On Resolving Shear Direction in Foliated Rocks Deformed by Simple Shear

On resolving shear direction in foliated rocks deformed by simple shear

ALLEN J. DENNIS Department of Physical Sciences, University of South Carolina, Aiken, South Carolina 29801 DONALD T. SECOR, JR. Department of Geological Sciences, University of South Carolina, Columbia, South Carolina 29208

ABSTRACT geneous simple shear. At a later point in the analysis, we show that our results can be extended to the more general case of heterogeneous simple We present a three-dimensional model for the formation of crenu- shear. Deformation is assumed to take place in a parallel-sided zone of lations in ductile shear zones, based on compatibility conditions, fixed width and orientation and infinite length. The zone boundary or wall infinitesimal-displacement equations, and our own field observations. is defined to be a planar interface between material deforming within the In many shear zones, at least two and probably three mesoscopic slip zone and undeformed material outside the zone. Compatibility is pre- systems are active. Crenulation slip is interpreted to compensate for served both within the shear zone and at the zone boundaries. An element the displacement component of foliation slip normal to the shear zone from within the shear zone is chosen for analysis (Fig. 2). This element is wall. In zones in which foliation slip is an important mode of deforma- assumed to be of a size such that deformation within it is homogeneous at tion, foliation and crenulation slip vectors do not necessarily lie in the a macroscopic scale. The displacement direction of the zone is assumed to same plane, nor are they necessarily perpendicular to crenulation be parallel to the Xi coordinate direction, and the X1-X2 plane is parallel axes. Solutions for the orientation of the crenulation plane, crenula- to the zone walls. Deformation within the element is assumed to be related tion slip vector, and magnitude of the crenulation slip shear strain to the operation of three mechanisms. belong to one of two possible solution sets, given the orientation of (1) Material in the element is assumed to be foliated. The foliation slipping foliation, the foliation slip vector, and the magnitude of slip on planes are oblique to all coordinate directions and to the zone walls (Fig. foliation. Hence, shear sense may be reliably interpreted from com- 2). Slip is occurring along foliation surfaces. Foliation slip is assumed to be posite planar fabrics in ductile shear zones, but more specifically, penetrative at both the macroscopic and mesoscopic scales and is modeled shear direction cannot. These results have broad implications in the in- as homogeneous simple shear with displacements parallel to foliation sur- terpretation of the kinematic significance of mineral lineations oblique faces and oblique to all coordinate directions (A, Fig. 2). The simple-shear to crenulation axes, and in the deduction of shear strain from angular displacements acting parallel to foliation are inclined to zone boundaries relations between planar fabric elements. (X|-X2 plane), and consequently foliation slip will rotate the boundaries and change the thickness of the zone. In order to maintain compatibility INTRODUCTION between material inside and outside the zone, we assume that the material inside undergoes a rigid-body rotation that restores the X! direction and In a recent paper (Dennis and Secor, 1987), we concluded that the X[-X2 plane to their original orientations (B, Fig. 2), and we assume crenulations develop in some shear zones in order to compensate for the that slip is occurring along crenulation axial surfaces in order to restore the displacement component of foliation slip normal to the zone walls, thereby zone to its original thickness and preserve a simple-shear deformation path preserving a simple-shear deformation path. In that paper, we presented a (C, Fig. 2). kinematic model in which the crenulation axes were perpendicular to the (2) Crenulation slip is assumed to be penetrative only at the macro- overall displacement direction of the zone (Fig. 1, la and lb). The present scopic scale and is modeled as a simple shear with displacements parallel paper is a generalization of the above model, in which the crenulation axes to crenulation axial surfaces and oblique to all coordinate directions. The are not necessarily perpendicular to the overall displacement direction. displacements associated with crenulation slip are inclined to zone bound- This generalized model provides a new explanation for puzzling occur- aries, and consequently crenulation slip will rotate the X[-X2 plane. In rences of crenulation axes oblique or subparallel to elongation lineations in order to maintain compatibility at the zone boundary, we assume that the some shear zones (Fig. 1, Ha, lib, Ilia, and Illb). The generalized model is material inside the zone experiences a second rigid-body rotation that applied in a future companion paper by P. E. Sacks and others to analyze restores the X[ direction and the Xj-X2 plane to their original orientations the displacement history of the Modoc shear zone in the eastern Piedmont (D, Fig. 2). The combined effects of foliation and crenulation slip and the of South Carolina and Georgia. associated rigid-body rotations yield a simple shear with the shear direction parallel to the X[-X2 plane. THE MODEL (3) The net effect of the operation of any additional deformation mechanisms (for example, intracrystalline translation glide and climb in In a theoretical analysis of the kinematics of shear zones, Ramsay and quartz) is assumed to be a simple shear penetrative at macroscopic and Graham (1970, p. 811 -812) concluded that "if the walls of the shear zone mesoscopic scales, with the displacement direction parallel to the X[-X2 are undeformed and volume change is unimportant, such zones can only plane. The combined effects of all of the above deformations (three simple be formed by the process of heterogeneous simple shear." Initially, in the shears and two rigid-body rotations) yield a simple shear in which the present paper, we consider a constant-volume zone of progressive homo- shear direction is parallel to X! (E, Fig. 2).

Geological Society of America Bulletin, v. 102, p. 1257-1267, 6 figs., 2 tables, September 1990.

