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An analysis of low stress creep data for coarse-grained pure lead
Volume 8, number
5
MATERIALS
LETTERS
June 1989
AN ANALYSIS OF LOW STRESS CREEP DATA FOR COARSE-GRAINED R.S. MISHRA
PURE LEAD
I, H. JONES and G.W. GREENWOOD
Division of Metals, School of Materials, University of Sheffield, Sheffield Sl SJD, UK Received
15 February
1989; in final form 13 March
1989
An analysis has been carried out of low stress creep data for pure lead at 298 to 323 K and grain sizes from 200 to 500 pm, as compiled by Gifkins and Snowden. The results suggest pipe-diffusion-controlled Harper-Dom creep to be the dominant creep mechanism for the data considered. This finding is consistent with the theoretical predictions.
1. Introduction At low stress g and intermediate to high temperature T,viscous creep (i.e. strain rate &a) is often observed. Creep mechanisms which conform to such a stress exponent of unity are diffusional creep [ l31 and Harper-Dorn (HD) creep [ 4 1. Although these mechanisms have the same stress dependence, they differ both in their grain size dependence and in expected creep rate. Unlike diffusional creep, HD creep is thought to be grain size independent and is expected to dominate in coarse-grained materials. Until recently, it was believed that HD creep is significant only for very coarse-grained materials (typically with grain sizes d> 500 pm) and at temperatures close to the melting point, = 0.9T, [ 5 1. Within the last ten years, however, experimental evidence has accumulated to suggest that HD creep could be ’ Present address: Defence Metallurgical Hyderabad,
Table 1 Experimental Material
Research
reports of Harper-Dorn Temperature
creep at intermediate TIT,
(K)
P-CO[8 1
Laboratory.
India.
dominant in materials with grain sizes as small as 125 pm and at temperatures as low as 0.35T, [6]. HD creep rate can be written in the general form [ 7 ] hD =&dEblW
(Dp +a,pD,blE
,
(1)
where AH,,is a dimensionless constant, E is Young’s modulus, b is Burgers vector, k is Boltzmann’s constant, T is the absolute temperature, D, is the lattice diffusivity, u, is the area around a dislocation core participating in pipe diffusion, p is the dislocation density and D, is the pipe diffusivity. The reported observations of HD creep at intermediate temperatures are summarised in table 1. It is worth noting that the observed activation energy for HD creep is close to that for lattice self-diffusion for P-Co [ 8 ] and a-Ti [IO], whereas it is similar to that for pipe self-diffusion for a-Fe [ 9 ] and a-Zr [ 6 1. This difference has been reconciled by Chokshi [ 71 in relation to effective diffusivity, which is the second term in parentheses in eq. ( 1). Gifkins and Snowden [ 111 have compiled low
temperatures Stress
Grain size
(MPa)
t (pm)
Activation energy
1039-I 127 823-943
0.59-0.64 0.40-0.54
i 5.3 <3.5
252-319 134-478
a-Ti [IO]
956-1088
0.50-0.56
< 1.7
143-443
1.0x 1o-9
::
a-Zr [6]
773-1023
0.35-0.48
< 1.8
127-342
8.0x
Q,
a) Calculated
using A HD=A;1Ja,p
7.4X 10-10 2.4x lO_‘“’
QP
a-Fe [9]
10-9”’
[ 71, where A’,, is the reported constant in original reference. 187
Volume 8, number
5
MATERIALS
stress creep data for pure lead. These data were obtained in the temperature range 298 to 323 K (O.SOT,,,to 0.537’,,,) and for grain sizes from 200 to 500 pm. They noted that the observed creep rates were a few orders of magnitude faster than the predicted diffusional creep rates and proposed a grain boundary sliding mechanism to account for this. However, Raj and Ashby [ 12 ] have shown that diffusional flow accommodated by grain boundary sliding and grain boundary sliding accommodated by diffusional flow are essentially the same process. The purpose of this note is to reanalyse the viscous creep regime data compiled by Gifkins and Snowden [ 111 to seek for any evidence of HD creep in lead, as both the temperature and grain size ranges involved correspond with the conditions under which HD creep has been observed in other pure metals at intermediate temperatures (see table 1).
