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On a new method of deformation calorimetry a reply to a letter by R. O. Williams
5qcript~ ~11iT\I.I,[IRGI<\
Vol. Printed
12, pp. 9 9 5 - 9 9 8 , 1978 in tile United States
ON A NEW METHOD
OF D E F O R M A T I O N
Pergamon [)Tess,
Inc
CALORIMETRY
A REPLY TO A L E T T E R BY R. O. WILLIA}4S
D. R~nnpagel, Institut Technische
Ch.
Schwink
A f~r Physik
Universit~t,
33 Braunschweig,
[Received September l,
Germany
197~)
In the p r e c e d i n g paper W i l l i a m s /I/ makes some critical remarks on our new, improved m e t h o d of d e f o r m a t i o n c a l o r i m e t r y /2,3/. He critizices we n e g l e c t e d the thermal isolation of the sample crystal and casts some doubts not nearer specified on the quality of the " s o p h i s t i c a t e d data analysis" which we apply. We take the o p p o r t u n i t y I) to present some c o n s i d e r a t i o n s and an e x p e r i m e n t c o n c e r n i n g the heat flow in the sample during the deformation, 2) to outline our e v a l u a t i o n p r o c e d u r e and to show its virtual simplicity and c o m p l e t e n e s s and 3) to d e s c r i b e two e x p e r i m e n t s c r i t i c a l l y testing the e v a l u a t i o n procedure and the m e t h o d as a whole. I. Thermal isolation and heat flow. We e x p e r i m e n t a l l y reduce, of course, the heat flow from the sample crystal to the surroundings as far as possible /3/: The d e f o r m a t i o n is p e r f o r m e d in a v a c u u m chamber (to suppress convection) and the crystal with shoulders at its ends is hanging in grips of stainless steel (to reduce heat conduction). However, the e s s e n t i a l point is the following: The density of heat sources during plastic d e f o r m a t i o n is u n i f o r m only w i t h i n a middle part of the barshaped crystal (4 mm ~), it begins to d e c r e a s e about 15 m m from each end (where the crystal cross section c o r r e s p o n d i n g l y remains larger) and becomes zero w i t h i n the parts of the crystal w h i ch are clamped by grips, b e c a u s e there no d e f o r m a t i o n at all occurs - i r r e s p e c t i v e of the type of c o n n e c t i o n b e t w e e n crystal and pull rods. The inevitably a p p r e c i a b l e heat c a p a c i t y of these parts of the crystal not taking part in the plastic d e f o r m a t i o n has the c o n s e q u e n c e that the main heat flow during the time of m e a s u r e m e n t occurs from the h o m o g e n e o u s l y d e f o r m i n g middle part of sample to its n o t - d e f o r m i n g ends (their heat c a p a c i t y being e s p e c i a l l y low by use of crystals with shoulders h a n g i n g in grips /3/), w h e r e a s the heat flow to the stainless steel pull rods during this time is much smaller. To prove that e x p e r i m e n t a l l y we have m e a s u r e d the t e m p e r a t u r e increase of a crystal during deformation, T(t), r e a l i z i n g two quite d i f f e r e n t heat conductivities b e t w e e n b e t w e e n crystal shoulders and grips. In the first e x p e r i m e n t the crystal was t h e r m a l l y isolated from the grips by ceramic plates of 3 mm thickness, in the second one, conversely, a heat c o n d u c t i o n paste was to g u a r a n t e e a very good heat contact. The T ( t ) - c u r v e s of both e x p e r i m e n t s v i r t u a l l y were the same which shows that there is no appreciable heat flow to the grips during the time of measurement.
995
996
REPLY TO LETTER
Vol.
12,
No.