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Figure 1. Three structural styles recognized in sheared anisotropic rocks. Ia. Crenulation axes (triangles) lie in a plane parallel to the zone wall and are oriented at a high angle to mineral lineation (con- toured at 10%, 20%, 30%, 40%; n = 45). Great circle indicates zone-wall orientation. Data from the Irmo shear zone (Fig. 9 of Dennis and Secor, 1987). Ib. Composite sketch of fabric elements in the Irmo shear zone. In this view, the crenulation axes are horizontal, and the right- lib hand face of the block is parallel to the shear zone boundary. Ila. Cren- ulation axes (triangles) lie in a plane parallel to the zone wall and are subparallel to mineral lineation (con- toured at 2%, 5%, 10%; n = 50). Data from the Brevard zone (Plate 5 of Bryant and Reed, 1970). lib. Com- posite sketch of fabric elements in the Brevard zone. In this view, the crenulation axes are horizontal, and the right-hand face of the block is parallel to the shear zone boundary. Ilia. In one lithology, crenulation axes (triangles) lie in a plane parallel to the shear zone wall. In a sec- Illa ond lithology, typically a felsic lllb orthogneiss, mineral lineation is oblique to subparallel with crenulation axes and may be oblique to a plane parallel to the zone wall (con- toured at 2.5%, 5%, 10%, 15%; n = 131). Data from the Modoc zone (P. E. Sacks and others, unpub. data). Mb. Sketch map of a part of the Modoc zone, eastern Pied- mont of South Carolina and adjacent Georgia. Wavy pattern, crenu- lated paragneisses; stipple, lineated orthogneisses.

In Figure 2, finite, temporary changes in the orientation of the shear the deformation are infinitesimal also permits some simplification essential zone boundary are illustrated. Under most circumstances, it seems unlikely in a subsequent mathematical analysis. that such changes would be compatible with the external boundary con- It is well known that a given deformation may be the result of an straints controlling the orientation of the zone. We therefore assume that infinite variety of deformation paths. Cause and effect considerations out- the deformations involved in the various stages of the deformation path are lined above suggest the sequence of deformations illustrated in Figure 2. In infinitesimal, producing only infinitesimal, temporary changes in the orien- the following analysis, we assume this deformation path. One consequence tation of the zone. We will show that under some circumstances, finite of the assumption that the displacements associated with each step are displacements may accumulate along foliation and crenulation as a result infinitesimal, however, is that the relative order in which the steps are of the superposition of a large number of cycles of infinitesimal deforma- taken becomes immaterial. Therefore, rigorous justification of the order of tion. The assumption that the displacements associated with each cycle of the steps illustrated in Figure 2 is unnecessary.

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Figure 2. Stylized development of lineations oblique or parallel to crenulation axes in a zone deforming by simple shear. This figure shows steps (A-E) in the progressive deformation of a cubic element centered on the origin. A volume of foliated material is sheared (A). The volume deforms by slip on foliation surfaces, and this slip is modeled as a simple shear. Dashed line encloses a part of the X,-X2 plane (shear zone wall). Note that a part of the element is rotated "beneath" the X1-X2 plane. The orientation of the zone is preserved by a rigid-body back rotation (B). This rotation restores the (strained) planes initially parallel to the zone wall to parallelism with the zone wall and restores the X] axis in the strained material to its pre-shearing orientation. Slip on inclined crenulation surfaces compensates for the displacement component of foliation slip normal to the zone wall (C). A second rigid-body back rotation is required because displacements associated with crenulation slip are inclined to the zone wall (D). Although the zone's orientation and thickness have been preserved by operations A-D, the overall shear direction is not necessarily parallel to Xj. A third simple shear (E) is therefore assumed, with displacements parallel to the zone wall. The product of transformations A-E is assumed to be a macroscopically homogeneous simple shear parallel to X].

TABLE 1. NOTATION CONVENTIONS AND SUMMARY OF VARIABLES USED IN THE MODEL We now analyze the deformation that takes place in each of the five displacement steps illustrated in Figure 2. Symbols and notation conven- (1) Roman capital letters designate transformation matrices. X| is the cartesian, Lagrangian coordinate system; X! is parallel to the shear direction, and X-j is perpendicular to the zone wall. tions used in the analysis are explained in Table 1. Each displacement step (2a) Lower-case Roman letters with two indices are individual transformation elements. is modeled as a coordinate transformation relating the positions of points (2b) Boldface lower-case Roman letters are direction cosine vectors of poles to planes. A lower-case Roman letter with a single index denotes a specific element of that vector. An exception will be that x will represent a sin- before and after the displacement episode. Roman-numeral superscripts gle point x's Lagrangian coordinates. (3) Greek capital letters are scalars. are used to indicate the consecutive position of points following each (4) A lower-case Greek letter is the direction cosine vector of a line. The letter with a single index specifies a par- transformation. For example, X2 refers to one of the coordinates of a point ticular element of the vector. 1 (5) Roman-numeral superscripts on x (for example, x") will indicate successive positions of x as A, B, C, prior to deformation, and X2" would refer to its value following the third D, E each act in turn on the point. A superscripted variable's values reflect its cumulative reorientation to that time; x11 indicates the position of x following its transformation BA x = x11. Similarly, Roman-numeral transformation or displacement step. The transformations involved in each superscripts used on other variables will indicate their changing values, as the transformations reorient them. of the five displacement steps shown in Figure 2 can be represented in (6) Indices vary from 1 to 3, and following the summation convention, a repeated (dummy) index in a given term indicates the sum of the term as the dummy varies from 1 to 3. matrix notation as follows.

