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On the analysis of stress relaxation data
Journal of Nuclear Materials 96 (1981) 178-186 0 North-Holland Publishing Company
ON THE ANALYSIS OF STRESS RELAXATION DATA F . POVOLO Con&i& National de Energia Atdmica, Dto. de Materiales, Au. de1 Libertador 8250, Buenos Aires, Comisi6n de Investigaciones Cientificas de la Provincia de Buenos Aires, Argentina
Received 19 May 1980
The various equations, used in the literature to describe the stress relaxation behaviour of metals and alloys, are shown in normalized stress versus strain rate plots, in such a way that the predictions of the different theoretical models can be compared directly. Finally, it is shown how to fit the experimental stress relaxation curves to a particular theoretical model and an example is given for load relaxation data taken in Zry4 at 673 K.
last equation is generally used for an analysis of the experimental o(t) curves. As pointed out by Dotsenko [4], however, the processing of the experimental relaxation curves is not always easy to perform and the use of a wrong equation might give serious errors in the calculated dislocation parameters. Hart [.5,6] has used a state variable approach for the interpretation of the stress relaxation curves. The important variables in this phenomenological model of plastic deformation are: stress, plastic strain rate, temperature and hardness. The experimental stress relaxation curves are considered in a log u versus log P diagram and not as stress versus time. The deformation model consists of essentially two parallel branches [S-7]. The stresses in this two branches are uf and u, respectively and their sum equals the applied stress u. At high homologous temperatures the constant hardness log u - log P curves can be represented by a flow law
1. Introduction Stress relaxation experiments are widely used to obtain information on the elementary dislocation processes responsible for the plastic deformation of solids. There are several reviews in the literature dealing with the analysis of stress relaxation experiments [l-3]. In a more recent one, the fundamental aspects of the stress relaxation in crystals and the determination of the activation parameters from the experimental data are analyzed by Dotsenko [4]. The theoretical expressions needed to relate the experimental data to elementary dislocation models are based on the equations [4] t = -(b/M),
(1)
i = cppbi7,
(2)
where ~5 is the measured stress variation rate, M the effective modulus of the sample and machine system, P the plastic strain rate, p the mobile dislocation density, b the Burgers vector, cpan orientation factor and U the average dislocation velocity. For stress relaxation, it is generally assumed that the mobile dislocation density is constant and with a particular dependence of the average dislocation velocity with the effective stress, a differential equation between CJand ir is obtained, by combining eqs. (1) and (2), which integrated gives the stress as a function of time. This
ln(u*/u,)
= (&*/a;)’ ,
(3)
where u* is a hardness parameter, h is a temperature Independent parameter and e* is a parameter which depends on temperature, heat treatment and deformation. In the region of stress and temperature where eq. (3) is applicable, ua approaches u, h approaches & and the measured log u -log 6 are concave downward. 178
F. Povolo/ On the analysisof stressrelaxationdata The curves for different hardness are related translation along a straight line in the log space. At low homologous temperatures, the hardness log u - log P curves are represented e = Ci’(o#r?i~
through u - log e constant by
)
(4)
where ri* is a rate parameter, or an effective stress,M is a constant and 311 the anelastic modulus. In the region where eq. (4) is applicable, the measured log u - log i curves are concave upward. It will be shown how all the proposed theoretical expressions, used to describe the stress relaxation behaviour of crystals, can be plotted in a normalized log u -log P diagram, in such a way that a direct comparison can be made between the predictions of the different theoretical expressions or with the experimental stress relaxation data.
179
with u* = kT/V* ;
e* = cppbvo ;
A = (OiV* + AGo)/kT. The strain rate sensitivity parameter, m = dln a/ dln e = dlog u/dlog P, is obtained, differentiating eq. (6) as m = l/[A t ln(&/P*)] .
(6.1)
Eqs. (6) and (6.1) are shown in fig. 1 for different values of A. 2.2. Viscous dislocation motion In this case BiY=Cbb, where B is the viscous drag constant. Fin eq. (2)
Substituting
P = pb’p(U - Oi)/B .
2. Theory
Rearranging When eq. (2) is combined with the various expressions proposed in the literature to relate the average dislocation velocity to the effective stress [4], different normalized log u versus log P relationships can be obtained.
