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Stress relaxation and mechanical equation of state in b.c.c. metals in monotonic loading
STRESS
RELAXATION AND IN B.C.C. METALS
MECHANICAL EQUATION OF IN iMONOTONIC LOADING*
H, YAMADAf
and
CHE-YU
STATE
Lit
Stress relaxation experiments in monotonic loading have been performed on niobium and z-iron of two purity levels. The stress-strain rate relationship obtained at constant structures suggests that these metals exhibit the behavior of mechanical equation of state in the absence of discontinuous flow. RELAXATIOS
DE L-4 COSTR_iISTE ET EQUATIOS YECASIQCE D’ETAT 1IET_%VS CC. SOUYIS ;r USE CN-%RGE JIOSOTOSE
D_%XS LES
Des experiences de relaxation de la contra&e SOILSune charge monotone ont 606 effectu&s sup du de d4formation obtenue pour niobium et du fer a de deux degrCs de pure+&. La relation contrainte-vitesse des strnotures con&antes suggere que ces m&aus prtjsentent un comportement correspondant B l’equation mgcanique d’8tat en l’absence de discontinuit6 dans l’&oulement plastique. SP_~SSCSGSREL_~SdTIOS
USD JIET,ILLES
MECH_~ISCHE BE1 MOSOTOSER
ZTjSTdSDSGLEICHUSG BELdSTUSG
IS
B.C.C.
,4n Siob- und z-Eisenproben zwei verschiedener Reinheiten tvurden SpannungsrelasationsesperiDie bei konstanten Strukturen gemessenen Verfestigunnsmente bei monotoner Belastung du~hgef~hrt. kurven deuten darauf hin, da13 diese Metalle die meohanische Zustandsgleic~~lng in Abxvesenheit &iskontinuierlichen Fliel3ens erfiillen. INTRODUCTION
crosshead is then hetd fixed. The stress relasation results from converting the elastic strain of the specimen to plastic strain. The stress vs time data yield stress-strain rate relations at various hardness (plastic strain) levels. The orerall accuracy of the data is estimated to be 5 per cent. The experiment is performed at 297K. For initial loading the cross-head speed is 0.02 in./min. The high purity niobium (99.999%) is electro-deposited niobium obtained from Vnion Carbide Corporation and is electron beam melted. The commercial purity niobium (99.9 %) is also obtained from Union Carbide coloration. Both types of niobium specimen are swaged and annealed (1100°C and 1 hr for high purity niobium and 1400’G and 5 min for commercial purit,y niobium). The grain size is SO jdrn for the former and is 120 ,um for the latter. a-iron of two purities are used. The ion-er purity a-iron is Ferrovac E (99.5%; Crucible Steel). The higher purity u-iron is decarburized (99.95 %, Glidden Xetals). They are also swaged and annealed (70092 and 10 min for Ferrovoc E and 700°C and 3 min for decarburized u-iron). The grain size of these specimens is 41 pm and 72 ,um respectively. The specimen is rod-shaped. Its diameter is & in. wit,h a 2 in. gage section.
Recent stress relaxation experiment.s in monotonic loading hare demonstrated the existence of mechanical equation of state in f.c.c. metals.‘lv2) This paper reports some of the experimental results on b.e.c. metals to show that these metals also exhibit the behavior of mechanical equation of state. The formulation of mechanical equation of state using the concept of hardness (structure) parameter is introduced by Hart.c3) Stress relaxation esperiments on aluminum,(“) nickel and TD nickel’21 have shon-n the applicability of his theory. In this work t.tvo b.c.c, metals, niobium and a-iron, are examined to test whether or not Hart’s theory can be extended to other crystal systems. The stress relaxation experiments generate also stress-strain rat,e data which cover a wide range of strain rate at essentially constant structure. These data can provide a useful phenomenological basis for t,he consideration of t.he deformation mechanism in solids. Room temperature experiments are performed in this work to avoid the effects of recovery. At this temperature t,he plastic deformation in niobium and in a-iron is controlled by grain matrix processes. EXPERIMENTS
The principle and the detailed experimental procedure of the st,ress relaxation experiment are giren in Refs. 1, 2 and 4. All the experiments in this work are performed in monotonic loading. During the stress relaxation experiments the specimen is first deformed to the desired plastic strain level. The
Typical log o-log B data at rarious plastic strain Ievels for high purity niobium are shorrn in Fig. 1. These curves are concaved upward. The cnrwzs for aluminum and nickel are found to be concared in the opposite direction. f1r2) The present curves are similar Taothe nickel data in that the slope of the curves decreases as the hardness (plastic strain) lerel increases.
* Received December 20, 192. Revised July 17, 1973. t Department of 3Iaterials Science and Engineering, Cornell University, Ithaca, Sew York 14830, T;.S.A. _ICTX
MET;1LLVRGIC,1,
VOL.
