- Email: [email protected]
Serrated flow under uniaxial and multiaxial stress and strain conditions
Scripta METALLURGICA et MATERIALIA
SERRATED
FLOW
Vol.
UNDER
25, pp. 2417-2422, 1991 Printed in the U.S.A.
UNIAXIAL
AND
MULTIAXIAL
STRESS
Pergamon Press plc All rights reserved
AND
STRAIN
CONDITIONS
Monika Fellner*, Erwin Pink~+ and Ewald Werner + *Erich-Schmid-Institut fur FestkSrperphysik, Osterreichische Akademie der Wissenschaften, +Institut f~r Metallphysik, Montanuniversit~t, Leoben, Austria (Received June 20, 1991)
Introduction Serrated flow has been discussed over the years almost exclusively in terms of uniaxial tension. There are, however, a few reports which indicate that the stress state affects the occurrence of serrations. In equibiaxial tension, for instance, serrations were said to be "less evident" (i). The effects of the stress state were also demonstrated to influence the appearance of stretcher-strain markings in deep - drawing processes (2). A rectangular strip of an alloyed aluminium sheet was indented by a ball while the strip was gripped by a locking ring. However, the width of the sheet sample was less than the diameter of the locking ring, so that the stress distribution within the indented sheet was not symmetrical. Deformation bands developed in areas which were not in contact with the punching ball, inclined at a certain angle to the principal stresses. But in the center of contact with the ball, markings were found to be perpendicular to the principal-stress direction. When the sheet sample was fully gripped by the locking ring, markings did not appear in the center areas where the stress state is mainly biaxial. Markings did, however, appear in the peripherical areas with positive radial and negative tangential stresses, resembling the uniaxial stress state in a tensile sample (2). These results give the impression that stress states deviating from the uniaxial state suppress serrated flow. Quite contrary is the often experienced behaviour that, in tensile tests, the deformation may be homogeneous throughout the initial, increasing, portion of the stress-strain curve when the stress state is uniaxial, and that serrations appear as soon as necking has set in, i.e. for triaxial stress states. It is also known from experiments that serrations can be confined to the increasing portion of the stress-strain curve which has no serrations after the maximum load point. In many other experiments both "pre-necking" and "necking" serrations appear during one test. All these serrations do not necessarily have to be of the same type. A- or B-serrations* on the increasing section of the load-extension curve may be followed by B-type serrations during necking (necking serrations always seem to be of type B). No systematic research has, to our knowledge, been carried out. This report deals with first results from comparing, under different stress states, the pre-necking serrations with the necking serrations. We only compare serrations of type B (which is essential because A- and B-type serrations are, under most testing conditions, significantly different in size). * A-type serrations are separated by long strain intervals, because they represent the initiation of the first deformation at one side of the sample, and B-serrations follow each other in close succession as the deformation spreads inhomogenously along the sample's gauge length (3,4).
2417 0036-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press plc
2418
SERRATED FLOW
Vol.
25, No. II
The results for the size of serrations in tensile tests are often plotted in their dependence on the nominal strain rate ~. which is the cross-head speed related to the initial gauge length. However, a characteristic of serrated flow is the severe localization of deformation in narrow bands (the so-called "stretcher-strain markings"). But localizing the d e f o r m a t i o n leads to an increase of the strain rate, and therefore the nominal strain rate is not the true strain rate of the d e f o r m a t i o n in the narrow band.
The Properties of Necking Serretions We analyzed necking serrations from tests with 15 mm-rods of an AIMg5alloy in the as-received (i.e. to some extent coldworked) condition, comparing the size of the load drops A L N. Except for a very short transition range, AL N is extremely regular, and rather constant as necking proceeds, indicating that Aa N (= A L N related to the true, i.e. progressively reduced, cross section) increases. The necking serrations are in general much more regular than the pre-necking serrations. Fig. 1 is a plot of the initial values of Ao N versus a nominal strain rate ~., obtained from the load drop which appear right after necking has started (divided by the true cross section at the end of uniform deformation). These curves resemble in principle those obtained previously for pre-necking stress drops A a m (3).
20
~ 0
6
\<> -\
~ \
\
-5
\ \ ~ , , X o \
\
6\
2 I
60-75%
v
o
\\\_~Oc -4
\.',
\48°C
20 -3
-2 log [~n ( s'l)]
FIG. 1. The stress-drop size of necking serrations of type B for as-received AIMgS, plotted in dependence of the nominal strain rate.
