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Stress relaxation in titanium
Scripta ~'4ETALLURG]CA
Vol. 8, pp. 739-742, 1974 P r i n t e d in the U n i t e d States
STRESS
RELAXATION
F. V. Ellis
Department
Press,
Pergamon
Inc.
IN T I T A N I U M
and C h e - Y u
Li
of M a t e r i a l s Science and E n g i n e e r i n g Cornell University Ithaca, New York 14850
(Received M a r c h
22,
1974)
Introduction Stress
relaxation
i n v e s t i g a t o r s I-3 the
theories
of d i s l o c a t i o n
recently
also
equation
of state 4.
a limited some
concept
to o b t a i n
time
dynamics.
data
stress
The p u r p o s e
relaxation
equation
relaxation
data
Discussion to show
by several
report
on
are used
of p l a s t i c
on t i t a n i u m
will
that
based
experiments
the c o n c e p t
of the p r e s e n t
and
reported
are d i s c u s s e d
relaxation
to support
data.
of state
have been
investigations
Stress
experimental
time.
stress
of p l a s t i c
the p r e v i o u s
on t i t a n i u m
of these
The e x i s t i n g
relaxation
longer
experiments
The results
cover
only
is to p r e s e n t
be made
using
the c o n c l u s i o n s
the
made by
i n v e s t i g a t o r s 2-3 may be questioned.
Experiments Room
temperature
stress
relaxation
experiments
are p e r f o r m e d
in the p r e s e n t
work.
The principle, p r o c e d u r e , and e q u i p m e n t of the e x p e r i m e n t have been 4 d e s c r i b e d p r e v i o u s l y . C o m m e r c i a l p u r i t y t i t a n i u m of a p p r o x i m a t e l y 99.7%
purity
is used.
annealed
two inches at 0.02 stress
Titanium
in v a c u u m
and a grain
in/min,
rods
for an hour
stress
stress given
load-time
particular single level
stress
specimen after
another
data
shown
strain
rate
4.
from each
strain
The
and
length
of
specimen
is p r e - s t r a i n e d
level
to each
strain)
prior
relaxation
a stress
run.
relaxation
shown each
Data
run.
Data
from runs
curves
stress 5-6
of p l a s t i c
hardness
739
from a
from one
(or p l a s t i c
are o b t a i n e d
equation
strain)
from
of state 4
log s t r e s s - l o g
should b e l o n g
to log
the a p p r o a c h the data
1-4 are o b t a i n e d
to a h i g h e r
to the c o n c e p t
These
1 using
represents
from runs
constant
run are c o n v e r t e d
in Fig.
curve
the s p e c i m e n
1 represent
relationships.
stress
figure
relaxation
According
in Fig.
and D i s c u s s i o n
rate plots
In this
by r e l o a d i n g
specimen.
curves
50~.
diameter
has a gage
run.
- log p l a s t i c in r e f e r e n c e
to 0.125"
specimen
(or p l a s t i c
Results The
swaged
The
size of a p p r o x i m a t e l y
to a d e s i r e d
relaxation
are arc cast, at 500°C.
plastic
to a p a r a m e t e r
family
740
STRESS R E L A X A T I O N
IN T I T A N I U M
Vol.
8, No.
of curves w i t h the a r b i t r a r y p a r a m e t e r b e i n g the h a r d n e s s p a r a m e t e r if a p l a s t i c e q u a t i o n of state exists.
A simple way to test the one p a r a m e t e r
family of curves is the p o s s i b i l i t y that these curves can be t r a n s l a t e d along a s t r a i g h t line to o v e r l a p each other.
It is seen from Fig.
these curves are n e a r l y a set of p a r a l l e l s t r a i g h t lines. s t r a i g h t lines can be t r a n s l a t e d p a r a l l e l to the log stress axis.
1 that
A set of p a r a l l e l
to o v e r l a p each o t h e r along a path w h i c h is The curves in Fig.
1 do show h o w e v e r slight
c u r v a t u r e w h i c h is w i t h i n the a c c u r a c y of the experiment.
U n f o r t u n a t e l y the
c u r v a t u r e of these curves does not show s u f f i c i e n t c h a r a c t e r i s t i c s the d e t e r m i n a t i o n of a unique t r a n s l a t i o n path.