and taking the logarithm,
log(u/cJ*) = log(1 + i/P’) )
(7)
with P’ = pb' p/Boi ;
Uie
U*=
m is given by
2.1. Thermally activated dislocation motion
nr = l/(1 + 6*/i), The relationship
for
used in this case is
iT= u. exp[-AG@)/kT]
,
Eqs. (7) and (7.1) are plotted in fig. 2. (5)
where u. is the preexponential factor, AC(C) the change in free energy, O= u - oi the effective stress, oi the internal stress, k the Boltzmann constant and T the absolute temperature. On assuming that the free energy changes linearly with the effective stress, i.e. AG@) = AGO - V’O , where V’ is the activation volume, the following relationship is obtained by substituting eq. (5) into eq. (2)
u/u* =A + ln(P/e*) ,
2.3. Empirical equations relating average dislocation velocity to effective stress (1) Johnston tionship u=K&
and Gilman
[8] obtained
the rela-
,
where K and m’ are material constants perature. Substituting into eq. (2),
at a given tem-
f = ppbK(o - Ui)m* or
e = cppb~o exp{ - [AGO - V*(U - ui)] /kT}. Taking the logarithm
(7.1)
u/u* = 1 + (i/Z)““’
of both sides and rearranging,
,
with (6)
P’ = VpbKaim*
;
U*=Ui*
63)
F. Povolo /On the analysisof stress relaxation data
180
Fig. 1. Plot of log(o/o*) versus log(h/d*) (full curves) according to eq. (6) and m versus log(c/i*) (broken curves) according to eq. (6.1), for different values of A.
m
isgiven by
na = (r/C)““*/[l
+ (e/6*)““*] m* .
where u. is a limiting velocity and D a characteristic drag stress. Substituting into eq. (2), (8.1)
i = lppbUeexp[-D/(o
- Oi)] .
Eqs. (8) and (8.1) are plotted in fig. 3 as log(o/o*) versus log(P/G*) and m versus log(k/k”), respectively. (2) Gilman [9] proposed the relationship
This equation can be expressed as
U = u. exp(-D/5) ,
with
u/o* = 1 - (~/Ui)/ln(~/~*) ,
4*=gpbVo
;
(9)
U*=Oi s
The strain rate sensitivity parameter is given by M = (~/uj)~ [ln(&?)]2 f 1 - (~/~)/ln(~~~*)] *
(99.1)
Eqs. (9) and (9.1) are plotted in fig. 4 for different values of D/q and u & Ui, 6 < I+*as required by the model. (3) Gillis and Sargent [ 101 used the relationship ti = @f(u - u. - &) u. exp [-(D + HE)/u] , where D, H, M, P and u. are constants, This equation can be written as Fig. 2. Plot of log{&*) versus log(;/~*) according to eq. (7) and m versus log@/&*) according to eq. (7.1).
e/I;* = [(u/u*) - A] exp(-u*/u)
,
(10)
F. Povoio /On the analysis of stress relaxation data
181
Fig. 3. Plot of lug(o/o*) WTSUS log(;/i*)(full curves) according to eq. (8) and m versus log(i/d*) (broken curves) according to eq. (83, for different values of m*.
with u* =D+He
;
ti* = ~u*(D
tally constant so that u*, 6’ and A can be considered as constants. The strain rate sens~ti~ty p~amet~r is given by
+ HE) ;
A = (iJo + fk)/(D +i%$ .
During each stress relaxation run, E remains practi-
1 -.4(0*/o)]
.
(10.1)
Eqs. (IO) and (10.1) are plotted in fig. 5 for different values of A. (4) An exponential relationship.between the average dislocation velocity and the effective stress has been sometime used to describe stress relaxation behaviour [2]. Since the hyperbolic sine include the exponenti~ as a particular case, the following relationship will be considered [ 1 l]
E g
m = [(o/u*) - A I/[ (u/u*) +
10
$09 Foe 0.7 0.6
V= Ua sinh]C(u -
Ui)]
,
OS
where uo and C are constants. Substituting into eq. G)>
0.L 03
a2
P = @YJbUe sinh [C(U - Ui)] ,
0.1
which can be written as
0 -7
-6
-5
4
'-3
-2
0 log csh
u/o* = 1 + (l/COi) sinh-’ (P/i’) .
Fig. 4. Plot of logfa.!u*) versuslog{iJ~*)(full curves) according to eq. (9) and m versus log(&*) (broken curves) accord-
with
ing to eq. (9.11, for different values of D/u~.
P*=cppbu,;
o*=q.
(11)
F. Povolo / On the analysis of stress relaxation data
182
1.4
bg
The strain rate sensitivity
(u/u’ ),m
parameter is given by
m = (e/f!*)(l/CUi)/ [ 1 + (C/i*)2] 1’2
t
X [ 1 + (l/Coi)
sinh-’ (e/e*)] .
(11.1)
Eqs. (11) and (11 .l) are plotted in fig. 6 for different values Of 1/CUi.
2.4. Analysis ofsleeswyk et al. [12] By making some assumptions these authors found a relationship between stress and strain rate of the type u/u0 = 1 - (cr/uO>ln(M@) ,
(12)
where u. and PO are the applied stress and the strain rate, respectively, at the beginning of the relaxation and (Yis a constant equal to da/din P. m is obtained by differentiating eq. (12) as
m = @/uo)/ [1 - (41~0) Wd0/41
.
(12.1)
Fig. 5. Plot of log(o/o*) versus log(i/E*) (full curves) according to eq. (10) and m versus log(i/i*) (broken curves) according to eq. (lOA), for different values of.4.