a pole to the foliation plane "x, "Xl'" a slip vector in the plane of foliation x 1 T magnitude of shear strain of lines parallel to a and a as a result of foliation slip simple shear A 2 = X2 to axis of rotation, foliation slip back rotation x i! magnitude of rotation, foliation slip back rotation _ 3_ b pole to the crenulation plane 0 slip vector in the plane of crenulation ¥ magnitude of shear strain of lines parallel to b and 0 as a result of crenulation slip simple shear Vl Xl»" A slip vector of third simple shear, in the X| X2 plane = •P magnitude of shear strain of lines parallel to X3 and K, as a result of the third simple shear B X2> x2" 7 magnitude of net shear strain of lines parallel to X and X3 prior to deformation 1 ( X3 x3"

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xi» xjin following transformation B, the sums of the 21, 31, and 32 elements of A

c x2" = x2m (3) and B must be zero: _x ni_ _x3n_ 3 a2ra! + aj3n = 0 (12)

nl = D x2 x2iv (4) a3ra! - co2ii = 0 (13)

x3i" _x3iv_

a3ra2 + a>ift = 0 (14) X,1V fxiVl ,V E X2 = x2v (5) Solving equations 12,13, and 14 for cot and substituting the results in x3iv_ x3v 11, we obtain

r We now derive expressions for the transformation matrices A-E. 1 a2ra! «3 *i

Consider a foliation surface obliquely inclined to all coordinate direc- -a2rai 1 a3ra2 = [B] (15)

tions. This foliation surface is represented by the equation -a3ra! -a3ra2 1

a,x, (6) An expression for the crenulation slip transformation is obtained by a derivation analogous to that leading to 10. where a; is the direction cosine vector of the pole to foliation, and n is the perpendicular distance from the foliation surface to the origin. A simple shear acts along the foliation, and a point P originally located at X; is 1 + ßi'^b," j8 n¥bin 1 + ß niPb2n ß IlVb " = [C] (16) displaced to x/. The shear strain of two originally perpendicular lines 2 2 2 3 foHVbi" ß "Vb2" 1 + ß "Vb " parallel to a; and the displacement direction will be 3 3 3

T = (displacement of P)/ (a;Xj) (7) where b;" is the direction cosine vector of the normal to the crenulation axial surfaces following transformation B, /?;" is the direction cosine vector From equation 7, the components of the displacement will be of the displacement direction in the crenulation axial surface, and ^ is the shear strain of lines parallel to bj" and /3j" as a consequence of crenulation «Aft (8) slip simple shear. Similarly, an expression for the transformation matrix of the second where a; is the direction cosine vector of the foliation slip direction. rigid-body rotation is obtained by a derivation analogous to that leading to Therefore the transformation equation for foliation slip is equation 15.

1 Xj = X; + «¡rajXj (9) 1 /y'^b," ß3"Vbi« 1 ß3nyb2n [D] (17)

11 u and its corresponding transformation matrix is -ftii^b! -ß3nVb2 1

1 + aiTai cqra2 «îT^ The transformation matrix for a simple shear (E) with displacement

«2 Tai 1 + a2ra2 «2^3 = [A] (10) direction lying in the Xi-X2 plane is «3 Ta] «3^2 1 + «3 Ta3

Foliation planes are obliquely inclined to both the shear zone wall 1 \2

displacement direction. The second coordinate transformation (B) is a where k\ and \2 are the direction cosines of the displacement direction rigid-body rotation that returns the shear zone wall and the overall dis- (X3 = 0), and $ is the shear strain of lines parallel to X3 and as a placement direction to their original orientations. If products of infinites- consequence of this third simple shear. imal terms are neglected, the transformation matrix for an infinitesimal If the net effect of this sequence of transformations is a simple shear

rigid-body rotation is with displacements parallel to Xt, the overall transformation matrix must be

1 -ÛJ3ÎI w2il ÛJ3ÎÎ 1 -wili [B] (11) 1 0 y -0J2O. CUjft 1 0 1 0 = [F] (19) 0 0 1 (Love, 1944, p. 69-70); to; is the direction cosine vector of the axis of rotation, and ft is the amount of rotation. If the shear zone wall and where 7 is the net shear strain of lines parallel to Xj and X3 prior to overall displacement direction are to return to their original orientations deformation.

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TABLE 2. RESULTS OF THE CALCULATION OF POSSIBLE CRENULATION SLIP SYSTEMS A set of equations constraining the variables that occur in the entries CORRESPONDING TO A SINGLE FOLIATION SLIP SYSTEM of matrices A-E is obtained by composing A-E and equating the product

toF. Crenulation slip Crenulation slip foliation system from system from slip system equation 29 equation 30 xi r 0.001 V 0.000524 ¥ 0.000624 x [E][D][C][B][A] = [F] 2 (20) ai 0.1392 bl -0.2773 bl -0.2062 a 0.0872 -0.1736 b2 (assumed) -0.1736 x 2 bj (assumed) 3 a3 0.9864 b3 0.9449 »3 0.9630 "1 0.7547 ßl 0.7236 ßl 0.8167 0.6356 h 0.6093 ß2 0.5115 These equations are «3 -4.1626 h 0.3243 »3 0.2671