The
to allow
d e t e r m i n a t i o n of a
u n i q u e t r a n s l a t i o n p a t h has b e e n p o s s i b l e for c o n s t a n t h a r d n e s s
log stress -
log p l a s t i c strain rate curves of a v a r i e t y of m a t e r i a l s as r e p o r t e d in r e f e r e n c e 4. homologous
In the same r e f e r e n c e it is d i s c u s s e d that in general at low
temperatures
the c o n s t a n t h a r d n e s s
log stress - log p l a s t i c strain
rate curves w i l l show a c u r v a t u r e c o n c a r v i n g u p w a r d and c o n c a r v i n g d o w n w a r d at high h o m o l o g o u s
temperatures.
The i n t e r m e d i a t e h o m o l o g o u s t e m p e r a t u r e s
range c o r r e s p o n d to the t r a n s i t i o n r e g i o n w h e r e the curves will show less characteristics. mediate homologous
It is p o s s i b l e that r o o m t e m p e r a t u r e t e m p e r a t u r e range for titanium.
t h e r e f o r e to d e t e r m i n e
falls in the inter-
It will be d i f f i c u l t
the t r a n s l a t i o n path.
S a r g e n t and C o n r a d 2 has used the slope of load vs log time plot to d e t e r m i n e the a c t i v a t i o n v o l u m e for p l a s t i c covers a p e r i o d of five minutes.
flow.
Their r e l a x a t i o n time
In their data range the load - log time
plot shows a s t r a i g h t line correlation.
The typical results of the p r e s e n t
w o r k in a p l o t of load vs log time are shown in Fig.
2.
It is seen that the
p r e s e n t data show a c o n t i n u o u s l y c h a n g i n g slope at longer r e l a x a t i o n times. The values of the a c t i v a t i o n volume o b t a i n e d by these i n v e s t i g a t i o r s may t h e r e f o r e be questioned. 3 R o d r i g u e z et al. p r o p o s e d several a r g u m e n t s in an a t t e m p t to use stress r e l a x a t i o n data to r a t i o n a l i z e m e c h a n i s m s
for p l a s t i c flow in titanium.
Some of their a r g u m e n t s d e p e n d on the fact that long time limit of load vs log time p l o t a p p r o a c h e s
a s t r a i g h t line.
line limit is shown by the p r e s e n t data
No e v i d e n c e of long time s t r a i g h t
(Fig. 2) w h i c h ~overs a r e l a x a t i o n
time one d e c a d e longer than that involved in the e x p e r i m e n t s of R o d r i g u e z et al.
The p r e s e n t w o r k u l t i l i z e s a d i g i t a l data a c q u i s i t i o n s y s t e m and good
t e m p e r a t u r e control of the entire t e s t i n g s y s t e m 4. Fig.
2 is w i t h i n e x p e r i m e n t a l
The c u r v a t u r e shown in
limit.
R o d r i g u e z et al. 3 p r e s e n t some other a r g u m e n t s relying on the experidP m e n t a l o b s e r v a t i o n that log ~-~ vs P plots give a s t r a i g h t line c o r r e l a t i o n dP w h e r e P is load and t is time. Since ~-~ is p r o p o r t i o n a l plastic strain rate 4, in Fig.
3 the log plastic
typical data of the p r e s e n t work.
strain rate is p l o t t e d vs stress b a s e d on S t r a i g h t line c o r r e l a t i o n is not observed.
The present results suggest that u n c e r t a i n i t y exists in the e x p e r i m e n t a l e v i d e n c e on t i t a n i u m used by these authors. In summary the p r e s e n t w o r k shows that the c o n s t a n t h a r d n e s s
log stress-
7
Vol.
8, No.
7
STRESS
RELAXATION
log plastic strain rate relationships
IN T I T A N I U M
741
for titanium obtained from room tem-
perature stress relaxation experiments exhibit behavior not inconsistent with the requirements of plastic equation of state.
The longer time stress
relaxation data suggest that some of the results reported in the literature based on shorter time data may required further examination. Acknowled@ement This work is supported by the National Science Foundation through the Materials Science Center at Cornell University.
The authors wish to
acknowledge B. Addis for specimen preparation. References i.
C.C.
2.
G. Sargent and H. Conrad,
Law and D. N. Beshers,
Scripts Met. 6, 635
3.
