Law of critical resolved shear stress

7.3.4 Law of critical resolved shear stress

very small and insufficient to cause deformation by This law states that slip takes place along a given slip

slipping.

plane and direction when the shear stress reaches a critical value. In most crystals the high symmetry of atomic arrangement provides several crystallographic

7.3.5 Multiple slip

equivalent planes and directions for slip (i.e. cph crys- tals have three systems made up of one plane contain-

The fact that slip bands, each consisting of many slip ing three directions, fcc crystals have twelve systems

lines, are observed on the surface of deformed crystals made up of four planes each with three directions,

shows that deformation is inhomogeneous, with exten- while bcc crystals have many systems) and in such

sive slip occurring on certain planes, while the crystal cases slip occurs first on that plane and along that

planes lying between them remain practically unde- direction for which the maximum stress acts (law 3

formed. Figures 7.12a and 7.12b show such a crystal above). This is most easily demonstrated by testing in

in which the set of planes shear over each other in the tension a series of zinc single crystals. Then, because

slip direction. In a tensile test, however, the ends of zinc is cph in structure only one plane is available for

a crystal are not free to move ‘sideways’ relative to the slip process and the resultant stress–strain curve

each other, since they are constrained by the grips of will depend on the inclination of this plane to the

the tensile machine. In this case, the central portion of the crystal is altered in orientation, and rotation of both

by chance during the process of single-crystal growth, the slip plane and slip direction into the axis of ten- and consequently all crystals will have different values

sion occurs, as shown in Figure 7.12c. This behaviour is more conveniently demonstrated on a stereographic

have different values of the flow stress as shown in projection of the crystal by considering the rotation of Figure 7.11a. However, because of the criterion of a

the tensile axis relative to the crystal rather than vice critical resolved shear stress, a plot of resolved shear

versa. This is illustrated in Figure 7.13a for the defor- stress (i.e. the stress on the glide plane in the glide

mation of a crystal with fcc structure. The tensile axis, direction) versus strain should be a common curve,

P , is shown in the unit triangle and the angles between within experimental error, for all the specimens. This

plot is shown in Figure 7.11b. respectively. The active slip system is the ⊲1 1 1⊳ plane The importance of a critical shear stress may be

and the [ 1 0 1] direction, and as deformation proceeds demonstrated further by taking the crystal which has

the change in orientation is represented by the point, P,

Figure 7.11 Schematic representation of (a) variation of stress versus elongation with orientation of basal plane and (b) constancy of revolved shear stress .

206 Modern Physical Metallurgy and Materials Engineering

Figure 7.12 (a) and (b) show the slip process in an unconstrained single crystal; (c) illustrates the plastic bending in a crystal gripped at its ends .

Figure 7.13 Stereographic representation of (a) slip systems in fcc crystals and (b) overshooting of the primary slip system .

moving along the zone, shown broken in Figure 7.13a, As slip occurs on the one system, the primary sys-

tem, the slip plane rotates away from its position of maximum resolved shear stress until the orientation of

Beyond this point, slip should occur equally on both the primary system and a second system (the conjugate system) ⊲1 1 1⊳ [0 1 1], since these two systems receive equal components of shear stress. Subsequently, during the process of multiple or duplex slip the lattice will rotate so as to keep equal stresses on the two active sys- tems, and the tensile axis moves along the symmetry

line towards [1 1 2]. This behaviour agrees with early

observations on virgin crystals of aluminium and cop- per, but not with those made on certain alloys, or pure metal crystals given special treatments (e.g. quenched from a high temperature or irradiated with neutrons). Results from the latter show that the crystal continues to slip on the primary system after the orientation has reached the symmetry line, causing the orientation to overshoot this line, i.e. to continue moving towards [1 0 1], in the direction of primary slip. After a certain amount of this additional primary slip the conjugate system suddenly operates, and further slip concen- trates itself on this system, followed by overshooting in the opposite direction. This behaviour, shown in Figure 7.13b, is understandable when it is remembered that slip on the conjugate system must intersect that on the primary system, and to do this is presumably more difficult than to ‘fit’ a new slip plane in the relatively undeformed region between those planes on which slip has already taken place. This intersection process is more difficult in materials which have a low stacking fault energy (e.g. ˛-brass).