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SIPA irradiation creep in a material with a preferred direction for dislocation Burgers vectors
Journal of Nuclear Materials 79 (1979) 180-183 o North-Holland Publishing Company
SIPA IRRADIATION CREEP IN A MATERIAL WITH A PREFERRED DISLOCATION BURGERS VECTORS
DIRECTION
FOR
G.W. LEWTHWAITE UKAEA, Dounreay Nuclear Power Development
Establishment,
Thurso, Caithness, Scotland, UK
Received 23 June 1978
The effect of establishing a preferred Burgers vector orientation by cold-working, on the SIPA creep rate of an otherwise isotropic material is examined. A modest increase only in the SIPA creep rate parallel to the notional cold-working direction is predicted, followed by a decline to zero creep rate achieved when all Burgers vectors are parallel to the cold-working direction. Large values of the ratio of the creep rates parallel and transverse to the cold-working direction may be obtained from suitable dislocation distributions.
1. Introduction
tal [6] with their Burgers vectors parallel to the crystal axes, which are also the principal stress axes. These dislocations are the only sinks for point-defects in the crystal and mutual recombination of the vacancies and interstitials produced during irradiation is ignored. The preferred dislocation Burgers vector orientation is simulated by relaxing the assumption that pX = p,, = pz = ip (pi is the length of dislocation with Burgers vector parallel to the L-direction. p is the total dislocation density) usually used in SIPA creep calculations for isotropic materials. The dislocation distribution used is pX =Bp,py=$p,pz=$pwith8t@t$=1. The general expression for the strain rate of a crystal such as the one described is [6]:
In some circumstances otherwise isotropic metals are known to respond anisotropically to irradiation after cold-working [ 11. The effects observed by Buckley [l] included an induced radiation-growth parallel to the cold-working direction and different irradiation creep coefficients parallel and transverse to the coldworking direction (the creep rate of a rolled strip was up to ten times higher parallel to the rolling direction than in the transverse direction) and he suggested that the observed phenomena arose because a preferred dislocation Burgers vector orientation developed during cold-working. In this note the influence of such a preference on the stress-induced preferential absorption (SIPA) mechanism of irradiation creep [2-41 is discussed. Whether or not irradiation creep coefficients are dependent on stressing direction in a cold-worked structure is of some importance to the operation of liquid metal cooled fast breeder reactors and in this context has been studied experimentally by Kenfield et al. [5].
f,,
= (Z;, pxY + Zfrz~xz)Wi -
+ Z,v, pxz> Dv Cv .
(1)
E,, is the strain rate in the x-direction, pxy and px. are the legnths of dislocation with Burgers vectors parallel to the x-axis and dislocation lines parallel to they and z axes respectively. D, and C, are the pointdefect diffusivities and concentrations respectively (a = I for interstitials and (Y= V for vacancies). Z& and Z’$ are the dislocation sink parameters for point defects (the subscripts indicate that Z&, for example, is the sink parameter for the dislocations p,). From
2. Analysis The model analysed is a simple one - dislocations are “allowed” to adopt six orientations in a single crys180
G. W.Lewthwaite / SZA irradiationcreep
rate theory
[7]: Z$pii ,
DaCa = K/c
(i #i),
(2)
in which K is the displacement damage rate. To reveal more clearly results deemed to be significant to the present purpose and also to simplify the algebra Z$ are set equal to one another now rather than later (this is a simplification frequently, but notalways, adopted elsewhere [4,7]). Eq. (1) then simplifies to give: &X = (Z~JJPxy + ZLPxz)I~z~jPij
(3)
= u/Eep .