“2, FEBRf;_4RY
1974
‘49
ACT-1
250
METALLITRGICA,
VOL.
The data for commercial purity niobium and decarburized x-iron exhibit the same feature as those shown in Fig. 1. The existence of mechanical equation of state can be tested by the scaling relation.(l*?) The scaling relation alloTs one to superpose any one of the log olog d curves by translating (4 log u, 4 log 9) onto any of the others of the same material in such a way that the overlapping segments of each curve match within experimental error. A master curve can therefore be constructed that each measured curve is entirely contained as a segment of the master curve and that. the required translations (4 log u, 4 log g) are related to satisfy the condition of one parameter family of curves.
22.
19i-1
-1
519
0
RUN
X 104
c,
RUN
* I05
0
RUN # 106
0 0
RUN # 107 RUN (t IO6
l
RUN Y 109
418 417 4 16
!
: 0 s 3 0
415-
3 A 0”
409406 407 406 i -II
-10
-9 LOG
-6 STRAIN
-7
-6 RATE
-5 (PER
-4
-3
SEC)
2. Master log cr-log d curve for high purity niobium constructed by superposing curves in Fig. 1 onto the curve for run number 104. FIG.
curve can actually extend the data for a given hardness to cover a wider strain rate range than that obtained from each individual stress relaxation run. (Seven decades vs five decades for high purity niobium.) The individual log b-log 8 curves for commercial purky niobium and decarburized u-iron are not presented. The master curve and t,he corresponding scaling relation for these materials are shown in Figs. 3-5(b, c)
420-
* UNLOADING a RELOADING 4051 -9
I -6 LOG
,
I
I
I -4
,;‘,A,,
&
&
-3
SEC1
FIG. 1. Log cs-log C curves for high purity niobium at various hardness (plastic strain) and room temperature. The straight line shown represents the scaling relation. The reloading log o-log E data are shown for run numbers 104 and 105.
The master curve for high purity niobium is shown in Fig. 2. It, is constructed by translating all the curves in Fig. 1 onto the curve of run number 104 along the path represented by the straight line. The scaling relation (translation path) is obtained by plotting the log o-log B values of the point,s of the same slope (d log o/d log g) from each curve in Fig. 1 and is shown in Fig. S(a). A similar scaling relation is found in aluminum however the slope of the t,ranslation path for aluminum is different from those obtaincd in the present work. Using the master curve (Fig. 2) and the scaling relation shown in Fig. 5(a) one can generate all the log a-log g curves at various hardness levels for high purity niobium (Fig. 1). It is seen that the master
431-
4304.29 -
4244.23422-
_
421
1
420
-9
LO;ESTRAtN -7
-6 RATE-&R&
-3
FIG. 3. Master log u-log d curve for commercial purity niobium at room temperature. It is constructed bysuperposing all the curves onto the curve for run number ‘79.
=
RUfY
ii
92
a
RUN
#
91
2
RUN
#
92
4.18-
4 17-
zi
0
RUN
++ 93
:
RUN
C
9s
l
RUN
#
95
4.16-
SLOPE
!
3
3
4.4i
:: c A
v, w”
it
4.3-
: m
cn 415s
c3 2
g
4.14-
8 J
4.13-
4.2
-7
I
I -9
-7
-6
STRAiN
RATE
-5
-4
(PER
SEC)
-3
FIG. 4. Jlwter log o-log 6 curve for clecarbnrized r-iron at room temperature. It is constructed by superposing all the curves onto the curve for run number 90.
1
I
I
4Sr Sl_OPE=0.l4
‘=
4.3-
-6
STRAIN
i
-5
RATE
(PER
-4
SEC)
3(c).
relat iou for clecarburizecl
x.iron.
respectively.
The master curve for commercial purity niobium is constructed by translating all the curves onto the curre of run number 79. The master curve for decarburized x-iron is constructed by tran&ting all the cur’ves into the curve of run number 90. The experimental results presented above suggest that these materials exhibit the behavior of mechanical equation of atate in monotonic loading.
.
i
2. Diulocatior~~ dpz.xic8
ctrrd iAerml
In a recent paper Gupta stress relaxation data u4ng
w” 42 i .
-do it
l
4.,,j,: -6
LOG
-7
-6
STRAiN
RATE
-5
-4
(PER
SEC)
FIG. .5(a). Scaling relation for high purity niobium. The data points represent the points ha.ving the same slope from curves in Fig. 1.
I
fLOPE=O.I2
4.4
.
1
J-
m 4.3 ~
/
4.2
/*
4.11 -9
-7
-8
LOG
STRAIN
RATE
-6
(PER
-5
SEC)
FICJ. 5(h). Scaling relation
for cummercial
purity
niobium.
