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A new approach to the temperature dependence of the yield stress in polycrystalline austenitic stainless steels
Materials Science and Engineering, 80 (1986) LS-L9
L5
Letter
A n e w approach to the temperature dependence of the yield stress in polycrystalline austenitic stainless steels
R. A. VARIN Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada)
K. KURZYDLOWSKI Institute of Materials Science and Engineering, Warsaw Technical University, 02-524 Warsaw, Narbutta 85 (Poland) K. TANGRI Metallurgical Sciences Laboratories, University of Manitoba, Winnipeg, Manitoba R3T 2N2 (Canada)
(Received July 25, 1985; in revised form December 24, 1985)
ABSTRACT The appearance o f a plateau followed by an abrupt decrease in the yield stress as a function o f test temperature observed in type 316 austenitic stainless steel is re-interpreted on the basis o f the relationship between the rate o f formation o f extrinsic grain boundary dislocations and the rate o f their spreading by core widening. The mathematical model derived gives good agreement with the experimental data.
1. INTRODUCTION It is n o w well recognized that both the yield and the flow stresses in single and polycrystals of metals and alloys drop relatively rapidly above approximately 0.5Tin where Tm is the melting temperature. The first suggestion that in polycrystals this p h e n o m e n o n was to some extent related to the processes occurring at the grain boundaries was put forward by Ball [ 1 ]. This idea was later challenged by Hirsch and Warrington [ 2 ] who performed experiments on aluminum and copper polycrystals 0025-5416/86/$3.50
and claimed to observe basically the same behaviour as for single crystals. Ever since then the models developed by other researchers to explain the above p h e n o m e n o n were focused solely on the behaviour of dislocations within the grains [ 3 - 5 ] although Gallagher [3] admitted in his paper that "grain boundary effects are liable to be important at elevated temperatures". Unfortunately, until quite recently there was no systematic study on the dependence of the elevated temperature deformation behaviour of polycrystals on grain size. The experiments by the above-mentioned researchers were limited to polycrystals with a singular grain size only. Very recently, Mannan et al. [6] clearly showed a dependence of the yield stress (measured at 0.2% p r o o f stress) on the grain size in a type 316 austenitic stainless steel deformed up to 1200 K. The grain sizes in the specimens were changed from 25 to 650 #m. Figure 1 shows the dependence of the yield stress (0.2% p r o o f stress normalized for Young's modulus) on the temperature for 25 and 650/~m grain sizes, replotted from the original data of Mannan et al. [6]. A plot for ultrafine-grained type 316 steel (grain size, about 2 pm) investigated in the present work is also included in Fig. 1. Three striking features are visible in Fig. 1. Firstly, there is a strong dependence of the "plateau" stress on grain size. Secondly, there is a substantial yield stress drop for a very small grain size and almost none for the 650 pm grain size, which as a matter of fact can be approximated to the behaviour of a single crystal. As seen, the yield stress curve is almost flat up to about 1220 K. Thirdly, the yield stress drop begins at a lower temperature with decreasing grain size. These three factors clearly show that the yielding, at least in an austenitic stainless steel, is strongly controlled by the grain boundaries especially for very small and medium grain sizes. Therefore, it is postulated here that the major mechanism responsible for the observed © Elsevier Sequoia/Printed in The Netherlands
L6
3.C
0
~\\\
200
400 600 800 TEMPERATURE(K)
10(30
1200
Fig. 1. Experimentally observed variation in the yield stress (0.2% proof stress) with temperature: ®, o, present work, d ~ 2 pro, e = 2.2 X 10 -4 s-l; [], data of Mannanetal. [6], d = 25 pro, ~ = 3 x 10 -4 s - 1 ; ~ , data of Mannan et al. [6], d = 650 pro, e = 3 X 10 -4 s-1. As ultrafine-grained steel (about 2 / l m grain size) exhibited inhomogeneous yielding behaviour (Liiders deformations at lower temperatures, the broken line also shows the 0.2% p r o o f stress values obtained by extrapolating the work-hardening region of the flow curve back to the elastic line. The yield stress curves for 25 and 650 pm grain sizes were replotted from the data of Mannan et al. [6].
behaviour is the annihilation of lattice dislocations entering grain boundaries (and possibly also nucleated at the boundaries) by spreading. An individual dislocation impinging on a random (general) grain boundary forms a socalled extrinsic grain b o u n d a r y dislocation (EGBD) which is thermally Unstable, however, and as such it may be easily incorporated into the grain b o u n d a r y when the temperature increases. This process is manifested in a gradual disappearance of the contrast of an EGBD when heating in situ in a transmission electron microscope [ 7 - 1 1 ] and it is c o m m o n l y called "the spreading of an EGBD". Now, considering the deformation process dynamically, we have on the one hand a continuous supply of lattice dislocations interacting with grain boundaries and forming EGBDs and on the other hand, if the temperature is high enough, a continuous spreading of newly formed EGBDs. The mechanical behaviour of polycrystalline material is therefore governed by the relative rates of these processes, i.e. the rate pEGBDfo~medof formation of EGBDs and the rate ~S6BDs~ea d of their spreading. It should be pointed o u t that b y "spreading" we assume here a complete relaxation of the EGBD's stress field. It is also assumed here that EGBD strain energy is completely dissi-
pated when its core width approaches the value of the order of two extinction distances in the transmission electron microscope which, in turn, corresponds to complete disappearance of EGBD contrast in the transmission electron microscope. The aim of the present work is to propose a formal mathematical description of the model presented and to establish the dislocation conditions at the grain boundaries leading to a drop in the yield stress at elevated temperatures.
