Theoretical treatment

7.6.2.1 Theoretical treatment

Figure 7.38 Dependence of flow stress on (dislocation The properties of a material are altered by cold-

density) 1 /2 for Cu, Ag and Cu–Al . working, i.e. deformation at a low temperature rel- ative to its melting point, but not all the properties are improved, for although the tensile strength, yield strength and hardness are increased, the plasticity and

l the mean distance between dislocations which is general ability to deform decreases. Moreover, the

, and ˛ a constant; in the Taylor model ˛ D physical properties such as electrical conductivity, den-

sity and others are all lowered. Of these many changes Cu–Al single crystals and polycrystalline Ag and Cu. in properties, perhaps the most outstanding are those

In his theory Taylor considered only a two- that occur in the mechanical properties; the yield stress

dimensional model of a cold-worked metal. However, of mild steel, for example, may be raised by cold work

because plastic deformation arises from the movement

of dislocation loops from a source, it is more Such changes in mechanical properties are, of

from 170 up to 1050 MN/m 2 .

appropriate to assume that when the plastic strain is course, of interest theoretically, but they are also

, N dislocation loops of side L (if we assume for of great importance in industrial practice. This is

convenience that square loops are emitted) have been because the rate at which the material hardens during

given off per unit volume. The resultant plastic strain deformation influences both the power required and the

is then given by

method of working in the various shaping operations,

D NL 2 b (7.24) while the magnitude of the hardness introduced

governs the frequency with which the component must

and l by

be annealed (always an expensive operation) to enable

further working to be continued. 1/2 ] D [1/4LN] (7.25) Since plastic flow occurs by a dislocation mecha-

l'

nism the fact that work-hardening occurs means that it Combining these equations, the stress–strain relation becomes difficult for dislocations to move as the strain

(7.26) increases. All theories of work-hardening depend on

this assumption, and the basic idea of hardening, put is obtained. Taylor assumed L to be a constant, i.e. forward by Taylor in 1934, is that some dislocations

the slip lines are of constant length, which results in a become ‘stuck’ inside the crystal and act as sources of internal stress which oppose the motion of other

Taylor’s assumption that during cold work the den- gliding dislocations.

sity of dislocations increases has been amply veri- One simple way in which two dislocations could

fied, and indeed the parabolic relationship between become stuck is by elastic interaction. Thus, two par-

stress and strain is obeyed, to a first approximation, allel edge dislocations of opposite sign moving on

in many polycrystalline aggregates where deformation parallel slip planes in any sub-grain may become stuck,

in all grains takes place by multiple slip. Experimen- as a result of the interaction discussed in Chapter 4.

tal work on single crystals shows, however, that the

G. I. Taylor assumed that dislocations become stuck work- or strain-hardening curve may deviate consider- after travelling an average distance, L, while the den-

ably from parabolic behaviour, and depends not only on crystal structure but also on other variables such as

due to the dislocations getting in each other’s way. crystal orientation, purity and surface conditions (see The flow stress is then the stress necessary to move

Figures 7.39 and 7.40).

a dislocation in the stress field of those dislocations The crystal structure is important (see Figure 7.39) in that single crystals of some hexagonal metals slip only on one family of slip planes, those parallel to

the basal plane, and these metals show a low rate of

Mechanical behaviour of materials 227 which would describe the complete stress–strain rela-

tionship is difficult, and consequently the present-day approach is to examine the various stages of hardening and then attempt to explain the mechanisms likely to give rise to the different stages. The work-hardening behaviour in metals with a cubic structure is more complex than in most other structures because of the variety of slip systems available, and it is for this rea- son that much of the experimental evidence is related to these metals, particularly those with fcc structures.