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 helicoi- dal cracks sweeps out less surface area than one parent crack twist- ing 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 direc- tion, 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 vari- ety 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 Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/93/12/1291/3444723/i0016-7606-93-12-1291.pdf by guest on 01 October 2021 1292 POLLARD AND OTHERS 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 atten- tion 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 Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/93/12/1291/3444723/i0016-7606-93-12-1291.pdf by guest on 01 October 2021 DILITANT ECHELON CRACKS 1293 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.