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Incorporation of peierls stress into deformation mechanism maps
ScriptaMetallur~ica.&I&t&h,
Vol. 33. No. 4, pp. 633-638.1995 Else-via Science Ltd copyli@o 1995 Ada tdeahgica Inc. FbtSdUItheUSAAllIi$ltSd 095671ccl95 S9.50 + .Gu
Peqpmon 0956-716X(95)00230-8
lNCORPORATION OF PEIERLS STRESS INTO DEFORMATION MECHANISM MAPS Jian N. Wang and T. G. Nieh Chemistry & Materials Science, L-370, Lawrence Livermore National Laboratory, P-0. Box 808, Liver-more, CA 94550, USA (Received March 1,1995) (Revised March 28,1995) Jntroduction Crystalline materials deform plastically by a number of alternative, often competing, mechanisms. For engineermg and geological applications, it is essential to know the conditions under which a specific mechanism is dominant over others. Deformation mechanism maps offer such information in a compact, diagrammatic form [l]. Existing deformation maps usually include me&anisms such as power law (I-L) dislocation creep aud diffusionalcreep. Harper-Dorn (H-D) creep [2] which is a Newtonian dislocation creep, and grain boundary sliding creep were scarcely considered because of unavailability of experimental data (e.g. [3-51). For dislocation creep, in a manner similar to diflbsional creep, the rate equation is also determined from experimental data. Consequently, existing mechanism maps were mainly constructed for specific materials under specific conditions. Recently, the knowledge of P-L creep and H-D creep has been greatly expanded, in particular, the transition t?om El-L creep to H-D creep. It is now possible to predict reasonably these two hinds of creep, and therefbre to conshuct a generalized deformation mechanism map, independent of materials. The objective of this paper is to construct such generalized mechanism maps. Transitions Between Creeo Regimes
Semi-theoretical rate equations for H-D creep and P-L creep have now become available. Since most experiments on metals, ceramics and silicates show that H-D creep has an activation energy equal to that for lattice self-diffusion, the general rate equation for this creep is of the form
(1) deformation, eij, given by: Ed= a$T, where ati is the second rank deformation, Ed,given by: Em= a$T, where ag is the second rank where b is the uniaxial strain rate in steady state, o is the uniaxial stress, G is the shear modulus, b is the length of the Burgers vector, k is Boltzmann’s constant, T is the absolute temperature, DL is the lattice difksion coe.tIicientof the rate controlling species, aud A, is a dimensionless constant having values from 1O-” to IO”. With the experimentally observed relationship between the dislocation density and the Peierls stress of the crystal, rp [6,7], A, in equation (1) for H-D creep controlled by dislocation climb under saturated conditions was derived as [8] 633
634
DEFORMATIONMECHANISMMAPS
Vol. 33, No. 4
1.4 x . (T~/G)~ AID=
(2)
-In (T*/G)
It has been shown that the strain rate in H-D creep predicted by equations (1) and (2) is within one order of magnitude of the experimentally measured strain rates for many crystalline materials [8]. The range of ZIG ofinterestisbetween4x1O6and5x1O”,whichcorrespondsto-In(rdG)=12.4-5.3.Taking-In(~dc> 5 8.9, A,, may be approximated by 2
(3) P-L dislocation creep, on the other hand, can be described by the Dom equation [9]:
(4)
where n is the stress exponrzt which genera@ varies fkom - 3 to 5, and A, is a dimensionless constant which ranges from unity to as large as 1016.Existing data indicate that the transition from P-L creep to H-D creep occurs at a stress u = fi zp [IO, 111. Based on this observation and the rate equation for H-D creep, a relationship between A, and n in equation (4) is obtained [ 121:
A PL =
(5)
which can be well approximated by 0-n)
I
=P APL
=
Oml
l
t z-
(6)
This relationship agrees very well with experimental data [ 121. Diffusional creep involves mass transport either through crystal lattice - Nabarro-Hening (N-H) creep [ 13,141 or along grain boundary - Coble creep [ 151. Over the temperature range where dislocation creep is lattice-dSkon controlled at higher stresses or larger grain sizes, N-H creep occurring at lower stresses and smaller grain sizes is dominant over Coble creep. The rate equation for N-H creep is [ 13,141 (7) where 0 is the atomic volume, d is the grain size, and A, is a dimensionless geometric constant. Taking ANH = 40 [ 161 and D = 0.7 b3, equation (7) can be rewritten as ie - 28 .!!$($)2.(;)
(8)
Vol. 33, No. 4
635
DEFORMATIONMECHANISMMAPS
The condition marking the transition between one mechanism and another can be determined as follows: for that between P-L creep and H-D creep [ 10, 111,
for that between H-D creep and N-H creep [ 171,
for that between P-L creep and N-H creep, o -E G
% G
16.7*
!-I 0b
6.51($[$*
when
n-3
(11)
when
n-4
(12)
(13) The transitions between H-D creep and P-L creep and betweem H-D creep and N-H creep expressed by equations (9) and (10) have been verified by experimental data [ 10, 1 1,171. Results and Discussion In principle, deformation mechanism maps can be constructed in a three dimensional space (u/G, d/b and
r/G>. To make a clear illustration, however, the maps are now constructed in a two dimensional space (cr/G and d/b) for d&ent values of rJG. Figures 1,2 and 3 are maps for the cases of n = 3,4 and 5, respectively. In the figures the creep regime boundaries are determined at rJG = 1OS’,106, 105, 1O-‘,1Om3, 10” and 10-l. Several key observations can be made from these maps:
1) The boundaries between H-D creep and others are such al&&d by changing r/G that this creep regime expands greatly with increasing riG. This observation indicates that H-D creep is rate-controlling in materials with high rr/G over wide ranges of conditions of o/G and d/b, as already concluded before [7, 8,131. However, the boundary between P-L creep and N-H creep is insignificantly affected by changing r/G. The magnitude of the stress exponent n only slightly intluences the boundary between P-L creep and N-H 2) creep. The&ore, n is not an import& factor determinin g the transition between these two kinds of creep. on the transitionsbetween any two regimes is through the term r/G, and thus 3) The effect ‘ofm is minor. This is a result of the fact that P-L creep, H-D creep, and N-H creep are all lattice-diffusion controlled and have the same activation energy which is equal to that for self-lattice diffusion. 4) These maps are useful in locating the regime boundaries for any crystalline material. This is because for a given material under a fured temperature, rr/G can be estimated from a theoretical equation by
636
DEFORMATIONMECHANISMMAPS
Vol. 33, No. 4
100 Zp/G = 10-l
\ 10-l
I’
10-Z
$/G
= 1 O-2 $,p/G=
~\
1O-3
10-3 (J
10-4
G 10-S zp/G
= 1 O-6
10-6 I
10-T
10-E
m 100
-i-
10’
102
103
104
105
106
107
108
109
10’0
d/b Figurel.Generalizeddeformationmechanismmapforthecaseofn=3.
following the procedures used in previous studies [8,11,17]. That is, rP should be determined as the stress required to glide a dislocation in the easiest slip system of the crystal in a polycrystalline material.
100
10’
102
103
104
105
106
10’
108
109
d/b Figure 2. Generalized deformation mecbahm
map for the caw of n = 4.
10’0
Vol. 33, No. 4
637
DEFORMATIONMFLZWISM MAPS
100
10’
102
103
104
105
106
10’
108
109
10’0
d/b Figure 3. Generalizeddeformationmechnism map for the case of n = 5.
