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Reply to the comments of novak and truchly on the growth of gas bubbles in a stressed medium and the application to stress enhanced swelling in alpha uranium
JOURNAL
REPLY GAS
OF NUCLEAR
TO
THE
BUBBLES
MATERIALS
32 (1969)
COMMENTS IN
A
OF
STRESSED
ENHANCED
0. NORTH-HOLLAND
363-364
NOVAK
AND
MEDIUM
TRUCHLY
AND
SWELLING
THE
IN ALPHA
PUBLISHING
ON
THE
APPLICATION
CO., AMSTERDAM
GROWTH TO
OF
STRESS
URANIUM
J. W. HARRISON UKAEA,
Metallurgy
D&&ion,
AERE,
Harwell,
Berkshire,
UK
Received 5 May 1969
The author accepts the comments of Novak and Truchly (see p. 362 of this issue) that Blackburn’s 1) expression gives an estimate of shear stresses in uranium metal arising from growth processes. However the conclusions reached in the original paper 2) still remain valid, for the following reasons. Consider the stress within a grain to be pure shear, an assumption which is physically sensible since the stress arises from a growth shear process. Owing to the impingeing neighbouring grains, surface tractions are created on the grain surfaces. We will assume that the tangential stresses are relaxed at the grain boundaries and consider only the normal tractions 334). These may be calculated by considering what distribution of normal tractions on the grain surface (assumed to be spherical) will give rise to. a state of pure shear stress within a grain. A calculation due to Herring 5) shows that in fact these normal tractions range from positive to negative over the spherical grain surface and are of the same magnitude as the volume average of the shear stresses within the grain. Only the positive tractions will lead to bubble growth and moreover these growing bubbles will lie on grain boundaries or other reasonably large angle boundaries. Now in the original paper calculations were made of bubble swelling in uranium metal and compared with Bellamy’s 6) experimental results. The initial bubble radius was taken from metallographical evidence but implicit in
the assumptions was the fact that bubble densities were of the order of lOis/cms. This indeed is of the same order of magnitude of the density of grain boundary bubbles to be expected if one uses the recent measurements of Hudson 7) of the total number of bubbles present in uranium metal. Two further points may be made. Firstly, from the remarks made above about the sign of the normal tractions created at the grain surfaces it is evident that there will be regions of the crystalline mass where over quite small distances there will be a sign change in the stress. This means that bubble growth will take place only on boundaries (including certain twin boundaries) oriented to give positive tractions and none on neighbouring boundaries where the stress is negative and suggests that there could be local large variations in bubble growth rates. There is certainly abundant metallographical evidence of this 69899) and this could also be the reason for the observation frequently observed. of aligned porosity Secondly, although we have assumed above that tangential grain surface stresses are relaxed this will only occur if the temperature is high enough. At Iow enough temperatures, when tangential stresses are not relaxed, pronounced tearing or grain distortion will occur. Again this is observed metallographically in uranium at temperatures of about 300 “C, the change to distortion free swelling occurring above this temperature at about 350 “C lo), where it is 363
364
J.
known
that
presumably
recovery stress
processes
relaxation
begin occurs
W.
HARRISON
and more
readily. A final remark may be made about the effect
References 1) W. S. Blackburn, Phil. Mag. 6 (1961) 503 2) J. W. Harrison, J. Nucl. Mat. 23 (1967) 139
3,
is
the
author’s
opinion
that
the
S. Ke,
Phys.
Rev.
19 (1948)
285;
71 (1947) 20 (1949)
533;
J. Appl.
274
King, Cahn and Chalmers, Nature 161 (1948) 652 C. Herring,
swelling
behaviour of pure uranium is worse than impure material because of recrystallisation processes sweeping bubbles together and on to twin and grain boundaries and producing a situation where a much larger fraction of bubbles reside on boundaries and hence suffer accelerated growth due to boundary stresses.
T.
Phys.
of added impurities. It is well known that quite small impurity additions markedly effect recrystallisation processes in uranium metal. It
J. Appl.
Phys.
21 (1950)
R. G. Bellamy,
J. E. Bainbridge
Jones, AERE-R
4427
B. Hudson,
J. Nucl.
437
and D. V. C.
(1963) Mat.
22 (1967)
121
7) 9
R. G. Bellamy, Uranium and Graphite Symposium
9)
C. L. Angerman
(London,
1962)
13 (1964)
lo)
S.
T.
and G. R. Caskey, J. Nucl. Mat.
