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Estimation of the volume fraction of hard and soft regions formed during high temperature creep
Materials Science and Engineering, Al25 (1990) L5-L9
L5
LeRer
Estimation of the volume fraction of hard and soft regions formed during high temperature creep FERDINAND DOBEg and ALENA ORLOVA
Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, ~i~kova 22, 61662 Brno (Czechoslovakia) (Received October 16, 1989; in revised form January 16, 1990)
Abstract Several new methods for estimating the volume fraction of hard and soft regions formed during high temperature creep of metallic materials are presented. These methods start from available microscopic (dislocation density, subgrain size, subgrain misorientation) and macroscopic (applied stress, steady state creep rate) data. The application of the proposed methods is illustrated by means of data obtained in creep testing of aluminium at 573 K. A markedly heterogeneous dislocation structure is frequently formed during plastic deformation of metals under various straining conditions. This structure can be rather well approximated to by considering it as a composite of two "phases": (i) cell walls or subgrain boundaries with high dislocation density and (ii) cell or subgrain interiors in which the dislocation density is relatively low. These components are usually termed hard and soft regions. In the last decade, the idea of a schematic description of real dislocation structures by two-component representation has acquired a favourable application in the modelling of the mechanical behaviour of metallic materials [1-7]. One of the crucial points of this modelling is the determination of quantitative characteristics (especially of the volume fractions) of the components. At low temperatures, cell structures with walls of non-negligible thickness are formed and thus it is possible to ascertain the volume fraction by direct microscopic observa0921-5093/90/$3.50
tion. In contrast, at high temperatures, the hard regions consist of planar dislocation networks to which it is hardly possible to ascribe an explicit width. Nix et al. [5] chose a constant thickness of 100 nm but they were aware of the arbitrariness of this proposal. Regarding the obvious importance of the problem we would like to propose and discuss several possible methods by which to estimate the volume fractions of hard and soft regions. The measured quantities pertinent to our problem are as follows: the steady state creep rate at constant applied stress o, the mean subgrain size L, the dislocation density Ps in subgrain interiors and the mean misorientation 0 of adjacent subgrains. We shall idealize the observed equiaxed substructure by a regular array of cubic soft regions with edge length L - h separated by a continuous network of hard regions with uniform thickness h. The thickness h and the volume fraction fn of hard regions are related by
h=L{l +(fH-1) 1/3}
(1)
This relation is not too sensitive to the chosen geometry and is also valid for other shapes of equiaxed structural elements. The local dislocation density in subboundaries (related to the volume of hard regions) can be evaluated from the substructure parameters by means of the relation
4(31/2)0 P u = bLfH
(2)
where b is the length of the Burgers vector and fu the volume fraction of hard regions. This equation corresponds to subboundaries built up by two systems of screw dislocations containing a 60 ° angle which are perhaps the most representative of subboundaries in crept f.c.c, metals. The total dislocation density is given by
p =f.p.
+fsps
(3)
where fs(--- 1 --fH) is the volume fraction of soft regions. © Elsevier Sequoia/Printed in The Netherlands
L6
Based on the above-mentioned measurable quantities, the following methods for evaluation of the volume fraction fn can be proposed.
Method 1 Both theoretical considerations and experimental data [6] have shown that a relation can be expected between the local dislocation density and a characteristic dimension of the respective region. Specifically, for the hard regions we can write tOH1/2h = P1
(4)
where the parameter P1 was found to be approximately equal to 4 [6]. This result follows from the studies of low temperature deformation of copper single crystals in which the cell walls had a non-negligible thickness. Care must be taken when considering the case of high temperature conditions. Combining eqns. (1), (2) and (4) we can numerically calculate the volume fraction fnFor a low volume fraction of hard regions, i.e. for L >>h, we obtain the simplified relation
A =
3(31/2)p~2b 40L
(5)
ble to assume that the extent of the relevant stress fields, i.e. the haft-thickness of the hard regions, is P2 times larger than this interdislocation distance. Since the total area of subboundaries (composed of two systems of dislocations) in a unit volume is 3/L, the interdislocation distance is 4 = 6/(PHfHL). The volume fraction of the hard regions is then
(
f.=l-
31/2p2b13
1
(8)
OL ]
or, in a simplified version valid for low fH,
3(3t/2)P2b fH =
OL
(9)
For Pz=(P2/2) 2 the result is identical to that obtained using method 1.