The major difference between conventional maps and the present ones is that the conventional maps [l] were con&w&d for individual materials whereas the present generalized ones are for all kinds of crystalline materials. Nevertheless, it should be noted that the maps shown here are applicable to creep only at high homologous temperatures where deformation tends to be lattice-dijIi.uion controlled. At relatively low temperatures, grain boundary diffusionalcreep (i.e., Cobe creep) and dislocation creep controlled by ditTusion along dislocation core may become important, and therefore should be considered, as in conventional maps. Acknowledment
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405Eng-48. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13.
H. J. Frost and M. F. Ashby, Defonnurion-mechanism Maps. Peqamon Press, Otiord (1982). J. Harper aud J. E. Dom,Acta metall 5,654 (1957). T. G. Lang&m,A Deh&matxiC. G. Sammis, Strength ofMetals andAlloys (e&ted by R C. Gifkias), p. 757. PerganxmPress, oxford (1982). 0. A Ruano, J. Wadsworth and 0. D. Sherby,Acta metall. 36.1117 (1988). 0. D. Sherby and J. Wadswortb,prOg. Muter. Sci. 33,169 (1989). J. N. Wang, Scrip. metall. mater. 29,1505 (1993). J. N. Wang Phikx. Mag. 71A, 115 (1995). J. N. Wang, and T. G. La&m, Acta met&L mater. 42.2487 (1994). A K Mukbqjee, J. E. Bii and J. E. Dam, Trans. Amer. Sot. Metals 62,155 (1969). J. N. Wang, Skip. met&! mater. 29,733 (1993). J. N. Wang and T. G. Nieh, Acfu met& mater. in press (1995). J. N. Wang and T. G. Nieh, Mater. Sci. Engng. A, in press (1995). Nabanq F. R N. in Report of a Conference on Strength of Solids. The Physical Society, Lomba, p. 75 (1948).
638
14. 15. 16. 17.
DEFORMATION MECHANISM MAPS
Herring,C. J. Appl. Phys.21,437 (1950). Cable,R L.J. Appl. Phys.34,1679 (1963). Harris,J. E.Metul Sci. J. 7, 1 (1973). J. N. Wang,Philos. Msg. 71A, 105 (1995).
Vol. 33, No. 4
Vol. 33. No. 4, pp. 633-638.1995 Else-via Science Ltd copyli@o 1995 Ada tdeahgica Inc. FbtSdUItheUSAAllIi$ltSd 095671ccl95 S9.50 + .Gu
Peqpmon 0956-716X(95)00230-8
lNCORPORATION OF PEIERLS STRESS INTO DEFORMATION MECHANISM MAPS Jian N. Wang and T. G. Nieh Chemistry & Materials Science, L-370, Lawrence Livermore National Laboratory, P-0. Box 808, Liver-more, CA 94550, USA (Received March 1,1995) (Revised March 28,1995) Jntroduction Crystalline materials deform plastically by a number of alternative, often competing, mechanisms. For engineermg and geological applications, it is essential to know the conditions under which a specific mechanism is dominant over others. Deformation mechanism maps offer such information in a compact, diagrammatic form [l]. Existing deformation maps usually include me&anisms such as power law (I-L) dislocation creep aud diffusionalcreep. Harper-Dorn (H-D) creep [2] which is a Newtonian dislocation creep, and grain boundary sliding creep were scarcely considered because of unavailability of experimental data (e.g. [3-51). For dislocation creep, in a manner similar to diflbsional creep, the rate equation is also determined from experimental data. Consequently, existing mechanism maps were mainly constructed for specific materials under specific conditions. Recently, the knowledge of P-L creep and H-D creep has been greatly expanded, in particular, the transition t?om El-L creep to H-D creep. It is now possible to predict reasonably these two hinds of creep, and therefbre to conshuct a generalized deformation mechanism map, independent of materials. The objective of this paper is to construct such generalized mechanism maps. Transitions Between Creeo Regimes
Semi-theoretical rate equations for H-D creep and P-L creep have now become available. Since most experiments on metals, ceramics and silicates show that H-D creep has an activation energy equal to that for lattice self-diffusion, the general rate equation for this creep is of the form
(1) deformation, eij, given by: Ed= a$T, where ati is the second rank deformation, Ed,given by: Em= a$T, where ag is the second rank where b is the uniaxial strain rate in steady state, o is the uniaxial stress, G is the shear modulus, b is the length of the Burgers vector, k is Boltzmann’s constant, T is the absolute temperature, DL is the lattice difksion coe.tIicientof the rate controlling species, aud A, is a dimensionless constant having values from 1O-” to IO”. With the experimentally observed relationship between the dislocation density and the Peierls stress of the crystal, rp [6,7], A, in equation (1) for H-D creep controlled by dislocation climb under saturated conditions was derived as [8] 633