C
---K’(a
.sft-~.s.~
and Li’jJ interpret the relationship -
their
Ql’
Jvhere d$/dt is the rate of stress relaxation and is related to plastic strain rate through the machine constant and the cross-sectional area of the specimen,‘“) K’ is a constant. ~~ is internal stress and 1 -_=y iii*
5
I
I
I
-8
LOG
5
,
Frr;. Scalinq
4.10[ -10
ii
I
I
LOG
1
-I = 2
1
;
4.11
: v,
I
-
4.1 -8
4 12-
kc m
= 0.83
* _ d In (0 dlnf
oi) .
The long time limit of log (-cZd/dt) x-s log t pives the value of m* which is assumed constant over the stress range involved in each stress relasation run. Typical log (--dc/dt) x-5log t plot for high purity niobium is shown in Fig. 6. The ralues of .L* for niobium are given in Table 1 for t-arious hardness (plastic strain) levels. The initial portion of the log (--da/&)log t data (up to lo” set) and the calculatecl w*‘s of the present work are in agreement l\-ith those of Gupta and Li.(jJ The data of this world extend, however, to much longer time and lead to interesting results in decarburized x-iron. The initial portion of the log (-~~~+Iog t plot of decarburized x-iron seems to behare similar1.v to the data of Gupta and Li(j*“) (Fig. 7). At longer time the
-I
I
I
0
I
,
3 (SEC)
LOG ,:,E FK.
6. Log
present dicted
( --da,‘&)
I
t
4
5
vs log t plot for high puritv
data deviate
from the strai,aht
by Gupt’a and Li. w
niobium.
FIG. 7. Log showing the
line limit pre-
Discussion
will be made
later as to t.he reason for this detiation. Recent,ly to f.c.c.
the application
metals
of Gupta and Li’s met,hod
has been found to be unsuccessful.(‘*“)
The present, result seems to suggest its limitation in b.c.c.
metals.
The UL* in equation Ior G-log g curves,(‘3)
(1) is related
1
*
Ii 1 1-F;
E)’
u-here Y=-.
creasing
suggests
log +lo,a
nal skess the slope
One may calculate slope
The stress
increase
n&h
in-
TABLE
Material
of the
straight
line
“/, pure Sh
99.999 99.999 99.999 99.999 99.999 99.999
p’, 7; y0 % 7:
7’9 SO 81 5’
99.9 99.9 99.9 99.9
pure pure pure pure
90 91 92 93 94 9.5
99.95 99.95 99.95 99.95 99.95 99.95
10.5 106 10; 108 109
% % % %
1
FI( %)
pure Xb pure Sb pure Sb
pure Sb
pure
Sb
Sb Sb Sb Sb
% pure Fe T< pure Fe :; pure Fe T,; pure Fe :d pure Fe “/, pure Fe
0.S 1.s 2.46 4.31 6.lti 8.99
area in terms
t: curves
and
dependence
of the activation
depending
than
applied
erea reflects
with dislocation
on the deformation
the interpretation
mechanism.
of this stress dependence
due to the possibility
one deformation
mechanism
which will be discussed
of
stress.
that
more
may be inr0lrerl
in the nest section.
stress is constant
The shape of the present euperito be consistent with this pre-
the deviation
the activation
of log ~-log
can be complicated
will
of
E curved of decarburized
the shape of energy barrier associated motion
the slope of the constant
of Gupta and
for the stress-dependence
of the log c-lop
cl In 8
curres
using the method
z-iron
or not iutcr-
z-iron.
During strain
the initial
(up
7.7 (51.5) 8.3 75 11:: 7.7 10.0
flow is observed
P.S.5 4.70 6.S 9.48
6.0 6.7 6.4 7.2
0.57 1.40 2.20 3.7’ 5.0s ti.BO
2.9 2.4 1.8 3.2 3 .5 3.5
strain).
where discontinuous
in commercial
flow occurs,
strain
E
(up
resion
the log ~-log 1 data
of mechanical
equation
of
in commercial
in this low plastic strain range as com-
pared to those given in Table Lopv o-log
dis-
purity nio-
and in Ferrovac
Higher values of ttb’*are obtained
purity niobium
strain
desired p&tic experiment?
In the plastic
do not exhibit, the behavior state.
to obtain
relaxation
to 2 per cent strain)
to 10 per cent m* (from Guptlz and Li’s method)
loading
level for the stress
continuous bium
104
calculated
in clecarburizerl
as to whether
Li can be used to account
However.
applied stress (0) if internal
However,
(Fig. 7) raises the question
d in 0
that
t
for a $x-en structure. mental curve seems diction.