11 11 a,rai + ß, ^! = 0 (21)

n l a2ra2 + ß2 ^W = 0 (22) and 21-23 can be used to determine values for all of the other dependent variables that characterize the crenulation slip system. Similarly, 32, 30, n n a3 ra3 + /?3 ¥b3 = 0 (23) 28, and 21-23 can be used to determine a different set of values for the n same arbitrarily chosen value of b2 . As an example, the above procedure

r(a,a2 + a2ai) + ^'V + fo'V) = 0 (24) has been used to calculate the two possible crenulation slip systems corre- H sponding to an assumed set of values for b2 , a;, a;, and T (Table 2). 1 1 + r(a2a3 + a3a2) + ^(fo'V + ft V) ^ = 0 (25) Apparently, compatibility considerations dictate that for a given foliation slip system, there exist two families of potential crenulation slip systems, ll l r(aia3 + <*3a,) + nßi W + ft'V) + X^ = 7 (26) each being characterized by one degree of freedom. In one of these families, the line of intersection between foliation and crenulation planes (that The direction cosine terms in equations 21-26 are also constrained by the is, the crenulation axis) is parallel to the shear zone wall, as indicated by Pythagorean theorem and by the fact that displacements associated with equation 29. In the second family of solutions, as indicated by 30, the slip deformations must lie within the corresponding slip planes. crenulation axis will generally be oblique to the shear zone wall. The The assumption that crenulations develop in a shear zone in order to crenulation orientation that actually develops in a given situation may be preserve a simple-shear path following foliation slip is of central impor- determined by factors not considered in our analysis, such as material tance in the preceding discussion. Therefore, we regard the quantities that properties and the applied stress system. In the general solutions to the characterize the crenulation slip system (b;11, ft11, as dependent varia- compatibility equations outlined above, the predicted crenulation plane bles. These variables are constrained by equations 21-24 together with orientations are oblique to all coordinate directions, and the slip directions are oblique to crenulation axes. bjV = 1 (27) If the two possible families of solutions indicated by equations 29 and 30 are simultaneously met, then /V'ft" = 1 (28) aj/a2 = ct\/a2 (33) Because there are seven variables that characterize crenulation slip, but only six constraining equations, there is a single degree of freedom in the and from equations 21 and 22 crenulation slip system. It is possible by substitution to obtain equations n u involving only two of the dependent variables. For example, j8i , jS2 , and Vb2 = ft/ft (34) ^ can be eliminated from equation 24 by using 21 and 22. This leads to two possible solutions. Equations 33 and 34 together with 29 and 30 define a single family of solutions to the compatibility equations in which the slip vector and the 11 II b! = a1b2 /a2 (29) poles to foliation and crenulation all lie in a single plane normal to the shear zone wall. In this special family of solutions, the crenulation axes are

bi" = «ib2"/«2 (30) parallel to the shear zone wall, as indicated by 29, and foliation and crenulation slip vectors are perpendicular to crenulation axes. This special n A relationship involving b2 and ¥ can be obtained by using equations family of solutions to the compatibility equations was previously analyzed 21-23, 28, and 29 in 27. by Dennis and Secor (1987). The three dependent variables (X[, X2, $) that characterize the third <3„ simple-shear transformation (E) are completely constrained by equations 25 and 26 together with Similarly, by using equations 21-23, 28, and 30 in 27, we obtain X;Xj = 1 (35) (32, To this point, we have analyzed the transformations taking place in a single cycle of infinitesimal deformation. The composition of transforma- n If a particular value is selected for b2 in equation 31, then 31, 29,28, tions A-E yields an infinitesimal simple shear in which the Xj-X2 coordi-

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nate plane is parallel to the shear plane, and X( is the overall displacement direction. If the above model is to be useful in understanding crenulation development in natural shear zones, it is necessary to postulate a large number of cycles of infinitesimal deformation, which, when summed, result in finite displacements along crenulation axial surfaces. If displacements from successive deformation cycles are to accumulate along crenulation surfaces, then the orientation of crenulation surfaces at the end of one cycle must be included in that family of planes that can serve as slip surfaces in subsequent deformation cycles. Equation 29 specifies that crenulation axes be parallel to the shear zone wall. Although infinitesimal rotations and strains occur in planes parallel to the shear zone wall during each deformation cycle, at the end of each cycle the deformations sum to a simple shear in which no rotations or strains are permitted in planes parallel to the shear zone wall. Therefore there can be no cumulative rotation or stretching of crenulation axes. The crenulation and foliation planes are progressively reoriented as finite strains accumulate, but the crenulation axes themselves do not rotate and must remain parallel to the shear zone wall. Hence the crenulation planes will always belong to that family of planes that satisfy the compatibility equations, and finite slips may accumulate along crenulation axial surfaces as the cumulative result of a large number of deformation cycles. Ulti- mately, it is possible that the crenulation planes may rotate into an orientation where the resolved shear stress is too low for continued slip, necessitating the development of a new set of crenulation planes in an orientation favorable for slip. The other possible family of solutions to the compatibility equations, expressed by equation 30, specifies that the line of intersection between crenulation axial surfaces and an imaginary plane perpendicular to the foliation slip vector be parallel to the shear zone wall. Compatibility constraints dictate that if finite slips are to accumulate along crenulation planes under the conditions specified by equation 30, the direction of the foliation slip vector must vary in such a way that 30 continues to be satisfied. Ultimately, because of the progressive rotation of material planes during finite strain, the resolved shear stresses in foliation or crenulation may be inadequate for continued slip in the requisite directions, and thereby necessitate the development of a new set of crenulation planes. The rotations of foliation and crenulation that occur during a single Figure 3. Heterogeneous simple shear in a shear zone can be coordinate transformation are obtained by inverting the transformation modeled as a series of homogeneously deforming layers parallel to (using Cramer's rule) and substituting the resulting expressions for the old shear zone walls. Within each layer, a combination of multiple slip coordinates into the equation for the plane of interest in its original orienta- systems results in homogeneous simple shear. Heterogeneous defor- tion (see also Flinn, 1978). Similarly, the progressive reorientation of mation can then be considered to be the result of differences in either foliation and crenulation during several transformations may be traced. In or both shear strain magnitude or displacement direction between this analysis, we assume that foliation surfaces are not reoriented at the individual homogeneously deforming layers. Adapted from Ramsay mesoscopic scale by crenulation slip because crenulations are not penetra- and Huber (1987, Fig. 26.17, p. 608). tive at the mesoscopic scale (see Dennis and Secor, 1987, and Dennis and others, 1987, for discussion). If products of infinitesimal quantities are neglected, the following expression for the direction cosine vector of the 11 pole to foliation following transformation E is obtained. [bi (1 - q2r a2 - tt3r a3)] b,v = (39) x/H n n [a, + r ai (a2a2 + «3*3) + ¥b, (a2ft, + »3/83")] 2 . y fe" (1 - «2r a[ /a2 - a3r a3)] = (36) bo* (40) vr N/H 2 _ [a2 - q2r at + q3r a2a3 - a^'^b," + a3ft"Vb2"] 2 [b3" - + b; ) -

