Acta metall, mater. Vol. 39, No. 7, pp. 1549-1554, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
CREEP DEFORMATION OF DUCTILE TWO-PHASE ALLOYS M . T A N A K A 1, T. S A K A K I 2 and H. H Z U K A 1 ZDepartment of Mechanical Engineering for Production, Akita University, 1-I, Tegatagakuen-cho, Akita 010 and 2Department of Mechanical Engineering, Faculty of Technology, Tokyo Metropolitan University, 2-1-1, Fukasawa, Setagaya-ku, Tokyo 158, Japan (Received 20 April 1990; in revised form 4 December 1990)
Ab~ract--A continuum mechanics model is developed to explain the creep deformation of ductile two-phase alloys. The model predicts that the transient creep is caused by the internal stresses in second phase and matrix resulting from the difference in creep strain between two phases induced by the strength difference, even if the inherent transient creep in both phases is not taken into account. The difference in creep strain between two phases in steady-state creep is analytically obtained for the alloys in which both second phase and matrix exhibit the exponential law, the power-law or the hyperbolic sine law creep. The continuum mechanics model gives the same values of steady-state creep rate as the constant creep rate model by McDanels and co-workers. The results of analysis based on the continuum mechanics model are compared with the experimental results. Rrginr---On drveloppe un modrle mrcanique de continuum pour expliquer la drformation par fluage d'alliages biphas~s ductiles. Le modrle prrvoit que le fluage transitoire est dO aux contraintes internes dans la seconde phase et dans la matrice, contraintes qui rrsultent de la diffrrence de la drformation de fluage entre les deux phases, cette diffrrence &ant induite par la diffrrence de rrsistance mrcanique, m~me lorsque le fluage transitoire dans les deux phases n'est pas pris en compte. La diffrrence de drformation entre les deux phases en fluage en rrgime permanent est obtenue analytiquement pour les alliages dans lesquels la fois la seconde phase et la matrice suivent une loi de fluage exponentielle, en puissance ou en sinus hyperbolique. Le modrle mrcanique du continuum donne les mSmes valeurs de la vitesse de fluage en rrgime permanent que le modrle fi vitesse de fluage constante de McDanels et coil Les rrsultats de l'analyse basre sur le modrle mrcanique du continuum sont comparrs aux donnres exprrimentales. Znsmnmenfasmng--Um der Kriechverformung der duktil-zweiph~isigen Legierungen zu erkl/iren, entwickelt sich ein kontinuummechanisches Modell. Das Modell sagt voraus, da$ das Obergangskriechen durch die inneren Spannungen, die durch den aus dem Festigkeitsunterschied erzeugten Kriechanspannungsunterschied zwischen der zweiten Phase und der Grundmasse in der Kriechverformung gebildet werden, verursacht werde, wenn das eigene ~bergangskriechen in den beiden Phasen auch nicht beriicksichtigt wird. Der Kriechanspannungsunterschied zwischen beiden Phasen in dem station/iren Kriechen analytisch berechnet sich ffir den Legierungen, in den die Abh/ingigkeit der Kriechgeschwindigkeit von Spannung in der beiden Phasen mit dem Exponentialgesetz, dem Potenzgesetz oder dem Hyperbolic-Sine-Gesetz beschrieben wird. Das kontinuummechanische Modell gibt die gleichen Werten der station/iren Krieehgeschwindigkeit als wie das Konstant-Kriechgeschwindigkeitmodell von McDanels und seinen Kollegen. Die Ergebnisse der Analyse mit dem kontinuummechanischen Modell wird mit den experimentalen Ergebnissen vergleicht.
I. INTRODUCTION Extensive studies have been made on the creep deformation in two-phase metallic materials including short fiber- or continuous fiber-reinforced composite materials [1-4] and superplastic alloys [5, 6]. The creep deformation of these materials is affected by several factors such as strength, volume fraction, shape and size of second phase [1-4, 7, 8] and experimental variables including stress and temperature [1, 3, 5, 9]. But, theoretical discussions have not been fully made on the effects of these factors on the creep deformation of the ductile twophase alloys. The stress (~) dependence o f steady-state creep rate (Es) in the high-temperature creep of metallic
materials is generally expressed in terms of the power law (~soc tr", n: stress exponent) or the exponential law [~s ocexp(Btr), B: materials constant] or the hyperbolic sine law (~socct[sinh(fla)]", ~t, #, n: materials constants) [9, 10]. In this study, a continuum mechanics model has been developed to explain the creep deformation of the ductile two-phase alloys, in which the creep rate in second phase and matrix exhibits the exponential law, the power law, or the hyperbolic sine law dependence on the applied stress. An analysis on the creep deformation is made on the several factors including the strength difference between two phases, volume fraction of the second phase, and the applied stress, based on the continuum mechanics model. The results of the calculations were compared with the experimental results.
