Formation and Interpretation of Dilatant Echelon Cracks

Formation and interpretation of dilatant echelon cracks

DAVID D. POLLARD | U.S. Geological Survey, Metilo Park, California 94025 PAUL SEGALL PAUL T. DELANEY U.S. Geological Survey, Flagstaff, Arizona 86001

ABSTRACT 1974, p. 82; Broek, 1974, p. 31-38), and echelon cracks in glass have been produced in laboratory experiments (Sommer, 1969). The The relative displacements of the walls of many veins, joints, ubiquity of echelon cracks attests to their importance in the fracture and dikes demonstrate that these structures are dilatant cracks. We process, but it suggests the possibility of several mechanisms of infer that dilatant cracks propagate in a principal stress plane, nor- origin. We focus on a particular mechanism that applies to some, mal to the maximum tensile or least compressive stress. Arrays of but not to all of the crack arrays mentioned above. echelon crack segments appear to emerge from the peripheries of The two principal views of fractures, in-plane and cross- some dilatant cracks. Breakdown of a parent crack into an echelon sectional, reveal different features of the echelon geometry. In the array may be initiated by a spatial or temporal rotation of the fracture plane (Fig. 1), we see smooth echelon-crack surfaces that remote principal stresses about an axis parallel to the crack propa- are bounded by rough steps and emerge from a single parent crack. gation direction. Near the parent-crack tip, a rotation of the local principal stresses is induced in the same sense, but not necessarily through the same angle. Incipient echelon cracks form at the parent-crack tip normal to the local maximum tensile stress. Further longitudinal growth along surfaces that twist about axes parallel to the propagation direction realigns each echelon crack into a remote principal stress plane. The walls of these twisted cracks may be idealized as helicoidal surfaces. An array of helicoidal cracks sweeps out less surface area than one parent crack twisting through the same angle. Thus, many echelon cracks grow from a single parent because the work done in creating the array, as measured by its surface area, decreases as the number of cracks increases. In cross sections perpendicular to the propagation direction, echelon cracks grow laterally, each crack overlapping its neighbors, until the mechanical interaction of adjacent cracks limits this growth. Dilation of each crack pinches the tips of adjacent cracks into an asymmetrical form and introduces local stresses that can cause lateral growth along a curving, sigmoidal path. Sigmoidal echelon cracks may link at tip-to-plane intersections, leaving a step in the through-going crack wall. The geometry of dilatant echelon cracks may be used to infer spatial or temporal changes in the orientation of principal stresses in the Earth.

INTRODUCTION

Dilatant echelon cracks are associated with fractures in a variety of materials and at scales ranging from microns to kilometres. Examples include cracks in test specimens of granite (Kranz, 1979), fringe cracks on joints (Hodgson, 1961), quartz-filled veins (Beach, 1975), segments of igneous dikes (Anderson, 1951, p. 56), and ground cracks in volcanic regions (Nakamura, 1970; Duffield, 1975). Other natural echelon cracks occur as crevasses in glacial ice (Meier, 1960), as tension gashes in shear zones (Ramsay, 1967, p. 88-91), as cracks in soil within active fault zones (Brown and oth- Figure 1. Surfaces of dilatant echelon cracks and the parent ers, 1967), and as traces of fault segments (Wallace, 1973). Cleavage crack. A: Photograph normal to joint in hand specimen of shale. B: fractures in metal crystals display an echelon geometry (Fellows, Photograph oblique to the joint.

Geological Society of America Bulletin, v. 93, p. 1291-1303, 16 figs., December 1982.

1291

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The association of echelon cracks with a single parent crack is well segments of a quartz vein (Fig. 3A) about 10 cm wide and 1 cm documented for both joints (Kulander and others, 1979) and sheet thick strike about 5° from the strike of the vein. Segments of a intrusions (Pollard and others, 1975). We emphasize the following minette dike (Fig. 3B) about 200 m wide and 2 m thick strike about geometric features observed in these examples and idealized in Fig- 6° from the strike of the dike. Noteworthy similarities between ure 2. (1) Cracks of similar width form along a narrow breakdown these two examples include the following. (1) Crack segments zone marking the periphery of the parent crack, (2) each crack is a within a particular array have comparable widths and thicknesses; twisted surface that systematically diverges from the orientation of (2) strikes of cracks systematically diverge from the strike of the the parent, and (3) echelon-crack length may be several times array; (3) cracks display a pronounced asymmetry in cross-sectional greater than width. One purpose of this paper is to analyze a break- form near their tips; and (4) the overlap of crack tips approximately down mechanism that produces echelon cracks with these geomet- equals the step between cracks. A second purpose of this paper is to ric characteristics from a parent crack. rationalize these common geometric features of echelon cracks in In cross section, the width, thickness, and relative orientation cross section. of discrete echelon segments are readily observed. We limit attention to echelon cracks oriented at small angles to the trend of their INITIATION OF ECHELON-CRACK ARRAYS array and having a small ratio of thickness to width. For example, Laboratory experiments have demonstrated that breakdown into many echelon-crack segments occurs when a parent crack, propagating in a principal stress plane, encounters a region in which the remote principal stresses have rotated about an axis parallel to the propagation direction (Sommer, 1969). The echelon cracks grow along twisting surfaces into the new principal stress plane. This concept has been applied by Pollard and others (1975) to interpret echelon igneous dike segments; by Ryan and Sammis (1978) to interpret cooling cracks in a basalt flow; and by Kulander and others (1979) to interpret joints. To study this process in more detail, we follow Gell and Smith (1967) and examine the stress field near a parent-crack tip where echelon cracks nucleate. The analysis applies to cracks in a brittle, elastic material propagating at low speeds where inertial forces are negligible.