A
i0g
(u/u’),
m
I
,
,
\
I
1.L -4
I
: \ :5
1I
1.2 _-
I
)
\
\ \ \ \ \
I
1.0 --
0.8--
0.6--
0.11 --
0.2~-----_-____ 0
7 -f+
0.01
1" -3
Fig. 6. Plot of log(u/o*) versus log(i/i*) eq. (ll.l), for different values of l/Cai.
(full curves) according to eq. (11) and m versus log(l/k*) (broken curves) according to
183
\\
‘\ \
\\
‘\ \ \
‘--.
--Ye? -.
Fig. 7. Plot of log(a/a*) versus log(;/i’) eq. (12.1), for different values of a/a*.
\ \
1.0 J
'p' \ \
0.6 --
-0.6
x
\
---_
(full curves) according to eq. (12) and m versus log(i/i*)
(broken curves) according to
';0.4 \ I \ \ !
W! *
z
Q-
-0.8 -1.0
?
! -2.0 ’
s
” u,
-i
!
0
1
Log(6 ! fi”) I
!
2
3
I
i
5
!
6
!
7
Fig. 8. Plot of log(a/o*) versus log(E/i*) (full curves) according to eq. (13) and m versus log(i/i’) eq. (13.1), for different values of A.
!
!
!,
8
9
10
J
(broken curves) according to
F. Povolo / On the analysis of stress relaxation data
184
Eqs. (12) and (12.1) are plotted in fig. 7 for different values of ofa* where u* = u. and &*= PO. 2.5. Hart’s phenomenological model / .5,6] At high homologous temperatures, applied with u = ua and & = P, so that u/u* = exp[-(e*/P)“]
,
and the strain rate sensitivity m = h(P*/i)” .
eq. (3) can be
(13) parameter
is (13.1)
Eqs. (13) and (13.1) are plotted in fig. 8 for several values of h. Finally, eq. (4) is similar to eq. (8) so that the curves shown in fig. 3 are valid in this case too.
3. Applications Fig. 9 shows some load relaxation data taken at 673 K, in Zry4 flat specimens [ 131 cutted with the tensile axis parallel to the rolling direction. The specimen was cold worked 64% by rolling and then stressrelieved at 813 K for 1 h. The results correspond to three succesive relaxations at increasing strains. The plastic strain rate was obtained from the stress variation rate by using eq. (1) [ 141 .M was calculated from the specimen dimensions and the constant of the machine measured as suggested by Povolo [ 151. The log u versus log P experimental curves are plotted in the same scale as the theoretical log(u/u*) versus log(i/e*) curves of figs. 1 to 8. Since dividing
&I T(M Pa)
by a constant means, in a logarithmic scale, a translation along the axis, the experimental curves of fig. 9 can be compared with the theoretical expressions by superimposing this figure to figs. 1 to 8 and matching (when this is possible) the experimental curves with the theoretical ones by translations along the axis (without rotations). If the matching is possible, the parameters u* and P* can be obtained from the coincidence of some value of log u and log P from the experimental curves with the corresponding log(u/u*) and log(&/i*) from the theoretical ones. The other parameters, for example A of fig. 1, are readed directly. By using this procedure, it can be easily seen that the data of fig. 9 can be matched, within experimental errors, to some of the curves of fig. 8, with the parameters given in table 1. The rest of the theoretical expressions, described in section 2, do not fit the experimental data. The theoretical curves calculated with eq. (13), with the parameters given in table 1, are compared with the experimental values of fig. 9 in fig. 10a. Fig. 10b shows a comparison between the m values predicted by eq. (13.1) and the experimental ones obtained by taking the derivative of the average curves shown in fig. 9. The agreement is fairly good in this case too, since a large error is introduced in the calculations of m, specially at both ends of the experimental average curves. In conclusion, it may be said that the log u - log P experimental curves shown in fig. 9 are well described by Hart’s phenomenological model for high homologous temperatures described by eq. (13). The physical meaning of the parameters is not clear, however, when eq. (13) is expressed as i =Au), the following relationship is obtained P = i* exp{ -(l/A)
ln[-ln(u/u*)]}
,
Table 1 Relaxation parameters obtained by fitting the experimental data of fig. 9 to the theoretical curves of fig. 8 Relaxation a* h e* number (s-l ) CmPa)
___
Fig. 9. Load relaxation data obtained, at 673 K, in three successive relaxations of a stress-relieved 2ry-Q flat specimen. First relaxation [full circles): E = 0.8%; second relaxation (open circles): E = 1.4%; third relaxation (crosses): E = 2.2%.
