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Constitutive description of creep in polygonized substructures
MATERIALS SCIENCE & ENGINEERING ELSEVIER
Materials Science and Engineering, A194 (1995) L5-L9
A
Letter
Constitutive description of creep in polygonized substructures Alena Orlovfi Institute of Physics of Materials, Academy of Sciences of the Czech Republic, 616 62 Brno, Czech Republic Received 21 February 1994; in revised form 16 June, 8 August, 28 September 1994
Abstract Constitutive equations of creep in polygonized substructures are formulated based on the composite model of heterogeneous dislocation structure. The suggested set of equations consists of a kinetic equation, which takes into account the elastic and plastic components of strain in hard and soft regions of the structure, and of two evolution equations, which describe the evolution of the local stress in soft regions and of the volume fraction of hard regions chosen as internal variables. As an illustration, the equations are applied to creep and dislocation structure data of a-iron discussed recently from another point of view. Keywords:Creep; Polygonized substructures
1. Introduction Constitutive equations [1,2] describe the kinetics of creep deformation by giving the strain rate as a function of external and internal variables. T h e external variables represent external conditions of the process (in uniaxial creep it is usually the applied stress and temperature) and the internal variables represent the influence of microstucture of the material in the course of creep. In a constitutive description of creep the latter fact is respected by evolution equations of relevant structural parameters. Polygonized substructures are frequent and characteristic for high-temperature creep in pure metals. T h e s e heterogeneous structures consist of regions of low dislocation density inside the subgrains, separated by narrow sub-boundary regions containing a relatively high dislocation density. This understanding of the substructures is a prerequisite for application of the twoc o m p o n e n t composite model of structure [3,4], which has been successful in explaining various aspects of static testing and creep (e.g. [5-7]). T h e model considers the dislocation structure as an agglomerate of hard sub-boundary regions bounding the soft regions of subgrain interiors. Mechanical properties of the agglomerate are then described in terms of local
stresses aH and a s and volume fractions fH and fs of the respective regions. Assumptions on the relationship between local stresses and local dislocation densities suggested by Mughrabi [8] and experimentally supported relationships between dislocation structure characteristics utilized by Dobe~ and Orlov~i [9] for evaluation of the volume fraction of hard regions allowed parameters of the model to be related to measurable dislocation structure data. H e n c e at present the dislocation structure evolution can be fairly easily expressed in terms of the composite model. As regards a description of the strain rate in the composite structure, the iso-strain rate model, suggesting the same value of the total strain rate in the whole composite structure, seemed to be plausible for a description of creep in a material in which the contiguity of regions should be conserved [10]. T h e total strain rate involves both the plastic and elastic components in the regions. Dobe~ [11,12] suggested that the N o r t o n relation of local plastic strain rates to local stresses may be accepted. T h e above-mentioned model and the listed assumptions were applied in an estimate of the average internal stress which can be measured by the strain transient dip test technique [13], and of its evolution reflecting the evolution of dislocation structure in normal
0921-5093/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSD1 0921-5093(94)02716-1
Letter / MaterialsScience and EngineeringA 194 (1995) L5-L9
L6
primary creep in a-iron [14]. This paper utilizes the formalism of the previous paper [14] for the formulation of constitutive equations of creep in a polygonized dislocation structure in terms of the composite model and of the experimentally found evolution of the microstructural parameters.
2. Kinetic equation of creep
(1)
+ o . / E = gs + as/E
where dots indicate time derivatives and E is Young's modulus. Utilizing the condition of mechanical equilibrium of the agglomerate under the applied stress o,
a=fHaH + fsas (2) and the complementarity of the volume fractions of hard and soft regions in the volume, (3)
fu+fs= 1 we can express the total creep rate g as
(4)
If for gH and gs Norton relationships to the respective local stresses are accepted, i.e.
and
as=As(as~E) "~
(5a, b)
are taken, Eq. (4) can be rewritten in the form
g = AH fH E - ,,. (~a--fHas + as )"" +As(a-fH)(as/E) "~ -(L/T.)