Stress State Required for Breakdown

Consider an elastic body subject to remote principal stresses aj" and 02, (tensile stress is positive) acting in the yz plane, and a crack of length 2a subject to pressure p lying in the xz plane with straight tips parallel to z (Fig. 4A). Breakdown into echelon segments is Figure 2. Idealized block diagram illustrating geometry of par- independent of the third remote principal stress, which acts in the ent and echelon cracks (Lutton, 1971). propagation direction, x. The parent crack initially grows and

® — quartz vein segment ——,

0 10 Figure 3. Cross sections of dila- CENTIMETERS tant echelon cracks. A: Map of part of a quartz-filled vein array from Carboniferous rocks near Black Pool, North Devon, England (Beach, 197S). B: Map of part of a minette dike near ® Ship Rock, New Mexico (Delaney minette dike segment and Pollard, 1981). The small en- closed areas along the contact contain breccia.

0 100 1 i _-•_ .... I METERS

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dilates in a plane perpendicular to the maximum tension or least quate because the stress distribution near the crack tip is dominated compression. In nature, aj^may change orientation spatially along by the geometric effect of the sharp tip and is not strongly depen- the propagation direction, or temporally as the parent crack inter- dent on where the shear is applied (Lawn and Wilshaw, 1975, p. 55). acts with adjacent structures.. This subjects the parent crack to shear A uniform rotation a of the remote stresses about the x axis

stresses. In our analysis, the effects of varying remote stresses on the introduces a normal stress and a shear stress of2 acting in the near-tip stress field are approximated by changing the orientation crack plane (Fig. 4A). Breakdown of the parent crack depends upon of afuniformly over the entire region. This approximation is ade- a and the applied stress ratio

R-(2p + a2°°)/(ar- erf), . OO p>-o i. (1)

The inequality relating p and afis required for the parent crack to be open. The numerator in equation 1 is the difference between the internal pressure and the remote mean stress or pressure, -(aj'c+ 2. The denominator in equation 1 is the remote maximum shear, (aj"- af)/2. Near the crack tip (Fig. 4B), the stress field is proportional to the stress-intensity factors (Lawn and Wilshaw, 1975, p. 51-56). The remote normal stress and internal pressure contribute to a

. mode / stress intensity, K[ - (ovy + p) V an, and the remote shear

stress introduces a mode ///stress intensity, Km = o°?z\[an. A crack subject only to mode / deformation will dilate and propagate in its own plane. Breakdown into echelon cracks is initiated by shear parallel to the crack tip, that is, the mode III deformation. The tendency for breakdown increases as the ratio of stress intensities

Km IK/ differs from zero. Writing a"y and a" in terms of a and the remote principal stresses by resolving stress in the xz plane, we have

Kin I Ki = (sin2 a)l(R + cos2a), R + cos2<* >0. (2) The inequality relating R and cos2a is required for the parent crack to remain open. Relations among the stress intensity ratio, rotation angle, and applied stress ratio are illustrated in Figure 5. For

Figure 4. Idealized block diagrams illustrating relations among applied stresses and cracks. A: Parent crack of length la and infi- nite width in z is loaded by pressure p and remote principal stresses af, o'i. Rotation a of principal stresses about x introduces

stresses o™y and acting on the crack plane. B: Small element 20 40 60 cut from near parent-crack tip with incipient echelon cracks of Rotation Angle , a (degrees) width 2b at angle fi to parent-crack plane loaded by maximum local tension o\„ Polar coordinates centered at parent-crack tip are r, 8. Figure 5. The ratio of mode III to mode / stress intensities Heavy arrows indicate relative motion of parent-crack walls when plotted versus angular rotation a of remote principal stresses for subject to mode / and mode /// deformation. several values of the stress ratio R.

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R > 10, we have Km IK; =sin2 a/R =0, and there is little tendency to break down for any rotation of the remote principal stresses. For a particular rotation angle, the stress intensity ratio increases as R decreases. For R- 1, we have K///IK/ = tana, and relatively small rotations of the principal stresses could initiate breakdown. The range of R in nature can be estimated from in situ stress measurements using hydraulic fracturing techniques. For a particular vertical fracture, two different values are calculated, one for lateral propagation and another for vertical propagation. From data compiled by Bredehoeft and others (1976) in the Piceance Basin of Colorado in the Cameron 702 well at seven depth intervals between 189 m and 453 m, we find 0.11 < R < 7.45 for lateral propagation and 0.29 < R < 6.50 for vertical propagation. Of the 14 values, 10 are in the range 0.30 < R < 0.90. From data compiled by Haimson (1978) for a well in the Michigan Basin at depths of 1,230 m, 2,850 m, and 3,660 m, we find 0.63 < R < 2.20 for lateral propagation and 0.56 < R < 0.62 for vertical propagation. Stress magnitudes clearly provide conditions conducive for breakdown at these two localities. Furthermore, fracture orientation at the well bore varied from N90°E at 405 m to N60°E at 457 m in Cameron 702, indicating a substantial rotation of principal stress directions about a vertical axis. Data from hydraulic fracturing tests in other regions suggest that stress states required for the formation of echelon cracks are common.