1 2 3
---3.74
x lo-lo 4.04 x 1o-‘o 7.05 x 1o-9
285.76 351.27 446.68
0.25 0.15 0.15
F. Povolo / On the anaiysisof stress relaxation data
185
mation variables u, P and E can be related by ~ogu=~de+~dlog~, where y is connected with the strain hardening coefficient and m is the strain rate sensitivity parameter already mentioned in the paper. During stress relaxation y is small so that with a good approximation m = dlog a/dlog P . The stress relaxation behaviour is completely characterized by the variables u, P and m. This is implicitly assumed, in the different theories, when eq. (2) is used with the assumption that the dislocation density remains constant and work hardening is negligeable. The stress relaxation curves, however, are generally considered in a stress versus time diagram. When eq. (5), for example, is viewed as u = u(t) the solution is
2.8
2.4
2.2
[I61 U(f)Ui=A(t+Q)-n 3
Log 6cs-‘,
2.0
I
-9
-8
I
1
-7
-0
I
/
1
-5
-4
I
Fig. 10. (a) Comparison
between the experiment& data of fii. 9 and those predicted by eq. (13) (full curves), with the parameters given in table 1. (b) Comparison between the m values obtained from the average curves of fii. 9 (fuil and open circles and crosses and those calculated with eq. (13.1) (full curves), with the parameters given in table 1. The relaxation number is indicated on each curve.
where n = l/(m* - l), A = [(m* - 1)&K]-” and II is an integration constant. When the stress dependence of the average disiocation density is of the type iT= u. exp(-D/7$ , which in a log CT- log e diagram leads to eq, (9), but in a o versus t diagram gives the differential equation -o(t) = Mpbquo exp [-D/$f)]
which is very difficult to visualize in terms of eq. (2) that relates the experimental data to elementary dislocation models. The data of fig. 9 will not be discussed further since they are given as an example of application of the method.
4. Discussion
(14)
.
(15)
This differential equation cannot be integrated to give u = u(t). Finally, eq. (5) and a linear dependence of AG from the effective stress give a u = u(t) relationship u(O)-u(~)=~ln~~t
I),
(16)
where u(O) is the stress at the beginning of the relaxation (t = 0), (Y= kT/V* and fi = Ml(O)far. e(O) is the plastic strain rate at the beginning of the relaxation (t = 0).
It is well known that the time is not a good variable for the description of the mechanical behaviour of crystals. As pointed out by Hart [S], in fact, the instantaneous mechanical state of a material is completely characterized by the stress and strain rate which is capable of sustaining at that moment. Furthermore, for small increments, the three defor-
It is evident that the predictions of eqs. (14)-( 16) cannot be directly compared in a u versus t diagram. The situation is completely different when the problem is considered in a log u -1ogP plot. Furthermore, the calculation of the different parameters is much easier in the last representation. In conclusion, it is suggested that the experimental
186
F. Povolo /On the a~atys~s of stress rekxation data
stress relaxation data should be considered in a log u -log C plot since this has more physical meaning than a u versus t representation. A direct comparison between the predictions of the different theoretical models is possible and the calculation of the different parameters is greatly simplified, with no additional assumptions. It must be pointed out that several orders of magnitude in a CTversus t plot does not necessarily means the same extension in a log CI- log &representation.
5. Conclusions It was shown that the different theoretical expressions proposed in the literature to describe the stress relaxation behaviour of crystals, can be plotted in a normalized log (I -log P plot in such a way that a direct comparison between the predictions of the different models can be made. Furthermore, the experimental stress relaxation curves can be easily compared with those given by the theoretical models and the evaluation of the different parameters is greatly simplified with respect to a stress versus time plot.
Acknowledgements This work was performed within the Special Intergovernamental Agreement between Argentina and the
Federal republic of Germany and was supported in part by the Comision de Investigaciones Cientffmas de la Provincia de Buenos Aires.
References [ 11 E.C. Aifantis and W.W. Gerberich,Mater. Sci. Eng. 21 (1975) 107. [2] G. Baur and P. Lehr, M&m. Sci. Rev. Mkt. 718 (1975) 551. [3] R. de Bat&, Reviews on the Reformation Behaviour of Materials, Vol. 1, No. 2 (1975). [4] V.I. Dotsenko, Phys. Status Solidi (b) 93 (1979) 1 I. [S] E.W. Hart, Acta Met. 18 (1970) 599. [6] E.W. Hart, J. Eng. Mater. Technol. 98 (1976) 193. [7] E.W. Hart, C.Y. Li, H. Yamada and G.L. Wire, Constitutive Equations in Plasticity, Ed. AS, Argon (MIT Press, Cambridge, 1975) p. 149. [B] W.G. Johnston and J.J. Gilman, J. Appl. Phys. 30 (1959) 139. [9] J.J. Gilman, Gust&. J. Phys. 13 (1960) 327. [IO] P.P. Gillis and G.A. Sargent, Scripta Met. 6 (1971) 451. [ll] AS. Krausz, Mater. Sci. Eng. 4 (1969) 193. fl2] A.W. Sleeswyk, G.H. Boersma, G. Hunt and D.J. Verel, in: Proc. 2nd Intern. Conf. on Strength of Metals and Alloys, Vol. I (ASM, 1970) p. 204. [ 131 F. Povoto and I. Alvarez, Load relaxation in &y-4 at 673 K, to be published. [14] F. Guiu and P.L. Pratt, Phys. Status Solidi 6 (1964) 111. [ 151 F. Povolo, J. Nucl. Mater. 78 (1978) 309. 1161 J.C.M. Li, Canad. J. Phys. 45 (1967) 493.