11
Thus, the only way to reduce the heat flow from the middle of the crystal during the m e a s u r i n g time is to use long samples. Originally, we used crystals of 80 mm length, but soon we changed to a length of 130 mm /2/, thereby e n l a r g i n g the h o m o g e n e o u s l y d e f o r m i n g part of the crystal by a factor of about two. 2. The e v a l u a t i o n procedure. We m e a s u r e the T ( t ) - c u r v e s with a rather high r e s o l u t i o n as well of t e m p e r a t u r e (~T~2.10-5K) as of time (at-~O,Is). That makes it p o s s i b l e to use the curves in their full length for the physical i n t e r p r e t a t i o n and for th@ control of the single steps of the e v a l u a t i o n procedure. A T ( t ) - c u r v e and its d e v i a t i o n from a straight line i m m e d i a t e l y allows to judge the influence of heat conduction. Before our work /2,3/ no T ( t ) - c u r v e as m e a s u r e d by a d e f o r m a t i o n c a l o r i m e t e r has been r e p o r t e d to our k n o w l e d g e and e x p l i c i t e l y discussed. But, that is the central p r o b l e m of q u a n t i t a t i v e d e f o r m a t i o n calorimetry. Each tensile d e f o r m a t i o n begins with a t h e r m o e l a s t i c cooling of the sample. The initial n e g a t i v e slope of the T ( t ) - c u r v e (Fig. I) is q u a n t i t a t i v e l y c o n n e c t e d with the initial increase of the stress, ~(t). In the regions of e l a s t i c - p l a s t i c t r a n s i t i o n and of s t a t i o n a r y plastic d e f o r m a t i o n the m e a s u r e d t e m p e r a t u r e has a t h e r m o e l a s t i c and a plastic contribution: Tmeas(t)
= T o + Tthermo(t)
+ Tpla(t)-
The t h e r m o e l a s t i c c o n t r i b u t i o n can be c a l c u l a t e d to a very good a p p r o x i m a t i o n as shown in /2,3/ (terms with factor E in equ. (13) of / 2 / ) . E v e n an exact c a l c u l a t i o n is p o s s i b l e using the second d e r i v a t i v e of ~(t) /4/. By s u b t r a c t i n g it t o g e t h e r with T o from Tmeas we get the pure plastic contribution, Tpla(t), from which the heat e v o l v i n g during deformation, the aim of the calorimetry, follows immediately.
E{t)
----<7 E t
I
G
FIG.
l
i
i
t
i
1
Example of a single e x p e r i m e n t c o n s i s t i n g of 6(t), ~(t) and Tmeas(t). Cu crystal in snage II. ~ = yield stress, ~ = average stress in the time interval of measurementlt~-t;1 c o r r e s p o n d i n g to a r e l a t i v e strain interval of about 3-10 -3 , T o = room temperature.
T
(The T ( t ) - c u r v e s after s t o p p i n g the e x p e r i m e n t are omitted.)
I I
/ ! t
at t e
Vo] .
1 2,
No.
11
RI!PIA ]'0 LE1TF{R
9U7
F o r its r e a l z z a ~ z o n we p r o c e e d t ~ a t w a y / 2 , 3 / : A p u r e l y e l a s t z c d e f o r m a t i o n of t h e c r y s t a l p r o d u c e s a c h a n g e of t e m p e r a t u r e w h i c h c a n be c a l c u l a t e d e x a c t l y in the a d i a b a t i c c a s e by t h e f o r m u l a for the t n e r m o e l a s t i c effect. A c o n s t a n t e l a s t i c s t r e s s r a t e w o u l d p r o d u c e a c o n s t a n t i n c r e a s e in T(t) ("test f u n c n i o n " ) . The real system will respon~ differently and yield a m~asuzable i n c r e a s e of T w l t h t i m e c a l l e d " r e s p o n s e f u n c t i o n " . W i t h its heft., the t e m p e r a t u r e increase durzng plastic defozmation car] be f u l l y d e s c r i b e d (equ. (13) in /2/) . S i n c e the p l a s t i c c o n t r i b u t i o n T p l a ( t ) is p r o p o r t i o n a l to the r e s p o n s e fiunction the c o r r e s p o n d i n g temperature s l o p e of p l a s t i c deformation, Tpla, c a n be c a l c u l a t e d . T h a t is the s i m p l e D r z n c i ~ l e of the a n a l y s zs. By t]~e irltrOQUCtlOn of the e x p e r i m e n t a l t e s t a n d r e s p o n s e ~ u n c t i o n all tile vazlous heat losses occumng in the e x p e r i m e n t are t a k e ~ i n t o a c c o u n t . T h u s , t h e p r o c e d u r e eo i p s o f u l f i l l s the r e q u i r e m e n t s of a c o m p l e t e treatn]ent of the beat conducclon p r o b l e m and a v o i d s the ap~)arentiy q r e a t d i f f i c u l t i e s of ~l~ze[y t h e o r e t i c a l calcu]ation w h i c h a r e p a r t l y our]in~-~d J n /I/. 