Orientation of Echelon Cracks

Several laboratory studies reviewed by Kobayashi and Four- ney (1978) indicate that tensile crack propagation in rock is accom- panied by the formation of microcracks in the region of stress concentration near the parent-crack tip. A similar phenomenon, namely, growth of incipient echelon cracks, may occur with introduction of a shear stress acting parallel to the parent-crack front. Erdogan and Sih (1963) proposed that cracks grow from a parent- crack tip perpendicular to the local maximum tension, a\. Thus, the orientation of incipient echelon cracks may be predicted by deter- mining the orientation of o\. Using polar coordinates r and 6 centered at the parent-crack tip (Fig. 4B), equations for the stress components in the region r « a and for a mixed-mode / and III loading are given by Lawn Figure 6. Incipient crack angle p plotted versus angular rota- and Wilshaw (1975, p. 53-54). These equations are simplified if we tion a of the remote principal stresses. A: Poisson's ratio is held restrict attention to the xz plane where 0 = 0°. The angle that a| constant and the stress ratio R is varied. B: The stress ratio R is held makes with the y axis is a function of the stress intensity ratio and constant and Poisson's ratio is varied. Poisson's ratio v. Although the signs of a and p are the same, substitution of 1 p = '¿tan" [Kml K,(Vi-v)l equation 2 into 3 reveals that their magnitudes are equal only for K,> 0. (3) certain combinations of the applied stress ratio and Poisson's ratio. Variations of P with a for constant Poisson's ratio and constant The inequality involving K/ assures an open parent crack. Equation applied stress ratio are illustrated in Figures 6A and 6B. The local 3 defines the orientation of a plane tangent to the surface that principal stress direction rotates through a smaller angle than the carries the maximum tensile stress and zero shear just in front of the remote direction, p < a, for large stress ratio and/or small Pois- parent-crack tip. We identify this as the incipient crack plane. For son's ratio. For small stress ratio and/or nearly incompressible Km = 0, we have P = 0, and incipient cracks grow in the parent- materials, the local stress rotates more than the remote stress, crack plane, as expected for pure dilational loading. For Km 0, P >a. incipient cracks grow at angle ¡3 to the parent-crack plane. Positive A continuous transition from parent to echelon cracks is com- Km is induced by a positive rotation a of the remote principal monly observed (Fig. 1). To initiate this smooth transition, the stresses and results in a positive angle p. This condition produces rotation of the local principal stress must be equal to or less than incipient cracks with a counterclockwise twist in orientation relative that of the remote principal stress in its first increment of change. If to the parent crack as viewed in Figure 4B. the local stress rotates more than the remote stress, incipient cracks

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new orientation, again in a principal plane. To explain why many echelon cracks of much smaller dimension than the parent crack tend to develop in this new orientation, we determined the energy required to form the cracks as a function of their number and geometry. A fundamental proposition of fracture mechanics is that this energy is proportional to the surface area created in an increment of crack extension (Broek, 1974, p. 15-17). To calculate the surface area, we constructed an analytic description of the echelon- crack surface. Data bearing on the three-dimensional geometry of echelon cracks are rare, but from our observations of basaltic dikes and joints, we have pieced together a qualitative description of changes in form as a continuous crack breaks down into numerous segments that twist into an echelon configuration. An echelon crack of width 2b that is straight in cross section (Fig. 3) can be generated by a straight line segment oriented perpendicular to the propagation direction. Extend an axis from the parent-crack tip oriented in the propagation direction and let the generating line twist through an angle w about this axis as it trans- lates in the propagation direction (Fig. 8A). The surface swept out is a helicoid, defined by the vector equation

r - (pu2) i -(w|Sini/2) j + (Micosis) k ,

-b +b, and 0 =Sw2 co. (5)

(Goetz, 1970, p. 103-166), where u\ and ui are curvilinear coordinates of the helicoidal surface that is coincident with the z axis for U2 = 0 and twists toward the y axis as «2 increases toward n/2. Position on a rectilinear generator that is perpendicular to x is defined by u\, and position on a circular helix with axis x is defined by «2- A family of side-by-side helicoidal surfaces, all originating from a single surface, represents an idealization of an echelon-crack array. Figure 7. The stress ratio R plotted versus Poisson's ratio. Three regions of differing crack behavior are defined by particular The ratio of change in position along x to change in angular relations between R and v. twist of the rectilinear generator is given by p. Thus, the rate of twist in a spatial, not temporal, sense is measured by 1/p. Data are will discontinuously abut the parent crack and form a rough break- almost completely lacking on the rate of twist. Although p is written down zone. Stated explicitly, smooth breakdown requires that as constant in equation 5, it appears to be a function of distance dp I da < 1 in the limit as a goes to zero. Substitution from equa- from the line of breakdown for some fractures. For the specimen in tions 2 and 3 shows that this condition implies Figure 1, the twist angle changes markedly within I cm of the breakdown zone, but it is roughly constant over the rest of the RHVi + v)H}A-v). (4) segment length. Equation 5 is used to derive the surface area S of a helicoid. For a Poisson's ratio typical of many rocks, v = 1/4, smooth The total surface area of the crack is twice the area of the helicoid breakdown occurs if R >3 (Fig. 7). Ranges of R estimated from in because there are two crack faces. Using the limits on u\ and «2 silu stress measurements include this value. For a very small Pois- specified above and the geometric relations indicated in Figure 8B, son's ratio typical of some rocks, smooth breakdown occurs if we find From equation 1, this requires that p>-02 ; that is, the