(4)
With the exception of f(u) and fii(v) the notation is the same as used previously [6]. Zb is the dislocation sink parameter for the stress-free situation and f(k*) and f@*) are functions of the elastic constants of the matrix and the point defects: f”(v) is a function of Poisson’s ratio v and the orientation of the dislocation relative to the applied stresses and values for f’i(v) for the six orientations used here have been given previously [6]. Using these values of f’i(v) gives: ZZy + ZL = Zb [2 - 2f(k‘)f(o)
+ 2fol*)f(o)(2
03
04
05
06
07
08
09
10
0
- 0 .
i.e. E,, is the strain per unit Here S,, =d(e,,)/d(Kt) displacement dose. The sink parameters are of the form [6,9]
f(o)
181
-
41 ,
Fig. 1. The ratio of the creep rate parallel to the cold-working direction G(P)) and the creep rate of an isotropic metal (t(I)) as a function of 8.
For metal with no preferred dislocation Burgers vector (hereafter referred to as an isotropic material), 8 = 4 = f, and these expressions give: t(1) = c,, = eyy = $Q*)
(2 - v)u/Eef’ ,
(8)
which is the strain rate derived previously [6] (@(I) is the tensile strain rate of an isotropic material). Suppose now that the cold-working direction is parallel to the x-axis and hence that the y-axis is transverse to this direction. Then, by an obvious extension of the notation used in C(I) we have: C(P) = 3,X 9 and C(T)= C,
.
and zbx + zbz = z;, -fol*>f(o)
+ z;, (2 -
To simulate cold-working
= z/J [2 - 2f(k’)f(u)
and the establishment
of a
41 .
With pii = pik, eq. (3) now becomes . exx = 11 -f(~‘)f(o)
+fol*M-(a)(2
- v)lO
x [l - f(k’)f(o) +foc*>f(o)(2
- v)@ - ;(G +
rl/w - 0 3 (5)
which, for f(u) << 1 and to first order in u/E, simplifies to: L exx = $3<1 - B)f@‘)f(u)(2 - v) . (6) If the tensile stress were applied parallel to the y-axis the corresponding expression for the strain rate (per dpa) would be: *
eyy = $#o
- dJ)fol*)f(~)(2
- v).
(7)
Fig. 2. The ratio of the creep rates parallel, P(P), and transverse, i(T), to the cold-working direction for B = 0.5 and 0.9 as a function of the parameter r.
182
G. W. Lewthwaite / SIA irradiation creep
preferred dislocation Burgers vector orientation, 6’> @= r$ with r < 1. Using these gives: @yv= ;W(l
W(1-w
+wf0l*M-(4(~ -4
set
(9)
and it is now possible to examine the ratios: C(P)/Z(I) = ;e< 1 - e) )
(10)
and O(P)/S(T) = e( 1 + #/r(l
+ er) ,
(11)
(both of which are equal to one for 0 = $J= 9 as required). <(P)/;(I) is shown in fig. 1 as a function of 0 and Z(P)/iZ(T) in fig. 2 as a function of r for 0 = i and e = 0.9.
3. Discussion Firstly, we see from eq. (10) and fig. 1 that the analysis predicts a shallow maximum in C(P) followed by a steady decline of G(P) to zero at 0 = 1. At the maximum (0 = 1) C(P)/P(I) = 1.125. This prediction contrasts sharply with Buckley‘s observations. He found, for example (see fig. 7 of [ 11) that the creep rate of 15% pre-strained y-uranium, during fissionfragment bombardment, was greater than twenty times the creep rate of annealed y-uranium. In real creep specimens such as Buckley’s the dislocation distributions will be more complicated than the simple one used here, but the model analysed is likely to exaggerate rather than moderate any effects and some other reason for the discrepancy appears to be required. Buckley, himself, suggested that a dislocation climbplus-glide mechanism of creep was required to understand the phenomena (the theory of SIPA creep postdates Buckley’s experiments). If such is the case then the analysis performed here does not refer to Buckley’s experiments and there is no conflict. It follows from this, of course, that SIPA is not the dominant mode of creep in these experiments and it remains to be established that SIPA is the dominant mode during irradiation in a fast-reactor. Alternatively some other essential feature could be missing from the analysis. One of the more obvious is mutual recombination which, strictly, is required in the above analysis otherwise the