ON THE ANALYSIS OF STRESS RELAXATION DATA F . POVOLO Con&i& National de Energia Atdmica, Dto. de Materiales, Au. de1 Libertador 8250, Buenos Aires, Comisi6n de Investigaciones Cientificas de la Provincia de Buenos Aires, Argentina
Received 19 May 1980
The various equations, used in the literature to describe the stress relaxation behaviour of metals and alloys, are shown in normalized stress versus strain rate plots, in such a way that the predictions of the different theoretical models can be compared directly. Finally, it is shown how to fit the experimental stress relaxation curves to a particular theoretical model and an example is given for load relaxation data taken in Zry4 at 673 K.
last equation is generally used for an analysis of the experimental o(t) curves. As pointed out by Dotsenko [4], however, the processing of the experimental relaxation curves is not always easy to perform and the use of a wrong equation might give serious errors in the calculated dislocation parameters. Hart [.5,6] has used a state variable approach for the interpretation of the stress relaxation curves. The important variables in this phenomenological model of plastic deformation are: stress, plastic strain rate, temperature and hardness. The experimental stress relaxation curves are considered in a log u versus log P diagram and not as stress versus time. The deformation model consists of essentially two parallel branches [S-7]. The stresses in this two branches are uf and u, respectively and their sum equals the applied stress u. At high homologous temperatures the constant hardness log u - log P curves can be represented by a flow law
1. Introduction Stress relaxation experiments are widely used to obtain information on the elementary dislocation processes responsible for the plastic deformation of solids. There are several reviews in the literature dealing with the analysis of stress relaxation experiments [l-3]. In a more recent one, the fundamental aspects of the stress relaxation in crystals and the determination of the activation parameters from the experimental data are analyzed by Dotsenko [4]. The theoretical expressions needed to relate the experimental data to elementary dislocation models are based on the equations [4] t = -(b/M),
(1)
i = cppbi7,
(2)
where ~5 is the measured stress variation rate, M the effective modulus of the sample and machine system, P the plastic strain rate, p the mobile dislocation density, b the Burgers vector, cpan orientation factor and U the average dislocation velocity. For stress relaxation, it is generally assumed that the mobile dislocation density is constant and with a particular dependence of the average dislocation velocity with the effective stress, a differential equation between CJand ir is obtained, by combining eqs. (1) and (2), which integrated gives the stress as a function of time. This
ln(u*/u,)
= (&*/a;)’ ,
(3)
where u* is a hardness parameter, h is a temperature Independent parameter and e* is a parameter which depends on temperature, heat treatment and deformation. In the region of stress and temperature where eq. (3) is applicable, ua approaches u, h approaches & and the measured log u -log 6 are concave downward. 178
F. Povolo/ On the analysisof stressrelaxationdata The curves for different hardness are related translation along a straight line in the log space. At low homologous temperatures, the hardness log u - log P curves are represented e = Ci’(o#r?i~
through u - log e constant by
)
(4)
where ri* is a rate parameter, or an effective stress,M is a constant and 311 the anelastic modulus. In the region where eq. (4) is applicable, the measured log u - log i curves are concave upward. It will be shown how all the proposed theoretical expressions, used to describe the stress relaxation behaviour of crystals, can be plotted in a normalized log u -log P diagram, in such a way that a direct comparison can be made between the predictions of the different theoretical expressions or with the experimental stress relaxation data.
179
with u* = kT/V* ;
e* = cppbvo ;
A = (OiV* + AGo)/kT. The strain rate sensitivity parameter, m = dln a/ dln e = dlog u/dlog P, is obtained, differentiating eq. (6) as m = l/[A t ln(&/P*)] .
(6.1)
Eqs. (6) and (6.1) are shown in fig. 1 for different values of A. 2.2. Viscous dislocation motion In this case BiY=Cbb, where B is the viscous drag constant. Fin eq. (2)
Substituting
P = pb’p(U - Oi)/B .
2. Theory
Rearranging When eq. (2) is combined with the various expressions proposed in the literature to relate the average dislocation velocity to the effective stress [4], different normalized log u versus log P relationships can be obtained.
and taking the logarithm,
log(u/cJ*) = log(1 + i/P’) )
(7)
with P’ = pb' p/Boi ;
Uie
U*=
m is given by
2.1. Thermally activated dislocation motion
nr = l/(1 + 6*/i), The relationship
for
used in this case is
iT= u. exp[-AG@)/kT]
,
Eqs. (7) and (7.1) are plotted in fig. 2. (5)
where u. is the preexponential factor, AC(C) the change in free energy, O= u - oi the effective stress, oi the internal stress, k the Boltzmann constant and T the absolute temperature. On assuming that the free energy changes linearly with the effective stress, i.e. AG@) = AGO - V’O , where V’ is the activation volume, the following relationship is obtained by substituting eq. (5) into eq. (2)
u/u* =A + ln(P/e*) ,
2.3. Empirical equations relating average dislocation velocity to effective stress (1) Johnston tionship u=K&
and Gilman
[8] obtained
the rela-
,
where K and m’ are material constants perature. Substituting into eq. (2),
at a given tem-
f = ppbK(o - Ui)m* or
e = cppb~o exp{ - [AGO - V*(U - ui)] /kT}. Taking the logarithm
(7.1)
u/u* = 1 + (i/Z)““’
of both sides and rearranging,
,
with (6)
P’ = VpbKaim*
;
U*=Ui*
63)
F. Povolo /On the analysisof stress relaxation data
180
Fig. 1. Plot of log(o/o*) versus log(h/d*) (full curves) according to eq. (6) and m versus log(c/i*) (broken curves) according to eq. (6.1), for different values of A.