E
which contains only two independent parameters, i.e. as and fH. Their changes in the course of creep may be expressed as functions of time t or creep strain e taken as the independent variable. If the latter variable is chosen and a parameter K = A s / A H is introduced, Eq. (6) changes to
(_
)nil
filE-"" afHaS+as
+K(1-fH)(os/E) "s
g =A n
(7) l+/
de ]
\ f.
as
- K ( o s / E ) "S
de
a_ast_ )nil
f i l E - " \~-~-H
as
+ K(1 -f~)(os/E) "S
(8) Unfortunately, the model gives no similar suggestion for the evolution of fH. Thus, as regards this parameter, we are obliged to make use of the indications from experimental data. In a previous paper [9], a relationship between fn and directly measured parameters of substructure, i.e. subgrain size d and misorientation 0, was suggested: (9)
where b is the Burgers' vector length and the numerical constant 5.2 is an empirical value derived from wider discussion of the data [9]. This value is at the lower bound of the interval of values of this constant (5.2, 20.8) which follow from various methods of fH estimation. The evaluation of fH from Eq. (9) in a previous paper [14] gave the volume fraction fH in the primary creep decreasing in accord with the empirical equation fH =f~[1 +Df e x p ( - Cfe)]
(6)
°-°s
de
fH = 5.2b/(Od)
g =fw~H +fses --flit o n / E - as~E)
gH=AH(aH/E) ""
The evolution of a s is given directly by the composite model Eq. (1). If we substitute Eqs. (7) and (5b) in Eq. (1), we can express the evolution equation of as in the form respecting the above-accepted assumptions:
d(as/E)
In the iso-strain rate composite model of creep, the demand of a compatible deformation of hard and soft regions, involving the elastic (oH/E , as~E) and the plastic (en, es) strain components, gives a condition for the total creep rate g as =
3. Evolution equations of internal variables
(10)
(with Dr= 1.132 and Cf=9.917) to the steady-state value f ~ = 0 . 0 1 9 4 . This equation or its differential form, which is of the type
dfH/de = - Cf( fH-- f~ )
(10a)
will be accepted as the evolution equation of the second internal variable fH.
4. Description of the creep curve
The constitutive Eqs. (7), (8) and (10) can be solved whenever the parameters K, A . , ns and nH are known. The result will be the g(e) dependence fitting the experimental creep curve. The choice of the parameters and the computational procedure was as follows.
Letter
/
The parameter K was evaluated with the assumption that the local stresses tend to adjust to saturated values in the instantaneous structure. The saturated value of a s would approach a value related uniquely to the local dislocation density Ps, a~a t : a M G b p ~ / 2 (1i) where a is a characteristic of dislocation-dislocation interaction, M the Taylor factor interrelating the tensile stress and shear stresses in the slip systems and G is the shear modulus. In previous work [14], an empirical dependence of a~at (evaluated with a = 0.2 and M = 3) on e was also found: oo + a ~ a t = a ~ { l + l / [ a s (Cos Do)]} (12) (with Co = 1.824 MPa -l and D o = 0 . 0 3 3 6 MPa -1) as an integral form of the equation do,at/de = - Co(a sat s - a~) 2
L7
Materials Science and Engineering A 194 (1995) L5-L 9
(12a)
with the steady-state value a~ = 23.5 MPa. A n assumption of approaching conditions of steady-state creep in the saturated state gives [11]
proved by computations with nine pairs of exponents n n and n s composed from values 1, 4 and 7 and with the pair n H = 4 . 7 2 and n s = 1 3 . 2 deduced from the previous analysis of steady-state creep data [14]. The calculated local stresses a s and a H are also only negligibly influenced by the choice of n H and n s. T h e computed data shown in the figures were obtained with the pair n H = n s = 4, which seems to be acceptable in accord with Evans et al. [15] and was used in similar analyses by Dobe~ (e.g. [12]). The constitutive description of the creep curve shown in Fig. 1 by means of the above equations yielded the following results. (i) The local strain rate gs in the soft region is slightly higher than the total strain rate g; the ratio gs/g decreases from ca. 1.04 at the very beginning of the curve to 1 (Fig. 2(a)). Thus, in view of Eq. (1), the elastic component of the strain rate is negligible in the soft region and deformation of the soft region is almost completely plastic in the course of the creep curve (Fig. 2(b)).