derived creep rate has a finite value at zero dislocation density! Also the possible influence of vacancy clusters produced at the site of displacement cascades [lo] and acting as recombination centres [ 1 l] is not included. The consequences arising from additional features such as these, which are expected to be more potent in an annealed metal of low dislocation density, will be examined elsewhere. Accepting that the present analysis refers to circumstances in which mutual recombination can be ignored and network dislocations are the dominant sink for irradiation produced pointdefects it can be concluded that large increases in the SIPA creep rate will not be produced by the establishment of a preferred Burgers vector by cold working Whether or not creep rates could be depressed below the isotropic material value depends upon whether values of 0 in excess of about 0.66 could be realized in practice (to reduce the creep rate by a factor of 2, for example, requires 0 = 0.87). Further it is not evident that such high values of 0 would persist during irradiation for there is ample evidence [12,13] that the network-dislocation density changes during irradiation and so, perhaps, too would any initial anisotropy. A situation could be envisaged where the creep rate was initially depressed but increased during irradiation as, through reorganization of the dislocation network and the nucleation, growth and subsequent entanglement of interstitial loops, 0 decreased towards 4. The ratio P(P)/;(T) [eq. (11) and fig. 21 is a function of both 0 and the parameter r. If r is equal to 1, appropriate perhaps to a tensile pre-strain parallel to the x-axis, then C(P)/@(T) approaches 2 as 6’--f 1. At 0 = k ie when O(P) has its maximum value, P(P)/@(T) is 15. Again this contrasts with Buckley’s observations, as mentioned in the introduction values of P(P)/C(T) of the order of ten were observed. Values such as these can be obtained from the model by assuming that r < 1; for example, if 0 = 4 and r = O.O5,O(P)/S(T) = 11 (fig. 2). Buckley’s suggestion of the evolution of a preferred Burgers vector direction during cold-working (fig. 9 of [ 11) is not inconsistent with the requirement that r be less than 1, but direct evidence is not available. Kentield et al.‘s experiments, at low dose levels, revealed no difference in the creep behaviour parallel and transverse to the cold-working direction. Such behaviour is consistent with values of r close to 1, but again there is no direct information regarding the dislocation distribution.
G. W. Lewthwaite / SIA irradiation creep
183
4. Conclusions
References
The effect of the establishment of a preferred Burgers vector orientation by prior cold-working on the SIPA creep rate of an otherwise isotropic metal has been examined. FOI circumstances where mutual recombination of the irradiation-produced pointdefects can be ignored and where the network dislocations are the dominant sinks for the defects, a modest increase only in the SIPA creep rate parallel to the cold-working direction is obtained, followed by a decline to zero creep rate when all Burgers vectors are parallel to the cold-working direction. The ratio of the creep rates parallel and transverse to the cold working direction is sensitive to the dislocation distribution and large values can be obtained from suitable distributions.
[l] S.N. Buckley, UKAEA report AERE-R5944 (1968) p. 541. [2] P.T. Healdand M.V. Speight, Phil. Mag. 29 (1974) 1075. [3] W.G. Wolfer and M. Ashkin, J. Appl. Phys. 46 (1975) 541. [4] R. Bullough and M.R. Hayns, J. Nucl. Mat. 57 (1975) 348. [5] T.A. Kenfield, H.J. Busboom and W.K. Appleby, Trans. ANS 22 (1975) 185. (61 G.W. Letihwaite, Phil. Mag., in press. [I] A.D. Brailsford and R. Bullough, J. Nucl. Mat. 44 (1972) 121. [8] P.T. Heald and M.V. Speight, Acta. Met. 23 (1975) 1390. [9] R. Bullough and J.R. Willis, Phil. Mag. 31 (1975) 855. [lo] B.L. Eyre and C.A. English, UKAEA report AERERI934 (1974) p. 239. [ 1 l] P.T. Heald and M.V. Speight, J. Nucl. Mat. 64 (1977) 139. [ 12ltJ.I. Bramman, C. Brown, C. Cawthome, E.J. Fulton and G. Linekar, UKAEA report AERE-RI934 (1974) p. 71. [ 131H.R. Brager, F.A. Garner and G.L. Guthrie, J. Nucl. Mat. 66 (1977) 301.