(‘1
G
/)I *
This equation
to the slope of
for tlecarhurizrrl z-irk)11 deviation of iong time limit preclictecl by Gupts and Li.‘“’
litnit at longer times observed
the
-_=1’
hardness
even
(--rln ~lt)~3 1021plot
i data
1.
of Ferrovac
E at large
(9-16o/0) are shown in Fig. 8.
plastic
These data also
show the deviation of the lon,a time straight line limit in the log (--do/&) x-s log t plot. These data ~ho\v verhigh stress sensitirity at. lower strain rates, i.e. the 10,a G-log 2 plot. is almost a horizontal
line.
The time dependent transient, effects suggested by Hart and Solomon”’ will lead to a highly stress sensitive behavior. In their case the transient effects are
Y~X_%D_%
4.541
’ -9
FIG. 5. Log
aXD
CHE-YITU
, I -a -7 LOG STRAIN
LI:
JIECH.XSIC_%L
/ # I -6 -5 -4 RATE (PER SEC)
G-log d cwtte
strain and room
for Ferrovac
I -3
E at, Q-160;,
temperw_we.
associated with the build-up of dislocation pile-ups. In the present work the high stress sensitivity observed in Ferrovac E may also be related with time dependent processes. The most likely possibility is the effects caused by impurity drag associated with dislocat.ion motion. These effects may occur in a lesser extent in niobium and in decarburized u-iron showi in Fig. ri. The master curve of decarburized x-iron seems also to show a transition in deformation process at lower strain rates. This possibility can be supported from the observation that the slope of the scaling relation, d In o/d In ij,, of niobium is close to the stress sensitivity (~8In b/d In 6) found in steady state creep experiments.(‘) Similar results are obtained also in aluminum,“) lead(*) and in stainless steels.(Q) But this correlat,ion is not found in decarburized a-iron(‘) (Figs. 6a, b vs 50). It is encouraging, however, that the behavior of mechanical equation of state is st.ill found in all three metals.
The transient effects resulting from the buiId-up of dislocation pile-ups mentioned above(l) have been associated with the stress relaxation data of aluminum, nickel and TD nickel obtained in t,he initial portion of the stress relaxation run (high stress end of the log Glog d curve). This part of the log a-log B curve can be identified from it-s high stress sensitivit.y and from the fact that it does not. fit well onto the master curve. The present data do not eshibit this type of behavior.
EQU_ITIOS
OF
YT_%TE
IS
B.C.C.
XETdLS
2.53
The log o-log (j data obtained during reloading after the stress relaxat,ion run can be used t,o test whether or not recovery has occurred.“) If during the stress reIaxation the structure of the specimen is unchanged, the reloading log a-log 6 data should retrace the log olog d curve obtained during stress relaxation. On the other band, if recovery has occurred, the reloading data will lie below the stress relasation curve. It is difficult to obtain accurate low strain rate data during reloacling. Typical reloading data are shown in Fig. 1 and are found to lie close to the stress relasation curve. Together with t,he fact that the plastic strain required t,o reload the specimen to its original hardness is less than lOwa, it may be concluded that. the amount of recovery during stress relaxation is small in the present work. In summary the present work has found: (a) In t.he absence of discont.inuous flon-, niobium and a-iron exhibit the behavior of mechanical equation of state in monotonic loading. (b) The shape of constant structure log g-log B curve for a-iron cannot be accounted for solely based on internal stress. (c) The transient effects resulting from the build-up of dislocation pile-ups are not found in the b.c.c. metals investigated in this work. ACKNOWLEDGENESTS
This work is supported by the Naterials Science Center of Cornell Universit.y, sponsored by SSF. The authors wish to espress their appreciation of B. Addis for specimen preparat,ion. REFERENCES 1. K. W. HART and H. D. SOLOMOS, Load relaxation of poly. crystalline high purity aluminum, dcta Xet. (to be published.) 2. H. Y_.uuD.% and CHE-YU LI, Stress relaxation and meohaniCal equation of state in nickel and TD nickel, Xet. Tran.s. (to be published.) 3. E. W. HOT, dcto ,Ilet. 18, 599 (1970). 4. D. LEE ~3 E. \V. H.UIT, Xes. Trans. 2, I2-l.5 (1971). j. I. G~PT_A and J. C. JL. LX, Xet. Trarw. 1, 13% (1970). 6. C. C. Law and D. S, BESIFXERS, Stress relesation in f.c.c. and h.c.p. metals, Scripta Xelet. (to be published.) 7. S. B~~.U~BR.UUSI_%X and J. C. 11. Lr, J. Xat. $ci. 5. ill (1910). 8. G. WIRE, H. Y_&x%D~ and CHE-Yc Lx, 1lechanical equation of state and grain boundary sliding in lead, _%EC Report No. COO-2172-Q. Cornell finiversitv (19i’7). (To be published.) 9. H. Yasra~a and CHE-Yu LI, Stress relaxation and mechanical equation of state in austenitic stainless steels, Xet. Trans. 4, 2133 (1973).