(13)
g~,t=g~,=g
3E-4
from which we obtain a-- a s + asat K=As/AH
-
(o~,,),,SE,~_ ~.
•~
(14)
2E-4
< c1£ z
at any g(e) point. Now for a chosen pair of n s and nil, Eq. (8) was integrated numerically by the Runge-Kutta method to obtain the corresponding as(e) dependence within the strain interval of the creep curve. Then the g(as, fH) dependence could be evaluated from Eq. (7) and fitted to the corresponding experimental curve by determining the scaling factor A H at any point of the creep curve. Parameters As and AH following from the above procedure are dependent on e, i.e. they contribute part of the strain dependence of the primary creep rate. A possible reason for this may be as follows: the explicit power laws for gs and gU with A s and A H c o n s t a n t are assumed [ 11,12] to be valid in the steady-state creep to which steady-state dislocation structures constituted under various applied stresses correspond. In the primary creep, however, the dislocation structure evolves towards the steady state under a given applied stress. A deviation of the instantaneous structure from the steady state structure, being a function of strain, may with representation of gs and es by Eqs. (5) be respected by strain-dependent A s and A HThe choice of the exponents n H and n s was not decisive for the calculated evolution of local strain rates gH and es with the growing strain. This was
1E-4
G~
T=873 K, o-= 7,3.5 MPo
\
0E~
0)1
0 )2
0.3
STRAIN
Fig. 1. Creep curve of a-iron under the conditions applied stress a = 73.5 MPa and temperature T = 873 K.
1.5
"•-•
~s/~
1.0
.G
(a)o.o 0.3
0.2 w ~'0.1 0.0 0.0
(b)
0.1
0.2
0.3
STRAIN
Fig. 2. Contributions of local strain rates gH and gs in hard and soft regions, respectively, to the total strain rate g (a) and corresponding local strains eHand es (b).
L8
Letter
/
Materials Science and Engineering A194 (1995) L5-L9
6000
(ii) The local strain rate gH in the hard region is lower than g. It starts from very low values and approaches ~ in the course of the curve (Fig. 2(a)). The difference between gu and ~ in view of Eq. (1) indicates an elastic strain rate component in the hard region which diminishes in the course of the primary creep while the process approaches the steady state. The strain eH reached by the plastic process is thus smaller than e s in the soft region (Fig. 2(b)). (iii) The evolutions of local stresses a s and aH are shown in Figs. 3 and 4, respectively. The calculated local stress as in the soft region and an in the hard region, which is obtained as a complement of as in the equation of mechanical equilibrium (Eq. (2)), show a considerable difference from the corresponding saturated values a~~t and a~' only at the very beginning of the creep curve, which decreases to negligible values at a strain less than 0.01. A pronounced elastic process of straining should therefore be expected only within this interval. (iv) The evolution of a s (and all) in the following creep process is given first of all by changes of the structure manifested by the evolution of a~"t (and a~t) and the changes of fH. Both a~"t and fiJ, in accord with empirical Eqs. (12) and (10), would reach their steady values a~ and f ~ at very high strains (e ~ o0). The attaining of a constant creep rate at e = 0.2 may therefore result more from an interference from other processes (tertiary creep) than from establishing of the structural steady state. For o H (and a~t), it is the value of fH which determines their values in relation to the values of as (and a~at) (Eq. (2)). Although values of the local stress in the sub-boundary region as high as 20-30 times the applied stress were observed (e.g. [16]), and can be supported also by computations of the sub-boundary stress field [17], the present values of aH and a~ t seem to be unacceptably high. From this point of view, the parameter fH evaluated from structure data by means of Eq. (9) might be underestimated. Note that, as illustrated by the dashed curves for as and aH in Figs. 3 and 4, which were computed with fh--- 2fH, that the application of higher values of fH will result in more realistic values of all, while the values of as would remain unchanged. Thus, probably a proportionality factor higher than the minimum value of 5.2 [9] should be adopted in Eq. (9) to obtain more realistic values of frJ.