SIPA IRRADIATION CREEP IN A MATERIAL WITH A PREFERRED DISLOCATION BURGERS VECTORS
DIRECTION
FOR
G.W. LEWTHWAITE UKAEA, Dounreay Nuclear Power Development
Establishment,
Thurso, Caithness, Scotland, UK
Received 23 June 1978
The effect of establishing a preferred Burgers vector orientation by cold-working, on the SIPA creep rate of an otherwise isotropic material is examined. A modest increase only in the SIPA creep rate parallel to the notional cold-working direction is predicted, followed by a decline to zero creep rate achieved when all Burgers vectors are parallel to the cold-working direction. Large values of the ratio of the creep rates parallel and transverse to the cold-working direction may be obtained from suitable dislocation distributions.
1. Introduction
tal [6] with their Burgers vectors parallel to the crystal axes, which are also the principal stress axes. These dislocations are the only sinks for point-defects in the crystal and mutual recombination of the vacancies and interstitials produced during irradiation is ignored. The preferred dislocation Burgers vector orientation is simulated by relaxing the assumption that pX = p,, = pz = ip (pi is the length of dislocation with Burgers vector parallel to the L-direction. p is the total dislocation density) usually used in SIPA creep calculations for isotropic materials. The dislocation distribution used is pX =Bp,py=$p,pz=$pwith8t@t$=1. The general expression for the strain rate of a crystal such as the one described is [6]:
In some circumstances otherwise isotropic metals are known to respond anisotropically to irradiation after cold-working [ 11. The effects observed by Buckley [l] included an induced radiation-growth parallel to the cold-working direction and different irradiation creep coefficients parallel and transverse to the coldworking direction (the creep rate of a rolled strip was up to ten times higher parallel to the rolling direction than in the transverse direction) and he suggested that the observed phenomena arose because a preferred dislocation Burgers vector orientation developed during cold-working. In this note the influence of such a preference on the stress-induced preferential absorption (SIPA) mechanism of irradiation creep [2-41 is discussed. Whether or not irradiation creep coefficients are dependent on stressing direction in a cold-worked structure is of some importance to the operation of liquid metal cooled fast breeder reactors and in this context has been studied experimentally by Kenfield et al. [5].
f,,
= (Z;, pxY + Zfrz~xz)Wi -
+ Z,v, pxz> Dv Cv .
(1)
E,, is the strain rate in the x-direction, pxy and px. are the legnths of dislocation with Burgers vectors parallel to the x-axis and dislocation lines parallel to they and z axes respectively. D, and C, are the pointdefect diffusivities and concentrations respectively (a = I for interstitials and (Y= V for vacancies). Z& and Z’$ are the dislocation sink parameters for point defects (the subscripts indicate that Z&, for example, is the sink parameter for the dislocations p,). From
2. Analysis The model analysed is a simple one - dislocations are “allowed” to adopt six orientations in a single crys180
G. W.Lewthwaite / SZA irradiationcreep
rate theory
[7]: Z$pii ,
DaCa = K/c
(i #i),
(2)
in which K is the displacement damage rate. To reveal more clearly results deemed to be significant to the present purpose and also to simplify the algebra Z$ are set equal to one another now rather than later (this is a simplification frequently, but notalways, adopted elsewhere [4,7]). Eq. (1) then simplifies to give: &X = (Z~JJPxy + ZLPxz)I~z~jPij
(3)
= u/Eep .
(4)
With the exception of f(u) and fii(v) the notation is the same as used previously [6]. Zb is the dislocation sink parameter for the stress-free situation and f(k*) and f@*) are functions of the elastic constants of the matrix and the point defects: f”(v) is a function of Poisson’s ratio v and the orientation of the dislocation relative to the applied stresses and values for f’i(v) for the six orientations used here have been given previously [6]. Using these values of f’i(v) gives: ZZy + ZL = Zb [2 - 2f(k‘)f(o)
+ 2fol*)f(o)(2
03
04
05
06
07
08
09
10
0
- 0 .
i.e. E,, is the strain per unit Here S,, =d(e,,)/d(Kt) displacement dose. The sink parameters are of the form [6,9]
f(o)
181
-
41 ,
Fig. 1. The ratio of the creep rate parallel to the cold-working direction G(P)) and the creep rate of an isotropic metal (t(I)) as a function of 8.