remote compressive stress in the crack plane must be less than the b internal pressure. The small compressibility of rock and the occur- *=0r/ dr2 - rence of applied stress ratios of > 1 provide conditions conducive 2 + for smooth breakdown in the Earth. For the extreme case of /V \/p W? iut du2 incompressible materials, v = 1/2, we find that P = n/4 for any = wp2 j (6/p) sj(blp)2 + I + incremental change in a, making a smooth breakdown impossible. ln[(6/p) + V(6/p)2 + 1] J. (6) Number of Echelon Cracks For a rate of twist that is very large, 1/p » 1, 5 -nb2, and the Because dilatant cracks propagate in a principal stress plane, a helicoid degenerates into a circular disk of radius b lying in the yz rotation of the remote principal stresses will cause a crack to seek a plane. For a rate of twist that is very small, 1/ p -0, S =2xb, and the

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r c

ecHf&FMX

Figure 8. Geometric features of the helicoidal crack surface. A: Components of the vector equation 5, contours of constant «i and u%, crack width 2b, and twist angle to are illustrated in relation to the Cartesian coordinate system x, y, z. B: An element of surface SA is bounded by vectors Srj and dri tangent to the helicoidal surface at u\, uj. Components of these vectors in the Cartesian system are indicated.

helicoid degenerates into a rectangular plane of length x and width No upper bound on the number of incipient cracks is indicated 2b lying in the parent-crack plane. by the analysis of fracture energy. However, the cases of a great The surface area Si of a single twisted crack of width 2nb is number of cracks of very small width are precluded if we consider

compared to the surface area Sn of n cracks each of width 2b (Fig. the following scenario for incipient crack growth. The heterogeneity 9). As the number of cracks increases in an echelon array of given of rock suggests that cracks of unequal width and spacing form near total width, the surface area swept out decreases and therefore the the parent-crack tip. Those cracks with the appropriate combina- work done to create the array decreases. For example, consider tion of width, location, orientation, and driving stress propagate echelon cracks that have extended a distance in x equal to their most rapidly and become the dominant cracks. Intervening cracks, half-width b. Then the parameter b\p in Figure 9 is equal to the caught in regions of decreased stress surrounding the dominant twist angle. Choosing w = irj25 -1°, the total area swept out by 50 cracks, do not develop into members of the echelon array. Thus, cracks is only 30% of that swept out by the parent crack twisting different crack-propagation rates in conjunction with local stress through this small angle. In general, breakdown into n cracks that relief by dominant cracks lead to the formation of a finite number twist into an echelon array requires less energy than if the .entire of echelon cracks. parent crack grows into the new orientation. Constraints on the growth of dominant cracks come from the

N.E. SHIR ROCK DIKE

HEORETICAL DIKE SEGMENT FORM 40

d) o 20 c o H— CO 0 40 60 80 100 Q Number of Cracks , n -20 1200 1400 1600 Figure 9. Relative surface areas of one heli- Distance Along Dike , m coidal crack of width 2nb and of n helicoidal cracks each of width 2b plotted versus n for sev- Figure 10. Echelon-crack shapes in cross section. Vertical scale is exaggerated to eral values of the ratio b/p, where 1/p is the spa- show dilational form. A: Map of dike segments from Ship Rock, New Mexico tial rate of twist. (Delaney and Pollard, 1981). B: Theoretical model of dike segments.

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stress state near the parent-crack tip. Subject to a constant local ments and the theoretical form. The local irregularities formed as driving stress a|, a particular dominant crack (Fig. 4B) would flowing magma removed blocks of host rock, and so they should increase in width 2b without bounds because the stress intensity at not conform to an elastic dilation (Delaney and Pollard, 1981). its tip increases as o\sJbn. However, this growth is limited by the Several geometric parameters of echelon cracks in cross section sharp decrease in driving stress away from the parent-crack tip. To can be measured in field and laboratory studies. Twist angles rarely evaluate this decrease, consider the crack-tip stress field just before exceed 30° for echelon cracks that emerge smoothly from a single echelon cracks develop. parent (Hodgson, 1961; Lutton, 1971; Pollard and others, 1975). Important relations between width 2b, twist cu, distance between Oij =[(°W + PVijiQ) + *#)] centers 2c or 2k, step 5, and overlap o are

s = csinto (8a) o-b-k-b - ccosco (8b) where/and g are trigonometric functions of order unity (Lawn and Wilshaw, 1975, p. 53-54). The term V a/2r in equation 7 shows how (Fig. 11). Very near the line of breakdown we expect to find b - c, the driving stress for incipient echelon cracks decays and how it so that equation 8b reduces to o - c(l-cosai). Data at some distance depends upon parent-crack length. from the line of breakdown (Fig. 3) show that overlap exceeds this estimate, so some lateral propagation of the vein and dike segments has occurred. Thus, width may increase somewhat with twist and distance from the parent crack.