RELAXATION AND IN B.C.C. METALS
MECHANICAL EQUATION OF IN iMONOTONIC LOADING*
H, YAMADAf
and
CHE-YU
STATE
Lit
Stress relaxation experiments in monotonic loading have been performed on niobium and z-iron of two purity levels. The stress-strain rate relationship obtained at constant structures suggests that these metals exhibit the behavior of mechanical equation of state in the absence of discontinuous flow. RELAXATIOS
DE L-4 COSTR_iISTE ET EQUATIOS YECASIQCE D’ETAT 1IET_%VS CC. SOUYIS ;r USE CN-%RGE JIOSOTOSE
D_%XS LES
Des experiences de relaxation de la contra&e SOILSune charge monotone ont 606 effectu&s sup du de d4formation obtenue pour niobium et du fer a de deux degrCs de pure+&. La relation contrainte-vitesse des strnotures con&antes suggere que ces m&aus prtjsentent un comportement correspondant B l’equation mgcanique d’8tat en l’absence de discontinuit6 dans l’&oulement plastique. SP_~SSCSGSREL_~SdTIOS
USD JIET,ILLES
MECH_~ISCHE BE1 MOSOTOSER
ZTjSTdSDSGLEICHUSG BELdSTUSG
IS
B.C.C.
,4n Siob- und z-Eisenproben zwei verschiedener Reinheiten tvurden SpannungsrelasationsesperiDie bei konstanten Strukturen gemessenen Verfestigunnsmente bei monotoner Belastung du~hgef~hrt. kurven deuten darauf hin, da13 diese Metalle die meohanische Zustandsgleic~~lng in Abxvesenheit &iskontinuierlichen Fliel3ens erfiillen. INTRODUCTION
crosshead is then hetd fixed. The stress relasation results from converting the elastic strain of the specimen to plastic strain. The stress vs time data yield stress-strain rate relations at various hardness (plastic strain) levels. The orerall accuracy of the data is estimated to be 5 per cent. The experiment is performed at 297K. For initial loading the cross-head speed is 0.02 in./min. The high purity niobium (99.999%) is electro-deposited niobium obtained from Vnion Carbide Corporation and is electron beam melted. The commercial purity niobium (99.9 %) is also obtained from Union Carbide coloration. Both types of niobium specimen are swaged and annealed (1100°C and 1 hr for high purity niobium and 1400’G and 5 min for commercial purit,y niobium). The grain size is SO jdrn for the former and is 120 ,um for the latter. a-iron of two purities are used. The ion-er purity a-iron is Ferrovac E (99.5%; Crucible Steel). The higher purity u-iron is decarburized (99.95 %, Glidden Xetals). They are also swaged and annealed (70092 and 10 min for Ferrovoc E and 700°C and 3 min for decarburized u-iron). The grain size of these specimens is 41 pm and 72 ,um respectively. The specimen is rod-shaped. Its diameter is & in. wit,h a 2 in. gage section.
Recent stress relaxation experiment.s in monotonic loading hare demonstrated the existence of mechanical equation of state in f.c.c. metals.‘lv2) This paper reports some of the experimental results on b.e.c. metals to show that these metals also exhibit the behavior of mechanical equation of state. The formulation of mechanical equation of state using the concept of hardness (structure) parameter is introduced by Hart.c3) Stress relaxation esperiments on aluminum,(“) nickel and TD nickel’21 have shon-n the applicability of his theory. In this work t.tvo b.c.c, metals, niobium and a-iron, are examined to test whether or not Hart’s theory can be extended to other crystal systems. The stress relaxation experiments generate also stress-strain rat,e data which cover a wide range of strain rate at essentially constant structure. These data can provide a useful phenomenological basis for t,he consideration of t.he deformation mechanism in solids. Room temperature experiments are performed in this work to avoid the effects of recovery. At this temperature t,he plastic deformation in niobium and in a-iron is controlled by grain matrix processes. EXPERIMENTS
The principle and the detailed experimental procedure of the st,ress relaxation experiment are giren in Refs. 1, 2 and 4. All the experiments in this work are performed in monotonic loading. During the stress relaxation experiments the specimen is first deformed to the desired plastic strain level. The
Typical log o-log B data at rarious plastic strain Ievels for high purity niobium are shorrn in Fig. 1. These curves are concaved upward. The cnrwzs for aluminum and nickel are found to be concared in the opposite direction. f1r2) The present curves are similar Taothe nickel data in that the slope of the curves decreases as the hardness (plastic strain) lerel increases.
* Received December 20, 192. Revised July 17, 1973. t Department of 3Iaterials Science and Engineering, Cornell University, Ithaca, Sew York 14830, T;.S.A. _ICTX
MET;1LLVRGIC,1,
VOL.