For metal with no preferred dislocation Burgers vector (hereafter referred to as an isotropic material), 8 = 4 = f, and these expressions give: t(1) = c,, = eyy = $Q*)
(2 - v)u/Eef’ ,
(8)
which is the strain rate derived previously [6] (@(I) is the tensile strain rate of an isotropic material). Suppose now that the cold-working direction is parallel to the x-axis and hence that the y-axis is transverse to this direction. Then, by an obvious extension of the notation used in C(I) we have: C(P) = 3,X 9 and C(T)= C,
.
and zbx + zbz = z;, -fol*>f(o)
+ z;, (2 -
To simulate cold-working
= z/J [2 - 2f(k’)f(u)
and the establishment
of a
41 .
With pii = pik, eq. (3) now becomes . exx = 11 -f(~‘)f(o)
+fol*M-(a)(2
- v)lO
x [l - f(k’)f(o) +foc*>f(o)(2
- v)@ - ;(G +
rl/w - 0 3 (5)
which, for f(u) << 1 and to first order in u/E, simplifies to: L exx = $3<1 - B)f@‘)f(u)(2 - v) . (6) If the tensile stress were applied parallel to the y-axis the corresponding expression for the strain rate (per dpa) would be: *
eyy = $#o
- dJ)fol*)f(~)(2
- v).
(7)
Fig. 2. The ratio of the creep rates parallel, P(P), and transverse, i(T), to the cold-working direction for B = 0.5 and 0.9 as a function of the parameter r.
182
G. W. Lewthwaite / SIA irradiation creep
preferred dislocation Burgers vector orientation, 6’> @= r$ with r < 1. Using these gives: @yv= ;W(l
W(1-w
+wf0l*M-(4(~ -4
set
(9)
and it is now possible to examine the ratios: C(P)/Z(I) = ;e< 1 - e) )
(10)
and O(P)/S(T) = e( 1 + #/r(l
+ er) ,
(11)
(both of which are equal to one for 0 = $J= 9 as required). <(P)/;(I) is shown in fig. 1 as a function of 0 and Z(P)/iZ(T) in fig. 2 as a function of r for 0 = i and e = 0.9.
3. Discussion Firstly, we see from eq. (10) and fig. 1 that the analysis predicts a shallow maximum in C(P) followed by a steady decline of G(P) to zero at 0 = 1. At the maximum (0 = 1) C(P)/P(I) = 1.125. This prediction contrasts sharply with Buckley‘s observations. He found, for example (see fig. 7 of [ 11) that the creep rate of 15% pre-strained y-uranium, during fissionfragment bombardment, was greater than twenty times the creep rate of annealed y-uranium. In real creep specimens such as Buckley’s the dislocation distributions will be more complicated than the simple one used here, but the model analysed is likely to exaggerate rather than moderate any effects and some other reason for the discrepancy appears to be required. Buckley, himself, suggested that a dislocation climbplus-glide mechanism of creep was required to understand the phenomena (the theory of SIPA creep postdates Buckley’s experiments). If such is the case then the analysis performed here does not refer to Buckley’s experiments and there is no conflict. It follows from this, of course, that SIPA is not the dominant mode of creep in these experiments and it remains to be established that SIPA is the dominant mode during irradiation in a fast-reactor. Alternatively some other essential feature could be missing from the analysis. One of the more obvious is mutual recombination which, strictly, is required in the above analysis otherwise the
derived creep rate has a finite value at zero dislocation density! Also the possible influence of vacancy clusters produced at the site of displacement cascades [lo] and acting as recombination centres [ 1 l] is not included. The consequences arising from additional features such as these, which are expected to be more potent in an annealed metal of low dislocation density, will be examined elsewhere. Accepting that the present analysis refers to circumstances in which mutual recombination can be ignored and network dislocations are the dominant sink for