m
isgiven by
na = (r/C)““*/[l
+ (e/6*)““*] m* .
where u. is a limiting velocity and D a characteristic drag stress. Substituting into eq. (2), (8.1)
i = lppbUeexp[-D/(o
- Oi)] .
Eqs. (8) and (8.1) are plotted in fig. 3 as log(o/o*) versus log(P/G*) and m versus log(k/k”), respectively. (2) Gilman [9] proposed the relationship
This equation can be expressed as
U = u. exp(-D/5) ,
with
u/o* = 1 - (~/Ui)/ln(~/~*) ,
4*=gpbVo
;
(9)
U*=Oi s
The strain rate sensitivity parameter is given by M = (~/uj)~ [ln(&?)]2 f 1 - (~/~)/ln(~~~*)] *
(99.1)
Eqs. (9) and (9.1) are plotted in fig. 4 for different values of D/q and u & Ui, 6 < I+*as required by the model. (3) Gillis and Sargent [ 101 used the relationship ti = @f(u - u. - &) u. exp [-(D + HE)/u] , where D, H, M, P and u. are constants, This equation can be written as Fig. 2. Plot of log{&*) versus log(;/~*) according to eq. (7) and m versus log@/&*) according to eq. (7.1).
e/I;* = [(u/u*) - A] exp(-u*/u)
,
(10)
F. Povoio /On the analysis of stress relaxation data
181
Fig. 3. Plot of lug(o/o*) WTSUS log(;/i*)(full curves) according to eq. (8) and m versus log(i/d*) (broken curves) according to eq. (83, for different values of m*.
with u* =D+He
;
ti* = ~u*(D
tally constant so that u*, 6’ and A can be considered as constants. The strain rate sens~ti~ty p~amet~r is given by
+ HE) ;
A = (iJo + fk)/(D +i%$ .
During each stress relaxation run, E remains practi-
1 -.4(0*/o)]
.
(10.1)
Eqs. (IO) and (10.1) are plotted in fig. 5 for different values of A. (4) An exponential relationship.between the average dislocation velocity and the effective stress has been sometime used to describe stress relaxation behaviour [2]. Since the hyperbolic sine include the exponenti~ as a particular case, the following relationship will be considered [ 1 l]
E g
m = [(o/u*) - A I/[ (u/u*) +
10
$09 Foe 0.7 0.6
V= Ua sinh]C(u -
Ui)]
,
OS
where uo and C are constants. Substituting into eq. G)>
0.L 03
a2
P = @YJbUe sinh [C(U - Ui)] ,
0.1
which can be written as
0 -7
-6
-5
4
'-3
-2
0 log csh
u/o* = 1 + (l/COi) sinh-’ (P/i’) .
Fig. 4. Plot of logfa.!u*) versuslog{iJ~*)(full curves) according to eq. (9) and m versus log(&*) (broken curves) accord-
with
ing to eq. (9.11, for different values of D/u~.
P*=cppbu,;
o*=q.
(11)
F. Povolo / On the analysis of stress relaxation data
182
1.4
bg
The strain rate sensitivity
(u/u’ ),m
parameter is given by
m = (e/f!*)(l/CUi)/ [ 1 + (C/i*)2] 1’2
t
X [ 1 + (l/Coi)
sinh-’ (e/e*)] .
(11.1)
Eqs. (11) and (11 .l) are plotted in fig. 6 for different values Of 1/CUi.
2.4. Analysis ofsleeswyk et al. [12] By making some assumptions these authors found a relationship between stress and strain rate of the type u/u0 = 1 - (cr/uO>ln(M@) ,
(12)
where u. and PO are the applied stress and the strain rate, respectively, at the beginning of the relaxation and (Yis a constant equal to da/din P. m is obtained by differentiating eq. (12) as
m = @/uo)/ [1 - (41~0) Wd0/41
.
(12.1)
Fig. 5. Plot of log(o/o*) versus log(i/E*) (full curves) according to eq. (10) and m versus log(i/i*) (broken curves) according to eq. (lOA), for different values of.4.
A
i0g
(u/u’),
m
I
,
,
\
I
1.L -4
I
: \ :5
1I
1.2 _-
I
)
\
\ \ \ \ \
I
1.0 --
0.8--
0.6--
0.11 --
0.2~-----_-____ 0
7 -f+
0.01
1" -3
Fig. 6. Plot of log(u/o*) versus log(i/i*) eq. (ll.l), for different values of l/Cai.