The nominal strain rate ~. in the diagram is affected by a further ambiguity introduced by another deformation localization due to necking, admittedly not as severely as that due to the formation of stretcher-strain markings. A correction would still be difficult because the extent of necking is not exactly defined, and has not been attempted. However, if we consider the severe localization in deformation bands which is connected with the load drop, and if we assume that the extent of localization is equal for the pre-necking and the necking serrations, then the error introduced due to necking is negligible. Under this assumption we may plot the results for necking serrations in the same way against the nominal strain rate ~. in order to later attempt a comparison with data for pre-necking serrations.
Vol. 25, NO.
ii
SERRATED
Uniaxial,
FLOW
2419
B i a x i a l and T r i a x i a l S t r e s s C o n d i t i o n s
In order to answer the question of how stress states other than uniaxial influence the serrations, we conducted experiments at room temperature and at 50 and 70°C under a strictly uniaxial stress state (during uniform deformation in tensile tests with round samples of 4 mm diameter) and under a biaxial stress state (using the sample shape shown in the insert of Fig. 3 giving plane-strain and plane-stress conditions (5-7)). The material used for all these tests was the alloy AIZn5Mgl (annealed at 450°C for 1 hour and quenched in water) of which the dynamic strain-aging behaviour is exactly known (8,9). If testing was not possible immediately after quenching, the samples were stored at -20°C. This alloy exhibits pre-necking and necking serrations at the same time, so that serrated flow under a triaxial stress state can also be studied. The results are plotted in the Figs. 2 and 3. Obvious in both diagrams is the significant difference between small pre-necking and large necking serrations. In fact there is a sudden change in the size of the stress drop as soon as the stress-strain curve decreases after having reached its maximum point (the stress-drop sizes were measured immediately before and after this point; the properties of a short transition portion can easily be neglected). This tendency is real, and not feigned due to the rate difference between uniform deformation and necking. This difference (the increase of the true strain rate) would anyway tend to reduce the load drop during necking (see the general trend in Fig. i). Samples with quadratic cross sections behave in very much the same way as round samples, even to the extent of exhibiting serrations of identical size. These results are not included in the diagrams. Also tested were samples with a rectangular cross section (7 x 2 mm), resembling sheet samples, which were milled out of the identical material as the round and plane-strain samples. Again the pre-necking serrations are smaller than the necking serrations, although the difference is not quite as clear because the scatter is partly large.
10
--2O
•
2
4
-6
[] o
o pre-necking • necking I
2
I
-5
1 -O
o
0.6
[]
•
i
-/~
0
-3 tog[ti,(s -1 )1
FIG. 2. The stress-drop size of prenecking and necking serrations at 20°C, measured on round samples of heattreated AIZn5Mgl.
0.4
[]
pre-necking necking
i
I
-5
-4
n
i
-3 log [~% (s~)]
FIG. 3. The stress-drop size of prenecking and necking serrations at 20°C, measured on plane-strain samples of heat treated AIZnSMgl.
2420
SERRATED FLOW
Vol.
25, No.ll
a.
1
# 1 -O
0,6 0,k
~
~
2
o R _.d o,__:_
o Plane Strain Rectangu[ar
u o
• Round • • Plane Strain
A~,I
1 • Rectangutar
I
I
I
-5
-4
-3
I
-5
[og[£'n(s"1)]
FIG. 4. The stress-drop size of prenecking serrations, measured at 20°C on round, rectangular and planestrain samples of AIZn5Mgl.
1
-4
\ I
-3 log [ En (s"I )]
FIG. 5. The s t r e s s - d r o p s i z e o f necking serrations, measured at 20°C on round, rectangular and planestrain samples of AIZnSMgl.
To illustrate the relation between the results from round, sheet-like and plane-strain samples we combined all the data obtained at 20°C in the Figs.4 and 5 (the trend is comparable at the testing temperatures 50 and 70°C). Both the pre-necking and necking serrations as determined from round and planestrain samples are distinctly different. This difference is much more pronounced in the case of the pre-necking serrations at low rates, for which the ~a-values of the plane-strain sample are at comparatively very low levels. The sizes of pre-necking serrations from sheet-like samples are between those of round and plane-strain samples (Fig. 4). For the case of necking serrations the tendency may be similar, but too severe scatter (Fig. 5) and a rather high limit on the low-rate side of the diagrams where the serrations disappear in the case of the sheet-like samples do not allow a definite conclusion.