deformation processes in hard and soft regions, and two evolution equations describe the local stress in soft regions and the volume fraction of hard regions in the course of creep. From results of the application to experimental data, it follows that the soft region of the substructure is fully plastic, whereas the hard region deforms elasto-plastically in the whole primary creep stage. Apart from a small initial interval of strain where the local stresses are substantially elasto-plastic, their development in the creep process is controlled first of all by the evolution of dislocation structure.
5. Conclusion
Acknowledgements
The two-component model of deformation in a heterogeneous dislocation structure resulted in the suggestion of constitutive equations of creep. The kinetic equation takes into account elastic and plastic
This work was supported by the Grant Agency of CSAS under Gram No. 24110. The author thanks her colleague Dr. Ferdinand Dobeg of the same Institute for valuable stimulating discussions.
2000
'
/
,
I
0., ~
~at
4000 IOOO U') O3 l..d cY
2000
Or 0.00
.0
oO,
i 0.01
0.02
0.11 O.L2 STRAIN
0.5
Fig. 3. Evolution of local stress as in the soft regions in the course of creep (fullline). The dashed line in the inset shows the evolution of its saturated value O~s ~t(Eq. (11)). The dashed curve in the main figure was calculated for f~' - 2fH, with fH given by Eqn. (9).
,~ 80
90
60
60 ~
4o
~?'
,\
.00
' ~
,
0,01
0.0,~
_A
200.0
0.11
0.2i
0.3
STRAIN ¢ Fig. 4. Evolution of local stress a H(full line) in the hard regions and its saturated value @At (dashed line in the inset) in the course of creep. The dashed curve in the main figure corresponds to fh = 2fu, with fH givenby Eq. (9).
Letter
/
Materials Science and Engineering A194 (1995) L5-L9
References [1] A.S. Argon (ed.), Constitutive Equations in Plasticity, MIT Press, Cambridge, MA, 1975. [2] R.N. Ghosh and M. McLean, Acta Metall. Mater., 40 (1992) 3075. [3] W.D. Nix and B. Ilschner, in Proc. 5th Int. Conf. on Strength of Metals and Alloys, Pergamon Press, Oxford, 1980, p. 1503. [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] W. Blum, S. Vogler, M. Biberger and A.K. Mukherjee, Mater. Sci. Eng., A l l 2 (1989) 93. [7] S. Vogler and W. Blum, Creep and Fracture of Engineering Materials and Structures, Institute of Metals, London, 1990, p. 81.
L9
[8] H. Mughrabi, Mater. Sci. Eng., 85 (1987) 15. [9] F. Dobe~ and A. Orlov~i, Mater. Sci. Eng., A125 (1990) L5. [10] A. Orlov~, E Dobe~ and J. (~adek, in U. Messersehmidt, E Appel, J. Heydenreich and V. Schmidt (eds.), Electron Microscopy in Plasticity and Fracture Research, AkademieVerlag, Berlin, 1990, p. 259. [11] E Dobe~, Scr. Metall. Mater., 25 (1991) 2303. [12] E Dobe~, Mater. Sci. Eng., A167(1993) 31. [13] C.N. Ahlquist and W.D. Nix, Scr. Metall., 3 (1969) 679. [14[ A. Orlov~i, Mater. Sci. Eng., A163 (1993) 61. [15] R.W. Evans, W.J.E Roach and B. Wilshire, Scr. Metall., 19 (1985) 999. [16] M. Morris and J.-L. Martin, Acta Metall., 32 (1984) 549. [17] A. Orlov~, Mater. Sci. Eng., 96 (1987) L11.