2 2 I1 1 [a3-a3r(ai +a2 )-fe ^(aibi' +a2b2")-$(ai\|+a2X2)] where H is the sum of squares of the bracketed quantities. a3V = (38) Vg EXTENSION TO HETEROGENEOUS SIMPLE SHEAR where G is the sum of the squares of the bracketed quantities. Similarly, the direction cosine vector of the pole to crenulation axial surfaces follow- The preceding analysis described a possible sequence of infinitesimal ing transformation E is steps that may occur within a particular element in a zone deforming in

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simple shear. The element was assumed to be of a size such that deforma- In order to precisely determine the magnitude and direction of overall tion within it could be considered to be homogeneous at a macroscopic displacement of a heterogeneous simple shear zone, it is necessary that the scale. We have outlined the requisite conditions for finite slips to accumu- infinitesimal homogeneous deformations be integrated through time and late on foliation and crenulation within this element as a result of the across the entire width of the zone. If a variety of shear sense indicators is superposition of a large number of cycles of infinitesimal deformation. If available (Simpson, 1984), it may be possible to approximate the overall the model is to be useful in understanding crenulation development in displacement vector. In general, such an approximation requires estima- natural shear zones, it is now necessary for us to generalize our results to a tion of the histories and spatial variations of the scalars T, and and of larger scale in which deformation is heterogeneous. the vectors a;, b;, a;, ft, and X;. Attempts to determine the magnitude of One way to model deformation in a heterogeneous shear zone is to the overall displacement solely from measurements of the final orientation divide the zone into a series of homogeneously deforming layers parallel to of foliation (for example, Ramsay and Graham, 1970, eq. 39) will be valid the shear zone walls (Ramsay and Graham, 1970, p. 799 ff.). Bulk hetero- only in situations where the slips on foliation and crenulation and the geneous deformation of the zone can then be considered to be the result of associated rigid-body back rotations are negligible. differences in shear strain magnitudes or displacement directions between individual, homogeneously deforming layers (Fig. 3). Compatibility is CRENULATION MORPHOLOGY AND ORIENTATION maintained within layers and between layers. A bulk heterogeneous simple shear results from varying shear strain magnitudes and/or shear directions Dennis and Secor (1987) introduced the idea that crenulation slip between each layer. Moreover, the character of the deformation in each compensates for the displacement component of foliation slip normal to layer may vary with time, and the deformation may occur in different time the shear zone wall and maintains a simple-shear deformation path. Den- intervals in different layers. The above model may therefore be applied to nis and Secor (1987) recognized two complementary crenulation mor- shear zones in which work-hardening or work-softening processes are phologies (Fig. 4). The orientation of slipping foliation controls which of operative (Means, 1984), as well as to polyphase reactivation of a shear these two morphologies develops locally in a simple-shear zone. In a zone. dextral shear zone, when the slipping foliation is oriented at an acute angle

Figure 4. Angular relationships between slipping foliation and the shear zone wall control the orientation and morphology of crenulation in the shear zone. Reprinted with permission, from Dennis and Secor (1987). a. Development of reverse-slip crenulations (RSC) compensates for displacements normal to shear zone walls in systems where foliation is inclined at a clockwise acute angle to the shear zone movement direction (Xi). b. Development of normal-slip crenulations (NSC) compensates for normal displacements where foliation is inclined at a counterclockwise acute angle to Xj. c. Shearing of a foliation perpendicular to the cumulative flattening direction. If during progressive simple shearing this foliation is rotated into an easy slip orientation, and slip on foliation is an important mode of strain in the shear zone, then NSC will develop, d. If the foliation is parallel to the shear zone walls, then no compensating mechanism is required.