Discussion There are two clear results: (a) during a tensile test, and when both prenecking and necking serrations appear, the pre-necking serrations are significantly smaller than the necking serrations, (b) the stress-drop sizes of both pre-necking and necking serrations are larger in tests with round tensile samples than in plane-strain tensile tests. These statements are derived from the diagrams which use, in the case of plane-strain samples, the equivalent stress a ~ = {0/3)/2}a~I (5,6). However, it is doubtful whether a ~ is a meaningful characteristic for the manifestations of inhomogeneous flow. Each stress drop is due to a localized shear event. In plane-strain samples, the normal direction to the shear or fracture path lying in the shear plane is, as in round or quadratic or rectangular samples, inclined by 90 ° to the applied-stress direction, so that the plane-strain conditions are no longer relevant. It may rather be the shear stress which should be considered. But assuming that the inclination of the shear planes is the same for all shear events, the applied stress may still be a stress
Vol.
25, No.
Ii
SERRATED
FLOW
2421
suitable for comparisons. (When shear occurs in the n e c k e d regions of the sample, the shear area has d e c r e a s e d so that the stress d r o p of the necking serration is still larger in c o m p a r i s o n to that of the p r e - n e c k i n g serrations). Since also the angles by w h i c h the shear (or fracture) p l a n e s are inclined to the applied-stress d i r e c t i o n are identical (45 ± 3°), it is clear that the applied stress is a s u f f i c i e n t l y good measure for c o m p a r i n g the data from all types of samples. However, not u s i n g the e q u i v a l e n t s t r e s s e s for p l a n e - s t r a i n samples but the applied stress instead changes b a r e l y the diagrams: the applied stress is hardly (about 13%) larger, which w o u l d not e l i m i n a t e the gap between the curves. The statements (a) and (b) are also d e r i v e d from the d i a g r a m s which use the nominal strain rate of u n i f o r m d e f o r m a t i o n as an abcissa, and not the absolute characteristics for the h i g h l y localized d e f o r m a t i o n . This kind of plot has often been u s e d in the past to d e m o n s t r a t e the c h a r a c t e r i s t i c s of inhomogeneous deformation, which was always r e s t r i c t e d to i n v e s t i g a t i n g the pre-necking conditions. One might argue that n e c k i n g c h a n g e s the conditions. The true strain rate w i t h i n the n a r r o w d e f o r m a t i o n band m a y be different depending on whether the d e f o r m a t i o n occurs in the p r e - n e c k i n g or in the necking region. As a consequence, the positions of the curves in the Figs 2 5 may have to be s h i f t e d along the rate axis, and t h e y may f i n a l l y superimpose (and eliminate any d i f f e r e n c e s due to an influence of the stress state). No experimental facts are k n o w n which could support or rule out such an effect. However, we m i g h t also assume that the extent of l o c a l i z a t i o n is equal for the pre-necking and the n e c k i n g serrations. If this a s s u m p t i o n is true we can plot, for reasons of comparison, the results for both the p r e - n e c k i n g and the necking load drops in the same diagram, i.e. v e r s u s the n o m i n a l strain rate ~. as it was done in the d i a g r a m s (despite the fact that this is not a meaningful true strain rate of the event). Then the d i f f e r e n c e b e t w e e n the curves must truly indicate an e f f e c t of the stress state. Let us consider this second possibility. One d i f f i c u l t y in interpreting the experimental r e s u l t s is that the stress drops do not m o n o t o n i c a l l y increase in size as the stress state changes from u n i a x i a l i t y over b i a x i a l i t y to a triaxial state (Fig. 6). For instance, changing over to a t r i a x i a l stress state when the round samples start to neck increases the s t r e s s - d r o p size significantly. However, the u n i a x i a l s t r e