1549
1550
TANAKA et al.: CREEP DEFORMATION OF DUCTILE TWO-PHASE ALLOYS 2. STRESS AND CREEP STRAIN
Let us consider the creep deformation of a ductile two-phase alloy, in which the second phase with an arbitrary shape and volume fraction of f a r e dispersed in the matrix phase. Both phases are assumed to be elastically and plastically isotropic. It is also assumed that creep strains are uniform in individual phases, namely e~3 = -2Eli = -2EI2 for the matrix (phase I) and e li 33 = - 2E]] = - 2E~ for the second phase (phase II) under an applied tensile stress, O-A. Components of creep strain difference between two phases are AE33 (=E~3--E~3), AEH (=E]I--E~]), and AE22 (=e122- E22u_--AEH). Internal stresses arise from the strain difference (AEv). The components of internal stresses averaged over the matrix (O-~j) and these for the second phase (O-~) can be calculated by Eshelby's the equivalent inclusion method [11-13] and Mori-Tanaka's average internal stress concept [14] and are expressed as O-~3 = f A E AE33 = - f A E x O-~l = O-~2= fBEAE33 = f B E x
(1)
O-~ = (1 - f ) A E x O-]i = O-i~= - ( 1 - T ) B E x
(2)
where x is A*33 and E is Young's modulus [E = 2/~(1 + v ), #: rigidity, v: Poisson's ratio]. A and B are functions of Eshelby's tensor [12, 13, 15, 16], elastic moduli of the matrix phase (E,/~ and v) and those of the second phase (E*,/z* and v*). Therefore, the actual stress acting on each phase, the equivalent stress ( O ' e ) and the deviatoric stress (O-~j) are
references [12, 13, 15,16]. The average creep strain of the two-phase alloy, Eo, is expressed as Eij ~"
( 1 -- f )e ]S + f i
~"
(5)
The above equations are also applicable to the materials in which the volume fraction of the strong phase is large enough to surround the weak phase, if we assume that the weak second phase is embedded in the strong matrix. This enables the calculations of stresses and creep strains in each phase and in the material to be made for arbitrary volume fractions of the strong phase. 3. ANALYSIS OF CREEP DEFORMATION BASED ON CONTINUUM MECHANICS MODEL
3.1. Basic equations The creep deformation of ductile two-phase alloys at a current time t can be calculated on the basis of the continuum mechanics model, in which the interaction of creep deformation between second phase and matrix is taken into account. The difference in creep strain arises from the strength difference between them, and results in the internal stresses in both phases. If the creep law for both phases are expressed as arbitrary functions, F t and F n, of equivalent stresses, a i, and alei, equivalent creep strains, El and cn, etc. at a current time t, the increments of equivalent creep strain, dE I and dEu, during an infinitesimal time interval dt are expressed as del = Fi(a~, ¢1. . . . ) dt dE n = F n( _oi i. , e. 11 . . . . ) dt.