Method 3 The stress a H acting in the hard regions has to be greater than that acting in the soft regions (as) to maintain a strain compatibility. From the conditions of mechanical equilibrium it follows that o = oHfH + Osfs
(10)
A similar procedure can naturally be applied to the soft regions. In this case we start from the equation
The local stress as can be related to the local dislocation density Ps by the relation [6]
pst/2(L - h)= P1
Os = asMGbps 1/2
(6)
The volume fraction of the hard regions is found to be
Simple reasoning supported by experimental data suggests a somewhat greater value of P~ (about 10) in this case [5]. As a matter of fact, eqn. (7) suffers from the strong dependence of the expected f~ values on the parameter P1 and is therefore practically non-applicable.
Method2 The hard region is, in fact, a spatial object contiguous to the subgrain boundaries within which the stress is substantially greater than that acting in the soft regions. The range of the elastic stress field of a small-angle boundary is proportional to the distance between the dislocations which constitute the boundary. It is quite plausi-
(11)
where G is the shear modulus, M is the orientation factor which interrelates the axial and the shear stresses, and a s is a constant slightly sensitive to the type of dislocation arrangement and crystal structure [8]. An analogous relation for the hard regions should take into account the planar character of the dislocation networks in these regions at high temperatures. Therefore we must replace the square root of the local dislocation density directly by the reciprocal distance between dislocations. If this distance is the same as in method 2 we obtain (using eqn. (2)) oH -
a HMGbpH Lfn _ 2 a HMGO 6 31/2
(12)
Formulae such as eqns. (11) and (12) were originally derived for the flow stress in a constant strain-rate test. If the saturation stage of a constant strain-rate test may be equated with the steady state of a creep test, the application of
L7 eqns. (11) and (12) is justifiable. Since the configuration dependence of a s and/or tz H is probably rather weak [8], we assume that a H = aS = a. As a result we obtain from eqns. ( 10)-(12)
a/( a M G b ) - ps 1/2 f . - 201(31/2b)- psi~ 2
(13)
The values of fH following from this equation will be designated as the results of method 3. In addition to this, the method can be modified as follows. The applied stress which drives dislocations during creep (where inertial effects can certainly be neglected) is usually divided into an internal and an effective component. The internal component surmounts the stress fields of neighbouring dislocations, while the effective stress compensates for the rate-dependent lattice friction. For agreement with this concept, eqns. (11) and (12) should be adapted for corresponding internal stresses. The stresses a H and a s are (from the viewpoint of this concept) applied stresses. For the calculation of fH it is then necessary to know the relevant stress components. At present these components cannot be found without further assumptions. Owing to the compatibility requirements, the long-range internal stresses A a H = a H - - a and A a s = o - a s are generated. Assuming now (in accordance with the above concept of internal and applied stresses) that it is not a s nor o H but the long-range internal stresses A a s and A a H which are related to their respective dislocation densities in a manner analogous to eqns. (11 ) and (12), i.e.