3. ~l'esz ex~ez±~[tents. The selfconsistency or tne e v a l u a t i o n p r o c e d u r e c a n be t e s t e d by a s e p a r a t e e x p e r z m e n t /5/. T h e c o u r s e of t e m p e r a t u r e ms s i m u l t a n e o u s ]y m e a s u r e d at d i f f e r e n t p o s i t i o n s of the sam['le wit}l s e v e r a l t h e r m i s t o r s Jl~(iepei~um~]tJy of e a c h ot~ler w l t ~ i n one p l a s t i c s t r a z n Jr~terval (~E~-3-IO -3) . Aftezwards witil the s a m e t h e r m z s t o r s at the s a m e p o s L t i o n s the l e s p o n s e functions (after an e l a s t z c defoz~tation] axe r e g i s t e r e d a n d w i t h t h e m the cozr~.~spondJng T m o a s ( t ) - c u r v e s are e v a ] u a t e d as o u t l i n e d abc)ve. T h e d i f f e r e n t resuLtinq %p]a(t)-curves c o i n c i d e to w i t h i n I{.. Tiiat p r o v e s the c o n s i s t e n c y of the p]:oe,:~dL,~[-o
E'J0
-3
~o,;
4-
50
•4"
t
FIG.
~"
2
E v a l u a t e d r e s u l t s of 12 s u b s e quent szngie e~periments
0
!cf. Fi~. 1). Eexp: ~ E and Qmla ~ T p l a a r e q ~ i t e independently determined (see text),
-2 -30
but strongly correlated allel course). 03
-lO
L
0.08
f
t
0.09
,
I
O.lO
t
E {7
(par-
998
REPLY TO LETTER
Vol.
12,
No.
11
Finally, the c o m b i n e d diagrams of Fig. 2 are to d e m o n s t r a t e the r e l i a b i l i t y of our m e t h o d as whole. A copper crystal of middle o r i e n t a t i o n is d e f o r m e d in 12 s u b s e q u e n t strain intervals in stage II, the values of the applied average stress of each interval, [, follow a straight line. The relative strain rate, ~ , is d e t e r m i n e d by a special t e n s i o m e t e r /3/ d i r e c t l y for the small m i d d l e part of the crystal where also T is m e a s u r e d to avoid influences from the e l a s t i c i t y of the machine and the grips. N e v e r t h e l e s s , we find fluctuations in 6 of about 5% from interval to interval a l t h o u g h the external speed of the pull rods is taken the same for the 1 2 - i n t e r v a l - s e r i e s and remains rather well c o n s t a n t within each single interval as ~ itself does, too. With the d e f o r m a t i o n energy e x p e n d e d per time, Eexp = ~ ' ~ , varies simultaneously. On the other hand, the heat p r o d u c e ~ per time within each strain interval of plastic deformation, Qpla = Cp. Tpla, (Cp = specific heat at constant. pressure) is d e t e r m i n e d [see above and /2,3y) c o m p l e t e l y independent of Eexp,i.e. of the m e c h a n i c a l quantities 6 and ~ . Fig. 2 shows that there is a very good c o r r e l a t i o n between Eex D and Qmla, both running nearly parallel. This result d e m o n s t r a t e s quite c o h v i n c i n ~ l y the r e l i a b i l i t y and quality of our e x p e r i m e n t a l and analytical procedure. We conclude with the remark that we a p p r e c i a t e R.O. W i l l i a m s in /I/ having brought to our knowledge his letter of 1964 /6/ where he first applied the d e f o r m a t i o n c a l o r i m e t r y to single crystals. References /I/ /2/ /3/ /4/ /5/ /6/
R.O. Williams, p r e c e d i n g paper D. R6nnpagel, Ch. Schwink, Acta Met. 26, 319 (19"78) D. R~nnpagel, G. Schulz, Thermochim. Acta 22, 289 (1978) D. R~nnpagel, u n p u b l i s h e d K.-H. Sch6nborn, Diploma work 1976, TU B r a u n s c h w e i g R.O. Williams, Acta Met. 12, 745 (1964)
Vol. Printed
12, pp. 9 9 5 - 9 9 8 , 1978 in tile United States
ON A NEW METHOD
OF D E F O R M A T I O N
Pergamon [)Tess,
Inc
CALORIMETRY
A REPLY TO A L E T T E R BY R. O. WILLIA}4S
D. R~nnpagel, Institut Technische
Ch.
