In this article we will discus about:- 1. Introduction to Polycrystal 2. Dislocation Motion in Polycrystals 3. Ductility.
Introduction to Polycrystal:
The study of the basic mechanisms of the plastic deformations is facilitated by the study of plastic behaviour of single crystals. Most of the engineering materials are, however, polycrystalline, which contain a large number of grains with varying orientations. The individual grains of the polycrystalline aggregate experience the restraining effect of the surrounding grains during plastic deformation.
Since the crystallographic orientation changes abruptly across a grain boundary, the orientation of the slip planes in adjoining grains is also different. The grain boundaries, thus, act as barriers to the movement of dislocations from one grain to another. This feature tends to increase the strength of polycrystals.
At temperatures below 0.4 Tm.p. (on absolute scale), the grain boundaries are quite strong and the crack propagates normally in trans-granular manner across the grains. At higher temperatures, however, grain-boundary-sliding can occur resulting in inter-granular fracture. Therefore, the beneficial effects of grain boundaries on strength are usually confined to temperatures below 0.4 Tm.p.
ADVERTISEMENTS:
In special cases, even at low temperatures, the structure of grain boundary itself may be of paramount importance. For example, brittle films may sometimes form around the grains such as cementite in hypereutectoid steels, and bismuth in copper. This may lead to premature inter-granular failure of ductile materials.
Deformation in Bi-Crystals:
The importance of the change in orientation across a grain boundary to the process of slip has been demonstrated by experiments on bi-crystals. It is observed that as the orientation difference between the two crystals decreases, the stress-strain curve changes progressively from polycrystalline to single crystal in behaviour.
ADVERTISEMENTS:
Specially-grown-specimens with grain-boundaries parallel to each other and perpendicular to the tensile axis have also been deformed. It is observed that the process of slip occurs more easily if the slip-band does not end against a boundary, Fig. 6.35. Consequently, wedge-shaped areas near the grain boundaries remain relatively undeformed.
On the other hand, portions away from the boundaries, where the slip-bands end on free surfaces, undergo extensive plastic flow and the cross-sectional area in these portions is reduced. In the slip-bands ending on free surfaces, successive movement of dislocations occurs up to the surface, and the dislocations continue to disappear by creating slip steps equal to their Burgers vector.
On the contrary, a slip-band ending against a grain boundary would result in the piling-up of an array of dislocations against the boundary, Fig. 6.35 (b). Since these dislocations
emitted from a source are of identical sign, they repel each other and inhibit continued piling-up of more and more dislocations. Extensive plastic deformation therefore does not take place in regions close to boundaries and it can be visualized that polycrystalline materials will exhibit enhanced strength.
Dislocation Motion in Polycrystals:
The movement of dislocations in certain favourably oriented grains of a polycrystal may be initiated at stresses as low as in suitably oriented single crystals: Since the shear stress is maximum at an angle of 45° to the tensile axis, the favourably oriented grains will exhibit slip plane and slip direction oriented around 45° to the tensile axis.
ADVERTISEMENTS:
Due to misorientation of slip planes across a large angle boundary, the dislocation cannot cross the boundary. The slip process is discontinuous being restricted to the boundaries of individual grains. The dislocations piling-up against a boundary create a high stress-concentration around the boundary.
When this stress rises to a high level, it initiates slip in the less favourably oriented adjoining grain. Arrival of dislocations, at a boundary, or movement of dislocations away from it changes the shape of the grain at a boundary. For example, arrival of a group of n positive edge dislocations on a boundary is analogous to deposition of a stack of extra half planes of thickness, nb above the slip plane at the boundary, Fig. 6.36 (b).
Since, it is experimentally found that many polycrystals undergoing plastic deformation do not develop voids at the grain boundaries, it is evident that individual grains deform in such a manner as to fill the space after deformation just as they did before stressing. Since grains are randomly oriented, it follows that each grain must be capable of undergoing any arbitrary deformation to maintain continuity across the grain boundaries.
