Mechanical behaviour of materials 205 The choice of slip plane in these metals is often influ ts basal plane perpendicular to the tensile enced by temperature and a preference is shown for is,ie.φ=0° ejecting [11 2 below Tm/4,(110) from Tm/4 to Tm/2 and contrast to its ter haviour. where it is brittle it [123] at high temperatures, where Tm is the melt- will now appear since the shear stress on the ng point. Iron often slips on all the slip planes at slip plane is only zero for a tensile test and not for a once in a common (11 1> slip direction, so that a bend test. On the other hand, if we take the crystal with slip line (i.e. the line of intersection of a slip plane its basal plane oriented parallel to the tensile axis (i.e surface of a crystal) takes 90 )this specimen will appear brittle whatever appearance. tress system is applied to it. For this crystal, although the shear force is large, owing to the large area of the 7.3.4 Law of critical resolved shear stres slip plane, A/ cos o, the resolved shear stress is always This law states that slip takes place along a given slip sling small and insufficient to cause deformation by stress reaches a slipper value. In most crystals the high symmetry of arrangement provides several crystallographic 7.3.5 Multiple slip equivalent planes and directions for slip (i. e. cph crys- The fact that slip bands, each consisting of many slip als have three systems made up of one plane contain- 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, hows that deformation is inhomogeneous, with exten while bcc crystals have many systems)and in such sive slip occurring on certain planes, while the crystal direction for which the maximum stress acts (aw3 formed. Figures 7. 12a and 7 12b show such a crystal 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 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 tensile axis. The value of the angle is determined 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 more conveniently demonstrated on a stereographic stress,(e. the stress on the glide plane in the glide ma is shown in the unit triangle m re Te .r than tie en have different values of the flow stress as shown in projection of the crystal by considering the rotation Figure 7. 11a. However, because of the criterion of a the tensile axis relative to the crystal rathe critical resolved shear stress, a plot of resolved shear versa. This is illustrated in Figure 7. 13a for the defor within experimental error, for all the specimens. This P and [101], and P and (11 1) are equal to a and p, plot is shown in Figure 7. 1 1b. respectively. The active slip system is the(11 1) plane The importance of a critical shear stress may be and the [101] direction, and as deformation proceeds demonstrated further by taking the crystal which has the change in orientation is represented by the point, P, shear strain Figure 7.11 Schematic representation of (a) variation of stress versus elongation with orientation of basal plane and (b) constancy of revolved shear stressThe choice of slip plane in these metals is often influ￾enced by temperature and a preference is shown for {l 1 2} below Tm/4, {110} from Tin~4 to Tin~2 and {l 2 3} at high temperatures, where Tm is the melt￾ing point. Iron often slips on all the slip planes at once in a common (l 1 1) slip direction, so that a slip line (i.e. the line of intersection of a slip plane with the outer surface of a crystal) takes on a wavy appearance. 7.3.4 Law of critical resolved shear stress This law states that slip takes place along a given slip plane and direction when the shear stress reaches a critical value. In most crystals the high symmetry of atomic arrangement provides several crystallographic equivalent planes and directions for slip (i.e. cph crys￾tals have three systems made up of one plane contain￾ing three directions, fcc crystals have twelve systems made up of four planes each with three directions, while bcc crystals have many systems) and in such cases slip occurs first on that plane and along that direction for which the maximum stress acts (law 3 above). This is most easily demonstrated by testing in tension a series of zinc single crystals. Then, because zinc is cph in structure only one plane is available for the slip process and the resultant stress-strain curve will depend on the inclination of this plane to the tensile axis. The value of the angle tp is determined by chance during the process of single-crystal growth, and consequently all crystals will have different values of tp, and the corresponding stress-strain curves will have different values of the flow stress as shown in Figure 7.1 l a. However, because of the criterion of a critical resolved shear stress, a plot of resolved shear stress (i.e. the stress on the glide plane in the glide direction) versus strain should be a common curve, within experimental error, for all the specimens. This plot is shown in Figure 7.1 lb. The importance of a critical shear stress may be demonstrated further by taking the crystal which has Mechanical behaviour of materials 205 its basal plane oriented perpendicular to the tensile axis, i.e. ~- 0 ~ and subjecting it to a bend test. In contrast to its tensile behaviour, where it is brittle it will now appear ductile, since the shear stress on the slip plane is only zero for a tensile test and not for a bend test. On the other hand, if we take the crystal with its basal plane oriented parallel to the tensile axis (i.e. tp = 90 ~ this specimen will appear brittle whatever stress system is applied to it. For this crystal, although the shear force is large, owing to the large area of the slip plane, A~ cos 4~, the resolved shear stress is always very small and insufficient to cause deformation by slipping. 7.3.5 Multiple slip The fact that slip bands, each consisting of many slip lines, are observed on the surface of deformed crystals shows that deformation is inhomogeneous, with exten￾sive slip occurring on certain planes, while the crystal planes lying between them remain practically unde￾formed. Figures 7.12a and 7.12b show such a crystal in which the set of planes shear over each other in the slip direction. In a tensile test, however, the ends of a crystal are not free to move 'sideways' relative to each other, since they are constrained by the grips of the tensile machine. In this case, the central portion of the crystal is altered in orientation, and rotation of both the slip plane and slip direction into the axis of ten￾sion occurs, as shown in Figure 7.12c. This behaviour is more conveniently demonstrated on a stereographic projection of the crystal by considering the rotation of the tensile axis relative to the crystal rather than vice versa. This is illustrated in Figure 7.13a for the defor￾mation of a crystal with fcc structure. The tensile axis, P, is shown in the unit triangle and the angles between P and [ 1 01], and P and (1 1 1) are equal to ~ and ~b, respectively. The active slip system is the (1 1 1) plane and the [ 10 1 ] direction, and as deformation proceeds the change in orientation is represented by the point, P, 15 ~ 30 ~ 60 ~ f i f f f f I ,, ,4,, .= , i elongation shear strain (a) (b) f / , ! Figure 7.11 Schematic representation of (a) variation of stress versus elongation with orientation of basal plane and (b) constancy of revoh,ed shear stress