s s i n g in a r o u n d sample (before necking sets in) results in stress drops larger than t h o s e o b t a i n e d in tests with a plane-strain, i.e. biaxial, stress state. This t e n d e n c y in p l a n e - s t r a i n samples to give the smallest stress drops is equally m a n i f e s t in the larger difference between p r e - n e c k i n g and n e c k i n g b e h a v i o u r (c.f. Fig. 3 in c o m p a r i s o n with the result of Fig. 2). Indeed these o b s e r v a t i o n s agree w i t h some state-ments in the literature, refered to in the introduction, that a b i a x i a l stress state impedes the a p p e a r a n c e of s e r r a t i o n s and p r e v e n t s the f o r m a t i o n of surface markings (1,2). Consider first the fact that serrations are larger under m u l t i a x i a l stress conditions. As we see in Fig. 7, a reduction of the d e f o r m a t i o n energy can be achieved by i n c r e a s i n g the strain rate when the n o m i n a l s t r a i n rate ~n is between the limits ~w~ and ~ . T h r o u g h l o c a l i z a t i o n of d e f o r m a t i o n and the resulting increase of the true local strain rate the a p p l i e d stress will be reduced. The r e s u l t i n g stress drop A o will d e c r e a s e from a m a x i m u m value ~a~= = u~= - u ~ for ~, = ~(a~=) to smaller values for ~ ( a ~ ) < ~. < % ( a ~ ) , in agreement w i t h all the e x p e r i m e n t a l results of the Figs 1 - 5. Can such a trend be achieved by c h a n g i n g the stress state? In p r i n c i p l e a t r i a x i a l stress state can m o d i f y the stress drop by way of a " h y d r o s t a t i c pressure" effect. The position of the d y n a m i c s t r a i n - a g i n g peak is d e t e r m i n e d by the diffusivity of the a l l o y i n g elements: these d i s s o l v e d elements w o u l d be retarded due to increased h y d r o s t a t i c pressure. But such p r e s s u r e e f f e c t s are negligible in metals for all p r a c t i c a l purposes. Also, since the d y n a m i c s t r a i n - a g i n g peak (and thus the r e g i m e of n e g a t i v e strain-rate sensitivity) is shifted towards
2422
SERRATED FLOW
"B6
r~
= 7xlO-s s-1 n pre-necking m necking
RD
,Q
Vol.
25, No, ii
RD
~MQx // PS // // //
\\ \\ \ \ ~
II
2
//
i
P5 2
// // // // // // // L//
\\~" \\
I I I I
\ \ / A \\ \\ \\ \\
I I I I
b I
n
-~ tog
\\
3
Axiotify of Sfress
FIG. 7. The schematic stress/strainrate dependence of an alloy with a superimposed dynamic strain-aging peak.
FIG. 6. The stress-drop size of prenecking and necking serrations for the various sample shapes tested at the strain rate of 7x10"5/s at 20°C, in dependence of the stress axiality (RD.. ..round, RA...rectangular, PS...plane strain).
lower rates, the pressure increase would lower Aa, and this is contrary to the observed growth. Therefore, the observed size differences in serrations cannot be an effect of the stress state only. The axiality of the deformation before necking depends on the type of sample used. While round and rectangular samples are tested under conditions of triaxial straint the deformation is confined to two axes in plane-strain samples. Thus, the impediment of pre-necking serrations in these samples may be due to the combination of biaxial stress and strain conditions.
A=knowledgement We are grateful to Dr. H. Matzner for his support in supplying us with the sample material, which was a gift of Aluminium Ranshofen GmbH.
Refsrsnoes i. B.A. Parker, Strength of Metals and Alloys (ICSMA 5), Vol. 2, p. 899, Edts P. Haasen, V. Gerold and G. Kostorz, Pergamon Press, London (1979) 2. D.S. Thompson, Intern. Automotive Engg. Congr. Detroit, Paper No. 770203, Soc. Automotive Engineers, Warrendale, Pa. (1977) 3. J.D. Baird, The Inhomogeneity of Plastic Deformation, p. 191, American Soc. Metals, Metals Park, OH (1973) 4. E. Pink and A. Grinberg, Acta metall. 30, 2135 (1982) 5. R. Hill, J. Mech. Phys. Solids 1, 19 (1952) 6. D.P. Clausing, Intern. J. Fract. Mech. 6, 71 (1970) 7. E. Werner, Z. Metallkd. 79, 585 (1988) 8. E. Pink and W.M. Webernig, Acta metal1. 35, 127 (1987) 9. E. Pink, Acta metall. 37, 1773 (1989)
SERRATED
FLOW
Vol.
