As remarked in Chapter 1, v. Goler & Sachs (1927) derived equations for equal double slip in f.c.c. crystals in tension with the loading axis on a symmetry line, and Taylor (1927a) derived comparable equations in compression (and also gave the equation of the unstretched cone for f.c.c. crystals). Equations applicable to any crystal class for these same symmetry conditions and for both tension and compression were developed by Bowen & Christian (1965), who presented formulas for various specific combinations of slip systems in f.c.c. and b.c.c. crystals. A general equation for the deformation gradient in (proportional) double slip of arbitrary relative amounts was first given in the work of Chin, Thurston, & Nesbitt (1966) mentioned previously. They carried the analysis no further, however, and applied the equation only to cases of equal, symmetric double slip in f.c.c. crystals.

Apparently the first explicit equations for rotation and stretch of a crystal material line in arbitrary (proportional) double slip in f.c.c. and b.c.c. crystals were developed by Shalaby & Havner (1978) (independently of the work of Chin, Thurston, & Nesbitt (1966)). The equations were illustrated for various nonsymmetric axis positions and relative amounts of slip. General equations for material line and areal vectors and both the finite deformation gradient and its inverse in arbitrary (proportional) double slip were derived in Havner (1979). Here we shall follow this last approach to the analysis of double slip in crystals.