(6)
O-¢3+ O-h = O-¢3 - f A E x O-I 2 ~-~71e =
O-]l = f B E x ( l / , f i ) x/(o-~, - O-h) ~ + [o-[~ - (o-¢~ + ~ ) 1 2 + [(O-¢3 + O-~3) - O-I,] 2 O-¢~ - f ( ~
+ S)Ex
(3)
= O-¢3 - f ~ E x
O-;i = _ 2O-~ --- -- 2O-~] = (2•3)
(O-A_ f K E x )
for the matrix, and O-¢3+ O-~13= O-¢3+ (1 - T ) A E x 11 _ _ O'22 --
a~] = --(1 - f ) B E x
II__ O-e - -
(1/N//2) J ( a I11- , , ~~l1,2 , + [ o - ~I|- ( o - l ~ + O-¢3+
[i ~ + [(~33 ~ + O-,~)_o-fi]~ ~33)] n
(1 - f )KEx
O-;7 = --2a m = --20-~I11= (2/3) {a¢3 + (1 - T ) K E x }
for the second phase, were K ( = A + B ) is a shape factor. The value of shape factor K is (7 - 5v)/[10(1 - v2)] for the spherical second phase of the same elastic moduli as those of the matrix [11]. The values in general cases have been given in
where the current equivalent creep strains are given by Ex=
dEt, =0
~II=
d~II. =0
(7)
TANAKA et al.: CREEP DEFORMATION OF DUCTILE TWO-PHASE ALLOYS Corresponding increments of creep strains are
The value of Ax (and x) is positive when the second phase is stronger than the matrix, and is negative when the former is the weaker phase•
dE~j -- ( 3 a ua / 2 a ,l) de I dE~J -- (3a ,IfU/2a,II ) de II.
(8)
Using equations (6), (7) and (8), one can calculate the mean creep rate of the two-phase alloy, ~u ( = dEu/dt ), to be given by
~.,j =
(1 - f )(dE~j/dt) + f(dE~/dt).
(9)
If F ~ and F n are determined so as to include inherent transient creep of each phase, we can exactly follow the creep of the two-phase alloy. For simplicity, the inherent transient creep is neglected and both phases are assumed to have only steady-state creep in the following calculations. If the creep deformation in both second phase and matrix exhibits the exponential law dependence on stress, the creep rates in both phases at a current time t are expressed as follows by equations (3), (4), (6), (7), (8) and (9)
E~3 = A[ exp(Bt aJ~) = A t exp {B, (a A~-fKEx)} (s -I )
•n ----All exp (BII a ~1) E33 = An exp{Bu[a3A3 + (1 - f ) K E x ] } ( s - ' ) ~33=(1 - - f ) E 3-]3 + fE.n33(S - ] )
(10)
w h e r e a l , AII, B I and BI! a r e materials constants. If
both phases show the power-law creep, the creep rates can be given by E~3 = K I ( O I ) m = K, (O'3A3--fKEx)'(S-') •II £33 =
Ktt(a*lI ) n
= rti{o "A + (l - f ) K E x } " ( s -l) E33 = (1 --f)E~3 + f E l I ( s - l )
(11)
where K~, KH, m and n are materials constants. If the alloys exhibit the hyperbolic sine creep, the creep rates are given by g ~3 = ~q[sinh(fll(71 )]m
3.2. Transient creep resulted from strength difference between second phase and matrix Creep deformation of two-phase alloys can be calculated by the following procedure on the basis of the continuum mechanics model. (1) The values of ~3 and E33 .n are calculated under an initial condition of EI __ - % 1I = 0 at t = 0, when a tensile stress, a3A3is applied [equations (10), (I1) or (12)1. (2) Strain increments in both phases, AE,~and AEi], and the increment of strain difference, Ax, are obtained by equations (13) and (14) on the basis of each creep law, provided that the creep rate in each phase is constant during a short time interval
At(t <
= ~, {sinh[fll (aA3 -- fKEx)]}m (S - l )
0.024
-n = ~tH{sinh(flHail)}, E33 = cqt(sinh {fla(aA3 + (1
,,
•
(12)
where at, ~11, ill, flll, m, and n are materials constants. Therefore, the strain increment in each phase during an infinitesimal time interval Ats is given by AE~3 = ~I3At
A ~ = i~At.
(13)
(14)
3
/;
/
//--
A':A" = 10- ' B"E:3x'O ' B'"E=IO-a'~/"
|
/,' ..... cor~aot~ep r~te modeL~.'" lo-
/,'
~:o~o. ,:o2o
"
.S.'"
C00,i.~ro mechlrfics model ~ ' _ 5
L/oooo ,s(2,,o- y" ]/,:, :a/E ,/'" r2xlO 0"010 ~/, 'zx
jf
The increment in creep strain difference between two phases, Ax, is expressed as Ax = ae~3 - a e ~ .
