Grain boundary contribution to creep

6.9.2 Grain boundary contribution to creep

In the creep of polycrystals at high temperatures, the grain boundaries themselves are able to play an important part in the deformation process due to the fact that they may (1) slide past each other or (2) create vacancies. Both processes involve an activation energy for diffusion and therefore may contribute to steady-state creep.

Grain boundary sliding during creep was inferred initially from the observation of steps at the boundaries, but the mechanism of sliding can be demonstrated on bi-crystals. Figure 6.64 shows a good example of grain boundary movement in a bi-crystal of tin, where the displacement of the straight grain boundary across its middle is indicated by marker scratches. Grain boundaries, even when specially produced for bi-crystal experiments, are not perfectly straight, and after a small amount of sliding at the boundary interface, movement will be arrested by protuberances. The grains are then locked, and the rate of slip will be determined by the rate of plastic flow in the protuberances. As a result, the rate

366 Physical Metallurgy and Advanced Materials

Figure 6.64 Grain boundary sliding on a bi-crystal tin (after Puttick and King, 1952; courtesy of Institute of Materials, Minerals and Mining).

of slip along a grain boundary is not constant with time, because the dislocations first form into piled- up groups, and later these become relaxed. Local relaxation may be envisaged as a process in which the dislocations in the pile-up climb towards the boundary. In consequence, the activation energy for grain boundary slip may be identified with that for steady-state creep. After climb, the dislocations are spread more evenly along the boundary, and are thus able to give rise to grain boundary migration, when sliding has temporarily ceased, which is proportional to the overall deformation.

A second creep process which also involves the grain boundaries is one in which the boundary acts as a source and sink for vacancies. The mechanism depends on the migration of vacancies from one side of a grain to another, as shown in Figure 6.65, and is often termed Herring–Nabarro creep, after the two workers who originally considered this process. If, in a grain of sides d under a stress σ , the atoms are transported from faces BC and AD to the faces AB and DC the grain creeps in the direction of the stress. To transport atoms in this way involves creating vacancies on the tensile faces

AB and DC and destroying them on the other compressive faces by diffusion along the paths shown. On a tensile face AB the stress exerts a force σb 2 (or σ 2/3 ) on each surface atom and so does work σb 2

× b each time an atom moves forward one atomic spacing b (or 1/3 ) to create a vacancy. The energy of vacancy formation at such a face is thus reduced to

(E f − σb 3 ) and the concentration of vacancies in equilibrium correspondingly increased to

c τ = exp[(−E f + σb 3 )/kT ] =c 0 exp(σb 3 /kT ). The vacancy concentration on the compressive faces

−σb 3 /kT ). Vacancies will therefore flow down the concentration gra- dient, and the number crossing a face under tension to one under compression will be given by Fick’s

will be reduced to c c =c 0 exp(

law as

φ = −D v d 2 (c T −c c )/αd,

where D v is the vacancy diffusivity and α relates to the diffusion length. Substituting for c T ,c c and

D = (D v c 0 b 3 ) leads to φ

= 2dD sinh(σb 3 /kT )/αb 3 .

Each vacancy created on one face and annihilated on the other produces a strain ε =b 3 / d 3 , so that the creep strain rate ˙ε = φ(b 3 / d 3 ). At high temperatures and low stresses this reduces to

(6.56) where the constant B

H = 2Dσb /α d −N 2 bf kT =B H −N Dσω/d kT ,

H –N ∼ 10.

Mechanical properties I 367 s

Vacancy conc c T

d Vacancy conc c C

(a)

(b) Figure 6.65 Schematic representation of Herring–Nabarro creep; with c T > c c vacancies flow

from the tensile faces to the longitudinal faces (a) to produce creep, as shown in (b).

In contrast to dislocation creep, Herring–Nabarro creep varies linearly with stress and occurs at T ≈ 0.8T m with σ

≈ 10 6 Nm −2 . The temperature range over which vacancy–diffusion creep is significant can be extended to much lower temperatures (i.e. T ≈ 0.5T m ) if the vacancies flow down

the grain boundaries rather than through the grains. Equation (6.56) is then modified for Coble or grain boundary diffusion creep, and is given by

(6.57) where ω is the width of the grain boundary. Under such conditions (i.e. T ≈ 0.5–0.6T m and low

˙ε Coble =B c D gb σ ω/kTd 3 ,

stresses), diffusion creep becomes an important creep mechanism in a number of high-technology situations, and has been clearly identified in magnesium-based canning materials used in gas-cooled reactors.