2. FORMAL DESCRIPTION OF EXTRINSIC G RA I N BOUNDARY DISLOCATION BEHAVIOUR DURING D EF O RMA TI O N AT ELEVATED TEMPERATURES
Recent experiments on austenitic stainless steel (type 316 [12] and 0.01wt.%C-19wt.%Cr18wt.%Ni [ 13 ] ) have shown that the kinetics of the spreading of EGBDs pre-existing in rand o m grain boundaries is best described by an equation of the general form
ta=ATaexp(AHb~
(1)
\RTd ]
where ta is the spreading time, Td the spreading temperature, A H b the activation enthalpy for grain boundary diffusion and A a constant. Equation (1) was derived theoretically by Johannesson and ThSlen [ 14 ], Lojkowski [15] and Lojkowski and Grabski [16]. The constant A in the Lojkowski and Grabski model has the following form: A -
11kSm a
G6~2Dob
(2)
where k is Boltzmann's constant, Sm the core width of an EGBD at which its image becomes invisible using transmission electron microscopy (equal to about t w o extinction distances [ 17 ] ), G the shear modulus, 5 the grain boundary width for diffusion (normally assumed to be 5 × 10 -l° m), $2 the atomic volume and D0 b the pre-exponential term in the expression for the grain boundary diffusivity (Db = D 0 b e x p ( - - z~r-Ib/RT)).
Let us assume for simplicity that the grains are cubes with sides equal to d (where d is the average grain size) and they deform with the strain rate ~g. In time dt the deformation is
L7 TABLE 1
d% = ~g dt. This d e f o r m a t i o n level m u s t be achieved b y the m o v e m e n t o f s o m e d e n s i t y Pm o f lattice d i s l o c a t i o n s such as
deg ~ pmbYc
(3)
w h e r e ~ ~ d in view o f t h e fact that, at yielding, grain b o u n d a r i e s act also as sources and g e n e r a t e d d i s l o c a t i o n s will travel across t h e grains. T h e t o t a l l e n g t h o f m o b i l e dislocations is o b v i o u s l y prod 3. T h u s t h e d e n s i t y o f E G B D s at t h e grain b o u n d a r y w h i c h is shared b y t w o grains can be expressed as pEGBD -- Prod3 3d 2
(4)
S u b s t i t u t i n g Pm f r o m eqn. (4) f o r ~ = d, we obtain pEGBD __ deg 3b
(5)
Experimentally observed temperatures of the yield stress drop (taken at 95% of the plateau stress value) and the ratios of the rate of spreading of extrinsic grain boundary dislocations to the rate of formation of extrinsic grain boundary dislocations Grain
Tdrop
size
(a t O. 95Gplateau)
(Urn)
(K)
2 25 40-60 125 270 650
850 960 1025 1040 1040 1030
AEGBD t P spread/ pEGBDformed
0.11 0.10 0.23-0.16 0.10 0.05 0.02
The following parameters were used for the calculations: ~ ' ~ b = 177 kJ mo1-1 [22];D0 b = 6.1 x 10-4 m 2 s - 1 [22];S m ~ 6 x 10-8 m [19]; ~ = 11.64 X 10-30 m 3 (zero on the assumption of a lattice parametera of 3.598 X 10-1° m [20]);G = 7.74 X 10 l° N m-2 [21].
Thus, EGBD ~EGBDformed
--
_
~
_
dt
I lOC
N
~g
(6)
-
3b
Similarly, the rate o f spreading o f E G B D s can be expressed as dS
pEGBDspread = pEGBD v --
(7)
•~
I
IO00 /
~
g
90C
! !
/
/
2
bc~ 80C
dt GRAIN
w h e r e pEGBDv (m m -3) is n o w expressed per u n i t v o l u m e and is n u m e r i c a l l y equal to p ~ . T h e f a c t o r dS/dt is t h e rate o f c o r e spreading o f an E G B D and is equal t o Sm/ta w h e r e Sm is t h e final core w i d t h and td (the spreading time f o r S~) can be easily c a l c u l a t e d f r o m eqns. (1) and (2).
3. RESULTS T h e ratios o f pEGBDslrtead/pEGBDformed were calculated e m p l o y i n g eqns. (6) and (7) f o r t h e various grain sizes investigated. T h e results are listed in Table 1 t o g e t h e r w i t h t h e t e m p e r a tures o f t h e yield stress d r o p . I n o r d e r to ensure u n i f o r m i t y , these t e m p e r a t u r e s were read f r o m t h e o0.0o2 = f(T) p l o t s at t h e values o f yield stress c o r r e s p o n d i n g t o 0.95 o f t h e plat e a u stress (5% d r o p f r o m t h e p l a t e a u stress).