_~
7
I
0020i-
-f)KEx)}y(s -~)
6~ = (1 - f ) ~ h +f~gg(s-b
1551
00
•
•
. ~ . ' • "."
_
-
-
5x16'
-
10 Time
20
30
(ks)
Fig. 1. Calculated creep curves of ductile two-phase alloys
based on the continuum mechanics model, where the second phase and the matrix exhibit the exponential law creep.
TANAKA et al.: CREEP DEFORMATION OF DUCTILE TWO-PHASE ALLOYS
1552
between the second phase and the matrix, although it does not appear in the calculation results based on the constant creep rate model. The transient creep is more prominent under the higher stresses. The slope of creep curves (creep rate) calculated by the continuum mechanics model approaches with time to the steady-state creep rate calculated by the constant creep rate model. The time to the steady-state creep increases with a decrease of applied stress. Figure 2 shows the effects of creep strength of the second phase on the configuration of the calculated creep curves on the two-phase alloys under a constant applied stress (gAffE = 2 x 10-5). The continuum mechanics model shows that the transient creep behavior is more remarkable when the strength difference between two phases is greater, that is, the difference in creep strain is larger. The time to the steady-state creep increases with increasing the strength difference between second phase and matrix. Figure 3 shows the change in the equivalent stresses in both second phase (tr],I) and matrix (a~) in the two-phase alloys described above during creep. The equivalent stress in each phase in the constant stress model can be calculated by putting E33 = ~13 ---- E3 3"11 = AI exp(Bza~ ) = An exp(Bll trn) and (7 A 33 = (1 - - f ) a ~ +f(r~ [. The continuum mechanics model shows that the equivalent stress in the strong second phase increases and that in the weak matrix decreases more rapidily with time under the higher applied stresses ((rA). The change in the stress values is larger under the higher applied stresses and in the alloys with greater strength difference between second phase and matrix. This is the reason that the transient creep is more prominent when the strength difference between two phases is larger and the applied stress is higher. These stress values approach with time to the steady-state values calculated by the constant creep rate model.
~ ,
0.030
-7 BrE=SXlO3 / , / A,=A,=IO, /,,' v=~3o, ,=o2o, ~,,~=zx~o ~ / , --continuum mechanicsmodel / , " -----constantcreepratem ~ , ~
o3
!~
0'0201
!~
//
,/numerals(50-lO3l:Bil'E "/
0.010 /
...........
0
10
20
9rE=SxlOp, BIrE=IO~, 9=-0.50, f=0.20 ~rals(~t~- 2x103): (~/E
4
[" 2x103 / - 1(~
5X10"4
2 .-. 0 ? o 12 [ '~
o
O~/E (~/E.._ continuummecilanicsn'~xlet ............. constantcreeprate model
10 /
~
~/~_-:-_-_~:__~_~,_.~_=~,_o'_ .
.
.
.
.
.
.
.
.
.
.
.
.
.
-
30
Time (,ks) Fig. 2. Effects of the creep strength of second phase on the configuration of creep curves calculated by the continuum mechanics model, where the second phase and the matrix exhibit the exponential law creep.
.
8
4
)
^,
~,~E=~x,o'/
I 0
\
1
10 20 Time (ks)
30
Fig. 3. The change in the equivalent stresses in second phase and matrix during creep calculated by the continuum mechanics model, where the second phase and the matrix exhibit the exponential law creep (A~= AH= 1 0 - 7 ) . The same results of the calculations were also obtained for the materials in which the stress dependence of the creep rate was described by other creep laws. 3.3. Steady-state creep in ductile two-phase alloys
In the steady-state creep, the creep rate of the ductile two-phase alloys, ~33, should be constant under a constant applied stress, a A d~33/dt = (d~33/dx)(dx/dt) = 0.
(15)
The first terms, d~33/dx, is not necessarily zero, because the internal stresses in the alloys are generally finite functions of the creep rate. Since the internal stresses in both second phase and matrix [equations (l) and (2)] are considered to be constant in the steady-state creep; dx /dt = dE~/dt - dE~3/dt = 0.