634
DEFORMATIONMECHANISMMAPS
Vol. 33, No. 4
1.4 x . (T~/G)~ AID=
(2)
-In (T*/G)
It has been shown that the strain rate in H-D creep predicted by equations (1) and (2) is within one order of magnitude of the experimentally measured strain rates for many crystalline materials [8]. The range of ZIG ofinterestisbetween4x1O6and5x1O”,whichcorrespondsto-In(rdG)=12.4-5.3.Taking-In(~dc> 5 8.9, A,, may be approximated by 2
(3) P-L dislocation creep, on the other hand, can be described by the Dom equation [9]:
(4)
where n is the stress exponrzt which genera@ varies fkom - 3 to 5, and A, is a dimensionless constant which ranges from unity to as large as 1016.Existing data indicate that the transition from P-L creep to H-D creep occurs at a stress u = fi zp [IO, 111. Based on this observation and the rate equation for H-D creep, a relationship between A, and n in equation (4) is obtained [ 121:
A PL =
(5)
which can be well approximated by 0-n)
I
=P APL
=
Oml
l
t z-
(6)
This relationship agrees very well with experimental data [ 121. Diffusional creep involves mass transport either through crystal lattice - Nabarro-Hening (N-H) creep [ 13,141 or along grain boundary - Coble creep [ 151. Over the temperature range where dislocation creep is lattice-dSkon controlled at higher stresses or larger grain sizes, N-H creep occurring at lower stresses and smaller grain sizes is dominant over Coble creep. The rate equation for N-H creep is [ 13,141 (7) where 0 is the atomic volume, d is the grain size, and A, is a dimensionless geometric constant. Taking ANH = 40 [ 161 and D = 0.7 b3, equation (7) can be rewritten as ie - 28 .!!$($)2.(;)
(8)
Vol. 33, No. 4
635
DEFORMATIONMECHANISMMAPS
The condition marking the transition between one mechanism and another can be determined as follows: for that between P-L creep and H-D creep [ 10, 111,
for that between H-D creep and N-H creep [ 171,
for that between P-L creep and N-H creep, o -E G
% G
16.7*
!-I 0b
6.51($[$*
when
n-3
(11)
when
n-4
(12)
(13) The transitions between H-D creep and P-L creep and betweem H-D creep and N-H creep expressed by equations (9) and (10) have been verified by experimental data [ 10, 1 1,171. Results and Discussion In principle, deformation mechanism maps can be constructed in a three dimensional space (u/G, d/b and
r/G>. To make a clear illustration, however, the maps are now constructed in a two dimensional space (cr/G and d/b) for d&ent values of rJG. Figures 1,2 and 3 are maps for the cases of n = 3,4 and 5, respectively. In the figures the creep regime boundaries are determined at rJG = 1OS’,106, 105, 1O-‘,1Om3, 10” and 10-l. Several key observations can be made from these maps:
1) The boundaries between H-D creep and others are such al&&d by changing r/G that this creep regime expands greatly with increasing riG. This observation indicates that H-D creep is rate-controlling in materials with high rr/G over wide ranges of conditions of o/G and d/b, as already concluded before [7, 8,131. However, the boundary between P-L creep and N-H creep is insignificantly affected by changing r/G. The magnitude of the stress exponent n only slightly intluences the boundary between P-L creep and N-H 2) creep. The&ore, n is not an import& factor determinin g the transition between these two kinds of creep. on the transitionsbetween any two regimes is through the term r/G, and thus 3) The effect ‘ofm is minor. This is a result of the fact that P-L creep, H-D creep, and N-H creep are all lattice-diffusion controlled and have the same activation energy which is equal to that for self-lattice diffusion. 4) These maps are useful in locating the regime boundaries for any crystalline material. This is because for a given material under a fured temperature, rr/G can be estimated from a theoretical equation by
636
DEFORMATIONMECHANISMMAPS
Vol. 33, No. 4
100 Zp/G = 10-l
\ 10-l
I’
10-Z
$/G
= 1 O-2 $,p/G=
~\
1O-3
10-3 (J
10-4
G 10-S zp/G
= 1 O-6
10-6 I
10-T
10-E
m 100
-i-
10’
102
103
104
105
106
107
108
109
10’0
d/b Figurel.Generalizeddeformationmechanismmapforthecaseofn=3.