Lateral Growth of Echelon Cracks

To illustrate the nature of lateral growth, we considered a three-crack array (Fig. 11) subject to uniform driving pressure p. Using only three cracks minimizes the computations, yet it provides adequate insight to generalize about an array with more members. Each crack is of width 2b\ crack centers are spaced at 2c; and angles Figure 11. Cross section of three echelon cracks indicating of twist are in the range 0° < to < 30°. Dilation of an isolated crack width lb, twist cu, distances between centers 2c and 2k, step 2s, and produces only a mode I stress intensity factor, K/ = p\Jbn. How- overlap 2o. Heavy arrows indicate relative motion of crack walls ever, mechanical interaction among several closely spaced echelon when subject to mode / and mode // deformation. cracks changes this dilation and introduces a local shear acting To demonstrate the decay of driving stress, consider the stress perpendicular to the crack tip. This shear (Fig. 11) contributes to a component of ay from equation 7 that would drive an incipient mode II stress intensity factor, Ku. crack at the tip of a parent crack subject to Km/ K/ =0.15. This Conditions for lateral-crack propagation along a straight crack component is the normal stress acting on a plane containing the x path are determined by the crack extension force G, which is equiv- axis and making angle /? = 15° to the y axis (Fig. 4B). The magni- alent to the energy available for an increment of crack growth tude of the normal stress drops by more than 50% from a point on (Broek, 1974, p. 115-126). Theoretical analyses have shown that a the xz plane to a point only 0.05a from this plane. The decay of remote compressive stress acting parallel to a crack may produce a driving stress is even more pronounced for greater values of Km IK/ straight crack path even in the presence of a mode II stress intensity and /3. Thus, dominant cracks will grow only to widths that are a (Cotterell and Rice, 1980; Karihaloo and others, 1980). Thus, where small fraction of the parent-crack length. adjacent echelon cracks have followed straight paths (Fig. 3), we infer the action of such a stress. Normalizing G by the crack exten- GROWTH AND FORM OF ECHELON CRACKS sion force for an isolated crack G„ we have IN CROSS SECTION G/Gi = (Kh Kj,)lp2b (9) Echelon cracks with small ratios of thickness to width may extend to lengths that are great relative to their width with little The crack extension force for the middle crack of the three-crack change in twist angle (Fig. 1). In these cases, a two-dimensional array is approximately equal to that for an isolated crack if elastic solution is adequate to analyze lateral-crack propagation, b\c < 0.5 (Fig. I2A). For greater values of 6/c, we find that G/ G,- stability, and the form of echelon cracks in cross sections that are increases due to crack interaction, reaching a maximum value near far from the breakdown zone. We use the Schwarz Alternating b = c. Adjacent cracks create stress fields that promote propagation Technique (Sokolnikoff, 1956, p. 318-326) to solve the elastic prob- during this constructive phase of interaction. However, for lem. The fundamental stress functions are those derived by Pollard overlapping tips, b/c> 1, each crack works against the dilation of and Holzhausen (1979) for uniform tractions acting on a patch of its neighbors and G/G, decreases to a value less than that for an the crack wall. This method closely approximates analytic solutions isolated crack. The maximum magnitude of G/G,- and the rate of for simple crack configurations (Segall and Pollard, 1980). The increase and decrease with b/c are greater for smaller twist angles. efficacy of our solution method for more complex arrays, such as The propagation criterion is that G must reach a critical value

an igneous dike, is demonstrated by comparing Figures 10A and Gc. Normalizing the critical value by G,-, we have 10B. If local irregularities in the dike contact are ignored, there is 2 1 excellent agreement between the observed form of the dike seg- Gel Gì = [2nGclcp ( 1 - »)] (b/cY (10)