(16)
This equation also shows ~ 3 = E33 -n = •33' •
/ ,"sx'o2
,
6 / f[
(17)
The calculation results in the previous Section 3.2 and equation (17) indicate that the continuum mechanics model gives the same values of the steady-state creep rate as the constant creep rate model by McDanels et ai. [1]. Putting Ax = 0 in equation (14) and using equations (10) and (13), one can calculate the explicit solution of the difference in creep strain in the steady-state creep, x,, in the materials whose matrix and second phase show the exponential law creep x, = {ln(A1/Au) - ( B n - Bi)a3~ }/
( { B [ f + Bii(1 - f ) } K E ) .
(18)
TANAKA et al.: CREEP DEFORMATION OF DUCTILE TWO-PHASE ALLOYS
o.ooe 873K
0.004
,7
0.0O2
V ~
/
-
0 0'~>1,
~
/
./5.,';/
territe' ~ ," " pearlite ,// ° ' 1 6 9 ~ 4 . 2 8 6 * ,7 ~ / ' ~ steels/7 / ~ ( e
-
/
,,/~r~rals(0.169-0,532) ume"-frac-tion'-o't
-
~
T-"'. ~:.'-
e×perimentel rosults --
. . . . . . 5. . Time
1553
constantc~ep rate model
10~
. . . .
15
(ks)
Fig. 4. Calculated creep curves based on the continuum mechanics model and the experimental results in ferrite-pearlite steels under a stress of 49 MPa at 873 K. If the individual phases of the two-phase alloys exhibit the power-law creep [equation (11)], the values of ¢r~'3and x, are expressed as functions of the steady-state creep rate (~,) of the alloys cfA = (1 - - f ) ( d , / K I )'/"+ f ( i , / K n )
TM
Xs = [(~s/Kll) 1" -- (gs/KI)I/m]/(KE).
(19) (20)
For the hyperbolic sine law creep [equation (12)], the
values of a A and xs are obtain as follows aA3 = (1 --f){(l/fll)sinh-l(~/~ti) l/'}
+ f{(1/fln)sinh-I(~JOtn) 1/"}
(21)
x~ = {(l/fln)sinh -1 ((-s,/O~ll) TM -- (1/fll)sinh '(is/oq)l/"}/(KE).
were also shown in this figure. The continuum mechanics model can explain to some extent the transient creep behavior of ferrite-pearlite steels, while both the constant creep rate and the continuum mechanics models give somewhat larger creep strains than the experimental results. Figure 5 shows the experimental results by McDanels and co-workers and the results of calculations based on the continuum mechanics model or McDanels et al.'s constant creep rate model on copper containing continuous W wires (aspect ratio: c/a = 650; radius of wire: a = 6.35 x 10 -5 m) crept at 1089 K [1]. The stress dependence of steady-state creep rate of each phase is expressed as ~3---7.01 x
(22)
Equation (19) is the same as the one obtained by McDanels et al. [1]. It was confirmed in this study that the constant creep rate model gives the same values of the steady-state creep rate as the continuum mechanics model.
~ i = 5.62 x lO-27(a~)6"Sa(s-~)(W)
105
4. C O M P A R I S O N OF R E S U L T S OF
1021
&
EXPERIMENTAL R E S U L T S
Cu-W wires 1089 K 0 " 0 0 exlx~'imental
10 0
Figure 4 shows the calculated creep curves based on the continuum mechanics model and the experimental results in ferrite-pearlite steels crept under a stress of 49 M P a at 873 K. The stress dependence of steady-state creep rate is given by ~3 = 1.10 x 10-6[sinh(0.0230ale)]s39(s - ' )
(25)
where Young's moduli and Poisson's ratios of Cu [18] and W [19] were used in the numerical calculations.
104' THEORETICAL CALCULATIONS WITH
IO-16(O'Ie)SA7(S-I)(Cu)
~
i d 2<
b ¢, 164 2 MPe
- 166
o ~45 MPa
(23)
16 8
(24)
16'2
for ferrite matrix and is expressed as •n _-£33
6.13 × 10 -9 exp(0.0568aI,1)(s -])
for pearlite phase. The physical constants used in the numerical calculations are E = 1.65 x 105 MPa and v = 0.34 [17], and the shape of the minor phase (pearlite phase in ferrite matrix or ferrite phase in pearlite matrix) is assumed to be spherical. The calculation results of the constant creep rate model AM 39/7--L
0 0.2 0.4 0.6 0.8 1.0 Volume fraction of W wires, t Fig. 5. Comparison of the calculation results based on the
continuum mechanics model or the constant creep rate model and the experimental results by McDanels and co-workers in W-wire reinforced copper crept at 1089 K
(aspect ratio of W wire: c/a--. 650; radius of W wire: a = 6.35 × 10 -s m).