&as = as' MGbps m AaH =
2an'MGO 3m
(14) (15)
we obtain (again for aH ' = as' ) p s 1/2
= 2 o/(31%)+ m,/2
(16)
At first sight, it is very questionable whether eqns. (14) and (15) are correct. The abovementioned long-range internal stresses should have the wavelength equal to the subgrain size, whereas the wavelength of the dislocationinduced internal stress is Ps-1/2 or 31/2b/(20) and is obviously smaller. Moreover, it is apparent that
the heterogeneization of the dislocation structure leads to a decrease in the dislocation density in the soft regions and, at the same time, to an increase in the difference a - a s which is contrary to eqn. (14). To make eqn. (16) consistent with eqns. (11) and (12) we must assume that the ratio fsas/(fHaH) of mechanical work done locally in soft and hard regions is equal to one. Under this assumption we obtain from eqn. (10) US _
or-- O S
OH
O H - - (7
Since 20/(31/2b)~ ps 1/2 and Lps 1/2= 10 [5], we have 5(31/2)b fH = - OL
(17)
Method 4 Nix and Ilschner [1] proposed equations for the plastic strain rate in the hard and the soft regions gH --
6"44Db°(~) kT
3 (18)
and gs=eso
exp - ~ - ~ s i n h / ~ - o
] ,
(19)
where D is the diffusion coefficient, gso = 1012 s-1, AF is the Helmholtz free energy associated with moving a dislocation past an obstacle and k T has its usual meaning. As a consequence of the compatibility condition in the steady state of a constant-stress creep test, the strain rates in both regions are the same and equal to the measured creep rate. For a given creep rate g and an applied stress a we can therefore calculate the stresses acting in both hard and soft regions, i.e. o n and a s. From eqn. (10) we can then calculate the volume fraction fH" This method was used in our previous paper [9], where it has also been shown that the above-mentioned model [1] can be successfully applied for the simulation of the measured internal stress in aluminium. This supports the correctness of eqns. (18) and (19). To illustrate the proposed methods we shall use the data obtained in a more extensive study of creep in aluminium of 99.99% purity at 573 K. The results of constant tensile stress creep tests
L8
and of parallel transmission electron microscopy investigations were published in previous papers [10, 11]. The choice of this stock of data was motivated by the fact that all the necessary substructure and creep data were measured on the same specimens. For the calculations the following values of parameters and constants were used: P l = 4 , P 2 = l , ct=0.2 [12], M--2, D = l . 9 3 x 1 0 -17 m 2 s -1 [13], G - - 2 1 7 0 0 MPa [13], b = 0.286 nm, AF-- 0.44 Gb 3 [3]. The results of the calculations are summarized in Fig. 1. The values of the parameters P1 and P2 have to be considered to be a first-order approximation. A detailed discussion of their influence on the results that can be obtained is given elsewhere [14]. Methods 1-3 give values of the volume fraction fH which increase with increasing applied stress. The results of methods 1 and 2 are in agreement with that which has been said above about the equivalence of these two methods for low volume fractions. Plausible results are also obtained for the modified method 3 expressed using eqn. (16) in spite of the objections raised against the correctness of eqns. (14) and (15). The possible explanation of this paradox consists probably in the fact that mechanical work done'in particular regions is approximately equal in each case. The experimental data published by Mughrabi [4] for copper single crystals deformed at low temperatures show that the ratio fs Os/(fH OH) does indeed tend to one. Reasonable results are also yielded by method 4. This method, which is not based on structure data, is the only one that gives fH values which decrease with increasing applied stress. The values of fH calculated for a constant thickness h-- 100 nm [5] are presented in Fig. 1 for comparison. The resulting values range from 3.5% to 5.0% and are between the results obtained for methods 2 and 4 and those obtained for method 1. From the viewpoint of the present results no quantitative objection can be made to the assessment of the hard region thickness proposed by Nix et al. [5]. At the same time, it is clear that the insertion of a constant thickness term equal to the dislocation core diameter, i.e. h = 2b, tends to give unrealistically low estimates of the volume fraction fH = 0.02%. Another method for estimating the volume fractions of the components in heterogeneous dislocation structures has been published by Ung~ir et al. [15]. This method employs an evalua-
0.15
,
,
,
,_z Z _o I--
0.12
/ 0 METHOD 3
,o-" Q09
/ •g,,.
o/ / d! \ . '"d' ,&h:100nm ~ f ~ , ~ / A__m...~...~ ._~... ~ . . . . . .E.o (~6) ~"~'~_ ..... ~METHOD 2 ~'~O~'~V'~' ~J ~ I "-v---"~-..-g ~ METHOD I 4
0.06 t..U ~: :::) J >O
,o METHOD1
/ o--iJ ~ ---lJ_ / /
0.03 0
0
5
10
15
20
25
STRESS ~ EMPo'I
Fig. 1. Stress dependence of calculated volume fraction of hard regions in aluminium.
tion of broadened line profiles of X-ray Bragg reflections. It was applied to copper single crystals deformed at room temperature. The resulting values of the volume fraction f n are greater than those found in the present work for high temperature creep in alumim'um. Considering the difference in substructures corresponding to the respective materials and homologous temperatures of deformation, the observed difference in fH is as expected. At present, no direct method exists for establishing fH values at high temperature creep. This means that we have no criterion by which to decide which of the proposed methods is best. It should be noted that methods 1-3 can lead to qualitatively the same result, according to which the volume fraction of hard regions is proportional to b/(OL ) with the proportionality constant ranging roughly from 5.2 to 20.8. To establish this constant more precisely is the next task which remains to be solved.