“2, FEBRf;_4RY
1974
‘49
ACT-1
250
METALLITRGICA,
VOL.
The data for commercial purity niobium and decarburized x-iron exhibit the same feature as those shown in Fig. 1. The existence of mechanical equation of state can be tested by the scaling relation.(l*?) The scaling relation alloTs one to superpose any one of the log olog d curves by translating (4 log u, 4 log 9) onto any of the others of the same material in such a way that the overlapping segments of each curve match within experimental error. A master curve can therefore be constructed that each measured curve is entirely contained as a segment of the master curve and that. the required translations (4 log u, 4 log g) are related to satisfy the condition of one parameter family of curves.
22.
19i-1
-1
519
0
RUN
X 104
c,
RUN
* I05
0
RUN # 106
0 0
RUN # 107 RUN (t IO6
l
RUN Y 109
418 417 4 16
!
: 0 s 3 0
415-
3 A 0”
409406 407 406 i -II
-10
-9 LOG
-6 STRAIN
-7
-6 RATE
-5 (PER
-4
-3
SEC)
2. Master log cr-log d curve for high purity niobium constructed by superposing curves in Fig. 1 onto the curve for run number 104. FIG.
curve can actually extend the data for a given hardness to cover a wider strain rate range than that obtained from each individual stress relaxation run. (Seven decades vs five decades for high purity niobium.) The individual log b-log 8 curves for commercial purky niobium and decarburized u-iron are not presented. The master curve and t,he corresponding scaling relation for these materials are shown in Figs. 3-5(b, c)
420-
* UNLOADING a RELOADING 4051 -9
I -6 LOG
,
I
I
I -4
,;‘,A,,
&
&
-3
SEC1
FIG. 1. Log cs-log C curves for high purity niobium at various hardness (plastic strain) and room temperature. The straight line shown represents the scaling relation. The reloading log o-log E data are shown for run numbers 104 and 105.
The master curve for high purity niobium is shown in Fig. 2. It, is constructed by translating all the curves in Fig. 1 onto the curve of run number 104 along the path represented by the straight line. The scaling relation (translation path) is obtained by plotting the log o-log B values of the point,s of the same slope (d log o/d log g) from each curve in Fig. 1 and is shown in Fig. S(a). A similar scaling relation is found in aluminum however the slope of the t,ranslation path for aluminum is different from those obtaincd in the present work. Using the master curve (Fig. 2) and the scaling relation shown in Fig. 5(a) one can generate all the log a-log g curves at various hardness levels for high purity niobium (Fig. 1). It is seen that the master
431-
4304.29 -
4244.23422-
_
421
1
420
-9
LO;ESTRAtN -7
-6 RATE-&R&
-3
FIG. 3. Master log u-log d curve for commercial purity niobium at room temperature. It is constructed bysuperposing all the curves onto the curve for run number ‘79.
=
RUfY
ii
92
a
RUN
#
91
2
RUN
#
92
4.18-
4 17-
zi
0
RUN
++ 93
:
RUN
C
9s
l
RUN
#
95
4.16-
SLOPE
!
3
3
4.4i
:: c A
v, w”
it
4.3-
: m
cn 415s
c3 2
g
4.14-
8 J
4.13-
4.2
-7
I
I -9
-7
-6
STRAiN
RATE
-5
-4
(PER
SEC)
-3
FIG. 4. Jlwter log o-log 6 curve for clecarbnrized r-iron at room temperature. It is constructed by superposing all the curves onto the curve for run number 90.
1
I
I
4Sr Sl_OPE=0.l4
‘=
4.3-
-6
STRAIN
i
-5
RATE
(PER
-4
SEC)
3(c).
relat iou for clecarburizecl
x.iron.
respectively.
The master curve for commercial purity niobium is constructed by translating all the curves onto the curre of run number 79. The master curve for decarburized x-iron is constructed by tran&ting all the cur’ves into the curve of run number 90. The experimental results presented above suggest that these materials exhibit the behavior of mechanical equation of atate in monotonic loading.
.
i
2. Diulocatior~~ dpz.xic8
ctrrd iAerml
In a recent paper Gupta stress relaxation data u4ng
w” 42 i .
-do it
l
4.,,j,: -6
LOG
-7
-6
STRAiN
RATE
-5
-4
(PER
SEC)
FIG. .5(a). Scaling relation for high purity niobium. The data points represent the points ha.ving the same slope from curves in Fig. 1.
I
fLOPE=O.I2
4.4
.
1
J-
m 4.3 ~
/
4.2
/*
4.11 -9
-7
-8
LOG
STRAIN
RATE
-6
(PER
-5
SEC)
FICJ. 5(h). Scaling relation
for cummercial
purity
niobium.
