Environmental Engineering Reference
In-Depth Information
where A is a constant and Q c is the activation energy corresponding to the
rate controlling mechanism. The activation energy indicated in Equation
[3.27] is the apparent activation energy. This is because the effect of tem-
perature on the elastic modulus is not included here and that could have
a substantial effect. 62 The true activation energy can be obtained from the
following equation
47
exp
Q
RT
σ
ε
A
[3.28 ]
=
.
E
Here E is temperature-dependent modulus of elasticity,
d
E
(
)
EE
TT
[3.29 ]
E
0
(
TT
.
T
d
T
The temperature-normalized stress (
σ
/ E ) term includes the effect of tem-
perature, and thus the value of
Q obtained from Equation [3.28] is the true
activation energy. The activation energy thus obtained has been found to be
equivalent to the lattice self-diffusion activation energy.
Q
Mechanisms of fi ve power-law creep
There are several models that have been proposed to describe the rate con-
trolling mechanism in the fi ve power-law creep regime. The general con-
sensus is that the fi ve power-law creep regime is diffusion-controlled. This
is evident from the equivalence between the activation energy for creep
and that for self diffusion. In addition, factors affecting self-diffusion such
as phase transformation, superimposed hydrostatic pressure, etc., simi-
larly infl uence creep-rates thereby rendering support to the fact that the
creep-rate is proportional to self-diffusivity ( D L ). Thus, all models that have
been proposed to explain the fi ve power-law creep regime are built around
the concept of a dislocation climb-controlled creep mechanism.
The earliest model to describe creep by dislocation climb was proposed
by Weertman 56, 63 who considered the creep processes to be a result of the
glide and climb of dislocations, with climb being the rate controlling process.
The glide motion of dislocations is impeded by long range stresses due to
dislocation interactions and the stresses are relieved by dislocation climb
and subsequent annihilation. The rate of dislocation climb is determined
by the concentration gradient existing between the equilibrium vacancy
concentration and the concentration in the region surrounding the climb-
ing dislocation. Creep strain, however, arises mainly through the glide of
dislocations. In the glide-climb model (Fig. 3.12) a dislocation produced by
the Frank-Read source glides a distance ' L ' until a barrier of height ' h ' is
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