(full curves) according to eq. (11) and m versus log(l/k*) (broken curves) according to
183
\\
‘\ \
\\
‘\ \ \
‘--.
--Ye? -.
Fig. 7. Plot of log(a/a*) versus log(;/i’) eq. (12.1), for different values of a/a*.
\ \
1.0 J
'p' \ \
0.6 --
-0.6
x
\
---_
(full curves) according to eq. (12) and m versus log(i/i*)
(broken curves) according to
';0.4 \ I \ \ !
W! *
z
Q-
-0.8 -1.0
?
! -2.0 ’
s
” u,
-i
!
0
1
Log(6 ! fi”) I
!
2
3
I
i
5
!
6
!
7
Fig. 8. Plot of log(a/o*) versus log(E/i*) (full curves) according to eq. (13) and m versus log(i/i’) eq. (13.1), for different values of A.
!
!
!,
8
9
10
J
(broken curves) according to
F. Povolo / On the analysis of stress relaxation data
184
Eqs. (12) and (12.1) are plotted in fig. 7 for different values of ofa* where u* = u. and &*= PO. 2.5. Hart’s phenomenological model / .5,6] At high homologous temperatures, applied with u = ua and & = P, so that u/u* = exp[-(e*/P)“]
,
and the strain rate sensitivity m = h(P*/i)” .
eq. (3) can be
(13) parameter
is (13.1)
Eqs. (13) and (13.1) are plotted in fig. 8 for several values of h. Finally, eq. (4) is similar to eq. (8) so that the curves shown in fig. 3 are valid in this case too.
3. Applications Fig. 9 shows some load relaxation data taken at 673 K, in Zry4 flat specimens [ 131 cutted with the tensile axis parallel to the rolling direction. The specimen was cold worked 64% by rolling and then stressrelieved at 813 K for 1 h. The results correspond to three succesive relaxations at increasing strains. The plastic strain rate was obtained from the stress variation rate by using eq. (1) [ 141 .M was calculated from the specimen dimensions and the constant of the machine measured as suggested by Povolo [ 151. The log u versus log P experimental curves are plotted in the same scale as the theoretical log(u/u*) versus log(i/e*) curves of figs. 1 to 8. Since dividing
&I T(M Pa)
by a constant means, in a logarithmic scale, a translation along the axis, the experimental curves of fig. 9 can be compared with the theoretical expressions by superimposing this figure to figs. 1 to 8 and matching (when this is possible) the experimental curves with the theoretical ones by translations along the axis (without rotations). If the matching is possible, the parameters u* and P* can be obtained from the coincidence of some value of log u and log P from the experimental curves with the corresponding log(u/u*) and log(&/i*) from the theoretical ones. The other parameters, for example A of fig. 1, are readed directly. By using this procedure, it can be easily seen that the data of fig. 9 can be matched, within experimental errors, to some of the curves of fig. 8, with the parameters given in table 1. The rest of the theoretical expressions, described in section 2, do not fit the experimental data. The theoretical curves calculated with eq. (13), with the parameters given in table 1, are compared with the experimental values of fig. 9 in fig. 10a. Fig. 10b shows a comparison between the m values predicted by eq. (13.1) and the experimental ones obtained by taking the derivative of the average curves shown in fig. 9. The agreement is fairly good in this case too, since a large error is introduced in the calculations of m, specially at both ends of the experimental average curves. In conclusion, it may be said that the log u - log P experimental curves shown in fig. 9 are well described by Hart’s phenomenological model for high homologous temperatures described by eq. (13). The physical meaning of the parameters is not clear, however, when eq. (13) is expressed as i =Au), the following relationship is obtained P = i* exp{ -(l/A)
ln[-ln(u/u*)]}
,
Table 1 Relaxation parameters obtained by fitting the experimental data of fig. 9 to the theoretical curves of fig. 8 Relaxation a* h e* number (s-l ) CmPa)
___
Fig. 9. Load relaxation data obtained, at 673 K, in three successive relaxations of a stress-relieved 2ry-Q flat specimen. First relaxation [full circles): E = 0.8%; second relaxation (open circles): E = 1.4%; third relaxation (crosses): E = 2.2%.