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where ¿u is the elastic shear modulus. Different constant values of this ratio are represented by dashed curves in Figure 12A. The normalized crack extension force must exceed values on a particular dashed curve for propagation. Consider a three-crack array shortly after breakdown with a small twist angle, to = 5°, and width- to-spacing ratio near one, 6/c= 0.95 (point A, Fig. 12A). If the propagation criterion is met and the pressure remains constant, the central crack will readily propagate to a greater width (point A', Fig. 12A) where b/c — 1.2 and the crack extension force falls below the critical value. A large array of such cracks will stabilize in a configuration where adjacent cracks just overlap one another. This , geometry is displayed in Figure 3 and in the data collected for veins by Beach (1975, Fig. 7) and for dikes by Delaney and Pollard (1981, Fig. 10A). Next consider a three-crack array with greater twist angle, to = 30°, and smaller width-to-spacing ratio, b/c =0.7 (point B, Fig. 12A). For spacing equivalent to the previous example, propagation requires a greater pressure, and stabilization of the central crack (point B', Fig. 12A) occurs after much greater overlap, b/c- 1.5. Echelon cracks with considerable overlap are illustrated by veins in sandstone mapped by Beach (1980, Fig. 20). The stabilizing effect of crack interaction demonstrated by Figure 12A provides a rationale for the ubiquity of echelon-crack arrays in nature. Not all echelon cracks are straight (Fig. 13), implying that lateral growth can proceed along a curving path. Three different Crack Width , b/c theories for out-of-plane propagation under mixed mode / and II loading have been compared to laboratory tests on rock by Ingraf- fea (1981). The qualitative form of crack paths may be deduced from the sign and magnitude of the ratio KJJI K/. The dependence of this ratio on the width-to-spacing ratio is given in Figure 12B. The ratio is nearly zero for small values of b/c where crack interaction is negligible. For greater values of ¿/c and small twist angles, the ratio is negative in sign and decreases to a minimum value near b/c = 1. For still greater values of b/c, the ratio sharply increases to large positive values. For lower twist angles, the magnitudes of the extreme values are greater, as are the rates of change of the stress intensity ratio with crack width. These results are consistent with the estimates of Swain and others (1974) and the numerical calculations of Yokobori and others (1971) for two echelon cracks. A positive mode II factor (right-lateral shear) is associated with a path that diverges to the right side of the crack plane for an observer looking toward the crack tip (Fig. 12B). A negative mode //factor (left-lateral shear) is associated with a path that diverges to the left. The angle that the path makes with the crack plane is greater for greater values of the ratio Ku\ K/. The small negative minimum followed by greater positive values of Knj K/ as crack width increases suggests that tips of echelon cracks at small twist angles should follow a path that curves gently away from and then sharply toward adjacent cracks. Such paths have been documented for small cracks in glass laboratory specimens, for quartz-feldspar veins, and for much larger igneous dikes (Fig. 13). Crack Width , b/c The behavior just described ensures that adjacent crack seg- Figure 12. Graphs illustrating the nature of echelon-crack ments will not grow together tip-to-tip, even if they are nearly propagation and path for the central crack of a three-crack array. coplanar (Swain and others, 1974). Indeed, divergence should be A: Crack extension force G normalized by that for an isolated crack more pronounced at smaller twist angles. Instead of the tips joining, Gj of width 2b, plotted versus the ratio of crack width to distance the cracks should link tip-to-plane, leaving a sharp step in the between centers 2c. Dotted curves are for constant values of nor- through-going crack surface. This process is illustrated by labora- malized critical crack extension force Gc. Points labled A, A' andB, tory experiments on three echelon pressurized cracks at a twist B' are discussed in the text. B: Ratio of mode II to mode / stress angle of 15° (Fig. 14A). At constant pressure, the lateral propaga- intensity factors plotted versus ratio b/c. Insets illustrate propagation rate first increased, then decreased as the tips began to overlap, tion path dependence on sign of this ratio. a behavior that is in agreement with Figure 12A. To make the

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® ^^ -V»

GRANITIC ROCK

, yf / /' ® /C! f^f / 25 cm

quartz-feldspar vein

MANCOS SHALE

250 m

minette dike

Figure 13. Examples of curved crack propagation paths for Figure 14. Examples of echelon-crack linkage. A: Three cracks echelon cracks. A: Two cracks in a laboratory specimen of glass at an initial twist angle of 15° in gelatin subject to interna) pressure. (Swain and Hagan, 1978). B: Two quartz-feldspar veins in the Cot- Dashed forms represent initial cracks. B: Initial twist angle of 45°. tonwood stock, Utah (Lawton, 1980). C: Two igneous dikes near C: Sketch of dikes from West Greenland (Escher and others, 1976). Ship Rock, New Mexico (Delaney and Pollard, 1981).

cracks link, we had to increase the internal pressure substantially. placements are greatly exaggerated compared to those expected in In nature, the process may terminate at any position due to defi- rock by choosing a pressure-to-modulus ratio ofp(l - v)jy. - 1/10. cient driving pressure. If linkage is accomplished, the step or offset Typical values for rock are on the order of 10"2 to 10"4. For a single in the crack wall remains to allow measurement of echelon crack crack (Fig. 15A), there are no shear displacements of the crack wall, widths. Natural examples of linkage are documented by Hurlbut and the form is symmetric about the major and minor axes. Break- and Griggs (1939), Lutton (1971), and Pollard and others (1975). down into three colinear cracks with bjc = 0.98 (Fig. 15B) produces At large twist angles, crack linkage may produce a very ornate a marked decrease in dilation, and end cracks are not symmetric structure (Figs. 14B, 14C). When one tip establishes a connection, about their minor axes. Data consistent with a reduction in crack stresses relax at the other tip, precluding further propagation and dilation upon breakdown into coplanar segments were collected for isolating that tip to one side of the through-going fracture. As many two igneous sills by Pollard and others (1975). For bjc = 1 and a echelon cracks link in this manner, a structure is produced which twist of 15° (Fig. 15C), obvious distortions of the crack profile has consistently facing, hook-like apophyses arranged along its develop near adjacent crack tips and the crack wall displaces in wall. The igneous dikes from West Greenland (Fig. 14C) have been shear. For bjc = 1.4 (Fig. 15D), adjacent crack tips take on an interpreted by Escher and others (1976) to have formed along a asymmetric form that characterizes this type of crack interaction shear zone subjected to high regional shear stress. However, our and is commonly observed in nature (Fig. 3). Pressure in each crack model experiment reproduced a very similar structure with only an works against the dilation of adjacent crack tips, causing an abrupt internal pressure. The curved propagation paths leading to the thinning and a convex inward curvature of adjacent crack walls. hooks and the unusual linkage are caused by the local shear stresses The grid intersections of Figure 15 may be thought of as point of crack interaction and do not imply a remote shear. markers in an outcrop near dilating cracks. They reveal that displacements are generally outward next to the crack sides and Deformation Fields Due to Echelon Cracks inward off the ends of a crack. Echelon-crack dilation results in a left-lateral shear displacement (Figs. 15C, 15D) of outer corners of Deformation fields in the vicinity of pressurized cracks are the grid parallel to the line connecting crack centers. This left- illustrated in Figure 15. The crack profiles and a deformed square lateral shear is also apparent in the distortion of grid lines running grid are determined from the elastic displacement field, but dis- perpendicular to the line between crack centers. We emphasize that