2. Analysis of viscous creep data for lead For the purpose of the present analysis, the linear strain rate versus stress regime data for commercial purity lead at 298 and 323 K have been considered. Fig. 1 shows ikTID,Eb plotted against a/E for commercial purity lead. Here the strain rate has been normalised using the pipe self-diffusion coefficient (D, value was taken from Frost and Ashby [ 13 ] ) .
LETTERS
June 1989
This choice is justified as it brings the data from both temperatures together. Linear regression of these data then gives ikT/D,Eb=
1.0x 10-‘2a/E)
T< (QP-Q,)lRln(Do,IDo,a,p),
0
n = 1.0
l
where R is the gas constant, QP and Q, are the activation energies for lattice and pipe self-diffusion, respectively, and Don and Dop are the corresponding frequency factors. The limiting grain size for the dominance of pipe-diffusion-controlled HD creep is given by
I. Normalised
1O-4
F /E
strain rate as a function of normalised stress u/E for pure lead from data compiled by Gifkins and Snowden
[ill. 188
(4)
0
10-171 Fig.
(3)
where Ace is a constant related to Coble creep [ 31 and Dg is grain boundary self-diffusivity. Using eqs. (3 ) and (4) we find that pipe-diffusion-controlled HD creep is expected to dominate below 397 K (0.66T,) and for grain sizes larger than 25 pm for pure lead with a dislocation density of 3 x 1012rnm2. This is consistent with the experimental conditions used for the study of lead and supports our conclusion that the data considered here represent HD creep controlled by pipe diffusion.
0 295-299 K l 323K
10-s
(2)
The constant 1.Ox 1O- ” in eq. (2) depends on the dislocation density of the material and in the pipediffusion-controlled HD creep regime is equal to AHDup~. Further, A,,=2pb2 [ 141, so we obtain 8p2b4= 1.0x lo-l2 consistent with a dislocation density of 3~10’~ mV2, taking b=3.49x10-‘” m [ 13 1. Since this value of the dislocation density is in accord with the expectation for recrystallised material it is concluded that the linear strain rate-stress regime in Gifkins and Snowden’s compilation can be accounted for in terms of HD creep. This conclusion is supported by predictions of the conditions for pipe diffusion to dominate in HD creep as shown below. Following Chokshi [ 71, the temperature below which pipe diffusion will control HD is given by
d> (b3ACoDelAHDappDp)“3,
/
.
3. Conclusions An analysis of low stress creep data for pure lead obtained at 298 to 323 K and for grain sizes from 200 to 500 pm (as compiled by Gifkins and Snowden
Volume 8, number
5
MATERIALS
[ 111) suggests that the dominant creep mechanism is Harper-Dorn creep controlled by pipe diffusion. This finding is consistent with the theoretical predictions.
References [ 1 ] F.R.N. Nabarro, Report on a Conference on the Strength of Solids (The Physical Society, London, 1948) p. 77. [ 21 C. Herring, J. Appl. Phys. 2 1 ( 1950) 437. (31 R.L. Coble, J. Appl. Phys. 34 (1963) 1679. [4] J. Harper and J.E. Dom, Acta Met. 5 (1957) 654.
LETTERS
June 1989
[ 51 T.G. Langdon and P. Yavari, Acta Met. 30 ( 1982) 88 1. [ 61 J. Novotny, J. Fiala and J. Cadek, Acta Met. 33 ( 1985) 905. [7] A.H. Chokshi, ScriptaMet. 19 (1985) 529. [8] G. Malakondaiah and P. Rama Rao, Mater. Sci. Eng. 52 (1982) 207. [9] J. Fiala, J. Novotny and J. Cadek, Mater. Sci. Eng. 60 ( 1983) 195. [lo] G. Malakondaiah and P. Rama Rao, Acta Met. 29 (1981) 1263. [ 111 R.C. Gitlcins and K.U. Snowden, Trans. Met. Sot. AIME 239 (1967) 910. [ 121 R. Raj and M.F. Ashby, Met. Trans. 2 ( 197 1) I 113. [ 131 H.J. Frost and M.F. Ashby, Deformation mechanism maps (Pergamon Press, Oxford, 1982) p. 2 1. [ 141 K.L. Murty, Mater. Sci. Eng. 14 ( 1974) 169.