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Figure 5. Dextral reverse-slip crenulation (RSC) in greenschist-facies, foliated, mafic lapilli tuff, South Caro- lina Piedmont (Irmo shear zone).

measured clockwise to the shear zone movement direction, equivalent to a axes and oblique to the shear zone wall. In dextral shear zones, this negative X3 component of a; or a3 < 0, movement along that foliation foliation will be oriented at an acute, counterclockwise angle to the shear normal to the shear zone boundaries must be compensated. Reverse-slip zone movement direction. Consequently, normal-slip crenulations will de- crenulations (RSC) transfer slip "up" to "higher" foliation planes. Reverse- velop at an acute, clockwise angle to the shear zone movement direction. slip crenulations are rootless, intrafolial folds of foliation with the same This model explains why rootless, intrafolial folds of foliation (RSC) are vergence as the main zone (Fig. 5). The rootless aspect of these folds is relatively uncommon in sheared granitic rocks, whereas NSC-type struc- controlled by microfaults that root in a foliation surface. The orientation of tures are typically abundant. RSC axial surfaces makes an acute counterclockwise angle to the shear When a shear zone develops in previously foliated and deformed zone movement direction, or b3 > 0. rock, the style and orientation of crenulations that develop at a particular When the slipping foliation is oriented at an acute counterclockwise location are determined by the local orientation of foliation and the local angle to the shear zone movement direction (a3 > 0) in a dextral shear direction of foliation slip. If the parameters characterizing foliation orienta- zone, movement normal to the shear zone boundary is compensated by tion and slip vary throughout the zone, then the parameters controlling normal-slip crenulations (NSC). Normal-slip crenulations offset foliation crenulation orientation and slip may also vary, and it is possible that RSC in a normal sense that is synthetic with that of the main zone (Fig. 6). NSC and NSC may each be well developed in different domains in the shear axial or slip surfaces are commonly observed to root in foliation. Exten- zone. sional crenulation cleavages (Piatt and Vissers, 1981), shear bands (White In the preceding analysis, we generalized the result of Dennis and and others, 1980), and crenulation axial surfaces in Type IIS-C mylonites Secor (1987) to that case in which the intersection of foliation and the (Lister and Snoke, 1984) may all be kinematically equivalent to NSC. zone wall is not perpendicular to the shear zone movement direction (a2 ^ NSC axial surfaces in dextral shear zones are oriented at an acute clock- 0). The implications of this analysis for the kinematic significance of wise angle to the shear zone movement direction, or b3 < 0. crenulation slip are surprising; not only does crenulation slip compensate Simple-shear deformation of initially isotropic rocks or those with a for the displacement component of foliation slip parallel to the X3 axis, but weak primary fabric may result in the development of a tectonic foliation also some part of that component parallel to X2. One result of this rela- along which slip may occur. This foliation will be parallel to crenulation tionship is that the line of intersection of foliation and crenulation axial

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Figure 6. Dextral normal-slip crenulations developed in foliated mudstone, South Carolina Piedmont (Irmo shear zone). Note that both foliation and crenulation are inclined to bedding in this outcrop. We interpret the more massive sandy units on either side of the mudstone to represent local boundaries of simple-shear deformation. The regionally / penetrative foliation is not so well developed in the sandy unit as in the mudstone, and normal-slip crenulations do not penetrate the sandy unit; slip on foliation is not an important deformation mechanism in the sandy unit.

surfaces does not necessarily lie in the X]-X2 plane, so that the crenulation APPLICATIONS TO SPECIFIC AREAS axis may be oblique to the shear zone wall. A second consequence is that the directions of slip on foliation and crenulation are not necessarily per- Having outlined a compatibility-based kinematic framework for the pendicular to crenulation axes. This requires that some indication of slip simple shearing of rocks in which slip on foliation is an important mode of direction within these planar fabric elements be deduced from field obser- deformation, we now apply the model to the three structural styles illus- vations. Evidence for the direction of slip locally may include mineral trated in Figure 1. lineations, porphyroblast tails, or striae interpreted to be transport linea- (1) Crenulation axes lie in a plane parallel to the zone wall and are tions on the slipping plane, an offset line (admittedly uncommon), or normal to the bulk shear direction. Mineral lineations are perpendicular to rotated tension gash veins of the sort described by Hudleston (1989). the crenulation axes. In this case, the zone wall-foliation intersection is Finally it must be acknowledged that because each family of crenulation normal to the shear direction of the zone (la, lb). slip system solutions is characterized by a single degree of freedom, with (2) Crenulation axes may lie in a plane parallel to the zone wall but progressive simple shearing the direction of slip within planes may change, are oblique to the bulk shear direction. Mineral lineation may be oblique and only the most recent increment of slip may be preserved or discerned or subparallel to crenulation axes. In this case, the zone wall-foliation in the field. intersection is oblique to the shear direction of the zone (Ila, lib). These notes do not decrease the value of composite planar fabrics as (3) In one lithology, crenulation axes lie in a plane parallel to the powerful indicators of shear sense in rocks deformed by simple shear, but zone wall, but not necessarily normal to the shear direction, whereas in must temper any evaluation of the bulk shear direction based on the another lithology, a strong mineral lineation fabric lies in a plane parallel geometry of fabric elements. to the zone wall. We interpret this third style to indicate partitioning of displacement components between an anisotropic unit and an initially MINERAL LINEATIONS isotropic unit (Ilia, 111b).