182
Konobeevsky
Conference
(1958)
et
6/P/2230
al.,
Second
Geneva
REPLY GAS
OF NUCLEAR
TO
THE
BUBBLES
MATERIALS
32 (1969)
COMMENTS IN
A
OF
STRESSED
ENHANCED
0. NORTH-HOLLAND
363-364
NOVAK
AND
MEDIUM
TRUCHLY
AND
SWELLING
THE
IN ALPHA
PUBLISHING
ON
THE
APPLICATION
CO., AMSTERDAM
GROWTH TO
OF
STRESS
URANIUM
J. W. HARRISON UKAEA,
Metallurgy
D&&ion,
AERE,
Harwell,
Berkshire,
UK
Received 5 May 1969
The author accepts the comments of Novak and Truchly (see p. 362 of this issue) that Blackburn’s 1) expression gives an estimate of shear stresses in uranium metal arising from growth processes. However the conclusions reached in the original paper 2) still remain valid, for the following reasons. Consider the stress within a grain to be pure shear, an assumption which is physically sensible since the stress arises from a growth shear process. Owing to the impingeing neighbouring grains, surface tractions are created on the grain surfaces. We will assume that the tangential stresses are relaxed at the grain boundaries and consider only the normal tractions 334). These may be calculated by considering what distribution of normal tractions on the grain surface (assumed to be spherical) will give rise to. a state of pure shear stress within a grain. A calculation due to Herring 5) shows that in fact these normal tractions range from positive to negative over the spherical grain surface and are of the same magnitude as the volume average of the shear stresses within the grain. Only the positive tractions will lead to bubble growth and moreover these growing bubbles will lie on grain boundaries or other reasonably large angle boundaries. Now in the original paper calculations were made of bubble swelling in uranium metal and compared with Bellamy’s 6) experimental results. The initial bubble radius was taken from metallographical evidence but implicit in
the assumptions was the fact that bubble densities were of the order of lOis/cms. This indeed is of the same order of magnitude of the density of grain boundary bubbles to be expected if one uses the recent measurements of Hudson 7) of the total number of bubbles present in uranium metal. Two further points may be made. Firstly, from the remarks made above about the sign of the normal tractions created at the grain surfaces it is evident that there will be regions of the crystalline mass where over quite small distances there will be a sign change in the stress. This means that bubble growth will take place only on boundaries (including certain twin boundaries) oriented to give positive tractions and none on neighbouring boundaries where the stress is negative and suggests that there could be local large variations in bubble growth rates. There is certainly abundant metallographical evidence of this 69899) and this could also be the reason for the observation frequently observed. of aligned porosity Secondly, although we have assumed above that tangential grain surface stresses are relaxed this will only occur if the temperature is high enough. At Iow enough temperatures, when tangential stresses are not relaxed, pronounced tearing or grain distortion will occur. Again this is observed metallographically in uranium at temperatures of about 300 “C, the change to distortion free swelling occurring above this temperature at about 350 “C lo), where it is 363
364
J.
known
that
presumably
recovery stress
processes
relaxation
begin occurs
W.
HARRISON
and more
readily. A final remark may be made about the effect
References 1) W. S. Blackburn, Phil. Mag. 6 (1961) 503 2) J. W. Harrison, J. Nucl. Mat. 23 (1967) 139
3,
is
the
author’s
opinion
that
the
S. Ke,
Phys.
Rev.
19 (1948)
285;
71 (1947) 20 (1949)
533;
J. Appl.
274
King, Cahn and Chalmers, Nature 161 (1948) 652 C. Herring,
swelling
behaviour of pure uranium is worse than impure material because of recrystallisation processes sweeping bubbles together and on to twin and grain boundaries and producing a situation where a much larger fraction of bubbles reside on boundaries and hence suffer accelerated growth due to boundary stresses.
T.
Phys.
of added impurities. It is well known that quite small impurity additions markedly effect recrystallisation processes in uranium metal. It
J. Appl.
Phys.
21 (1950)
R. G. Bellamy,
J. E. Bainbridge
Jones, AERE-R
4427
B. Hudson,
J. Nucl.
437
and D. V. C.
(1963) Mat.
22 (1967)
121
7) 9
R. G. Bellamy, Uranium and Graphite Symposium
9)
C. L. Angerman
(London,
1962)
13 (1964)
lo)
S.
T.
and G. R. Caskey, J. Nucl. Mat.
182
Konobeevsky
Conference
(1958)
et
6/P/2230
al.,
Second
Geneva