UNDER
25, pp. 2417-2422, 1991 Printed in the U.S.A.
UNIAXIAL
AND
MULTIAXIAL
STRESS
Pergamon Press plc All rights reserved
AND
STRAIN
CONDITIONS
Monika Fellner*, Erwin Pink~+ and Ewald Werner + *Erich-Schmid-Institut fur FestkSrperphysik, Osterreichische Akademie der Wissenschaften, +Institut f~r Metallphysik, Montanuniversit~t, Leoben, Austria (Received June 20, 1991)
Introduction Serrated flow has been discussed over the years almost exclusively in terms of uniaxial tension. There are, however, a few reports which indicate that the stress state affects the occurrence of serrations. In equibiaxial tension, for instance, serrations were said to be "less evident" (i). The effects of the stress state were also demonstrated to influence the appearance of stretcher-strain markings in deep - drawing processes (2). A rectangular strip of an alloyed aluminium sheet was indented by a ball while the strip was gripped by a locking ring. However, the width of the sheet sample was less than the diameter of the locking ring, so that the stress distribution within the indented sheet was not symmetrical. Deformation bands developed in areas which were not in contact with the punching ball, inclined at a certain angle to the principal stresses. But in the center of contact with the ball, markings were found to be perpendicular to the principal-stress direction. When the sheet sample was fully gripped by the locking ring, markings did not appear in the center areas where the stress state is mainly biaxial. Markings did, however, appear in the peripherical areas with positive radial and negative tangential stresses, resembling the uniaxial stress state in a tensile sample (2). These results give the impression that stress states deviating from the uniaxial state suppress serrated flow. Quite contrary is the often experienced behaviour that, in tensile tests, the deformation may be homogeneous throughout the initial, increasing, portion of the stress-strain curve when the stress state is uniaxial, and that serrations appear as soon as necking has set in, i.e. for triaxial stress states. It is also known from experiments that serrations can be confined to the increasing portion of the stress-strain curve which has no serrations after the maximum load point. In many other experiments both "pre-necking" and "necking" serrations appear during one test. All these serrations do not necessarily have to be of the same type. A- or B-serrations* on the increasing section of the load-extension curve may be followed by B-type serrations during necking (necking serrations always seem to be of type B). No systematic research has, to our knowledge, been carried out. This report deals with first results from comparing, under different stress states, the pre-necking serrations with the necking serrations. We only compare serrations of type B (which is essential because A- and B-type serrations are, under most testing conditions, significantly different in size). * A-type serrations are separated by long strain intervals, because they represent the initiation of the first deformation at one side of the sample, and B-serrations follow each other in close succession as the deformation spreads inhomogenously along the sample's gauge length (3,4).
2417 0036-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press plc
2418
SERRATED FLOW
Vol.
25, No. II
The results for the size of serrations in tensile tests are often plotted in their dependence on the nominal strain rate ~. which is the cross-head speed related to the initial gauge length. However, a characteristic of serrated flow is the severe localization of deformation in narrow bands (the so-called "stretcher-strain markings"). But localizing the d e f o r m a t i o n leads to an increase of the strain rate, and therefore the nominal strain rate is not the true strain rate of the d e f o r m a t i o n in the narrow band.
The Properties of Necking Serretions We analyzed necking serrations from tests with 15 mm-rods of an AIMg5alloy in the as-received (i.e. to some extent coldworked) condition, comparing the size of the load drops A L N. Except for a very short transition range, AL N is extremely regular, and rather constant as necking proceeds, indicating that Aa N (= A L N related to the true, i.e. progressively reduced, cross section) increases. The necking serrations are in general much more regular than the pre-necking serrations. Fig. 1 is a plot of the initial values of Ao N versus a nominal strain rate ~., obtained from the load drop which appear right after necking has started (divided by the true cross section at the end of uniform deformation). These curves resemble in principle those obtained previously for pre-necking stress drops A a m (3).
20
~ 0
6
\<> -\
~ \
\
-5
\ \ ~ , , X o \
\
6\
2 I
60-75%
v
o
\\\_~Oc -4
\.',
\48°C
20 -3
-2 log [~n ( s'l)]
FIG. 1. The stress-drop size of necking serrations of type B for as-received AIMgS, plotted in dependence of the nominal strain rate.