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Figure IS. Theoretical deformation fields in the vicinity of pressurized straight cracks. Elastic displacements of a square grid are greatly exaggerated owing to a pressure-to-elastic modulus ratio of 1/10. A: One crack with width equal to array width. B: Three coplanar cracks with width to spacing, b/c - 0.98. C: Three echelon cracks at twist angle, a> = 15°, and width to spacing, b/c = 1. D: Three echelon cracks of twist angle, to = 15° and width to spacing, b/c = 1.4.

the loading involves no applied shear either on the crack walls or in are expected for the few outermost cracks. For example, dilation at the far field; these shear displacements are entirely due to the rela- the middle of the end crack of an 11-member array (Fig. 16) is only tive configurations of the cracks and the grid. Figure 15 also dem- 50% of that for the center crack. The end cracks exhibit a decrease onstrates how the acute deformation associated with echelon-crack in dilation as the number of cracks increases. The end cracks are tips decays from the tip region. At distances greater than one-half of unique in that crack interaction at the outer tips does not jeopardize that between crack centers, the deformation is similar to that propagation. Thus, an end crack may grow to much greater widths induced by coplanar cracks, or by a single crack of width Inc. than an interior crack if the driving pressure is sufficient. Dilation of an isolated crack of the same width as one of the Generalization of Results to Crack Arrays echelon cracks is shown in Figure 16 to demonstrate that mechanical interaction of closely spaced echelon cracks may greatly enhance Although the results just discussed for three echelon cracks are dilation. At incomplete exposures, only one or a few echelon cracks qualitatively applicable if there are additional members in the array, of a large array may crop out. An assumption that this crack or the various quantities will change in magnitude. We illustrate these small array behaved in isolation of the other cracks would lead to changes in Figure 16 for odd numbers of echelon cracks at twist an overestimate of the ratio of driving pressure to elastic stiffness. angle w = 5°, and with b/c = 1.1. The total width of the array For example, if the crack were near the center of an 11-member (parent-crack width) is held constant at 2A and the number of array (Fig. 16), this ratio would be more than 5 times its true value. echelon cracks is varied from 1 to 15. As number increases, the Clearly, the total array width and number of individual cracks are width of individual cracks decreases to maintain a constant ratio required for accurate calculations. b/c. Dilation at the middle of the center crack decreases relative to that for 1 crack of width 2A, reaching 50% for an 11-crack array. DISCUSSION The greatest change in dilation occurs between the 1- and 3-crack arrays, and very little change is noted for additional members >11. We have analyzed echelon cracks that grow from a single dilat- Our analysis shows that most of the cracks in an array will ing parent crack because of a change in orientation of the applied dilate like the center one. However, great quantitative differences loads. This change, a spatial or temporal rotation of the remote

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1.0 Al

< ÛO Figure 16. Dilation of central N 0.8 co and end cracks in- echelon array is plotted versus number of cracks. o Dilation is normalized by dilation of 0.6 single crack of width 14 equal to JD width of entire array. The twist angle ¡5 Center w = 5° and ratio of crack width to a>> Crack spacing b/c - 1.1. Also shown is the 0.4 dilation of a single isolated crack-of. a> width lb. cr End 0.2 Crack