1 2 3
---3.74
x lo-lo 4.04 x 1o-‘o 7.05 x 1o-9
285.76 351.27 446.68
0.25 0.15 0.15
F. Povolo / On the anaiysisof stress relaxation data
185
mation variables u, P and E can be related by ~ogu=~de+~dlog~, where y is connected with the strain hardening coefficient and m is the strain rate sensitivity parameter already mentioned in the paper. During stress relaxation y is small so that with a good approximation m = dlog a/dlog P . The stress relaxation behaviour is completely characterized by the variables u, P and m. This is implicitly assumed, in the different theories, when eq. (2) is used with the assumption that the dislocation density remains constant and work hardening is negligeable. The stress relaxation curves, however, are generally considered in a stress versus time diagram. When eq. (5), for example, is viewed as u = u(t) the solution is
2.8
2.4
2.2
[I61 U(f)Ui=A(t+Q)-n 3
Log 6cs-‘,
2.0
I
-9
-8
I
1
-7
-0
I
/
1
-5
-4
I
Fig. 10. (a) Comparison
between the experiment& data of fii. 9 and those predicted by eq. (13) (full curves), with the parameters given in table 1. (b) Comparison between the m values obtained from the average curves of fii. 9 (fuil and open circles and crosses and those calculated with eq. (13.1) (full curves), with the parameters given in table 1. The relaxation number is indicated on each curve.
where n = l/(m* - l), A = [(m* - 1)&K]-” and II is an integration constant. When the stress dependence of the average disiocation density is of the type iT= u. exp(-D/7$ , which in a log CT- log e diagram leads to eq, (9), but in a o versus t diagram gives the differential equation -o(t) = Mpbquo exp [-D/$f)]
which is very difficult to visualize in terms of eq. (2) that relates the experimental data to elementary dislocation models. The data of fig. 9 will not be discussed further since they are given as an example of application of the method.
4. Discussion
(14)
.
(15)
This differential equation cannot be integrated to give u = u(t). Finally, eq. (5) and a linear dependence of AG from the effective stress give a u = u(t) relationship u(O)-u(~)=~ln~~t
I),
(16)
where u(O) is the stress at the beginning of the relaxation (t = 0), (Y= kT/V* and fi = Ml(O)far. e(O) is the plastic strain rate at the beginning of the relaxation (t = 0).
It is well known that the time is not a good variable for the description of the mechanical behaviour of crystals. As pointed out by Hart [S], in fact, the instantaneous mechanical state of a material is completely characterized by the stress and strain rate which is capable of sustaining at that moment. Furthermore, for small increments, the three defor-
It is evident that the predictions of eqs. (14)-( 16) cannot be directly compared in a u versus t diagram. The situation is completely different when the problem is considered in a log u -1ogP plot. Furthermore, the calculation of the different parameters is much easier in the last representation. In conclusion, it is suggested that the experimental
186
F. Povolo /On the a~atys~s of stress rekxation data
stress relaxation data should be considered in a log u -log C plot since this has more physical meaning than a u versus t representation. A direct comparison between the predictions of the different theoretical models is possible and the calculation of the different parameters is greatly simplified, with no additional assumptions. It must be pointed out that several orders of magnitude in a CTversus t plot does not necessarily means the same extension in a log CI- log &representation.
5. Conclusions It was shown that the different theoretical expressions proposed in the literature to describe the stress relaxation behaviour of crystals, can be plotted in a normalized log (I -log P plot in such a way that a direct comparison between the predictions of the different models can be made. Furthermore, the experimental stress relaxation curves can be easily compared with those given by the theoretical models and the evaluation of the different parameters is greatly simplified with respect to a stress versus time plot.
Acknowledgements This work was performed within the Special Intergovernamental Agreement between Argentina and the
Federal republic of Germany and was supported in part by the Comision de Investigaciones Cientffmas de la Provincia de Buenos Aires.
References [ 11 E.C. Aifantis and W.W. Gerberich,Mater. Sci. Eng. 21 (1975) 107. [2] G. Baur and P. Lehr, M&m. Sci. Rev. Mkt. 718 (1975) 551. [3] R. de Bat&, Reviews on the Reformation Behaviour of Materials, Vol. 1, No. 2 (1975). [4] V.I. Dotsenko, Phys. Status Solidi (b) 93 (1979) 1 I. [S] E.W. Hart, Acta Met. 18 (1970) 599. [6] E.W. Hart, J. Eng. Mater. Technol. 98 (1976) 193. [7] E.W. Hart, C.Y. Li, H. Yamada and G.L. Wire, Constitutive Equations in Plasticity, Ed. AS, Argon (MIT Press, Cambridge, 1975) p. 149. [B] W.G. Johnston and J.J. Gilman, J. Appl. Phys. 30 (1959) 139. [9] J.J. Gilman, Gust&. J. Phys. 13 (1960) 327. [IO] P.P. Gillis and G.A. Sargent, Scripta Met. 6 (1971) 451. [ll] AS. Krausz, Mater. Sci. Eng. 4 (1969) 193. fl2] A.W. Sleeswyk, G.H. Boersma, G. Hunt and D.J. Verel, in: Proc. 2nd Intern. Conf. on Strength of Metals and Alloys, Vol. I (ASM, 1970) p. 204. [ 131 F. Povoto and I. Alvarez, Load relaxation in &y-4 at 673 K, to be published. [14] F. Guiu and P.L. Pratt, Phys. Status Solidi 6 (1964) 111. [ 151 F. Povolo, J. Nucl. Mater. 78 (1978) 309. 1161 J.C.M. Li, Canad. J. Phys. 45 (1967) 493.