Interaction of dislocations with…
Case 1: Point defects
We’ll start with a vacancy, then generalize. Recapping:
u
r
= C/r
2
; C needs to be determined some other fashion.
u
θ
= u
φ
= 0
e
rr
= du
r
/dr = 2C/r
3
e
θθ
= e
φφ
= u
r
/r = C/r
3
∆
= 0
σ
rr
= 2
μ
C/r
3
= 2
σ
θθ
= 2
σ
φφ
Note that
∆
= 0, i.e. there is no volume change for a vacancy. This seems like a
contradiction since a vacancy “logically” has less volume. We have to be a little careful
about this.
Looking again at a vacancy, consider that we create it by cutting out material inside some
radius R to create an atomic size void. Assuming that the atoms want to come closer
together (reverse signs for the other way) we model this as equivalent to some negative
pressure –P. Similar to the precipitate problem, the void will contract until
σ
rr
matches
this pressure, i.e.
σ
rr
–P = 0 or
P = 2
μ
C/R
3
…. Which we can solve for C if P is known.
By doing this we have reduced the radius from R to R(1+u
r
), i.e. the outer wall has
moved in by R*u
r
.
Suppose next that the vacancy sits in a region with some hydrostatic pressure P
H
, where
P
H
is negative for compression, positive for expansion. I will assume that the vacancy is
small enough that we don’t need to be concerned about variations of the hydrostatic
pressure across it, i.e. it is essentially a point. There will be an extra work term for
moving the outer wall in by R*u
r
of
W =  R*u
r
*P
H
(Sign sanity check; u
r
is negative, contraction with our origin in the center of the vacancy
so an expansion has a positive work.) We can lump R*u
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 Winter '10
 MatSci
 Force, Crystallographic defect, interstitials, Pierls Stess

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