1554
TANAKA et al.: CREEP DEFORMATION OF DUCTILE TWO-PHASE ALLOYS
In these materials, the strong W wires are assumed to be uniformly embeded in the copper matrix up to a W wire volume fraction of about 0.7. The calculated values of steady-state creep rate based on both mechanistic models agree well with the experimental values.
5. CONCLUSIONS A continuum mechanics model was developed to explain the creep deformation of the ductile two-phase alloys. The results of the analysis based on the model were compared with the experimental results. 1. The continuum mechanics model predicted that the transient creep occurred owing to the internal stresses in both second phase and matrix arising from the strength difference between two phases, even if the inherent transient creep in both phases was not taken into account. The transient creep could not be predicted by McDanels et al.'s model. 2. In the steady-state creep of the ductile two-phase alloys, in which the individual phases exhibited the power law, the exponential law, or the hyperbolic sine law creep, the internal stresses and the steady-state creep rate of the alloys can be obtained analytically by the continuum mechanics model as functions of the volume fraction and the shape of the second phase, the applied stress and the materials constants of both phases. 3. The continuum mechanics model gave the same values of the steady-state creep rate as the constant creep rate model by McDanels et al. But, the continuum mechanics model can simulate to some extent the transient creep behavior observed in the ductile two-phase alloys.
REFERENCES
I. D. L. McDanels, R. A. Signorelli and J. W. Weeton, ASTM STP 427, p. 124. ASTM, Philadelphia, Pa (1967). 2. A. Kelly and W. R. Tyson, 3. Mech. Phys. Solids 14, 177 (1966). 3. E. G. Ellison and B. Harris, Appl. Mater Res. 5, 33 (1966). 4. N. S. Stoloff, in Alloys and Microstrutural Design (edited by J. K. Tien and G. S. Ansell) p. 65. Academic Press, New York (1976). 5. M. Suery and B. Baudelet, in Creep of Engineering Materials and Structures (edited by G. Bernasconi and G. Piatti), p. 47. Applied Science, London (1982). 6. T. G. Langdon, in Creep and Fracture of Engineering Materials and Structures (edited by B. Wilshire and D. R. J. Owen), p. 141. Pineridge Press, Swansea (1981). 7. D. D. Pearson, B. H. Kear and F. D. Lemkey, in Creep and Fracture of Engineering Materials and Structures (edited by B. Wilshire and D. R. J. Owen), p. 213. Pineridge Press, Swansea (1981). 8. R. Lagneborg, in Creep and Fatigue in High Temperature Alloys (edited by J. Bressers), p. 41 Applied Science, London (1981). 9. F. Garofalo, Fundamentals of Creep and Creep-rupture in Metals (translated by M. Adachi), p. 50. Maruzen, Tokyo (1968). 10. V. Lupinc, in Creep and Fatigue in High Temperature Alloys (edited by J. Bressers), p. 14. Applied Science, London (1981). 11. J. D. Eshelby, Proc. R. Soc. A241, 376 (1957). 12. T. Mura and T. Mori, Micromechanics 23 (1976). 13. T. Mura, Micromechanics of Defects in Solids. Martinus Nijhoff, The Hague (1982). 14. T. Mori and K. Tanaka, Acta metall. 21, 571 (1973). 15. K. Tanaka and T. Mori, Acta metall, lg, 931 (1970). 16. K. Tanaka, K. Wakashima and T. Mori, J. Mech. Phys. Solids 21, 207 (1973). 17. Technical Data, Elastic Moduli of Metallic Materials, p. 137. Japan Soc. Mech. Engrs, Tokyo (1980). 18. Technical Data, Elastic Moduli of Metallic Materials, p. 157. Japan Soc. Mech. Engrs Tokyo (1980). 19. Technical Data, Elastic Moduli of Metallic Materials, p. 207. Japan Soc. Mech. Engrs, Tokyo. (1980).