Schwink
A f~r Physik
Universit~t,
33 Braunschweig,
[Received September l,
Germany
197~)
In the p r e c e d i n g paper W i l l i a m s /I/ makes some critical remarks on our new, improved m e t h o d of d e f o r m a t i o n c a l o r i m e t r y /2,3/. He critizices we n e g l e c t e d the thermal isolation of the sample crystal and casts some doubts not nearer specified on the quality of the " s o p h i s t i c a t e d data analysis" which we apply. We take the o p p o r t u n i t y I) to present some c o n s i d e r a t i o n s and an e x p e r i m e n t c o n c e r n i n g the heat flow in the sample during the deformation, 2) to outline our e v a l u a t i o n p r o c e d u r e and to show its virtual simplicity and c o m p l e t e n e s s and 3) to d e s c r i b e two e x p e r i m e n t s c r i t i c a l l y testing the e v a l u a t i o n procedure and the m e t h o d as a whole. I. Thermal isolation and heat flow. We e x p e r i m e n t a l l y reduce, of course, the heat flow from the sample crystal to the surroundings as far as possible /3/: The d e f o r m a t i o n is p e r f o r m e d in a v a c u u m chamber (to suppress convection) and the crystal with shoulders at its ends is hanging in grips of stainless steel (to reduce heat conduction). However, the e s s e n t i a l point is the following: The density of heat sources during plastic d e f o r m a t i o n is u n i f o r m only w i t h i n a middle part of the barshaped crystal (4 mm ~), it begins to d e c r e a s e about 15 m m from each end (where the crystal cross section c o r r e s p o n d i n g l y remains larger) and becomes zero w i t h i n the parts of the crystal w h i ch are clamped by grips, b e c a u s e there no d e f o r m a t i o n at all occurs - i r r e s p e c t i v e of the type of c o n n e c t i o n b e t w e e n crystal and pull rods. The inevitably a p p r e c i a b l e heat c a p a c i t y of these parts of the crystal not taking part in the plastic d e f o r m a t i o n has the c o n s e q u e n c e that the main heat flow during the time of m e a s u r e m e n t occurs from the h o m o g e n e o u s l y d e f o r m i n g middle part of sample to its n o t - d e f o r m i n g ends (their heat c a p a c i t y being e s p e c i a l l y low by use of crystals with shoulders h a n g i n g in grips /3/), w h e r e a s the heat flow to the stainless steel pull rods during this time is much smaller. To prove that e x p e r i m e n t a l l y we have m e a s u r e d the t e m p e r a t u r e increase of a crystal during deformation, T(t), r e a l i z i n g two quite d i f f e r e n t heat conductivities b e t w e e n b e t w e e n crystal shoulders and grips. In the first e x p e r i m e n t the crystal was t h e r m a l l y isolated from the grips by ceramic plates of 3 mm thickness, in the second one, conversely, a heat c o n d u c t i o n paste was to g u a r a n t e e a very good heat contact. The T ( t ) - c u r v e s of both e x p e r i m e n t s v i r t u a l l y were the same which shows that there is no appreciable heat flow to the grips during the time of measurement.
995
996
REPLY TO LETTER
Vol.
12,
No.