Obviously a geometric or compatibility analysis does not yield the Figure 1. Ia, lb criteria necessary to distinguish transport, stretching, and intersection lineations in the field. Our analysis does, however, suggest that mineral We interpret sheared rocks with lineation normal to crenulation axes lineation and crenulation may both be products of a single, bulk simple- to indicate slip on a foliation in which intersection with the shear zone wall shear deformation involving intracrystalline translation gliding and climb, is normal to the shear direction. This style of shear zone deformation is as well as slip on foliation and crenulation surfaces, and that stretching and equivalent to that presented by Dennis and Secor (1987). Crenulation axes intersection lineations may be oblique to each other and to the shear zone are normal to the shear direction. Slip vectors in the foliation and crenula- wall. It is not necessary that slip on foliation and crenulation axial surfaces tion define a plane normal to the shear zone wall, and perpendicular to be orthogonal to the line of intersection of those planes. crenulation axes. Slip on foliation and crenulation surfaces is consistent Our field observations and the published data discussed in the next with the shear sense of the zone. Crenulation slip occurs on surfaces section indicate that in some cases, there are no clear overprinting relation- inclined to the zone wall and preserves a simple-shear deformation path. ships between crenulation and mineral lineation on either the regional or The mineral lineation may represent the actual slip vector on a foliation or mesoscopic scales. We believe that mineral lineations, whether they result crenulation surface. Alternatively, the lineation may represent the maxi- from unequidimensional, single grains or elongate, polyminerallic aggre- mum principal axis of strain of a penetrative simple shear with shear plane gates, may represent local slip vectors in slipping folia or crenulation axial parallel to the zone wall and shear direction parallel to the shear direction surfaces, or a penetrative stretching lineation associated with transforma- of the zone (transformation E; Dennis and Secor, 1987). With progressive tion E. Geometric relationships between intersection lineations and elonga- simple shearing, this lineation direction will approach the bulk shear direction lineations in a bulk simple shear deforming by three slip systems is tion. We note that scatter of crenulation axes out of the plane of the shear complex and will vary spatially in a heterogeneous displacement field. zone wall and oblique to mineral lineation (Fig. 1, Ia) may be a result of

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local variations in the orientation of slipping foliation with respect to the and structures in those units will effectively be translated. There is some zone wall. evidence to suggest that this is the case in the Alleghanian Modoc zone of As noted above, foliation that slips may form during the shearing, or the South Carolina Piedmont (P. E. Sacks and others, unpub. data). Syn- the shearing may reactivate a pre-existing foliation. Examples of shear- shearing orthogneisses contain a strong mineral lineation that is oblique to related crenulations occurring in rocks with little or no primary fabric reverse-slip crenulation axes found in paragneisses. This interpretation prior to deformation would include the intersection of a splay of the Irmo implies that the Modoc zone is the surface trace of an oblique-slip fault. A shear zone with the Clouds Creek granite (Secor and others, 1986), the strike-slip component of Modoc shearing is taken up in the orthogneisses, orthogneisses of the South Armorican shear zone (Berthe and others, 1979), whereas an oblique-slip component of the deformation is taken up in the and the Roses mylonite in the axial zone of the Pyrenees and the Anza paragneisses. Borrego mylonite of southern California (Simpson and others, 1982; In this third structural style, there is a partitioning of displacement Simpson, 1984). Examples of shear-related crenulations forming in aniso- components between two rock types: a foliated paragneiss in which slip on tropic rocks in which the intersection of foliation and the zone wall is foliation is an important deformation mechanism, and an initially isotropic inferred to be normal to the shear direction include the Irmo shear zone, (?) orthogneiss in which the effects of the third simple shear (transforma- the thrust plane at Linville Falls (Bryant and Reed, 1961, p. 8; 1970), and tion E) are localized. Hollister and Crawford (1986), Burg and others the Sesia-derived Insubric mylonites (Schmid and others, 1987). (1984), and Cambray (1988) provided examples of other zones that may be responding to a sub-simple shearing in this way. Figure 1. Da, lib OTHER POSSIBLE SOLUTIONS We interpret sheared rocks with lineation oblique or parallel to crenulation axes to indicate slip on a foliation for which the intersection with It is not our intent to discount the possibility of reactivation of a the shear zone wall is oblique to the shear direction. Crenulation axes may pre-existing zone with a different sense, nor the possibility of high shear or may not be parallel to the shear zone wall, nor do they necessarily strains, nor the possibility of strongly inhomogeneous simple shear in some parallel the foliation-zone wall intersection. Crenulation slip compensates zones. A great deal of incontrovertible evidence supports the notion that in for the displacement components of foliation slip normal to the shear zone zones of high shear strain, mineral lineation parallels transport direction. movement direction. The effects of a third simple-shear system (trans- Cases include the Hercynian lie de Groix subduction-related complex formation E) may also be apparent. Elongation lineations occurring (Cobbold and Quinquis, 1980); the Monte Rosa nappe of the western oblique or parallel to crenulation axes are interpreted to represent local slip Alps (Lacassin, 1987; Lacassin and Mattauer, 1985); the Main Central vectors in planar fabric elements or to track the maximum principal axis of thrust of the Himalayas (Brunei, 1986); the Ben Hope thrust in the strain in the third simple shear. northwestern Scottish Highlands (Hobbs and others, 1976, p. 276; Barr Examples of shear zones with mineral lineations oblique or parallel to and others, 1986); the greenschist nappes on Shikoku, southwestern Japan crenulation axes include many famous zones the resolved shear directions (Faure, 1985); and the Lake George thrust in the southern Adirondacks of which continue to be debated. These zones include the southern Appa- (McLelland, 1985,1986) among others. Evidence from many other zones lachian Brevard fault zone in western North and South Carolina (Bryant is less conclusive (for example, Brevard fault zone). The model we have and Reed, 1970; Edelman and others, 1987; Bobyarchick and others, described provides a mechanism for the formation of shear-related crenu- 1988), the Bartlett's Ferry-Goat Rock fault system in Alabama and west- lations and oblique to subparallel mineral lineations under relatively small ern Georgia (Schamel and others, 1980; Hooper and Hatcher, 1988; Stel- (<4) bulk shear strains in zones of homogeneous simple shear. tenpohl, 1988), the Modoc zone in the eastern South Carolina Piedmont Ridley (1986) has recently presented an alternate model to explain (Secor and others, 1986; P. E. Sacks and others, unpub. data), and the the coincidence of mineral lineation and shear-related crenulation axes. He Insubric mylonites derived from Canavese metasedimentary rocks of the did not consider a bulk simple-shear path with the displacement of a point southern Alps (Schmid and others, 1987) among others. proportional to the distance of the point from the origin parallel to X3, but These zones share several characteristics. (1) They cut obliquely a composite of thrust shear and wrench shear. Thrust shear is defined to be across strongly anisotropic rocks, and that anisotropy may be a tectonic a simple shear with shear plane parallel to the X1-X3 plane; wrench shear foliation, a compositional layering, or both. (2) Reasonable conclusions is another simple shear with shear plane parallel to the XJ-XJ plane. about shear direction using a variety of shear-sense criteria have included Therefore, the displacement of a point is proportional to the distance of the either or both dip-slip and strike-slip components; that is, in some cases, point from the origin along both the X2 and X3 axes. It is an immediate deductions of shear sense have been inconclusive. (3) The structures are consequence of this specification that the maximum principal axis of strain within or bound the metamorphic core of the orogenic belt; that is, thermal is oblique to the Xj-X3 plane. Ridley discussed fold orientation and conditions may enhance mechanical anisotropy. (4) Finally, these zones morphology as a function of the ratio of wrench to thrust shear. His results are major orogenic features that may have been repeatedly reactivated. neither preclude nor support ours. Because Ridley's bulk deformation path varies from simple shear, researchers must judge the applicability of either Figure 1. ma, Illb model to individual zones.