References 1 W. D. Nix and B. Ilschner, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, Vol. 3, Pergamon, Oxford, 1980,p. 1503. 2 H. Mughrabi, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, Vol. 3, Pergamon, Oxford, 1980, p. 1615. 3 W. D. Nix, J. C. Gibeling and K. P. Fuchs, Mechanical testing for deformation model development, ASTM Spec. Tech. Publ., 765 (1982) 301. 4 H. Mughrabi, Acta Metall., 31 (1983) 1367. 5 W. D. Nix, J. C. Gibeling and D. A. Hughes, Metall. Trans. A, 16 (1985) 2215. 6 H. Mughrabi, Mater. Sci. Eng., 85 (1987) 15. 7 W. Blum, S. Vogler, M. Biberger and A. K. Mukherjee, Mater. Sci. Eng., Al12 (1989) 93. 8 F. E Lavrentev, Mater. Sci. Eng., 46 (1980) 191.
L9 9 F. Dobe~, Czech. J. Phys. B, 38 (1988) 524. 10 A. Orlowi, Z. Tobolov~i and J. ~adek, Philos. Mag., 26 (1972) 1263. 11 Z. Tobolov~iand J. ~adek, Philos. Mag., 26 (1972) 1419. 12 G. Saada, Acta Metall., 8 (1960) 200.
13 H. J. Frost and M. E Ashby, Deformation Mechanism Maps, Pergamon, Oxford, 1982. 14 F. Dobe~ and A. Odowi, submitted to Metall. Mater. 15 T. Ung~ir, H. Mughrabi, D. R6nnpagel and M. Wilkens, Acta Metall., 32 (1984) 333.
L5
LeRer
Estimation of the volume fraction of hard and soft regions formed during high temperature creep FERDINAND DOBEg and ALENA ORLOVA
Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, ~i~kova 22, 61662 Brno (Czechoslovakia) (Received October 16, 1989; in revised form January 16, 1990)
Abstract Several new methods for estimating the volume fraction of hard and soft regions formed during high temperature creep of metallic materials are presented. These methods start from available microscopic (dislocation density, subgrain size, subgrain misorientation) and macroscopic (applied stress, steady state creep rate) data. The application of the proposed methods is illustrated by means of data obtained in creep testing of aluminium at 573 K. A markedly heterogeneous dislocation structure is frequently formed during plastic deformation of metals under various straining conditions. This structure can be rather well approximated to by considering it as a composite of two "phases": (i) cell walls or subgrain boundaries with high dislocation density and (ii) cell or subgrain interiors in which the dislocation density is relatively low. These components are usually termed hard and soft regions. In the last decade, the idea of a schematic description of real dislocation structures by two-component representation has acquired a favourable application in the modelling of the mechanical behaviour of metallic materials [1-7]. One of the crucial points of this modelling is the determination of quantitative characteristics (especially of the volume fractions) of the components. At low temperatures, cell structures with walls of non-negligible thickness are formed and thus it is possible to ascertain the volume fraction by direct microscopic observa0921-5093/90/$3.50
tion. In contrast, at high temperatures, the hard regions consist of planar dislocation networks to which it is hardly possible to ascribe an explicit width. Nix et al. [5] chose a constant thickness of 100 nm but they were aware of the arbitrariness of this proposal. Regarding the obvious importance of the problem we would like to propose and discuss several possible methods by which to estimate the volume fractions of hard and soft regions. The measured quantities pertinent to our problem are as follows: the steady state creep rate at constant applied stress o, the mean subgrain size L, the dislocation density Ps in subgrain interiors and the mean misorientation 0 of adjacent subgrains. We shall idealize the observed equiaxed substructure by a regular array of cubic soft regions with edge length L - h separated by a continuous network of hard regions with uniform thickness h. The thickness h and the volume fraction fn of hard regions are related by
h=L{l +(fH-1) 1/3}
(1)
This relation is not too sensitive to the chosen geometry and is also valid for other shapes of equiaxed structural elements. The local dislocation density in subboundaries (related to the volume of hard regions) can be evaluated from the substructure parameters by means of the relation
4(31/2)0 P u = bLfH
(2)
where b is the length of the Burgers vector and fu the volume fraction of hard regions. This equation corresponds to subboundaries built up by two systems of screw dislocations containing a 60 ° angle which are perhaps the most representative of subboundaries in crept f.c.c, metals. The total dislocation density is given by
p =f.p.
