Influence of grain boundaries on plasticity

6.4.11 Influence of grain boundaries on plasticity

It might be thought that when a stress is applied to a polycrystalline metal, every grain in the sample deforms as if it were an unconstrained single crystal. This is not the case, however, and the fact that the aggregate does not deform in this manner is indicated by the high yield stress of polycrystals compared with that of single crystals. This increased strength of polycrystals immediately poses the question – is the hardness of a grain caused by the presence of the grain boundary or by the orientation difference of the neighboring grains? It is now believed that the latter is the case but that the structure of the grain boundary itself may be of importance in special circumstances, such as when brittle films, due to bismuth in copper or cementite in steel, form around the grains or when the grains slip past each other along their boundaries during high-temperature creep. The importance of the orientation change across a grain boundary to the process of slip has been demonstrated by experiments on ‘bamboo’- type specimens, i.e. where the grain boundaries are parallel to each other and all perpendicular to the axis of tension. Initially, deformation occurs by slip only in those grains most favorably oriented, but later spreads to all the other grains as those grains which are deformed first work harden. It is then found that each grain contains wedge-shaped areas near the grain boundary, as shown in Figure 6.30a, where slip does not operate, which indicates that the continuance of slip from one grain to the next is difficult. From these observations it is natural to enquire what happens in a completely polycrystalline metal where the slip planes must in all cases make contact with a grain boundary. It will be clear that the polycrystalline aggregate must be stronger because, unlike the deformation of bamboo-type samples, where it is not necessary to raise the stress sufficiently high to operate those slip planes which made contact with a grain boundary, all the slip planes within any grain of a polycrystalline aggregate make contact with a grain boundary, but nevertheless have to be operated. The importance

322 Physical Metallurgy and Advanced Materials

Slip systems not operating

Grain boundary in these areas

(a) Slip band

dr Source

Grain boundary

(b)

Figure 6.30 (a) Grain-boundary blocking of slip. (b) Blocking of a slip band by a grain boundary. of the grain size on a strength is emphasized by Figure 6.29b, which shows the variation in lower

yield stress, σ y , with grain diameter, 2d, for low-carbon steel. The smaller the grain size, the higher the yield strength according to a relation of the form

σ y =σ i + kd −1/2 , (6.18) where σ i is a lattice friction stress and k a constant usually denoted k y to indicate yielding. Because

of the difficulties experienced by a dislocation in moving from one grain to another, the process of slip in a polycrystalline aggregate does not spread to each grain by forcing a dislocation through the boundary. Instead, the slip band which is held up at the boundary gives rise to a stress concentration at the head of the pile-up group of dislocations which acts with the applied stress and is sufficient to trigger off sources in neighboring grains. If τ i is the stress a slip band could sustain if there were no resistance to slip across the grain boundary, i.e. the friction stress, and τ the higher stress sustained by a slip band in a polycrystal, then (τ −τ i ) represents the resistance offered by the boundary, which reaches a limiting value when slip is induced in the next grain. The influence of grain size can be explained if the length of the slip band is proportional to d, as shown in Figure 6.30(b). Thus, since the stress concentration a short distance r from the end of the slip band is proportional to (d/4r) 1/2 , the maximum shear stress at a distance r ahead of a slip band carrying an applied stress τ in a polycrystal is given by (τ −τ i )[d/4r] 1/2 and lies in the plane of the slip band. If this maximum stress has to reach

a value τ max to operate a new source at a distance r, then (τ −τ i )[d/4r] 1/2 =τ max

or, rearranging, τ =τ i + (τ max 2r 1/2 )d −1/2 ,

which may be written as

τ =τ i +k s d −1/2 .

Mechanical properties I 323

0 5 10 d ⫺1/2 (mm)⫺1/2

Figure 6.31 Schematic diagram showing the grain-size dependence of the yield stress for crystals of different crystal structure.

It then follows that the tensile flow curve of a polycrystal is given by σ = m(τ i +k s d −1/2 ),

(6.19) where m is the orientation factor relating the applied tensile stress σ to the shear stress, i.e. σ = mτ. For

a single crystal the m-factor has a minimum value of 2 as discussed, but in polycrystals deformation occurs in less favorably oriented grains and sometimes (e.g. hexagonal, intermetallics, etc.) on ‘hard’ systems, and so the m-factor is significantly higher. From equation (6.18) it can be seen that σ i = mτ i and k = mk s .

While there is an orientation factor on a macroscopic scale in developing the critical shear stress within the various grains of a polycrystal, so there is a local orientation factor in operating a dislocation source ahead of a blocked slip band. The slip plane of the sources will not, in general, lie in the plane

of maximum shear stress, and hence τ max will need to be such that the shear stress, τ c , required to operate the new source must be generated in the slip plane of the source. In general, the local orientation factor dealing with the orientation relationship of adjacent grains will differ from the

macroscopic factor of slip plane orientation relative to the axis of stress, so that τ max 1 = ′ 2 m τ c . For simplicity, however, it will be assumed m ′ = m and hence the parameter k in the Petch equation is given by k

=m 2 τ c r 1/2 .

It is clear from the above treatment that the parameter k depends essentially on two main factors. The first is the stress to operate a source dislocation, and this depends on the extent to which the dislocations are anchored or locked by impurity atoms. Strong locking implies a large τ c and hence

a large k; the converse is true for weak locking. The second factor is contained in the parameter m, which depends on the number of available slip systems. A multiplicity of slip systems enhances the possibility for plastic deformation and so implies a small k. A limited number of slip systems available would imply a large value of k. It then follows, as shown in Figure 6.31, that for (1) fcc metals, which have weakly locked dislocations and a multiplicity of slip systems, k will generally be small, i.e. there is only a small grain size dependence of the flow stress, for (2) cph metals, k will

be large because of the limited slip systems, and for (3) bcc metals, because of the strong locking, k will be large.

Each grain does not deform as a single crystal in simple slip, since, if this were so, different grains would then deform in different directions with the result that voids would be created at the grain boundaries. Except in high-temperature creep, where the grains slide past each other along their

324 Physical Metallurgy and Advanced Materials boundaries, this does not happen and each grain deforms in coherence with its neighboring grains.

However, the fact that the continuity of the metal is maintained during plastic deformation must mean that each grain is deformed into a shape that is dictated by the deformation of its neighbors. Such behavior will, of course, require the operation of several slip systems, and von Mises has shown that to allow this unrestricted change of shape of a grain requires at least five independent shear modes. The deformation of metal crystals with cubic structure easily satisfies this condition so that the polycrystals of these metals usually exhibit considerable ductility, and the stress–strain curve generally lies close to that of single crystals of extreme orientations deforming under multiple slip conditions. The hexagonal metals do, however, show striking differences between their single-crystal and polycrystalline behavior. This is because single crystals of these metals deform by a process of basal plane slip, but the three shear systems (two independent) which operate do not provide enough independent shear mechanisms to allow unrestricted changes of shape in polycrystals. Consequently, to prevent gaps opening up at grain boundaries during the deformation of polycrystals, some additional shear mechanisms, such as non-basal slip and mechanical twinning, must operate. Hence, because the resolved stress for non-basal slip and twinning is greater than that for basal-plane slip, yielding in a polycrystal is prevented until the applied stress is high enough to deform by these mechanisms.