Materials Science and Engineering, A194 (1995) L5-L9
A
Letter
Constitutive description of creep in polygonized substructures Alena Orlovfi Institute of Physics of Materials, Academy of Sciences of the Czech Republic, 616 62 Brno, Czech Republic Received 21 February 1994; in revised form 16 June, 8 August, 28 September 1994
Abstract Constitutive equations of creep in polygonized substructures are formulated based on the composite model of heterogeneous dislocation structure. The suggested set of equations consists of a kinetic equation, which takes into account the elastic and plastic components of strain in hard and soft regions of the structure, and of two evolution equations, which describe the evolution of the local stress in soft regions and of the volume fraction of hard regions chosen as internal variables. As an illustration, the equations are applied to creep and dislocation structure data of a-iron discussed recently from another point of view. Keywords:Creep; Polygonized substructures
1. Introduction Constitutive equations [1,2] describe the kinetics of creep deformation by giving the strain rate as a function of external and internal variables. T h e external variables represent external conditions of the process (in uniaxial creep it is usually the applied stress and temperature) and the internal variables represent the influence of microstucture of the material in the course of creep. In a constitutive description of creep the latter fact is respected by evolution equations of relevant structural parameters. Polygonized substructures are frequent and characteristic for high-temperature creep in pure metals. T h e s e heterogeneous structures consist of regions of low dislocation density inside the subgrains, separated by narrow sub-boundary regions containing a relatively high dislocation density. This understanding of the substructures is a prerequisite for application of the twoc o m p o n e n t composite model of structure [3,4], which has been successful in explaining various aspects of static testing and creep (e.g. [5-7]). T h e model considers the dislocation structure as an agglomerate of hard sub-boundary regions bounding the soft regions of subgrain interiors. Mechanical properties of the agglomerate are then described in terms of local
stresses aH and a s and volume fractions fH and fs of the respective regions. Assumptions on the relationship between local stresses and local dislocation densities suggested by Mughrabi [8] and experimentally supported relationships between dislocation structure characteristics utilized by Dobe~ and Orlov~i [9] for evaluation of the volume fraction of hard regions allowed parameters of the model to be related to measurable dislocation structure data. H e n c e at present the dislocation structure evolution can be fairly easily expressed in terms of the composite model. As regards a description of the strain rate in the composite structure, the iso-strain rate model, suggesting the same value of the total strain rate in the whole composite structure, seemed to be plausible for a description of creep in a material in which the contiguity of regions should be conserved [10]. T h e total strain rate involves both the plastic and elastic components in the regions. Dobe~ [11,12] suggested that the N o r t o n relation of local plastic strain rates to local stresses may be accepted. T h e above-mentioned model and the listed assumptions were applied in an estimate of the average internal stress which can be measured by the strain transient dip test technique [13], and of its evolution reflecting the evolution of dislocation structure in normal
0921-5093/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSD1 0921-5093(94)02716-1
Letter / MaterialsScience and EngineeringA 194 (1995) L5-L9
L6
primary creep in a-iron [14]. This paper utilizes the formalism of the previous paper [14] for the formulation of constitutive equations of creep in a polygonized dislocation structure in terms of the composite model and of the experimentally found evolution of the microstructural parameters.