A A Isolated - Crack _L i I I I I I I I L 5 7 9 II 13 15 17

Number of Echelon Cracks , n

principal stresses about an axis parallel to propagation, introduces a propagation and alternating steps serve to distinguish this class of rotation of the local principal stresses near the parent-crack tip. Ech- cracks. Propagation of finger-like cracks into a region with the elon cracks form normal to the local maximum tension and propa- remote stresses at a new orientation may generate echelon cracks as gate along twisted surfaces, thus realigning the crack faces to the re- each finger twists into the new principal stress plane. mote principal stress plane. The transition from parent to echelon Echelon cracks may initiate from isolated flaws rather than cracks may be smooth or discontinuous, depending upon the applied from a common parent. If these flaws are localized along a fault stress ratio and Poisson's ratio. Examination of joints, veins, and ig- (Brown and others, 1967) or shear zone (Ramsay, 1967), the neous dikes in outcrop has convinced us that this echelon-crack form echelon cracks form perpendicular to the local cj| and therefore and mode of origin are common. Nevertheless, the penchant for ech- strike at large angles to the zone. Finite shear strains may cause elon cracks to form over a great range of length scales, under dispa- considerable distortion and rotation of the cracks and may lead to rate environmental conditions, and in a great variety of materials, sigmoidal shapes in cross section (Beach, 1975). However, we have suggests that several mechanisms for their formation may exist. shown how a local shear stress induced by crack dilation creates Here, we briefly discuss different classes of echelon cracks, their curved crack paths and leads to sigmoidal shapes in absence of any mode of origin, and possible ambiguities in their interpretation. remote shear loading. Future research should focus on methods to Some grain-scale echelon cracks in polycrystalline materials distinguish these two mechanisms for the formation of sigmoidal abut a parent crack with a discontinuous change,in orientation at echelon cracks.' the line of breakdown. They exhibit little or no twist beyond this The proximity of echelon fringe joints to bedding surfaces in rough breakdown zone, which coincides with a grain boundary sedimentary rock has promoted the interpretation that the joints between strongly anisotropic crystals. A change in orientation of form because of these surfaces. However, a surface is not funda- the cleavage planes about an axis parallel to propagation produces mental to their origin because echelon fringe cracks commonly are the discontinuous breakdown (Lawn and Wilshaw, 1975, p. 112). found on joints at great distances from the nearest bedding surface. The controlling influence of anisotropy of the solid serves to distin- Because bedding surfaces are mechanical discontinuities in the rock guish this class of cracks. This is not a common mechanism in rock mass, we should expect a joint to encounter a varying state of stress at the outcrop scale, where anisotropy is less pronounced and twist as it approaches such a surface. The association of fringe joints and boundaries are rare. bedding surfaces implies that this variation includes a rotation of Breakdown into segments may occur in the absence of a the principal stresses about an axis oriented perpendicular to remote shear, and therefore without any twist of the segments, as in bedding. the finger-like crack growth on grain boundaries in metals (Fields The role of shear stress during crack propagation has gener- and Ashby, 1976) and at the periphery of igneous sills (Pollard and ated considerable discussion in the geologic literature with reference others, 1975). Initially, this produces an array of coplanar cracks. to joints, veins, and igneous dikes. For example, hackle marks Subsequent propagation of individual cracks slightly out of the which may develop into echelon fringe joints are considered diag- parent-crack plane due to structural inhomogeneities produces nostic of shear loading on the parent joint by Gash (1971). Some steps in the crack surface and an echelon appearance in cross sec- echelon vein arrays are interpreted as forming in shear zones, tion (Pollard, 1978). However, the offsets do not have a systematic oblique to the principal stresses (Beach, 1975). Escher and others sense, as cracks alternate from one side to the other of the parent- (1976) have used the asymmetric character of some echelon dikes to crack plane, and the cracks are not twisted surfaces. The in-plane infer the action of high shear stress parallel to the dikes. However,

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the opening displacements observed at the walls of many joints, Menlo Park, California, including Joe Andrews, Gary Mavko, and veins, and dikes are conclusive evidence that these are dilatant Art McGarr, have provided useful comments. We especially thank cracks. We infer that they form in a principal stress plane with no Eileen Herrstrom and Kristi Morrison for their help in constructing shear loading. Shear stresses do play a role during breakdown of a the laboratory apparatus and in carrying out the experiments on parent crack and during lateral growth of echelon cracks along pressurized cracks in gelatin. Through the efforts of Bill Brace, the curved paths. However, the cracks respond to shear stress by turn- Department of Earth and Planetary Sciences at Massachusetts ing into a principal stress plane. Institute of Technology was able to provide Pollard with an office Echelon cracks that exhibit features consistent with the mode and a beautiful view of the Charles River during the final stage of of origin considered in this paper may be used to make inferences preparation of this manuscript. about the state of stress. The sense of angular rotation a of the remote principal stresses is determined by the sign of the twist angle REFERENCES CITED co of the cracks. Although angles a and co are not necessarily equal for the initiation of breakdown, it is likely that echelon cracks will Anderson, E. M., 1951, The dynamics of faulting and dyke formation with propagate into a configuration symmetric with the remote stresses applications to Britain: London, England, Oliver and Boyd, 206 p. at some distance from the line of breakdown. Thus, the twist magni- Beach, A., 1975, The geometry of en-echelon vein arrays: Tectonophysics, v. 28, p. 245-263. tude indicates the magnitude of rotation of the remote principal 1980, Numerical models of hydraulic fracturing and the interpretation stresses. The angle between a segment and the parent crack provides of syntectonic veins: Journal of Structural Geology, v. 2, p. 425-438. a measure of rotation of the principal stress axes along the direction Bredehoeft, J. D., Wolff, R. G., Keys, W. S„ and Shuter, E., 1976, Hydrau- of fracture propagation. Angles among segments provide a measure lic fracturing to determine the regional in situ stress field, Piceance of the variability of principal stress orientation across the propaga- Basin, Colorado: Geological Society of America Bulletin, v. 87, p. 250-258. tion direction. 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M., and Nemat-Nasser, S., 1980, Crack kinkin» under nonsymmetric loading: Engineering Fracture Mechanics, v. 13, manuscript. Many of our colleagues at the U.S. Geological Survey, p. 879-888.

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