+fsps
(3)
where fs(--- 1 --fH) is the volume fraction of soft regions. © Elsevier Sequoia/Printed in The Netherlands
L6
Based on the above-mentioned measurable quantities, the following methods for evaluation of the volume fraction fn can be proposed.
Method 1 Both theoretical considerations and experimental data [6] have shown that a relation can be expected between the local dislocation density and a characteristic dimension of the respective region. Specifically, for the hard regions we can write tOH1/2h = P1
(4)
where the parameter P1 was found to be approximately equal to 4 [6]. This result follows from the studies of low temperature deformation of copper single crystals in which the cell walls had a non-negligible thickness. Care must be taken when considering the case of high temperature conditions. Combining eqns. (1), (2) and (4) we can numerically calculate the volume fraction fnFor a low volume fraction of hard regions, i.e. for L >>h, we obtain the simplified relation
A =
3(31/2)p~2b 40L
(5)
ble to assume that the extent of the relevant stress fields, i.e. the haft-thickness of the hard regions, is P2 times larger than this interdislocation distance. Since the total area of subboundaries (composed of two systems of dislocations) in a unit volume is 3/L, the interdislocation distance is 4 = 6/(PHfHL). The volume fraction of the hard regions is then
(
f.=l-
31/2p2b13
1
(8)
OL ]
or, in a simplified version valid for low fH,
3(3t/2)P2b fH =
OL
(9)
For Pz=(P2/2) 2 the result is identical to that obtained using method 1.
Method 3 The stress a H acting in the hard regions has to be greater than that acting in the soft regions (as) to maintain a strain compatibility. From the conditions of mechanical equilibrium it follows that o = oHfH + Osfs
(10)
A similar procedure can naturally be applied to the soft regions. In this case we start from the equation
The local stress as can be related to the local dislocation density Ps by the relation [6]
pst/2(L - h)= P1
Os = asMGbps 1/2
(6)
The volume fraction of the hard regions is found to be
Simple reasoning supported by experimental data suggests a somewhat greater value of P~ (about 10) in this case [5]. As a matter of fact, eqn. (7) suffers from the strong dependence of the expected f~ values on the parameter P1 and is therefore practically non-applicable.
Method2 The hard region is, in fact, a spatial object contiguous to the subgrain boundaries within which the stress is substantially greater than that acting in the soft regions. The range of the elastic stress field of a small-angle boundary is proportional to the distance between the dislocations which constitute the boundary. It is quite plausi-
(11)
where G is the shear modulus, M is the orientation factor which interrelates the axial and the shear stresses, and a s is a constant slightly sensitive to the type of dislocation arrangement and crystal structure [8]. An analogous relation for the hard regions should take into account the planar character of the dislocation networks in these regions at high temperatures. Therefore we must replace the square root of the local dislocation density directly by the reciprocal distance between dislocations. If this distance is the same as in method 2 we obtain (using eqn. (2)) oH -
a HMGbpH Lfn _ 2 a HMGO 6 31/2
(12)
Formulae such as eqns. (11) and (12) were originally derived for the flow stress in a constant strain-rate test. If the saturation stage of a constant strain-rate test may be equated with the steady state of a creep test, the application of
L7 eqns. (11) and (12) is justifiable. Since the configuration dependence of a s and/or tz H is probably rather weak [8], we assume that a H = aS = a. As a result we obtain from eqns. ( 10)-(12)
a/( a M G b ) - ps 1/2 f . - 201(31/2b)- psi~ 2
(13)
The values of fH following from this equation will be designated as the results of method 3. In addition to this, the method can be modified as follows. The applied stress which drives dislocations during creep (where inertial effects can certainly be neglected) is usually divided into an internal and an effective component. The internal component surmounts the stress fields of neighbouring dislocations, while the effective stress compensates for the rate-dependent lattice friction. For agreement with this concept, eqns. (11) and (12) should be adapted for corresponding internal stresses. The stresses a H and a s are (from the viewpoint of this concept) applied stresses. For the calculation of fH it is then necessary to know the relevant stress components. At present these components cannot be found without further assumptions. Owing to the compatibility requirements, the long-range internal stresses A a H = a H - - a and A a s = o - a s are generated. Assuming now (in accordance with the above concept of internal and applied stresses) that it is not a s nor o H but the long-range internal stresses A a s and A a H which are related to their respective dislocation densities in a manner analogous to eqns. (11 ) and (12), i.e.
