Resolved shear stress

7.3.2 Resolved shear stress

All working processes such as rolling, extrusion, forg- ing etc. cause plastic deformation and, consequently, these operations will involve the processes of slip or twinning outlined above. The stress system applied during these working operations is often quite com- plex, but for plastic deformation to occur the presence of a shear stress is essential. The importance of shear

Figure 7.9 Slip and twinning in a crystal .

204 Modern Physical Metallurgy and Materials Engineering stresses becomes clear when it is realized that these

A consideration of the tensile test in this way shows stresses arise in most processes and tests even when

that it is shear stresses which lead to plastic defor- the applied stress itself is not a pure shear stress. This

mation, and for this reason the mechanical behaviour may be illustrated by examining a cylindrical crys-

exhibited by a material will depend, to some extent, tal of area A in a conventional tensile test under a

on the type of test applied. For example, a ductile uniaxial load P. In such a test, slip occurs on the

material can be fractured without displaying its plastic slip plane, shown shaded in Figure 7.10, the area of

properties if tested in a state of hydrostatic or triax- ial tension, since under these conditions the resolved

normal to the plane OH and the axis of tension. The shear stress on any plane is zero. Conversely, materials applied force P is spread over this plane and may be

which normally exhibit a tendency to brittle behaviour resolved into a force normal to the plane along OH,

in a tensile test will show ductility if tested under con- P

ditions of high shear stresses and low tension stresses. line of greatest slope in the slip plane and the force

In commercial practice, extrusion approximates closely P

to a system of hydrostatic pressure, and it is common (force/area) is made up of two stresses, a normal stress

2 ⊲P/A⊳ for normally brittle materials to exhibit some ductility cos tending to pull the atoms apart, and a when deformed in this way (e.g. when extruded).

over each other. In general, slip does not take place down the line

7.3.3 Relation of slip to crystal structure

of greatest slope unless this happens to coincide with An understanding of the fundamental nature of plastic the crystallographic slip of direction. It is necessary,

deformation processes is provided by experiments on therefore, to know the resolved shear stress on the slip

single crystals only, because if a polycrystalline sample plane and in the slip direction. Now, if OT is taken

is used the result obtained is the average behaviour of to represent the slip direction the resolved shear stress

all the differently oriented grains in the material. Such will be given by

experiments with single crystals show that, although the resolved shear stress is a maximum along lines of greatest slope in planes at 45 ° to the tensile axis, slip occurs preferentially along certain crystal planes and

formula is written more simply as directions. Three well-established laws governing the slip behaviour exist, namely: (1) the direction of slip

is almost always that along which the atoms are most closely packed, (2) slip usually occurs on the most

the axis of tension. It can be seen that the resolved closely packed plane, and (3) from a given set of slip shear stress has a maximum value when the slip plane

planes and directions, the crystal operates on that sys- is inclined at 45 ° to the tensile axis, and becomes

tem (plane and direction) for which the resolved shear smaller for angles either greater than or less than 45 ° .

stress is largest. The slip behaviour observed in fcc When the slip plane becomes more nearly perpendic-

metals shows the general applicability of these laws,

since slip occurs along h1 1 0i directions in f1 1 1g that the applied stress has a greater tendency to pull

° ⊳ it is easy to imagine

planes. In cph metals slip occurs along h1 1 2 0i direc- the atoms apart than to slide them. When the slip

tions, since these are invariably the closest packed, plane becomes more nearly parallel to the tensile axis

but the active slip plane depends on the value of the

45 ° ⊳ the shear stress is again small but in this axial ratio. Thus, for the metals cadmium and zinc, c/a is 1.886 and 1.856, respectively, the planes of

is correspondingly large. greatest atomic density are the f0 0 0 1g basal planes and slip takes place on these planes. When the axial ratio is appreciably smaller than the ideal value of c/a D 1.633 the basal plane is not so closely packed, nor so widely spaced, as in cadmium and zinc, and other slip planes operate. In zirconium ⊲c/a D 1.589⊳ and titanium ⊲c/a D 1.587⊳, for example, slip takes

place on the f1 0 1 0g prism planes at room temperature and on the f1 0 1 1g pyramidal planes at higher tem- peratures. In magnesium the axial ratio ⊲c/a D 1.624⊳ approximates to the ideal value, and although only basal slip occurs at room temperature, at temperatures above 225 °

C slip on the f1 0 1 1g planes has also been observed. Bcc metals have a single well-defined close- Figure 7.10 Relation between the slip plane, slip direction

packed h1 1 1i direction, but several planes of equally and the axis of tension for a cylindrical crystal .

high density of packing, i.e. f1 1 2g, f1 1 0g and f1 2 3g.

Mechanical behaviour of materials 205 The choice of slip plane in these metals is often influ-

its basal plane oriented perpendicular to the tensile enced by temperature and a preference is shown for

° , and subjecting it to a bend test. In f1 1 2g below T m / 4, f1 1 0g from T m / 4 to T m / 2 and

contrast to its tensile behaviour, where it is brittle it f1 2 3g at high temperatures, where T m is the melt-

will now appear ductile, since the shear stress on the ing 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 h1 1 1i 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. with the outer surface of a crystal) takes on a wavy

° ) this specimen will appear brittle whatever appearance.

stress system is applied to it. For this crystal, although the shear force is large, owing to the large area of the