189
5
MATERIALS
LETTERS
June 1989
AN ANALYSIS OF LOW STRESS CREEP DATA FOR COARSE-GRAINED R.S. MISHRA
PURE LEAD
I, H. JONES and G.W. GREENWOOD
Division of Metals, School of Materials, University of Sheffield, Sheffield Sl SJD, UK Received
15 February
1989; in final form 13 March
1989
An analysis has been carried out of low stress creep data for pure lead at 298 to 323 K and grain sizes from 200 to 500 pm, as compiled by Gifkins and Snowden. The results suggest pipe-diffusion-controlled Harper-Dom creep to be the dominant creep mechanism for the data considered. This finding is consistent with the theoretical predictions.
1. Introduction At low stress g and intermediate to high temperature T,viscous creep (i.e. strain rate &a) is often observed. Creep mechanisms which conform to such a stress exponent of unity are diffusional creep [ l31 and Harper-Dorn (HD) creep [ 4 1. Although these mechanisms have the same stress dependence, they differ both in their grain size dependence and in expected creep rate. Unlike diffusional creep, HD creep is thought to be grain size independent and is expected to dominate in coarse-grained materials. Until recently, it was believed that HD creep is significant only for very coarse-grained materials (typically with grain sizes d> 500 pm) and at temperatures close to the melting point, = 0.9T, [ 5 1. Within the last ten years, however, experimental evidence has accumulated to suggest that HD creep could be ’ Present address: Defence Metallurgical Hyderabad,
Table 1 Experimental Material
Research
reports of Harper-Dorn Temperature
creep at intermediate TIT,
(K)
P-CO[8 1
Laboratory.
India.
dominant in materials with grain sizes as small as 125 pm and at temperatures as low as 0.35T, [6]. HD creep rate can be written in the general form [ 7 ] hD =&dEblW
(Dp +a,pD,blE
,
(1)
where AH,,is a dimensionless constant, E is Young’s modulus, b is Burgers vector, k is Boltzmann’s constant, T is the absolute temperature, D, is the lattice diffusivity, u, is the area around a dislocation core participating in pipe diffusion, p is the dislocation density and D, is the pipe diffusivity. The reported observations of HD creep at intermediate temperatures are summarised in table 1. It is worth noting that the observed activation energy for HD creep is close to that for lattice self-diffusion for P-Co [ 8 ] and a-Ti [IO], whereas it is similar to that for pipe self-diffusion for a-Fe [ 9 ] and a-Zr [ 6 1. This difference has been reconciled by Chokshi [ 71 in relation to effective diffusivity, which is the second term in parentheses in eq. ( 1). Gifkins and Snowden [ 111 have compiled low
temperatures Stress
Grain size
(MPa)
t (pm)
Activation energy
1039-I 127 823-943
0.59-0.64 0.40-0.54
i 5.3 <3.5
252-319 134-478
a-Ti [IO]
956-1088
0.50-0.56
< 1.7
143-443
1.0x 1o-9
::
a-Zr [6]
773-1023
0.35-0.48
< 1.8
127-342
8.0x
Q,
a) Calculated
using A HD=A;1Ja,p
7.4X 10-10 2.4x lO_‘“’
QP
a-Fe [9]
10-9”’
[ 71, where A’,, is the reported constant in original reference. 187
Volume 8, number
5
MATERIALS
stress creep data for pure lead. These data were obtained in the temperature range 298 to 323 K (O.SOT,,,to 0.537’,,,) and for grain sizes from 200 to 500 pm. They noted that the observed creep rates were a few orders of magnitude faster than the predicted diffusional creep rates and proposed a grain boundary sliding mechanism to account for this. However, Raj and Ashby [ 12 ] have shown that diffusional flow accommodated by grain boundary sliding and grain boundary sliding accommodated by diffusional flow are essentially the same process. The purpose of this note is to reanalyse the viscous creep regime data compiled by Gifkins and Snowden [ 111 to seek for any evidence of HD creep in lead, as both the temperature and grain size ranges involved correspond with the conditions under which HD creep has been observed in other pure metals at intermediate temperatures (see table 1).