C
---K’(a
.sft-~.s.~
and Li’jJ interpret the relationship -
their
Ql’
Jvhere d$/dt is the rate of stress relaxation and is related to plastic strain rate through the machine constant and the cross-sectional area of the specimen,‘“) K’ is a constant. ~~ is internal stress and 1 -_=y iii*
5
I
I
I
-8
LOG
5
,
Frr;. Scalinq
4.10[ -10
ii
I
I
LOG
1
-I = 2
1
;
4.11
: v,
I
-
4.1 -8
4 12-
kc m
= 0.83
* _ d In (0 dlnf
oi) .
The long time limit of log (-cZd/dt) x-s log t pives the value of m* which is assumed constant over the stress range involved in each stress relasation run. Typical log (--dc/dt) x-5log t plot for high purity niobium is shown in Fig. 6. The ralues of .L* for niobium are given in Table 1 for t-arious hardness (plastic strain) levels. The initial portion of the log (--da/&)log t data (up to lo” set) and the calculatecl w*‘s of the present work are in agreement l\-ith those of Gupta and Li.(jJ The data of this world extend, however, to much longer time and lead to interesting results in decarburized x-iron. The initial portion of the log (-~~~+Iog t plot of decarburized x-iron seems to behare similar1.v to the data of Gupta and Li(j*“) (Fig. 7). At longer time the
-I
I
I
0
I
,
3 (SEC)
LOG ,:,E FK.
6. Log
present dicted
( --da,‘&)
I
t
4
5
vs log t plot for high puritv
data deviate
from the strai,aht
by Gupt’a and Li. w
niobium.
FIG. 7. Log showing the
line limit pre-
Discussion
will be made
later as to t.he reason for this detiation. Recent,ly to f.c.c.
the application
metals
of Gupta and Li’s met,hod
has been found to be unsuccessful.(‘*“)
The present, result seems to suggest its limitation in b.c.c.
metals.
The UL* in equation Ior G-log g curves,(‘3)
(1) is related
1
*
Ii 1 1-F;
E)’
u-here Y=-.
creasing
suggests
log +lo,a
nal skess the slope
One may calculate slope
The stress
increase
n&h
in-
TABLE
Material
of the
straight
line
“/, pure Sh
99.999 99.999 99.999 99.999 99.999 99.999
p’, 7; y0 % 7:
7’9 SO 81 5’
99.9 99.9 99.9 99.9
pure pure pure pure
90 91 92 93 94 9.5
99.95 99.95 99.95 99.95 99.95 99.95
10.5 106 10; 108 109
% % % %
1
FI( %)
pure Xb pure Sb pure Sb
pure Sb
pure
Sb
Sb Sb Sb Sb
% pure Fe T< pure Fe :; pure Fe T,; pure Fe :d pure Fe “/, pure Fe
0.S 1.s 2.46 4.31 6.lti 8.99
area in terms
t: curves
and
dependence
of the activation
depending
than
applied
erea reflects
with dislocation
on the deformation
the interpretation
mechanism.
of this stress dependence
due to the possibility
one deformation
mechanism
which will be discussed
of
stress.
that
more
may be inr0lrerl
in the nest section.
stress is constant
The shape of the present euperito be consistent with this pre-
the deviation
the activation
of log ~-log
can be complicated
will
of
E curved of decarburized
the shape of energy barrier associated motion
the slope of the constant
of Gupta and
for the stress-dependence
of the log c-lop
cl In 8
curres
using the method
z-iron
or not iutcr-
z-iron.
During strain
the initial
(up
7.7 (51.5) 8.3 75 11:: 7.7 10.0
flow is observed
P.S.5 4.70 6.S 9.48
6.0 6.7 6.4 7.2
0.57 1.40 2.20 3.7’ 5.0s ti.BO
2.9 2.4 1.8 3.2 3 .5 3.5
strain).
where discontinuous
in commercial
flow occurs,
strain
E
(up
resion
the log ~-log 1 data
of mechanical
equation
of
in commercial
in this low plastic strain range as com-
pared to those given in Table Lopv o-log
dis-
purity nio-
and in Ferrovac
Higher values of ttb’*are obtained
purity niobium
strain
desired p&tic experiment?
In the plastic
do not exhibit, the behavior state.
to obtain
relaxation
to 2 per cent strain)
to 10 per cent m* (from Guptlz and Li’s method)
loading
level for the stress
continuous bium
104
calculated
in clecarburizerl
as to whether
Li can be used to account
However.
applied stress (0) if internal
However,
(Fig. 7) raises the question
d in 0
that
t
for a $x-en structure. mental curve seems diction.