P. Rodriguez, P. Dasgupta, S. L. Mannan, Samuel, Scripta Met. [, 671 (1973).
4.
E . W . Hart, Che-Yu Li, H. Yamada, and G. L. Wire, "Phenomenological Theory: A Guide to Constitutive Relations and Fundamental Deformation Properties" to be published in "Constitutive Equations in Plasticity" edited by A. Argon (MIT Press. planned for 1974).
Scripta Met. ~, 43
5.0
I
I
S. S. Vagardi,
I
I
.,.../"
4.9
,/
/
S" /
s •
(1972).
(1969).
~
and K. G.
I
4
..
3
.
6
,.
n
B, o°
o~
,/ /
4.8
y°
n-
I-
/"
(..9 0 _1
s 5
/
°~."
/
S ~° "
o
4.7
46
-8
I
I
I
I
I
-7
-6
-5
-4
-5
LOG
STRAIN
RATE
SEC- I )
Fig. 1 LOglo(Stress) vs lOglO(strain rate) data for titanium at room temperature and several strain levels.
742
STRESS
RELAXATION
20C
IN T I T A N I U M
I
I
Vol.
I
~5c
8, No.
I
S
_J
S
rF £3 < o
50 c D
aG c ao o
o
o~O 0
I i
0
LOG
Fig. Typical
L
1
5
4
2
load r e l a x a t i o n
for t i t a n i u m
vs l O g l 0 ( t i m e )
I I
T
-5-
.."
" -6-
.
.."
•°• ••
.Z"
j,, •
•°
3
4
,.
/ s.~*"
.-"
I
6
•
""
/
o•• . , f •
2
,:"
/
/
uJ
I
5
..:
o u] - 4 o9
data
at r o o m t e m p e r a t u r e .
-3
Z
I 2 TIME (SEC)
""''
...
" ""
•
~"
•
°. ••••
-7-
0 _1 - B 4 0
I 50
I 60
I
STRESS
Fig. LOgl0(strain
rate)
room temperature
7O
I 80
for t i t a n i u m
at
(PSI)
3
vs s t r e s s d a t a
and s e v e r a l
plastic
strain
levels.
90
7
Vol. 8, pp. 739-742, 1974 P r i n t e d in the U n i t e d States
STRESS
RELAXATION
F. V. Ellis
Department
Press,
Pergamon
Inc.
IN T I T A N I U M
and C h e - Y u
Li
of M a t e r i a l s Science and E n g i n e e r i n g Cornell University Ithaca, New York 14850
(Received M a r c h
22,
1974)
Introduction Stress
relaxation
i n v e s t i g a t o r s I-3 the
theories
of d i s l o c a t i o n
recently
also
equation
of state 4.
a limited some
concept
to o b t a i n
time
dynamics.
data
stress
The p u r p o s e
relaxation
equation
relaxation
data
Discussion to show
by several
report
on
are used
of p l a s t i c
on t i t a n i u m
will
that
based
experiments
the c o n c e p t
of the p r e s e n t
and
reported
are d i s c u s s e d
relaxation
to support
data.
of state
have been
investigations
Stress
experimental
time.
stress
of p l a s t i c
the p r e v i o u s
on t i t a n i u m
of these
The e x i s t i n g
relaxation
longer
experiments
The results
cover
only
is to p r e s e n t
be made
using
the c o n c l u s i o n s
the
made by
i n v e s t i g a t o r s 2-3 may be questioned.
Experiments Room
temperature
stress
relaxation
experiments
are p e r f o r m e d
in the p r e s e n t
work.
The principle, p r o c e d u r e , and e q u i p m e n t of the e x p e r i m e n t have been 4 d e s c r i b e d p r e v i o u s l y . C o m m e r c i a l p u r i t y t i t a n i u m of a p p r o x i m a t e l y 99.7%
purity
is used.
annealed
two inches at 0.02 stress
Titanium
in v a c u u m
and a grain
in/min,
rods
for an hour
stress
stress given
load-time
particular single level
stress
specimen after
another
data
shown
strain
rate
4.
from each
strain
The
and
length
of
specimen
is p r e - s t r a i n e d
level
to each
strain)
prior
relaxation
a stress
run.
relaxation
shown each
Data
run.
