ON ORIENTATION CHANGES ACCOMPANYING SLIP AND

" The purpose of this note is to make a few observations on several analyses of lattice rotation by slip and by twinning in deformation.

Several widely used analyses appear to predict different magnitudes and directions of rotation, but these differences are due only to the reference systems used. Although one analysis or another may appear to be appropriate for a given form of loading, this is because of the reference system natural to the geometry of loading rather than to the stress state producing slip.
Three analyses are considered below. All assume uniformly distributed slip. To facilitate comparison, only slip on a single system will be considered, although these basic analyses may be used under conditions of multiple slip. The Schmid "tension" analysis 1,2 describes the lattice reorientation relative to any physical line in the material. If the initial and present values of the angle between this line and the slip direction are designated l and l, respectively, where de dR/R, I is the crystallographic shear strain and is the angle between the reference axis and the slip plane normal. The reference line approaches the slip direction; the axis of rotation being normal to both the slip direction and the reference line. This analysis is commonly applied to long single crystals deformed in tension, with the tensile axis serving as the physical reference line and the strain being that along the tensile axi s.
Taylor "compression" analysis describes the lattice rotation relative to any physical plane in the material. If the strain is compressive (h < h ), the refero ence plane normal approaches the slip plane normal; the axis of rotation being lying in both the reference plane and slip plane. This analysis is commonly applied to the compression of flat wide crystals where the crystal faces in contact with the compression platens are the natural reference planes.
The difference in these two analyses is illustrated in the two dimensional illustration in two-dimensional illustration of the different rotations due to slip. A physical line, d, which was initially parallel to x suffers one rotation. The normal to a physical plane, n, which was also initially parallel to x, suffers a different rotation. deformation, the direction, d, and the normal, n, to the plane AC are both aligned with a crystallographic direction, x. After deformation, however, n and d are no longer parallel, having suffered different rotations.
It should be emphasized that the stress state causing slip is not intrinsic to either analysis; rather it is the geometry of the specimen and its constraints that make one or the other analysis more appropriate. Unfortunately, this has not been recognized in many theories of texture formation. The tensile axis of a long crystal will naturally remain parallel to its longest direction. During the compression of a flat crystal, the plane initially in contact with the compression platens will remain normal to the compression axis. However, one can conceive of tensile deformation of a short fat crystal where the Taylor "compression" analysis would be more appropriate, or the compression of a long thin crystal where the Schmid analysis would be more appropriate. Both analyses are special cases of a general treatment of finite strains by Chin et aZ. , that covers multiple slip as well as single slip. Theories of texture development are concerned with the lattice rotation within grains of a polycrystal. If the grains are extended in a direction in which they are already very elongated, the Schmid analysis seems appropriate. Conversely, if the grains are already pancake-shaped and subjected to further compression, the Taylor analysis would be better. In many cases, however, the grains are more or less equiaxed, and are constrained on all sides and directions by neighboring grains. Neither analysis is quite appropriate.
The mathematical analysis of rotations describes the rotation of particles in a body with infinitesimal strain: is the rotation about an axis a 3 normal to the where i3 is the shear displacement parallel i and j directions, u l to i occurring over a distance x. in the j direction.

3
The conventional shear strain, defined by u. u.
If i and j are the slip direction and slip plane normal, respectively, 9u'/xi3 0 so, d 1/2 dy (9) and the rotation occurs about an axis normal to both the slip direction and the slip plane normal. An interesting aspect of this analysis is that the rotation is referred to no single external plane or direction. For this reason, and because its predictions are between those of the first two analyses, it appears to be more appropriate for equiaxed grains. The axis of rotation predicted by the three analyses are shown in Figure 2   It is widely recognized that when a material undergoes twinning, the twinned regions undergo a large and discrete lattice reorientation relative to the untwinned regions. However, most workers have ignored the fact Figure 3.
Illustration of the rotations described by the three analyses for single slip. 3A illustrates the rotation of a physical direction toward the slip direction.
3B shows the rotation of a normal to a physical plane. The mathematical rotations are represented in 3C. that twinning also causes a gradual rotation of both the twinned and untwinned regions relative to the external coordinates.
If the twinned regions are fine and uniformly distributed, this gradual rotation is a continuous function of the macroscopic shear strain, where f is the volume fraction twinned and S is the discrete twinning shear strain. The magnitude of this rotation is exactly the same as that which would occur if the shear strain, y, were produced by uniform slip with the same plane and direction of shear.
Consider the special case of a line initially at an angle 1 to the twinning direction and 90 Io to the twinning plane. Its angle, l, to the twinning direction after twinning will be given by  (001) will be reoriented to lie parallel to (ii0) of the twin (see Figure 4).   Figure 4). A long crystal with an initial tensile axis of [111] would be converte_d by 100% twinning to a [ll4] orientation rather than [115]. CONCLUSION In applying the concepts of lattice rotation produced by slip and twinning to predicting crystallographic textures, careful attention must be paid to the reference system used. Orientation change due to twinning have two components; one from the discrete reorientation of the twinning plus one from the gradual rotation due to the shear.