11
Thus, the only way to reduce the heat flow from the middle of the crystal during the m e a s u r i n g time is to use long samples. Originally, we used crystals of 80 mm length, but soon we changed to a length of 130 mm /2/, thereby e n l a r g i n g the h o m o g e n e o u s l y d e f o r m i n g part of the crystal by a factor of about two. 2. The e v a l u a t i o n procedure. We m e a s u r e the T ( t ) - c u r v e s with a rather high r e s o l u t i o n as well of t e m p e r a t u r e (~T~2.10-5K) as of time (at-~O,Is). That makes it p o s s i b l e to use the curves in their full length for the physical i n t e r p r e t a t i o n and for th@ control of the single steps of the e v a l u a t i o n procedure. A T ( t ) - c u r v e and its d e v i a t i o n from a straight line i m m e d i a t e l y allows to judge the influence of heat conduction. Before our work /2,3/ no T ( t ) - c u r v e as m e a s u r e d by a d e f o r m a t i o n c a l o r i m e t e r has been r e p o r t e d to our k n o w l e d g e and e x p l i c i t e l y discussed. But, that is the central p r o b l e m of q u a n t i t a t i v e d e f o r m a t i o n calorimetry. Each tensile d e f o r m a t i o n begins with a t h e r m o e l a s t i c cooling of the sample. The initial n e g a t i v e slope of the T ( t ) - c u r v e (Fig. I) is q u a n t i t a t i v e l y c o n n e c t e d with the initial increase of the stress, ~(t). In the regions of e l a s t i c - p l a s t i c t r a n s i t i o n and of s t a t i o n a r y plastic d e f o r m a t i o n the m e a s u r e d t e m p e r a t u r e has a t h e r m o e l a s t i c and a plastic contribution: Tmeas(t)
= T o + Tthermo(t)
+ Tpla(t)-
The t h e r m o e l a s t i c c o n t r i b u t i o n can be c a l c u l a t e d to a very good a p p r o x i m a t i o n as shown in /2,3/ (terms with factor E in equ. (13) of / 2 / ) . E v e n an exact c a l c u l a t i o n is p o s s i b l e using the second d e r i v a t i v e of ~(t) /4/. By s u b t r a c t i n g it t o g e t h e r with T o from Tmeas we get the pure plastic contribution, Tpla(t), from which the heat e v o l v i n g during deformation, the aim of the calorimetry, follows immediately.
E{t)
----<7 E t
I
G
FIG.
l
i
i
t
i
1
Example of a single e x p e r i m e n t c o n s i s t i n g of 6(t), ~(t) and Tmeas(t). Cu crystal in snage II. ~ = yield stress, ~ = average stress in the time interval of measurementlt~-t;1 c o r r e s p o n d i n g to a r e l a t i v e strain interval of about 3-10 -3 , T o = room temperature.
T
(The T ( t ) - c u r v e s after s t o p p i n g the e x p e r i m e n t are omitted.)
I I
/ ! t
at t e
Vo] .
1 2,
No.
11
RI!PIA ]'0 LE1TF{R
9U7
F o r its r e a l z z a ~ z o n we p r o c e e d t ~ a t w a y / 2 , 3 / : A p u r e l y e l a s t z c d e f o r m a t i o n of t h e c r y s t a l p r o d u c e s a c h a n g e of t e m p e r a t u r e w h i c h c a n be c a l c u l a t e d e x a c t l y in the a d i a b a t i c c a s e by t h e f o r m u l a for the t n e r m o e l a s t i c effect. A c o n s t a n t e l a s t i c s t r e s s r a t e w o u l d p r o d u c e a c o n s t a n t i n c r e a s e in T(t) ("test f u n c n i o n " ) . The real system will respon~ differently and yield a m~asuzable i n c r e a s e of T w l t h t i m e c a l l e d " r e s p o n s e f u n c t i o n " . W i t h its heft., the t e m p e r a t u r e increase durzng plastic defozmation car] be f u l l y d e s c r i b e d (equ. (13) in /2/) . S i n c e the p l a s t i c c o n t r i b u t i o n T p l a ( t ) is p r o p o r t i o n a l to the r e s p o n s e fiunction the c o r r e s p o n d i n g temperature s l o p e of p l a s t i c deformation, Tpla, c a n be c a l c u l a t e d . T h a t is the s i m p l e D r z n c i ~ l e of the a n a l y s zs. By t]~e irltrOQUCtlOn of the e x p e r i m e n t a l t e s t a n d r e s p o n s e ~ u n c t i o n all tile vazlous heat losses occumng in the e x p e r i m e n t are t a k e ~ i n t o a c c o u n t . T h u s , t h e p r o c e d u r e eo i p s o f u l f i l l s the r e q u i r e m e n t s of a c o m p l e t e treatn]ent of the beat conducclon p r o b l e m and a v o i d s the ap~)arentiy q r e a t d i f f i c u l t i e s of ~l~ze[y t h e o r e t i c a l calcu]ation w h i c h a r e p a r t l y our]in~-~d J n /I/. 