Data
from runs
curves
stress 5-6
of p l a s t i c
hardness
739
from a
from one
(or p l a s t i c
are o b t a i n e d
equation
strain)
from
of state 4
log s t r e s s - l o g
should b e l o n g
to log
the a p p r o a c h the data
1-4 are o b t a i n e d
to a h i g h e r
to the c o n c e p t
These
1 using
represents
from runs
constant
run are c o n v e r t e d
in Fig.
curve
the s p e c i m e n
1 represent
relationships.
stress
figure
relaxation
According
in Fig.
and D i s c u s s i o n
rate plots
In this
by r e l o a d i n g
specimen.
curves
50~.
diameter
has a gage
run.
- log p l a s t i c in r e f e r e n c e
to 0.125"
specimen
(or p l a s t i c
Results The
swaged
The
size of a p p r o x i m a t e l y
to a d e s i r e d
relaxation
are arc cast, at 500°C.
plastic
to a p a r a m e t e r
family
740
STRESS R E L A X A T I O N
IN T I T A N I U M
Vol.
8, No.
of curves w i t h the a r b i t r a r y p a r a m e t e r b e i n g the h a r d n e s s p a r a m e t e r if a p l a s t i c e q u a t i o n of state exists.
A simple way to test the one p a r a m e t e r
family of curves is the p o s s i b i l i t y that these curves can be t r a n s l a t e d along a s t r a i g h t line to o v e r l a p each other.
It is seen from Fig.
these curves are n e a r l y a set of p a r a l l e l s t r a i g h t lines. s t r a i g h t lines can be t r a n s l a t e d p a r a l l e l to the log stress axis.
1 that
A set of p a r a l l e l
to o v e r l a p each o t h e r along a path w h i c h is The curves in Fig.
1 do show h o w e v e r slight
c u r v a t u r e w h i c h is w i t h i n the a c c u r a c y of the experiment.
U n f o r t u n a t e l y the
c u r v a t u r e of these curves does not show s u f f i c i e n t c h a r a c t e r i s t i c s the d e t e r m i n a t i o n of a unique t r a n s l a t i o n path.
The
to allow
d e t e r m i n a t i o n of a
u n i q u e t r a n s l a t i o n p a t h has b e e n p o s s i b l e for c o n s t a n t h a r d n e s s
log stress -
log p l a s t i c strain rate curves of a v a r i e t y of m a t e r i a l s as r e p o r t e d in r e f e r e n c e 4. homologous
In the same r e f e r e n c e it is d i s c u s s e d that in general at low
temperatures
the c o n s t a n t h a r d n e s s
log stress - log p l a s t i c strain
rate curves w i l l show a c u r v a t u r e c o n c a r v i n g u p w a r d and c o n c a r v i n g d o w n w a r d at high h o m o l o g o u s
temperatures.
The i n t e r m e d i a t e h o m o l o g o u s t e m p e r a t u r e s
range c o r r e s p o n d to the t r a n s i t i o n r e g i o n w h e r e the curves will show less characteristics. mediate homologous
It is p o s s i b l e that r o o m t e m p e r a t u r e t e m p e r a t u r e range for titanium.
t h e r e f o r e to d e t e r m i n e
falls in the inter-
It will be d i f f i c u l t
the t r a n s l a t i o n path.
S a r g e n t and C o n r a d 2 has used the slope of load vs log time plot to d e t e r m i n e the a c t i v a t i o n v o l u m e for p l a s t i c covers a p e r i o d of five minutes.
flow.
Their r e l a x a t i o n time
In their data range the load - log time
plot shows a s t r a i g h t line correlation.
The typical results of the p r e s e n t
w o r k in a p l o t of load vs log time are shown in Fig.
2.
It is seen that the
p r e s e n t data show a c o n t i n u o u s l y c h a n g i n g slope at longer r e l a x a t i o n times. The values of the a c t i v a t i o n volume o b t a i n e d by these i n v e s t i g a t i o r s may t h e r e f o r e be questioned. 3 R o d r i g u e z et al. p r o p o s e d several a r g u m e n t s in an a t t e m p t to use stress r e l a x a t i o n data to r a t i o n a l i z e m e c h a n i s m s
for p l a s t i c flow in titanium.