&as = as' MGbps m AaH =
2an'MGO 3m
(14) (15)
we obtain (again for aH ' = as' ) p s 1/2
= 2 o/(31%)+ m,/2
(16)
At first sight, it is very questionable whether eqns. (14) and (15) are correct. The abovementioned long-range internal stresses should have the wavelength equal to the subgrain size, whereas the wavelength of the dislocationinduced internal stress is Ps-1/2 or 31/2b/(20) and is obviously smaller. Moreover, it is apparent that
the heterogeneization of the dislocation structure leads to a decrease in the dislocation density in the soft regions and, at the same time, to an increase in the difference a - a s which is contrary to eqn. (14). To make eqn. (16) consistent with eqns. (11) and (12) we must assume that the ratio fsas/(fHaH) of mechanical work done locally in soft and hard regions is equal to one. Under this assumption we obtain from eqn. (10) US _
or-- O S
OH
O H - - (7
Since 20/(31/2b)~ ps 1/2 and Lps 1/2= 10 [5], we have 5(31/2)b fH = - OL
(17)
Method 4 Nix and Ilschner [1] proposed equations for the plastic strain rate in the hard and the soft regions gH --
6"44Db°(~) kT
3 (18)
and gs=eso
exp - ~ - ~ s i n h / ~ - o
] ,
(19)
where D is the diffusion coefficient, gso = 1012 s-1, AF is the Helmholtz free energy associated with moving a dislocation past an obstacle and k T has its usual meaning. As a consequence of the compatibility condition in the steady state of a constant-stress creep test, the strain rates in both regions are the same and equal to the measured creep rate. For a given creep rate g and an applied stress a we can therefore calculate the stresses acting in both hard and soft regions, i.e. o n and a s. From eqn. (10) we can then calculate the volume fraction fH" This method was used in our previous paper [9], where it has also been shown that the above-mentioned model [1] can be successfully applied for the simulation of the measured internal stress in aluminium. This supports the correctness of eqns. (18) and (19). To illustrate the proposed methods we shall use the data obtained in a more extensive study of creep in aluminium of 99.99% purity at 573 K. The results of constant tensile stress creep tests
L8
and of parallel transmission electron microscopy investigations were published in previous papers [10, 11]. The choice of this stock of data was motivated by the fact that all the necessary substructure and creep data were measured on the same specimens. For the calculations the following values of parameters and constants were used: P l = 4 , P 2 = l , ct=0.2 [12], M--2, D = l . 9 3 x 1 0 -17 m 2 s -1 [13], G - - 2 1 7 0 0 MPa [13], b = 0.286 nm, AF-- 0.44 Gb 3 [3]. The results of the calculations are summarized in Fig. 1. The values of the parameters P1 and P2 have to be considered to be a first-order approximation. A detailed discussion of their influence on the results that can be obtained is given elsewhere [14]. Methods 1-3 give values of the volume fraction fH which increase with increasing applied stress. The results of methods 1 and 2 are in agreement with that which has been said above about the equivalence of these two methods for low volume fractions. Plausible results are also obtained for the modified method 3 expressed using eqn. (16) in spite of the objections raised against the correctness of eqns. (14) and (15). The possible explanation of this paradox consists probably in the fact that mechanical work done'in particular regions is approximately equal in each case. The experimental data published by Mughrabi [4] for copper single crystals deformed at low temperatures show that the ratio fs Os/(fH OH) does indeed tend to one. Reasonable results are also yielded by method 4. This method, which is not based on structure data, is the only one that gives fH values which decrease with increasing applied stress. The values of fH calculated for a constant thickness h-- 100 nm [5] are presented in Fig. 1 for comparison. The resulting values range from 3.5% to 5.0% and are between the results obtained for methods 2 and 4 and those obtained for method 1. From the viewpoint of the present results no quantitative objection can be made to the assessment of the hard region thickness proposed by Nix et al. [5]. At the same time, it is clear that the insertion of a constant thickness term equal to the dislocation core diameter, i.e. h = 2b, tends to give unrealistically low estimates of the volume fraction fH = 0.02%. Another method for estimating the volume fractions of the components in heterogeneous dislocation structures has been published by Ung~ir et al. [15]. This method employs an evalua-
0.15
,
,
,
,_z Z _o I--
0.12
/ 0 METHOD 3
,o-" Q09
/ •g,,.