following the procedures used in previous studies [8,11,17]. That is, rP should be determined as the stress required to glide a dislocation in the easiest slip system of the crystal in a polycrystalline material.
100
10’
102
103
104
105
106
10’
108
109
d/b Figure 2. Generalized deformation mecbahm
map for the caw of n = 4.
10’0
Vol. 33, No. 4
637
DEFORMATIONMFLZWISM MAPS
100
10’
102
103
104
105
106
10’
108
109
10’0
d/b Figure 3. Generalizeddeformationmechnism map for the case of n = 5.
The major difference between conventional maps and the present ones is that the conventional maps [l] were con&w&d for individual materials whereas the present generalized ones are for all kinds of crystalline materials. Nevertheless, it should be noted that the maps shown here are applicable to creep only at high homologous temperatures where deformation tends to be lattice-dijIi.uion controlled. At relatively low temperatures, grain boundary diffusionalcreep (i.e., Cobe creep) and dislocation creep controlled by ditTusion along dislocation core may become important, and therefore should be considered, as in conventional maps. Acknowledment
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405Eng-48. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13.
H. J. Frost and M. F. Ashby, Defonnurion-mechanism Maps. Peqamon Press, Otiord (1982). J. Harper aud J. E. Dom,Acta metall 5,654 (1957). T. G. Lang&m,A Deh&matxiC. G. Sammis, Strength ofMetals andAlloys (e&ted by R C. Gifkias), p. 757. PerganxmPress, oxford (1982). 0. A Ruano, J. Wadsworth and 0. D. Sherby,Acta metall. 36.1117 (1988). 0. D. Sherby and J. Wadswortb,prOg. Muter. Sci. 33,169 (1989). J. N. Wang, Scrip. metall. mater. 29,1505 (1993). J. N. Wang Phikx. Mag. 71A, 115 (1995). J. N. Wang, and T. G. La&m, Acta met&L mater. 42.2487 (1994). A K Mukbqjee, J. E. Bii and J. E. Dam, Trans. Amer. Sot. Metals 62,155 (1969). J. N. Wang, Skip. met&! mater. 29,733 (1993). J. N. Wang and T. G. Nieh, Acfu met& mater. in press (1995). J. N. Wang and T. G. Nieh, Mater. Sci. Engng. A, in press (1995). Nabanq F. R N. in Report of a Conference on Strength of Solids. The Physical Society, Lomba, p. 75 (1948).
638
14. 15. 16. 17.
DEFORMATION MECHANISM MAPS
Herring,C. J. Appl. Phys.21,437 (1950). Cable,R L.J. Appl. Phys.34,1679 (1963). Harris,J. E.Metul Sci. J. 7, 1 (1973). J. N. Wang,Philos. Msg. 71A, 105 (1995).
Vol. 33, No. 4