Kulander, Barton and Dean1979; Müller & Dahm, Reference Müller and Dahm2000), en échelon segmentation may occasionally indicate high velocities (Cramer, Wanner & Gumbsch, Reference Cramer, Wanner and Gumbsch2000). This is because the en échelon breakdown is dependent on the ratio of mode III/mode I (Sommer, Reference Sommer1969), and when this ratio decreases, rare, rapid fracturing may occur. In this connection, we still do not know how to identify the critical conditions that would result in en échelon segmentation or in hackles. For instance, the case shown by Bahat, Grossenbacher & Karasaki (Reference Bahat, Grossenbacher and Karasaki1999), and Figure 3c herein, needs to be further investigated along this line.
(4) In another example, Peter Bankwitz (pers. comm.) observed different relationships of KIII/KI and plume morphology versus layer thickness (in non-sedimentary rocks) from the one mentioned in Section 4.b. This intriguing difference needs to be examined.
(5) We still do not know how to identify the mirror boundaries for fringes of the ‘continuous breakdown type’ and ‘transitional’ styles. Perhaps some guidance may come from distorted fractographies of certain ceramics and single crystals, which often have irregular or discontinuous (‘zig zag’ or ‘tongue’) boundaries (e.g. Rice, Reference Rice, Frechette, Course and Burdick1974).
6. Conclusions
This paper focuses on three new subjects: (1) the mirror plane and criteria for elucidating the mirror boundaries, (2) a new method of calculating KI and v in sedimentary rocks, and estimating the fracture velocities of four joints in two fracture provinces, and (3) arguing that the treatment of the fracture mechanics of jointing in sedimentary rocks must be carried out in conjunction with the possible variability of some six geological parameters.
A key criterion in characterizing the mirror plane is identifying the mirror boundary that separates the mirror plane and the fringe. There is a clear distinction between two en échelon segmentation types in the fringe, the ‘discontinuous breakdown type’, and the ‘continuous breakdown type’. The present study applies only to the discontinuous breakdown type.
Four criteria for elucidating mirror boundaries on joints may be useful: (1) morphological, (2) angular, (3) sense of stepping changes across mirror boundaries and (4) the ratio of mirror plane radius/the radius of the critical flaw, which should be around 15.
There is a correlation between the fracture conditions characteristic of the various genetic jointing groups, the joint fracture velocities and the corresponding induced fractographies. Often slow plumes are relatively short, show periodicity and typically exhibit superposition of arrest marks. On the other hand, faster plumes are longer, and show no superposition of arrest marks.
The tensile stress intensity, KI, and the velocity of joint propagation, v, are two basic fracture mechanic properties that represent the sum of mechanical parameters that stem from the overall, local geological conditions, some of which influence the jointing process. Therefore, when correlating KI and v conditions to plume characteristics in different fracture provinces, various geological influential parameters have to be taken into account: (1) layer thickness, (2) joint genetics, (3) remote and local stresses, (4) pore pressure, (5) different lithologies and (6) inter-dependent relationships between the influences of various parameters.
The difficulty in reaching fracture toughness (KIc) conditions in thin sedimentary layers where plane stress conditions prevail minimizes the likelihood of hackle formation in them. This is an important reason why hackles are almost unknown in these rocks. In granites, on the other hand, hackles are more likely to occur, where plane strain conditions are more common. The results show that the joints considered in this study from the Appalachian Plateau, USA, and the Syrian Arc in Israel never reached the KIc conditions.
Acknowledgements
We are grateful for the most useful comments made by Peter Bankwitz and J. L. Simón on an earlier version of this manuscript. Jiří Žák, an unknown referee and the editor helped to improve the paper significantly. Yoav Borenstein and Or Bialik provided technical help. This study was supported by the Earth Sciences Administration, the Ministry of Energy and Infrastructure.