Discussion of cases in which crenulation axes are oblique or parallel CONCLUSIONS to mineral lineation offers an opportunity to discuss a third case. If the third slip system (transformation E) acts in a "softer" rock unit, which (1) When stretching lineations are perpendicular to crenulation axes, deforms independently of the units around it, then the orientations of the axes lie in a plane parallel to the shear zone wall, and the overall fabric elements in those units will not be affected by the third slip system, transport direction is approximately perpendicular to the crenulation axes.

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Dennis, A. J., and Secor, D. T., 1987, A model for the development of crenulations in shear zones with applications from (2) When stretching lineations are oblique or parallel to crenulation the southern Appalachian Piedmont: Journal of Structural Geology, v. 9, p. 809-817. axes, the axes may or may not lie in a plane parallel to the shear zone wall, Dennis, A. J., Sacks, P. E., and Maher, H. D., 1987, Nature of the late Alleghanian strike-slip deformation in the eastern Southern Carolina Piedmont: The Irmo shear zone, in Secor, D. T., Jr., ed., Anatomy of the Alleghanian orogeny and the overall transport direction may be oblique to both stretching as seen from the Piedmont of South Carolina and Georgia: Columbia, South Carolina Geological Survey, Carolina Geological Society Guidebook for 1987, p. 49-66. lineations and crenulation axes. F/telmp^ s. H,, Liu, A., and Hatcher, R. D., 1987, The Brevard zone in South Carolina and adjacent areas: An (3) Oblique stretching lineations and crenulation axes can form si- Alleghanian, orogen-scale, dextral shear zone reactivated as a thrust fault: Journal of Geology, v. 95, p. 793-806. Faure, M., 198S, Mierotectonic evidence for eastward ductile shear in the Jurassic orogen of southwest Japan: Journal of multaneously during a single bulk simple-shear deformation. Structural Geology, v. 7, p. 175-186. Flinn, D., 1978, Construction and computation of three-dimensional progressive deformations: Geological Society of (4) Estimation of the magnitude of the overall displacement of a London Journal, v. 135, p. 291-305. shear zone solely from measurements of foliation orientation relative to the Hobbs, B. E, Means, W. D., and Williams, P. F., 1976, Outline of structural geology: New York, Wiley, 571 p. 1982, The relationship between foliation and strain: An experimental investigation: Journal of Structural Geology, zone wall using the technique of Ramsay and Graham (1970) is valid only v. 4, p. 411-428. Hollister, L. S., and Crawford, M. L., 1986, Melt-enhanced deformation: A major tectonic process: Geology, v. 14, if the slips along foliation and crenulation axial surfaces and associated p. 558-561. rigid-body back rotations are negligible. Hooper, R. J., and Hatcher, R. D., 1988, Pine Mountain terrane, a complex window in the Georgia and Alabama Piedmont; evidence from the eastern termination: Geology, v. 16, p. 307-310. (5) Composite planar fabrics are useful in estimating the sense of Hudleston, P. J., 1989, The association of folds and veins in shear zones: Journal of Structural Geology, v. 11, p. 949-957. Lacassin, R., 1987, Kinematics of ductile shear from the outcrop to crustal scale in the Monte Rosa Nappe, Western Alps: shear but cannot be used to precisely determine the shear direction. Tectonics, v. 6, p. 69-88. Lacassin, R., and Mattauer, M., 1985, Kilometre-scale sheath fold at Mattmark and implications for transport direction in the Alps: Nature, v. 316, p. 739-742. ACKNOWLEDGMENTS Lister, G. S., and Williams, P. F., 1979, Fabric development in shear zones: Theoretical controls and observed phenomena: Journal of Structural Geology, v. 1, p. 283-298. Love, A.E.H., 1944, Mathematical treatise on the theory of elasticity: New York, Dover, 643 p. McLelland, J. M., 1984, The origin of ribbon lineation within the southern Adirondacks, U.S.A.: Journal of Structural This work has been supported by National Science Foundation Geology, v. 6, p. 147-157. Grants EAR85-08123 and EAR88-03833 to Donald T. 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