2. Kinetic equation of creep
(1)
+ o . / E = gs + as/E
where dots indicate time derivatives and E is Young's modulus. Utilizing the condition of mechanical equilibrium of the agglomerate under the applied stress o,
a=fHaH + fsas (2) and the complementarity of the volume fractions of hard and soft regions in the volume, (3)
fu+fs= 1 we can express the total creep rate g as
(4)
If for gH and gs Norton relationships to the respective local stresses are accepted, i.e.
and
as=As(as~E) "~
(5a, b)
are taken, Eq. (4) can be rewritten in the form
g = AH fH E - ,,. (~a--fHas + as )"" +As(a-fH)(as/E) "~ -(L/T.)
E
which contains only two independent parameters, i.e. as and fH. Their changes in the course of creep may be expressed as functions of time t or creep strain e taken as the independent variable. If the latter variable is chosen and a parameter K = A s / A H is introduced, Eq. (6) changes to
(_
)nil
filE-"" afHaS+as
+K(1-fH)(os/E) "s
g =A n
(7) l+/
de ]
\ f.
as
- K ( o s / E ) "S
de
a_ast_ )nil
f i l E - " \~-~-H
as
+ K(1 -f~)(os/E) "S
(8) Unfortunately, the model gives no similar suggestion for the evolution of fH. Thus, as regards this parameter, we are obliged to make use of the indications from experimental data. In a previous paper [9], a relationship between fn and directly measured parameters of substructure, i.e. subgrain size d and misorientation 0, was suggested: (9)
where b is the Burgers' vector length and the numerical constant 5.2 is an empirical value derived from wider discussion of the data [9]. This value is at the lower bound of the interval of values of this constant (5.2, 20.8) which follow from various methods of fH estimation. The evaluation of fH from Eq. (9) in a previous paper [14] gave the volume fraction fH in the primary creep decreasing in accord with the empirical equation fH =f~[1 +Df e x p ( - Cfe)]
(6)
°-°s
de
fH = 5.2b/(Od)
g =fw~H +fses --flit o n / E - as~E)
gH=AH(aH/E) ""
The evolution of a s is given directly by the composite model Eq. (1). If we substitute Eqs. (7) and (5b) in Eq. (1), we can express the evolution equation of as in the form respecting the above-accepted assumptions:
d(as/E)
In the iso-strain rate composite model of creep, the demand of a compatible deformation of hard and soft regions, involving the elastic (oH/E , as~E) and the plastic (en, es) strain components, gives a condition for the total creep rate g as =
3. Evolution equations of internal variables
(10)
(with Dr= 1.132 and Cf=9.917) to the steady-state value f ~ = 0 . 0 1 9 4 . This equation or its differential form, which is of the type
dfH/de = - Cf( fH-- f~ )
(10a)
will be accepted as the evolution equation of the second internal variable fH.
4. Description of the creep curve
The constitutive Eqs. (7), (8) and (10) can be solved whenever the parameters K, A . , ns and nH are known. The result will be the g(e) dependence fitting the experimental creep curve. The choice of the parameters and the computational procedure was as follows.
Letter
/
The parameter K was evaluated with the assumption that the local stresses tend to adjust to saturated values in the instantaneous structure. The saturated value of a s would approach a value related uniquely to the local dislocation density Ps, a~a t : a M G b p ~ / 2 (1i) where a is a characteristic of dislocation-dislocation interaction, M the Taylor factor interrelating the tensile stress and shear stresses in the slip systems and G is the shear modulus. In previous work [14], an empirical dependence of a~at (evaluated with a = 0.2 and M = 3) on e was also found: oo + a ~ a t = a ~ { l + l / [ a s (Cos Do)]} (12) (with Co = 1.824 MPa -l and D o = 0 . 0 3 3 6 MPa -1) as an integral form of the equation do,at/de = - Co(a sat s - a~) 2
L7
Materials Science and Engineering A 194 (1995) L5-L 9
(12a)
with the steady-state value a~ = 23.5 MPa. A n assumption of approaching conditions of steady-state creep in the saturated state gives [11]
proved by computations with nine pairs of exponents n n and n s composed from values 1, 4 and 7 and with the pair n H = 4 . 7 2 and n s = 1 3 . 2 deduced from the previous analysis of steady-state creep data [14]. The calculated local stresses a s and a H are also only negligibly influenced by the choice of n H and n s. T h e computed data shown in the figures were obtained with the pair n H = n s = 4, which seems to be acceptable in accord with Evans et al. [15] and was used in similar analyses by Dobe~ (e.g. [12]). The constitutive description of the creep curve shown in Fig. 1 by means of the above equations yielded the following results. (i) The local strain rate gs in the soft region is slightly higher than the total strain rate g; the ratio gs/g decreases from ca. 1.04 at the very beginning of the curve to 1 (Fig. 2(a)). Thus, in view of Eq. (1), the elastic component of the strain rate is negligible in the soft region and deformation of the soft region is almost completely plastic in the course of the creep curve (Fig. 2(b)).