Some of their a r g u m e n t s d e p e n d on the fact that long time limit of load vs log time p l o t a p p r o a c h e s
a s t r a i g h t line.
line limit is shown by the p r e s e n t data
No e v i d e n c e of long time s t r a i g h t
(Fig. 2) w h i c h ~overs a r e l a x a t i o n
time one d e c a d e longer than that involved in the e x p e r i m e n t s of R o d r i g u e z et al.
The p r e s e n t w o r k u l t i l i z e s a d i g i t a l data a c q u i s i t i o n s y s t e m and good
t e m p e r a t u r e control of the entire t e s t i n g s y s t e m 4. Fig.
2 is w i t h i n e x p e r i m e n t a l
The c u r v a t u r e shown in
limit.
R o d r i g u e z et al. 3 p r e s e n t some other a r g u m e n t s relying on the experidP m e n t a l o b s e r v a t i o n that log ~-~ vs P plots give a s t r a i g h t line c o r r e l a t i o n dP w h e r e P is load and t is time. Since ~-~ is p r o p o r t i o n a l plastic strain rate 4, in Fig.
3 the log plastic
typical data of the p r e s e n t work.
strain rate is p l o t t e d vs stress b a s e d on S t r a i g h t line c o r r e l a t i o n is not observed.
The present results suggest that u n c e r t a i n i t y exists in the e x p e r i m e n t a l e v i d e n c e on t i t a n i u m used by these authors. In summary the p r e s e n t w o r k shows that the c o n s t a n t h a r d n e s s
log stress-
7
Vol.
8, No.
7
STRESS
RELAXATION
log plastic strain rate relationships
IN T I T A N I U M
741
for titanium obtained from room tem-
perature stress relaxation experiments exhibit behavior not inconsistent with the requirements of plastic equation of state.
The longer time stress
relaxation data suggest that some of the results reported in the literature based on shorter time data may required further examination. Acknowled@ement This work is supported by the National Science Foundation through the Materials Science Center at Cornell University.
The authors wish to
acknowledge B. Addis for specimen preparation. References i.
C.C.
2.
G. Sargent and H. Conrad,
Law and D. N. Beshers,
Scripts Met. 6, 635
3.
P. Rodriguez, P. Dasgupta, S. L. Mannan, Samuel, Scripta Met. [, 671 (1973).
4.
E . W . Hart, Che-Yu Li, H. Yamada, and G. L. Wire, "Phenomenological Theory: A Guide to Constitutive Relations and Fundamental Deformation Properties" to be published in "Constitutive Equations in Plasticity" edited by A. Argon (MIT Press. planned for 1974).
Scripta Met. ~, 43
5.0
I
I
S. S. Vagardi,
I
I
.,.../"
4.9
,/
/
S" /
s •
(1972).
(1969).
~
and K. G.
I
4
..
3
.
6
,.
n
B, o°
o~
,/ /
4.8
y°
n-
I-
/"
(..9 0 _1
s 5
/
°~."
/
S ~° "
o
4.7
46
-8
I
I
I
I
I
-7
-6
-5
-4
-5
LOG
STRAIN
RATE
SEC- I )
Fig. 1 LOglo(Stress) vs lOglO(strain rate) data for titanium at room temperature and several strain levels.
742
STRESS
RELAXATION
20C
IN T I T A N I U M
I
I
Vol.
I
~5c
8, No.
I
S
_J
S
rF £3 < o
50 c D
aG c ao o
o
o~O 0
I i
0
LOG
Fig. Typical
L
1
5
4
2
load r e l a x a t i o n
for t i t a n i u m
vs l O g l 0 ( t i m e )
I I
T
-5-
.."
" -6-
.
.."
•°• ••
.Z"
j,, •
•°
3
4
,.
/ s.~*"
.-"
I
6
•
""
/
o•• . , f •
2
,:"
/
/
uJ
I
5
..:
o u] - 4 o9
data
at r o o m t e m p e r a t u r e .
-3
Z
I 2 TIME (SEC)
""''
...
" ""
•
~"
•
°. ••••
-7-
0 _1 - B 4 0
I 50
I 60
I
STRESS
Fig. LOgl0(strain
rate)
room temperature
7O
I 80
for t i t a n i u m
at
(PSI)
3
vs s t r e s s d a t a
and s e v e r a l
plastic
strain
levels.
90
7