Instantaneous offset of marker lines
Slip along a fault causes a local heterogeneous displacement and stress field and can be studied by elementary elastic crack theory (Pollard and Segall, 1987). We investigate the two-dimensional, instantaneous deformation of two perpendicular marker lines, which are offset along a slip surface of finite length embedded in an elastic medium, following the procedure outlined by Grasemann et al. (2005). We restrict our discussion of far field boundary conditions to the end members of homogeneous deformation, i.e. pure and simple shear (Fig. 3). The sense of slip is controlled by the orientation of the slip surface with respect to the principal stress directions dividing the range of possible orientations into four quadrants, two of which have a dextral and the other two have a sinistral slip. A slip surface parallel to the principal stress directions does not experience instantaneous shear strain. The rotational behavior of the slip surface and the marker lines is controlled by the orientation of the principal stress directions (σ1 and σ3) with the eigenvectors of the deformation tensor (a1 and a3), which are directions of no instantaneous rotation (Bobyarchick, 1986). We restrict this discussion to marker lines, which are parallel and perpendicular to a1.
Figure 3. Instantaneous offset of two perpendicular marker layers
Instantaneous offset of two perpendicular marker layers along a slip surface and the orientation of the fabric elements with respect to the kinematic axes in a) pure shear and b) simple shear.
Coaxial deformation (i.e. pure shear) has two perpendicular eigenvectors, which are parallel to the principal stress direction (Fig. 3a). A slip surface, which is oblique to the stretching eigenvector a1 will instantaneously rotate into the direction of a1, i.e. the fabric attractor. Two material lines, which are parallel and perpendicular to the fabric attractor, do not rotate instantaneously. However, the line parallel to a1 will stretch and the line parallel to a3 will shorten. At 45° and 135° with respect to a1 the slip surface experiences its maximum instantaneous shear strain and rotation rate.
Simple shear has only one direction which is not instantaneously rotating, i.e. the fabric attractor parallel to the shear zone boundary (Fig. 3b). Fabric elements of any other orientation will rotate into the shear plane and therefore the mylonitic foliation of strongly strained rocks is considered to represent the orientation of the flow plane. The minimum and maximum principal stress directions are oriented at 45° and 135° respectively (measured from the fabric attractor counterclockwise). Slip surfaces with orientations larger than 135° and less than 45° will slip synthetically while the sense of slip between 45° and 135° is antithetic. At 45° and 135° the slip surfaces experience no instantaneous shear strain. Material lines perpendicular to the flow plane have the maximum instantaneous shear strain and rotation rate.
This short discussion about slip surface and two perpendicular material lines can be easily extended to general shear deformation and marker lines of any arbitrary orientation. For narrowing shear zones, σ1 will vary between 90° (pure shear) and 135° (simple shear). For broadening shear zones, σ1 will vary between 135° (simple shear) and 180°. In the following we extend this discussion to finite deformation including the heterogeneous perturbation strain generated by the slip plane including a homogeneous background strain.