(13)
g~,t=g~,=g
3E-4
from which we obtain a-- a s + asat K=As/AH
-
(o~,,),,SE,~_ ~.
•~
(14)
2E-4
< c1£ z
at any g(e) point. Now for a chosen pair of n s and nil, Eq. (8) was integrated numerically by the Runge-Kutta method to obtain the corresponding as(e) dependence within the strain interval of the creep curve. Then the g(as, fH) dependence could be evaluated from Eq. (7) and fitted to the corresponding experimental curve by determining the scaling factor A H at any point of the creep curve. Parameters As and AH following from the above procedure are dependent on e, i.e. they contribute part of the strain dependence of the primary creep rate. A possible reason for this may be as follows: the explicit power laws for gs and gU with A s and A H c o n s t a n t are assumed [ 11,12] to be valid in the steady-state creep to which steady-state dislocation structures constituted under various applied stresses correspond. In the primary creep, however, the dislocation structure evolves towards the steady state under a given applied stress. A deviation of the instantaneous structure from the steady state structure, being a function of strain, may with representation of gs and es by Eqs. (5) be respected by strain-dependent A s and A HThe choice of the exponents n H and n s was not decisive for the calculated evolution of local strain rates gH and es with the growing strain. This was
1E-4
G~
T=873 K, o-= 7,3.5 MPo
\
0E~
0)1
0 )2
0.3
STRAIN
Fig. 1. Creep curve of a-iron under the conditions applied stress a = 73.5 MPa and temperature T = 873 K.
1.5
"•-•
~s/~
1.0
.G
(a)o.o 0.3
0.2 w ~'0.1 0.0 0.0
(b)
0.1
0.2
0.3
STRAIN
Fig. 2. Contributions of local strain rates gH and gs in hard and soft regions, respectively, to the total strain rate g (a) and corresponding local strains eHand es (b).
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Letter
/
Materials Science and Engineering A194 (1995) L5-L9
6000
(ii) The local strain rate gH in the hard region is lower than g. It starts from very low values and approaches ~ in the course of the curve (Fig. 2(a)). The difference between gu and ~ in view of Eq. (1) indicates an elastic strain rate component in the hard region which diminishes in the course of the primary creep while the process approaches the steady state. The strain eH reached by the plastic process is thus smaller than e s in the soft region (Fig. 2(b)). (iii) The evolutions of local stresses a s and aH are shown in Figs. 3 and 4, respectively. The calculated local stress as in the soft region and an in the hard region, which is obtained as a complement of as in the equation of mechanical equilibrium (Eq. (2)), show a considerable difference from the corresponding saturated values a~~t and a~' only at the very beginning of the creep curve, which decreases to negligible values at a strain less than 0.01. A pronounced elastic process of straining should therefore be expected only within this interval. (iv) The evolution of a s (and all) in the following creep process is given first of all by changes of the structure manifested by the evolution of a~"t (and a~t) and the changes of fH. Both a~"t and fiJ, in accord with empirical Eqs. (12) and (10), would reach their steady values a~ and f ~ at very high strains (e ~ o0). The attaining of a constant creep rate at e = 0.2 may therefore result more from an interference from other processes (tertiary creep) than from establishing of the structural steady state. For o H (and a~t), it is the value of fH which determines their values in relation to the values of as (and a~at) (Eq. (2)). Although values of the local stress in the sub-boundary region as high as 20-30 times the applied stress were observed (e.g. [16]), and can be supported also by computations of the sub-boundary stress field [17], the present values of aH and a~ t seem to be unacceptably high. From this point of view, the parameter fH evaluated from structure data by means of Eq. (9) might be underestimated. Note that, as illustrated by the dashed curves for as and aH in Figs. 3 and 4, which were computed with fh--- 2fH, that the application of higher values of fH will result in more realistic values of all, while the values of as would remain unchanged. Thus, probably a proportionality factor higher than the minimum value of 5.2 [9] should be adopted in Eq. (9) to obtain more realistic values of frJ.