The nominal strain rate ~. in the diagram is affected by a further ambiguity introduced by another deformation localization due to necking, admittedly not as severely as that due to the formation of stretcher-strain markings. A correction would still be difficult because the extent of necking is not exactly defined, and has not been attempted. However, if we consider the severe localization in deformation bands which is connected with the load drop, and if we assume that the extent of localization is equal for the pre-necking and the necking serrations, then the error introduced due to necking is negligible. Under this assumption we may plot the results for necking serrations in the same way against the nominal strain rate ~. in order to later attempt a comparison with data for pre-necking serrations.
Vol. 25, NO.
ii
SERRATED
Uniaxial,
FLOW
2419
B i a x i a l and T r i a x i a l S t r e s s C o n d i t i o n s
In order to answer the question of how stress states other than uniaxial influence the serrations, we conducted experiments at room temperature and at 50 and 70°C under a strictly uniaxial stress state (during uniform deformation in tensile tests with round samples of 4 mm diameter) and under a biaxial stress state (using the sample shape shown in the insert of Fig. 3 giving plane-strain and plane-stress conditions (5-7)). The material used for all these tests was the alloy AIZn5Mgl (annealed at 450°C for 1 hour and quenched in water) of which the dynamic strain-aging behaviour is exactly known (8,9). If testing was not possible immediately after quenching, the samples were stored at -20°C. This alloy exhibits pre-necking and necking serrations at the same time, so that serrated flow under a triaxial stress state can also be studied. The results are plotted in the Figs. 2 and 3. Obvious in both diagrams is the significant difference between small pre-necking and large necking serrations. In fact there is a sudden change in the size of the stress drop as soon as the stress-strain curve decreases after having reached its maximum point (the stress-drop sizes were measured immediately before and after this point; the properties of a short transition portion can easily be neglected). This tendency is real, and not feigned due to the rate difference between uniform deformation and necking. This difference (the increase of the true strain rate) would anyway tend to reduce the load drop during necking (see the general trend in Fig. i). Samples with quadratic cross sections behave in very much the same way as round samples, even to the extent of exhibiting serrations of identical size. These results are not included in the diagrams. Also tested were samples with a rectangular cross section (7 x 2 mm), resembling sheet samples, which were milled out of the identical material as the round and plane-strain samples. Again the pre-necking serrations are smaller than the necking serrations, although the difference is not quite as clear because the scatter is partly large.
10
--2O
•
2
4
-6
[] o
o pre-necking • necking I
2
I
-5
1 -O
o
0.6
[]
•
i
-/~
0
-3 tog[ti,(s -1 )1
FIG. 2. The stress-drop size of prenecking and necking serrations at 20°C, measured on round samples of heattreated AIZn5Mgl.
0.4
[]
pre-necking necking
i
I
-5
-4
n
i
-3 log [~% (s~)]
FIG. 3. The stress-drop size of prenecking and necking serrations at 20°C, measured on plane-strain samples of heat treated AIZnSMgl.
2420
SERRATED FLOW
Vol.
25, No.ll
a.
1
# 1 -O
0,6 0,k
~
~
2
o R _.d o,__:_
o Plane Strain Rectangu[ar
u o
• Round • • Plane Strain
A~,I
1 • Rectangutar
I
I
I
-5
-4
-3
I
-5
[og[£'n(s"1)]
FIG. 4. The stress-drop size of prenecking serrations, measured at 20°C on round, rectangular and planestrain samples of AIZn5Mgl.
1
-4
\ I
-3 log [ En (s"I )]
FIG. 5. The s t r e s s - d r o p s i z e o f necking serrations, measured at 20°C on round, rectangular and planestrain samples of AIZnSMgl.
To illustrate the relation between the results from round, sheet-like and plane-strain samples we combined all the data obtained at 20°C in the Figs.4 and 5 (the trend is comparable at the testing temperatures 50 and 70°C). Both the pre-necking and necking serrations as determined from round and planestrain samples are distinctly different. This difference is much more pronounced in the case of the pre-necking serrations at low rates, for which the ~a-values of the plane-strain sample are at comparatively very low levels. The sizes of pre-necking serrations from sheet-like samples are between those of round and plane-strain samples (Fig. 4). For the case of necking serrations the tendency may be similar, but too severe scatter (Fig. 5) and a rather high limit on the low-rate side of the diagrams where the serrations disappear in the case of the sheet-like samples do not allow a definite conclusion.
