Deformation Mechanisms: Granular Flow - Materials Science for Structural Geology - page 230

Rutter ( 1983 ) has proposed a similar relationship but that of Pharr and Ashby
( 1983 ) has a more complicated stress dependence.

3. Accommodation by solution transfer along grain boundaries, reaction con-
trolled. In the notation of Sect. 5.6 the accommodation strain rate will be

e a ¼ C R V m bk
RT

ð r 1 r 3 Þ a

ð 7 : 15 Þ

d 0

where again d 0 ¼ C 3 d ; C 3 \1 as for ( 8.11 ), leading,through ( 8.9 ) and ( 8.10 ), to

r 1 r 3
d C 3 tan w C R V m bk

C 1 C 2

e ¼ ð 7 : 16 Þ

RT

4. Accommodation by dislocation glide. The absence of subgrain structure in su-
perplastically deforming materials suggests that any dislocation sources will tend
to be within the grain boundaries (the subgrain size being, in effect, greater than
the grainsize). Therefore, the density of dislocation sources and hence the mobile
dislocation density q may reasonably be assumed to be proportional to the grain
boundary area or the square of the grain size d. The mobile dislocation density
may, in addition, depend on the rate at which dislocations can be extracted from
the grain boundaries, as from a potential trough; that is, at low stresses ( Sect.
3.2.4 , Eq. 3.12d with F ¼ sbl Þ; q may also be postulated to be proportional to the
local stress r 1 r 3

ð Þ a , then the Orowan relation 6.4b leads to

ð r 1 r 3 Þ a

e a / ð r 1 r 3 Þ a

d 2

ð 7 : 17 Þ

Þ a ð r 1 r 3

exp Q
RT

or e a ¼ C 4

d 2

where Q is the sum of the activation energies for dislocation nucleation and
dislocation motion and C 4 is a constant of dimensions m 6 N -2 . If the motion were
limited by the rate of climb of the dislocations in the receiving grain boundary,
dissipating pile-ups there, then the second part of the activation energy could be
equal to that for grain boundary diffusion, making Q somewhat greater than for
grain boundary diffusion. However, there are obviously other possibilities and no
firm predictions about the level of the activation energy can be made at this stage.
Substituting ( 7.17 ) and ( 7.10 )in( 7.9 ) leads to the macroscopic flow law

r 1 r 3

Þ 2

e ¼ C 1 C 2

tan w C 4 exp Q

ð

ð 7 : 18 Þ

RT d 2

The exact value of the stress exponent in ( 7.18 ) is somewhat problematical
because of the tenuous nature of the assumptions made above about the stress