irradiation produced pointdefects it can be concluded that large increases in the SIPA creep rate will not be produced by the establishment of a preferred Burgers vector by cold working Whether or not creep rates could be depressed below the isotropic material value depends upon whether values of 0 in excess of about 0.66 could be realized in practice (to reduce the creep rate by a factor of 2, for example, requires 0 = 0.87). Further it is not evident that such high values of 0 would persist during irradiation for there is ample evidence [12,13] that the network-dislocation density changes during irradiation and so, perhaps, too would any initial anisotropy. A situation could be envisaged where the creep rate was initially depressed but increased during irradiation as, through reorganization of the dislocation network and the nucleation, growth and subsequent entanglement of interstitial loops, 0 decreased towards 4. The ratio P(P)/;(T) [eq. (11) and fig. 21 is a function of both 0 and the parameter r. If r is equal to 1, appropriate perhaps to a tensile pre-strain parallel to the x-axis, then C(P)/@(T) approaches 2 as 6’--f 1. At 0 = k ie when O(P) has its maximum value, P(P)/@(T) is 15. Again this contrasts with Buckley’s observations, as mentioned in the introduction values of P(P)/C(T) of the order of ten were observed. Values such as these can be obtained from the model by assuming that r < 1; for example, if 0 = 4 and r = O.O5,O(P)/S(T) = 11 (fig. 2). Buckley’s suggestion of the evolution of a preferred Burgers vector direction during cold-working (fig. 9 of [ 11) is not inconsistent with the requirement that r be less than 1, but direct evidence is not available. Kentield et al.‘s experiments, at low dose levels, revealed no difference in the creep behaviour parallel and transverse to the cold-working direction. Such behaviour is consistent with values of r close to 1, but again there is no direct information regarding the dislocation distribution.
G. W. Lewthwaite / SIA irradiation creep
183
4. Conclusions
References
The effect of the establishment of a preferred Burgers vector orientation by prior cold-working on the SIPA creep rate of an otherwise isotropic metal has been examined. FOI circumstances where mutual recombination of the irradiation-produced pointdefects can be ignored and where the network dislocations are the dominant sinks for the defects, a modest increase only in the SIPA creep rate parallel to the cold-working direction is obtained, followed by a decline to zero creep rate when all Burgers vectors are parallel to the cold-working direction. The ratio of the creep rates parallel and transverse to the cold working direction is sensitive to the dislocation distribution and large values can be obtained from suitable distributions.
[l] S.N. Buckley, UKAEA report AERE-R5944 (1968) p. 541. [2] P.T. Healdand M.V. Speight, Phil. Mag. 29 (1974) 1075. [3] W.G. Wolfer and M. Ashkin, J. Appl. Phys. 46 (1975) 541. [4] R. Bullough and M.R. Hayns, J. Nucl. Mat. 57 (1975) 348. [5] T.A. Kenfield, H.J. Busboom and W.K. Appleby, Trans. ANS 22 (1975) 185. (61 G.W. Letihwaite, Phil. Mag., in press. [I] A.D. Brailsford and R. Bullough, J. Nucl. Mat. 44 (1972) 121. [8] P.T. Heald and M.V. Speight, Acta. Met. 23 (1975) 1390. [9] R. Bullough and J.R. Willis, Phil. Mag. 31 (1975) 855. [lo] B.L. Eyre and C.A. English, UKAEA report AERERI934 (1974) p. 239. [ 1 l] P.T. Heald and M.V. Speight, J. Nucl. Mat. 64 (1977) 139. [ 12ltJ.I. Bramman, C. Brown, C. Cawthome, E.J. Fulton and G. Linekar, UKAEA report AERE-RI934 (1974) p. 71. [ 131H.R. Brager, F.A. Garner and G.L. Guthrie, J. Nucl. Mat. 66 (1977) 301.