Discussion There are two clear results: (a) during a tensile test, and when both prenecking and necking serrations appear, the pre-necking serrations are significantly smaller than the necking serrations, (b) the stress-drop sizes of both pre-necking and necking serrations are larger in tests with round tensile samples than in plane-strain tensile tests. These statements are derived from the diagrams which use, in the case of plane-strain samples, the equivalent stress a ~ = {0/3)/2}a~I (5,6). However, it is doubtful whether a ~ is a meaningful characteristic for the manifestations of inhomogeneous flow. Each stress drop is due to a localized shear event. In plane-strain samples, the normal direction to the shear or fracture path lying in the shear plane is, as in round or quadratic or rectangular samples, inclined by 90 ° to the applied-stress direction, so that the plane-strain conditions are no longer relevant. It may rather be the shear stress which should be considered. But assuming that the inclination of the shear planes is the same for all shear events, the applied stress may still be a stress
Vol.
25, No.
Ii
SERRATED
FLOW
2421
suitable for comparisons. (When shear occurs in the n e c k e d regions of the sample, the shear area has d e c r e a s e d so that the stress d r o p of the necking serration is still larger in c o m p a r i s o n to that of the p r e - n e c k i n g serrations). Since also the angles by w h i c h the shear (or fracture) p l a n e s are inclined to the applied-stress d i r e c t i o n are identical (45 ± 3°), it is clear that the applied stress is a s u f f i c i e n t l y good measure for c o m p a r i n g the data from all types of samples. However, not u s i n g the e q u i v a l e n t s t r e s s e s for p l a n e - s t r a i n samples but the applied stress instead changes b a r e l y the diagrams: the applied stress is hardly (about 13%) larger, which w o u l d not e l i m i n a t e the gap between the curves. The statements (a) and (b) are also d e r i v e d from the d i a g r a m s which use the nominal strain rate of u n i f o r m d e f o r m a t i o n as an abcissa, and not the absolute characteristics for the h i g h l y localized d e f o r m a t i o n . This kind of plot has often been u s e d in the past to d e m o n s t r a t e the c h a r a c t e r i s t i c s of inhomogeneous deformation, which was always r e s t r i c t e d to i n v e s t i g a t i n g the pre-necking conditions. One might argue that n e c k i n g c h a n g e s the conditions. The true strain rate w i t h i n the n a r r o w d e f o r m a t i o n band m a y be different depending on whether the d e f o r m a t i o n occurs in the p r e - n e c k i n g or in the necking region. As a consequence, the positions of the curves in the Figs 2 5 may have to be s h i f t e d along the rate axis, and t h e y may f i n a l l y superimpose (and eliminate any d i f f e r e n c e s due to an influence of the stress state). No experimental facts are k n o w n which could support or rule out such an effect. However, we m i g h t also assume that the extent of l o c a l i z a t i o n is equal for the pre-necking and the n e c k i n g serrations. If this a s s u m p t i o n is true we can plot, for reasons of comparison, the results for both the p r e - n e c k i n g and the necking load drops in the same diagram, i.e. v e r s u s the n o m i n a l strain rate ~. as it was done in the d i a g r a m s (despite the fact that this is not a meaningful true strain rate of the event). Then the d i f f e r e n c e b e t w e e n the curves must truly indicate an e f f e c t of the stress state. Let us consider this second possibility. One d i f f i c u l t y in interpreting the experimental r e s u l t s is that the stress drops do not m o n o t o n i c a l l y increase in size as the stress state changes from u n i a x i a l i t y over b i a x i a l i t y to a triaxial state (Fig. 6). For instance, changing over to a t r i a x i a l stress state when the round samples start to neck increases the s t r e s s - d r o p size significantly. However, the u n i a x i a