o/ / d! \ . '"d' ,&h:100nm ~ f ~ , ~ / A__m...~...~ ._~... ~ . . . . . .E.o (~6) ~"~'~_ ..... ~METHOD 2 ~'~O~'~V'~' ~J ~ I "-v---"~-..-g ~ METHOD I 4
0.06 t..U ~: :::) J >O
,o METHOD1
/ o--iJ ~ ---lJ_ / /
0.03 0
0
5
10
15
20
25
STRESS ~ EMPo'I
Fig. 1. Stress dependence of calculated volume fraction of hard regions in aluminium.
tion of broadened line profiles of X-ray Bragg reflections. It was applied to copper single crystals deformed at room temperature. The resulting values of the volume fraction f n are greater than those found in the present work for high temperature creep in alumim'um. Considering the difference in substructures corresponding to the respective materials and homologous temperatures of deformation, the observed difference in fH is as expected. At present, no direct method exists for establishing fH values at high temperature creep. This means that we have no criterion by which to decide which of the proposed methods is best. It should be noted that methods 1-3 can lead to qualitatively the same result, according to which the volume fraction of hard regions is proportional to b/(OL ) with the proportionality constant ranging roughly from 5.2 to 20.8. To establish this constant more precisely is the next task which remains to be solved.
References 1 W. D. Nix and B. Ilschner, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, Vol. 3, Pergamon, Oxford, 1980,p. 1503. 2 H. Mughrabi, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, Vol. 3, Pergamon, Oxford, 1980, p. 1615. 3 W. D. Nix, J. C. Gibeling and K. P. Fuchs, Mechanical testing for deformation model development, ASTM Spec. Tech. Publ., 765 (1982) 301. 4 H. Mughrabi, Acta Metall., 31 (1983) 1367. 5 W. D. Nix, J. C. Gibeling and D. A. Hughes, Metall. Trans. A, 16 (1985) 2215. 6 H. Mughrabi, Mater. Sci. Eng., 85 (1987) 15. 7 W. Blum, S. Vogler, M. Biberger and A. K. Mukherjee, Mater. Sci. Eng., Al12 (1989) 93. 8 F. E Lavrentev, Mater. Sci. Eng., 46 (1980) 191.
L9 9 F. Dobe~, Czech. J. Phys. B, 38 (1988) 524. 10 A. Orlowi, Z. Tobolov~i and J. ~adek, Philos. Mag., 26 (1972) 1263. 11 Z. Tobolov~iand J. ~adek, Philos. Mag., 26 (1972) 1419. 12 G. Saada, Acta Metall., 8 (1960) 200.
13 H. J. Frost and M. E Ashby, Deformation Mechanism Maps, Pergamon, Oxford, 1982. 14 F. Dobe~ and A. Odowi, submitted to Metall. Mater. 15 T. Ung~ir, H. Mughrabi, D. R6nnpagel and M. Wilkens, Acta Metall., 32 (1984) 333.