3. ~l'esz ex~ez±~[tents. The selfconsistency or tne e v a l u a t i o n p r o c e d u r e c a n be t e s t e d by a s e p a r a t e e x p e r z m e n t /5/. T h e c o u r s e of t e m p e r a t u r e ms s i m u l t a n e o u s ]y m e a s u r e d at d i f f e r e n t p o s i t i o n s of the sam['le wit}l s e v e r a l t h e r m i s t o r s Jl~(iepei~um~]tJy of e a c h ot~ler w l t ~ i n one p l a s t i c s t r a z n Jr~terval (~E~-3-IO -3) . Aftezwards witil the s a m e t h e r m z s t o r s at the s a m e p o s L t i o n s the l e s p o n s e functions (after an e l a s t z c defoz~tation] axe r e g i s t e r e d a n d w i t h t h e m the cozr~.~spondJng T m o a s ( t ) - c u r v e s are e v a ] u a t e d as o u t l i n e d abc)ve. T h e d i f f e r e n t resuLtinq %p]a(t)-curves c o i n c i d e to w i t h i n I{.. Tiiat p r o v e s the c o n s i s t e n c y of the p]:oe,:~dL,~[-o
E'J0
-3
~o,;
4-
50
•4"
t
FIG.
~"
2
E v a l u a t e d r e s u l t s of 12 s u b s e quent szngie e~periments
0
!cf. Fi~. 1). Eexp: ~ E and Qmla ~ T p l a a r e q ~ i t e independently determined (see text),
-2 -30
but strongly correlated allel course). 03
-lO
L
0.08
f
t
0.09
,
I
O.lO
t
E {7
(par-
998
REPLY TO LETTER
Vol.
12,
No.
11
Finally, the c o m b i n e d diagrams of Fig. 2 are to d e m o n s t r a t e the r e l i a b i l i t y of our m e t h o d as whole. A copper crystal of middle o r i e n t a t i o n is d e f o r m e d in 12 s u b s e q u e n t strain intervals in stage II, the values of the applied average stress of each interval, [, follow a straight line. The relative strain rate, ~ , is d e t e r m i n e d by a special t e n s i o m e t e r /3/ d i r e c t l y for the small m i d d l e part of the crystal where also T is m e a s u r e d to avoid influences from the e l a s t i c i t y of the machine and the grips. N e v e r t h e l e s s , we find fluctuations in 6 of about 5% from interval to interval a l t h o u g h the external speed of the pull rods is taken the same for the 1 2 - i n t e r v a l - s e r i e s and remains rather well c o n s t a n t within each single interval as ~ itself does, too. With the d e f o r m a t i o n energy e x p e n d e d per time, Eexp = ~ ' ~ , varies simultaneously. On the other hand, the heat p r o d u c e ~ per time within each strain interval of plastic deformation, Qpla = Cp. Tpla, (Cp = specific heat at constant. pressure) is d e t e r m i n e d [see above and /2,3y) c o m p l e t e l y independent of Eexp,i.e. of the m e c h a n i c a l quantities 6 and ~ . Fig. 2 shows that there is a very good c o r r e l a t i o n between Eex D and Qmla, both running nearly parallel. This result d e m o n s t r a t e s quite c o h v i n c i n ~ l y the r e l i a b i l i t y and quality of our e x p e r i m e n t a l and analytical procedure. We conclude with the remark that we a p p r e c i a t e R.O. W i l l i a m s in /I/ having brought to our knowledge his letter of 1964 /6/ where he first applied the d e f o r m a t i o n c a l o r i m e t r y to single crystals. References /I/ /2/ /3/ /4/ /5/ /6/
R.O. Williams, p r e c e d i n g paper D. R6nnpagel, Ch. Schwink, Acta Met. 26, 319 (19"78) D. R~nnpagel, G. Schulz, Thermochim. Acta 22, 289 (1978) D. R~nnpagel, u n p u b l i s h e d K.-H. Sch6nborn, Diploma work 1976, TU B r a u n s c h w e i g R.O. Williams, Acta Met. 12, 745 (1964)