l s t r e s s i n g in a r o u n d sample (before necking sets in) results in stress drops larger than t h o s e o b t a i n e d in tests with a plane-strain, i.e. biaxial, stress state. This t e n d e n c y in p l a n e - s t r a i n samples to give the smallest stress drops is equally m a n i f e s t in the larger difference between p r e - n e c k i n g and n e c k i n g b e h a v i o u r (c.f. Fig. 3 in c o m p a r i s o n with the result of Fig. 2). Indeed these o b s e r v a t i o n s agree w i t h some state-ments in the literature, refered to in the introduction, that a b i a x i a l stress state impedes the a p p e a r a n c e of s e r r a t i o n s and p r e v e n t s the f o r m a t i o n of surface markings (1,2). Consider first the fact that serrations are larger under m u l t i a x i a l stress conditions. As we see in Fig. 7, a reduction of the d e f o r m a t i o n energy can be achieved by i n c r e a s i n g the strain rate when the n o m i n a l s t r a i n rate ~n is between the limits ~w~ and ~ . T h r o u g h l o c a l i z a t i o n of d e f o r m a t i o n and the resulting increase of the true local strain rate the a p p l i e d stress will be reduced. The r e s u l t i n g stress drop A o will d e c r e a s e from a m a x i m u m value ~a~= = u~= - u ~ for ~, = ~(a~=) to smaller values for ~ ( a ~ ) < ~. < % ( a ~ ) , in agreement w i t h all the e x p e r i m e n t a l results of the Figs 1 - 5. Can such a trend be achieved by c h a n g i n g the stress state? In p r i n c i p l e a t r i a x i a l stress state can m o d i f y the stress drop by way of a " h y d r o s t a t i c pressure" effect. The position of the d y n a m i c s t r a i n - a g i n g peak is d e t e r m i n e d by the diffusivity of the a l l o y i n g elements: these d i s s o l v e d elements w o u l d be retarded due to increased h y d r o s t a t i c pressure. But such p r e s s u r e e f f e c t s are negligible in metals for all p r a c t i c a l purposes. Also, since the d y n a m i c s t r a i n - a g i n g peak (and thus the r e g i m e of n e g a t i v e strain-rate sensitivity) is shifted towards
2422
SERRATED FLOW
"B6
r~
= 7xlO-s s-1 n pre-necking m necking
RD
,Q
Vol.
25, No, ii
RD
~MQx // PS // // //
\\ \\ \ \ ~
II
2
//
i
P5 2
// // // // // // // L//
\\~" \\
I I I I
\ \ / A \\ \\ \\ \\
I I I I
b I
n
-~ tog
\\
3
Axiotify of Sfress
FIG. 7. The schematic stress/strainrate dependence of an alloy with a superimposed dynamic strain-aging peak.
FIG. 6. The stress-drop size of prenecking and necking serrations for the various sample shapes tested at the strain rate of 7x10"5/s at 20°C, in dependence of the stress axiality (RD.. ..round, RA...rectangular, PS...plane strain).
lower rates, the pressure increase would lower Aa, and this is contrary to the observed growth. Therefore, the observed size differences in serrations cannot be an effect of the stress state only. The axiality of the deformation before necking depends on the type of sample used. While round and rectangular samples are tested under conditions of triaxial straint the deformation is confined to two axes in plane-strain samples. Thus, the impediment of pre-necking serrations in these samples may be due to the combination of biaxial stress and strain conditions.
A=knowledgement We are grateful to Dr. H. Matzner for his support in supplying us with the sample material, which was a gift of Aluminium Ranshofen GmbH.
Refsrsnoes i. B.A. Parker, Strength of Metals and Alloys (ICSMA 5), Vol. 2, p. 899, Edts P. Haasen, V. Gerold and G. Kostorz, Pergamon Press, London (1979) 2. D.S. Thompson, Intern. Automotive Engg. Congr. Detroit, Paper No. 770203, Soc. Automotive Engineers, Warrendale, Pa. (1977) 3. J.D. Baird, The Inhomogeneity of Plastic Deformation, p. 191, American Soc. Metals, Metals Park, OH (1973) 4. E. Pink and A. Grinberg, Acta metall. 30, 2135 (1982) 5. R. Hill, J. Mech. Phys. Solids 1, 19 (1952) 6. D.P. Clausing, Intern. J. Fract. Mech. 6, 71 (1970) 7. E. Werner, Z. Metallkd. 79, 585 (1988) 8. E. Pink and W.M. Webernig, Acta metal1. 35, 127 (1987) 9. E. Pink, Acta metall. 37, 1773 (1989)