ADVERTISEMENTS:
The principles of theory of deformation say that total strain in plastic deformation can be expressed in terms of six independent components:
ɛxx, ɛyy, ɛzz, ϒxy, ϒzx and ϒyz. Since plastic flow occurs virtually without a change in volume, i.e.,
ɛxx + ɛyy+ ɛzz = 0 …(6.47)
Thus, plastic deformation can be defined in terms of five independent strain components as only two of the three strain components of equation (6.47) need be known. The glide of dislocations which belong to a specific slip-system produces shear strain only in the glide direction on the slip plane.
ADVERTISEMENTS:
Consequently, five independent slip systems are required for general slip. The condition for a slip-system being independent of others is that its operation produces a change in shape that cannot be produced by a combination of slip on two or more of the remaining systems. This is known as Von-Mises criterion.
The multiple-slip therefore has to occur in every grain of the polycrystal if each grain is to remain coherent with its neighbours. Occurrence of multiple-slip is associated with prolific dislocation intersections and thus results in a high rate of strain-hardening. The stress-strain curve of polycrystal therefore differs greatly from the flow-stress curve of single crystals, where only one slip system is active.
However, when a single crystal is oriented in such a manner that multiple-slip is occurring right from the beginning, the single crystal curves tend to show some resemblance to polycrystalline curves (compare in Fig. 6.37). Exact equivalence in stress levels and slopes of the curves however is not possible because the individual grains of polycrystal are stressed in a rather irregular way, unlike uniform stress being applied to single crystals.
In polycrystal, each grain is found to deform unevenly due to the fact that it faces a different grain (with different orientation) across each boundary. Moreover, the grain size also limits the length of the slip lines or slip-bands. The sizes of the grains have been found to have a pronounced effect on the mechanical properties. The yield strength of polycrystalline materials usually exhibits the following relationship to grain size (Hall-Petch relationship).
σLYP = σi + kyd– ½ …(6.48)
where, σLYP is the lower yield stress, i.e., the stress required to propagate Luder band along the specimen, σi, and ky are constants, and d is the average grain diameter.
Ductility of Polycrystalline Materials:
The number of independent slip systems available in a number of metals are summarised in Table 6.8. In FCC crystals, the slip occurs on {111} <11̅0> and thus, there are 12 physically different slip systems. It can be seen that only five of them are independent. The four slip planes of FCC system are ABC, ABD, ACD, and BCD as illustrated in Fig. 6.38. Each has three slip directions lying in the plane. For example, consider slip plane ABC with indices (111). The slip along [1̅10] can also be produced by the combination of slip on (111) [101̅] and (111) [01̅1] systems.
a/2 [101̅] + a/2 [01̅1] → a/2 [11̅0] …(6.49)
Now, choose any five slip directions, and it can be easily shown that slip on all twelve systems can be represented by them. The number of independent slip-systems can also be calculated mathematically.
Thus, a general strain in FCC crystals can be produced by multiple-slip. Since the critical resolved shear stress required for slip is also low in FCC crystals, they exhibit extensive ductility in polycrystalline state.
In case of BCC metals too, the requisite number of independent slip systems are available and a general strain can be produced by multiple-slip.
For HCP crystals, when the slip is occurring in the basal plane, only two independent slip-systems are available. Even when an additional set of slip systems {101̅0} <112̅0> is operating, the number of slip-systems amounts to only four. In [101̅1} <112̅0>, there are four independent slip-systems, but the changes in shape that can be produced by {101̅1} <112̅0> slip can be shown to be exactly identical to those produced by the combination of (0001) <112̅0> and {101̅0} <112̅0> systems.
Thus, even when all these three systems are simultaneously operating, the effective number of independent slip-systems is only four. The condition for general slip is thus not fulfilled, and it is seen that the extension parallel to the hexagonal axis is not possible. In a few HCP crystals, such as cadmium, an additional pyramidal-mode of slip {112̅2} <112̅3> can also operate.
The combination of basal and pyramidal slip satisfies the requirement of five independent slip systems, and thus, renders polycrystalline cadmium very ductile at ambient temperatures. Other HCP poly-crystals exhibit semi-brittle behaviour under comparable conditions of testing.