Synthetic slip plane, a hybrid kind of calcite twin data in dynamic analysis

Graphical abstract The synthetic slip plane is defined as the linear combination of a pair of twinned and untwinned e-planes in a single calcite crystal. This new kind of calcite e-twin data can be used together with the twinned e-plane to better determine stress in dynamic analysis.


Method details
Calcite e-twin data have been inverted for stress in rocks since the pioneering work of Turner [1]. This becomes an important tool for structural geologists to quantify stress in the upper crust during geological history [2,3]. Various methods of this dynamic analysis have been developed. They are based upon two kinds of e-twin data: twinned e-planes [1,4,5], and both twinned e-planes and untwinned e-planes [6][7][8][9][10][11][12]. In this paper we introduced a new kind of e-twin data, synthetic slip plane, as the linear combination of a pair of twinned and untwinned e-planes in a single calcite crystal. It can be used together with the twinned e-plane to better determine stress in dynamic analysis.

Synthetic slip plane
Calcite mechanical twinning occurs along the e slip system (Fig. 1a), when the resolved shear stress on the system, t, exceeds the critical resolved shear stress, t c . Let us consider a pair of twinned (e 1 ) and untwinned (e 2 ) e-planes in a calcite crystal with an upright c-axis (Fig. 1b),   1. (a) The e-planes and gliding directions in a single calcite with an upright c-axis. The synthetic slip planes from (b) a pair of twinned (e 1 ) and untwinned (e 2 ) e-planes and (c) a pair of twinned (e 1 ) and untwinned (e 3 ) e-planes, respectively, in the crystal. Great circles represent e-planes or synthetic slip planes. Short lines mark the gliding directions or the slip lines, and arrows indicate the gliding or slip senses, for instance, reverse in this figure. Lower-hemisphere, equal-area projection.
Weighting these inequalities and subtracting the second inequality from the first inequality, wn ð1Þ ss ð1Þ À 1 À w ð Þn ð2Þ ss ð2Þ ! 0 ð2Þ where w is weight, and w 2 0:5 1 ½ . Recasting the above inequality in Fry's [13] where v ð0Þ is the synthetic datum vector in the six-dimensional stress space, s is the stress vector, and T is matrix transposition. v ð0Þ ¼ v where the discriminant, D 21 , is expressed as follows, Whether the discriminant is positive or negative in sign requires additional work. Take as an example the calcite crystal in Fig. 1a, where n ð1Þ 2 ¼ s where the discriminant, D 31 , has a positive sign.
According to Eqs. (5), (6) and (8), there are four possible solutions of s ð0Þ and n ð0Þ :   4) for n ð0Þ and s ð0Þ , we have a seemingly unreal situation of n ð0Þ ¼ n ð1Þ =s ð1Þ and s ð0Þ ¼ s ð1Þ =n ð1Þ for the former pair, and a real situation of n ð0Þ ¼ n ð1Þ and s ð0Þ ¼ s ð1Þ for the latter pair. That is to say, the latter pair rather than the former pair is accepted as the real solution pair. Furthermore, for the latter pair only the solution with a larger similarity between n ð0Þ and n ð1Þ is considered as a unique solution, in the presence of the permutation of n ð0Þ and s ð0Þ in v ð0Þ . Such unique solutions for w 6 ¼ 1 are displayed in Fig. 1b and listed in Table 1.
In a similar way, for a pair of twinned (e 1 ) and untwinned (e 3 ) e-planes in Fig. 1a, we derive the unique solution of n ð0Þ and s ð0Þ for a varying weight (Fig. 1c and Table 1).

Applications
Numerous methods of dynamic analysis have been developed to infer stress from calcite e-twins measured at universal stage. They are based upon two kinds of e-twin data: twinned e-planes [1,4,5,8,12], and both twinned e-planes and untwinned e-planes [6,7,[9][10][11]. They may be categorized as graphic [1] and numerical [4,[6][7][8][9][10][11][12]. The synthetic slip plane devised in this paper is a hybrid kind of e-twin data, the linear combination of a pair of twinned and untwinned e-planes in a single calcite crystal. It is applicable to graphic and numerical dynamic analysis.

Turner's [1] C-T method
Turner [1] first developed the graphic method that determines the compression (C) and tension (T) axes in an auxiliary plane containing the normal to the twinned e-plane and the gliding line on the plane for each e-twin. This method is directly applicable to the synthetic slip plane. The use of the synthetic plane helps further restrain the extents of the compression and tension axes (Fig. 2).

McKenzie's [14] right dihedral method
The right dihedral method for fault data [14] is non-strictly applied to the twinned e-planes (see [5]). The incorporation of the synthetic slip planes into it further reduces the feasible fields of the maximum and minimum principal stress axes. Two examples are taken from a single calcite with an upright c-axis (Fig. 1a), for simplicity. The first example has one twinned (e 1 ) and two untwinned (e 2 and e 3 ) e-planes (Fig. 2a), and the second example possesses two twinned (e 1 and e 3 ) and one untwinned (e 2 ) e-planes (Fig. 2b). The feasible fields have a decreasing extent with the decrease in the weight and the increase in data number (Fig. 3). For w ¼ 0:5, the maximum principal stress axis has a Table 1 List of synthetic slip planes in Fig. 1b-c. "0/26.5 00 represents dip direction/dip angle for planar data, and bearing/plunge for linear data. Reverse or normal slip sense means the presence of reverse or normal slip component along the slip line.  Fig. 2. Lower-hemisphere, equal-area projection of the compression (C) and tension (T) axes through applying the C-T method [1] to two examples in a single calcite crystal: (a) one twinned e-plane and two synthetic slip planes, and (b) two twinned e-planes and two synthetic slip planes.
relatively narrow range of bearing and a relatively wide range of plunge for the first example (Fig. 3c) and a relatively wide range of bearing and a relatively narrow range of plunge for the second example (Fig. 3f).

Numerical dynamic analysis
Various methods about numerical dynamic analysis have been developed to obtain either the incomplete deviatoric stress or the reduced stress [4,5] or, more significantly, the complete deviatoric stress [6][7][8][9][10][11][12] through solving the optimum problems about the fit or misfit between the calculated and measured twinned and/or untwinned e-planes. The latter methods require the additional  knowledge of the value of the critical resolved shear stress to determine the maximum differential stress, as different from the former methods. However, the synthetic slip plane can be incorporated into only the former methods, for example, Spang's [4] method. Listed in Table 2 are the results by applying Spang's [4] method to examples shown in Fig. 2. For the first example, the maximum and minimum principal stresses are within the feasible fields obtained using the right dihedral method (Fig. 3a-c), regardless of the weight. The intermediate principal stresses have a constant orientation. With the decrease in the weight, the maximum or minimum principal stress has an increasing or decreasing plunge, and the stress ratio decreases.
For the second example, the maximum principal stress is always within the feasible fields (Fig. 3df); so is the minimum principal stress only for a larger value of weight. The maximum principal stresses have a constant orientation. With the decrease in the weight, the intermediate or minimum principal stress has a decreasing or increasing plunge, and the stress ratio decreases.

Concluding remarks
In this paper the synthetic slip plane is defined as the linear combination of a pair of twinned and untwinned e-planes in a single calcite crystal. It is independent of either twinned e-plane or untwinned e-plane. This auxiliary slip plane can be used together with the twinned e-planes to better determine stress in both graphic dynamic analysis and numerical dynamic analysis. However, in the latter analysis, this new kind of data is at present incorporated into only the existing methods that solve for the reduced stress.