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 direc- tion 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. 1257 Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/102/9/1257/3381013/i0016-7606-102-9-1257.pdf by guest on 29 September 2021 1258 DENNIS AND SECOR lb Figure 1. Three structural styles recognized in sheared aniso- tropic 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 paral- lel to the shear zone wall. In a sec- Illa ond lithology, typically a felsic lllb orthogneiss, mineral lineation is oblique to subparallel with crenula- tion 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 adja- cent 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. Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/102/9/1257/3381013/i0016-7606-102-9-1257.pdf by guest on 29 September 2021 FOLIATED ROCKS DEFORMED BY SIMPLE SHEAR 1259 Figure 2. Stylized development of linea- tions oblique or parallel to crenulation axes in a zone deforming by simple shear. This figure shows steps (A-E) in the progressive defor- mation of a cubic element centered on the origin. A volume of foliated material is sheared (A).