deformation processes in hard and soft regions, and two evolution equations describe the local stress in soft regions and the volume fraction of hard regions in the course of creep. From results of the application to experimental data, it follows that the soft region of the substructure is fully plastic, whereas the hard region deforms elasto-plastically in the whole primary creep stage. Apart from a small initial interval of strain where the local stresses are substantially elasto-plastic, their development in the creep process is controlled first of all by the evolution of dislocation structure.
5. Conclusion
Acknowledgements
The two-component model of deformation in a heterogeneous dislocation structure resulted in the suggestion of constitutive equations of creep. The kinetic equation takes into account elastic and plastic
This work was supported by the Grant Agency of CSAS under Gram No. 24110. The author thanks her colleague Dr. Ferdinand Dobeg of the same Institute for valuable stimulating discussions.
2000
'
/
,
I
0., ~
~at
4000 IOOO U') O3 l..d cY
2000
Or 0.00
.0
oO,
i 0.01
0.02
0.11 O.L2 STRAIN
0.5
Fig. 3. Evolution of local stress as in the soft regions in the course of creep (fullline). The dashed line in the inset shows the evolution of its saturated value O~s ~t(Eq. (11)). The dashed curve in the main figure was calculated for f~' - 2fH, with fH given by Eqn. (9).
,~ 80
90
60
60 ~
4o
~?'
,\
.00
' ~
,
0,01
0.0,~
_A
200.0
0.11
0.2i
0.3
STRAIN ¢ Fig. 4. Evolution of local stress a H(full line) in the hard regions and its saturated value @At (dashed line in the inset) in the course of creep. The dashed curve in the main figure corresponds to fh = 2fu, with fH givenby Eq. (9).
Letter
/
Materials Science and Engineering A194 (1995) L5-L9
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L9
[8] H. Mughrabi, Mater. Sci. Eng., 85 (1987) 15. [9] F. Dobe~ and A. Orlov~i, Mater. Sci. Eng., A125 (1990) L5. [10] A. Orlov~, E Dobe~ and J. (~adek, in U. Messersehmidt, E Appel, J. Heydenreich and V. Schmidt (eds.), Electron Microscopy in Plasticity and Fracture Research, AkademieVerlag, Berlin, 1990, p. 259. [11] E Dobe~, Scr. Metall. Mater., 25 (1991) 2303. [12] E Dobe~, Mater. Sci. Eng., A167(1993) 31. [13] C.N. Ahlquist and W.D. Nix, Scr. Metall., 3 (1969) 679. [14[ A. Orlov~i, Mater. Sci. Eng., A163 (1993) 61. [15] R.W. Evans, W.J.E Roach and B. Wilshire, Scr. Metall., 19 (1985) 999. [16] M. Morris and J.-L. Martin, Acta Metall., 32 (1984) 549. [17] A. Orlov~, Mater. Sci. Eng., 96 (1987) L11.