(‘1
G
/)I *
This equation
to the slope of
for tlecarhurizrrl z-irk)11 deviation of iong time limit preclictecl by Gupts and Li.‘“’
litnit at longer times observed
the
-_=1’
hardness
even
(--rln ~lt)~3 1021plot
i data
1.
of Ferrovac
E at large
(9-16o/0) are shown in Fig. 8.
plastic
These data also
show the deviation of the lon,a time straight line limit in the log (--do/&) x-s log t plot. These data ~ho\v verhigh stress sensitirity at. lower strain rates, i.e. the 10,a G-log 2 plot. is almost a horizontal
line.
The time dependent transient, effects suggested by Hart and Solomon”’ will lead to a highly stress sensitive behavior. In their case the transient effects are
Y~X_%D_%
4.541
’ -9
FIG. 5. Log
aXD
CHE-YITU
, I -a -7 LOG STRAIN
LI:
JIECH.XSIC_%L
/ # I -6 -5 -4 RATE (PER SEC)
G-log d cwtte
strain and room
for Ferrovac
I -3
E at, Q-160;,
temperw_we.
associated with the build-up of dislocation pile-ups. In the present work the high stress sensitivity observed in Ferrovac E may also be related with time dependent processes. The most likely possibility is the effects caused by impurity drag associated with dislocat.ion motion. These effects may occur in a lesser extent in niobium and in decarburized u-iron showi in Fig. ri. The master curve of decarburized x-iron seems also to show a transition in deformation process at lower strain rates. This possibility can be supported from the observation that the slope of the scaling relation, d In o/d In ij,, of niobium is close to the stress sensitivity (~8In b/d In 6) found in steady state creep experiments.(‘) Similar results are obtained also in aluminum,“) lead(*) and in stainless steels.(Q) But this correlat,ion is not found in decarburized a-iron(‘) (Figs. 6a, b vs 50). It is encouraging, however, that the behavior of mechanical equation of state is st.ill found in all three metals.
The transient effects resulting from the buiId-up of dislocation pile-ups mentioned above(l) have been associated with the stress relaxation data of aluminum, nickel and TD nickel obtained in t,he initial portion of the stress relaxation run (high stress end of the log Glog d curve). This part of the log a-log B curve can be identified from it-s high stress sensitivit.y and from the fact that it does not. fit well onto the master curve. The present data do not eshibit this type of behavior.
EQU_ITIOS
OF
YT_%TE
IS
B.C.C.
XETdLS
2.53
The log o-log (j data obtained during reloading after the stress relaxat,ion run can be used t,o test whether or not recovery has occurred.“) If during the stress reIaxation the structure of the specimen is unchanged, the reloading log a-log 6 data should retrace the log olog d curve obtained during stress relaxation. On the other band, if recovery has occurred, the reloading data will lie below the stress relasation curve. It is difficult to obtain accurate low strain rate data during reloacling. Typical reloading data are shown in Fig. 1 and are found to lie close to the stress relasation curve. Together with t,he fact that the plastic strain required t,o reload the specimen to its original hardness is less than lOwa, it may be concluded that. the amount of recovery during stress relaxation is small in the present work. In summary the present work has found: (a) In t.he absence of discont.inuous flon-, niobium and a-iron exhibit the behavior of mechanical equation of state in monotonic loading. (b) The shape of constant structure log g-log B curve for a-iron cannot be accounted for solely based on internal stress. (c) The transient effects resulting from the build-up of dislocation pile-ups are not found in the b.c.c. metals investigated in this work. ACKNOWLEDGENESTS
This work is supported by the Naterials Science Center of Cornell Universit.y, sponsored by SSF. The authors wish to espress their appreciation of B. Addis for specimen preparat,ion. REFERENCES 1. K. W. HART and H. D. SOLOMOS, Load relaxation of poly. crystalline high purity aluminum, dcta Xet. (to be published.) 2. H. Y_.uuD.% and CHE-YU LI, Stress relaxation and meohaniCal equation of state in nickel and TD nickel, Xet. Tran.s. (to be published.) 3. E. W. HOT, dcto ,Ilet. 18, 599 (1970). 4. D. LEE ~3 E. \V. H.UIT, Xes. Trans. 2, I2-l.5 (1971). j. I. G~PT_A and J. C. JL. LX, Xet. Trarw. 1, 13% (1970). 6. C. C. Law and D. S, BESIFXERS, Stress relesation in f.c.c. and h.c.p. metals, Scripta Xelet. (to be published.) 7. S. B~~.U~BR.UUSI_%X and J. C. 11. Lr, J. Xat. $ci. 5. ill (1910). 8. G. WIRE, H. Y_&x%D~ and CHE-Yc Lx, 1lechanical equation of state and grain boundary sliding in lead, _%EC Report No. COO-2172-Q. Cornell finiversitv (19i’7). (To be published.) 9. H. Yasra~a and CHE-Yu LI, Stress relaxation and mechanical equation of state in austenitic stainless steels, Xet. Trans. 4, 2133 (1973).