2. Analysis of viscous creep data for lead For the purpose of the present analysis, the linear strain rate versus stress regime data for commercial purity lead at 298 and 323 K have been considered. Fig. 1 shows ikTID,Eb plotted against a/E for commercial purity lead. Here the strain rate has been normalised using the pipe self-diffusion coefficient (D, value was taken from Frost and Ashby [ 13 ] ) .
LETTERS
June 1989
This choice is justified as it brings the data from both temperatures together. Linear regression of these data then gives ikT/D,Eb=
1.0x 10-‘2a/E)
T< (QP-Q,)lRln(Do,IDo,a,p),
0
n = 1.0
l
where R is the gas constant, QP and Q, are the activation energies for lattice and pipe self-diffusion, respectively, and Don and Dop are the corresponding frequency factors. The limiting grain size for the dominance of pipe-diffusion-controlled HD creep is given by
I. Normalised
1O-4
F /E
strain rate as a function of normalised stress u/E for pure lead from data compiled by Gifkins and Snowden
[ill. 188
(4)
0
10-171 Fig.
(3)
where Ace is a constant related to Coble creep [ 31 and Dg is grain boundary self-diffusivity. Using eqs. (3 ) and (4) we find that pipe-diffusion-controlled HD creep is expected to dominate below 397 K (0.66T,) and for grain sizes larger than 25 pm for pure lead with a dislocation density of 3 x 1012rnm2. This is consistent with the experimental conditions used for the study of lead and supports our conclusion that the data considered here represent HD creep controlled by pipe diffusion.
0 295-299 K l 323K
10-s
(2)
The constant 1.Ox 1O- ” in eq. (2) depends on the dislocation density of the material and in the pipediffusion-controlled HD creep regime is equal to AHDup~. Further, A,,=2pb2 [ 141, so we obtain 8p2b4= 1.0x lo-l2 consistent with a dislocation density of 3~10’~ mV2, taking b=3.49x10-‘” m [ 13 1. Since this value of the dislocation density is in accord with the expectation for recrystallised material it is concluded that the linear strain rate-stress regime in Gifkins and Snowden’s compilation can be accounted for in terms of HD creep. This conclusion is supported by predictions of the conditions for pipe diffusion to dominate in HD creep as shown below. Following Chokshi [ 71, the temperature below which pipe diffusion will control HD is given by
d> (b3ACoDelAHDappDp)“3,
/
.
3. Conclusions An analysis of low stress creep data for pure lead obtained at 298 to 323 K and for grain sizes from 200 to 500 pm (as compiled by Gifkins and Snowden
Volume 8, number
5
MATERIALS
[ 111) suggests that the dominant creep mechanism is Harper-Dorn creep controlled by pipe diffusion. This finding is consistent with the theoretical predictions.
References [ 1 ] F.R.N. Nabarro, Report on a Conference on the Strength of Solids (The Physical Society, London, 1948) p. 77. [ 21 C. Herring, J. Appl. Phys. 2 1 ( 1950) 437. (31 R.L. Coble, J. Appl. Phys. 34 (1963) 1679. [4] J. Harper and J.E. Dom, Acta Met. 5 (1957) 654.
LETTERS
June 1989
[ 51 T.G. Langdon and P. Yavari, Acta Met. 30 ( 1982) 88 1. [ 61 J. Novotny, J. Fiala and J. Cadek, Acta Met. 33 ( 1985) 905. [7] A.H. Chokshi, ScriptaMet. 19 (1985) 529. [8] G. Malakondaiah and P. Rama Rao, Mater. Sci. Eng. 52 (1982) 207. [9] J. Fiala, J. Novotny and J. Cadek, Mater. Sci. Eng. 60 ( 1983) 195. [lo] G. Malakondaiah and P. Rama Rao, Acta Met. 29 (1981) 1263. [ 111 R.C. Gitlcins and K.U. Snowden, Trans. Met. Sot. AIME 239 (1967) 910. [ 121 R. Raj and M.F. Ashby, Met. Trans. 2 ( 197 1) I 113. [ 131 H.J. Frost and M.F. Ashby, Deformation mechanism maps (Pergamon Press, Oxford, 1982) p. 2 1. [ 141 K.L. Murty, Mater. Sci. Eng. 14 ( 1974) 169.
189