SIZE
[Izrn)
Fig. 2. Variation in the yield stress drop temperature (both calculated employing eqns. (6) and (7) and experimental data) with grain size: 8, this work; o, data of Mannan et al. [6 ]; - - -, theoretical, p E G B D s p r e a d ---0.1pEGBDormed, ~ = 3 X 10-4 s-1.
I t is clearly seen t h a t the d r o p o c c u r s w h e n the rate o f spreading is a p p r o x i m a t e l y equal t o 0.1 o f t h e rate o f f o r m a t i o n : pEGBDsl~ead
~
0,1~5EGBDfo~a
(8)
Figure 2 shows t h e d e p e n d e n c e o f t h e yield stress d r o p t e m p e r a t u r e o n t h e grain size calc u l a t e d on t h e a s s u m p t i o n t h a t eqn. (8) is fulfilled t o g e t h e r with e x p e r i m e n t a l p o i n t s obt a i n e d f r o m t h e plots o0.0o2 = f(T). It is clearly seen t h a t t h e t h e o r e t i c a l curve fits t h e exp e r i m e n t a l p o i n t s very well f o r grain sizes up t o a b o u t 150 p m . A t larger grain sizes t h e
L8 TABLE 2 Experimentally observed temperatures corresponding t o 0.1Oplatea u a n d t h e r a t i o s o f t h e r a t e o f s p r e a d i n g o f e x t r i n s i c grain b o u n d a r y d i s l o c a t i o n s t o t h e r a t e o f f o r m a t i o n o f e x t r i n s i c grain b o u n d a r y d i s l o c a t i o n s
Grain size
Temperature at O.10plateau
(urn)
(K)
2 25 40-60 125 270 650
~ ~ ~ ~
1220 1250 1250 1280 1280 1280
AEGBD t ~.EGBDSPread/ P formed
148 13 8-5.4 4 2 1
drop temperature becomes almost independent of grain size. Now, another important question arises: what is the ratio of the rate o f spreading to the rate of formation of EGBDs when the strengthening effect of grain boundaries disappears at higher temperatures? Table 2 shows the values of the temperature for which the yield stress is equal to merely 10% of the plateau stress (90% drop from the plateau stress) and the calculated values of the rate of spreading to the rate of formation of EGBDs. It is seen that the strengthening by the grain boundaries becomes negligible when the rate of spreading becomes slightly higher than the rate of formation of EGBDs. In the ultrafinegrained structure this ratio is unusually high. This result will be discussed later.
4. D I S C U S S I O N
First o f all, it must be pointed o u t that evidence is provided in the present work which clearly indicates that grain b o u n d a r y processes are largely responsible for the drop in the yield stress at elevated temperatures. However, this effect seems to decrease for very large grain sizes (say, above a b o u t 150/~m for an austenitic stainless steel), and the yield stress drop becomes independent of grain size. It is therefore suggested that, because Hirsch and Warrington [2] used polycrystalline samples o f aluminum with a very large grain size (400 #m), they were not able to find any substantial difference between the behaviour of single crystals and so-called "polycrystals".
However, as shown by Gallagher [3] (see Fig. 4 in his paper) there is a considerable difference between the drop temperature for polycrystals (about 700 K - 0.52Tin) and that for single crystals (about 1220 K - 0.9Tin) of copper. The presence of plateaus followed by sudden drops in yield stress as a function of deformation temperature in austenitic stainless steel is not interpreted clearly in the literature. Jenkins and Smith [22] attributed these phenomena to a dynamic-strain-aging effect (manifested by serrations in the stress-strain curve) due to carbon-vacancy pairs between 473 and 673 K and due to chromium a t o m dislocation interactions for effects above 673 K. However, they observed the same plateau and sudden drop in yield stress (0.2% offset) in pure Fe-35wt.%Ni alloy which revealed no sign of dynamic strain aging. Michel e t al. [23] investigated the high temperature deformation (up to 1089 K) of t y p e 316 austenitic stainless steel and concluded that the observed plateau and sudden drop in ultimate tensile strength at about 0.5Tin were a result of the change in the character of the dislocation substructure from a cell structure to a subgrain structure. This explanation which might be reasonable for deformation levels larger than a few per cent is completely inadequate to explain plateaus and drops in the yield stress (0.2% offset) where the formation of any significant a m o u n t of substructure is hardly expected. Also, any dynamic-strainaging effects seem to be quite improbable for deformations e as low as 0.002. For example, at the strain rate ~ of 3 × 10 -4 s -1 employed by Mannan e t al. [6] the time required to attain e = 0.2% is only a b o u t 7 s. However, Barnby's results [24] indicate that a time of a b o u t 16 h at 623 K is required to obtain such a locking which results in only a small yield point in type 316 stainless steel. The whole effect seems to be rather weak. Thus, it is unlikely that any substantial diffusion can occur within about 7 s at elevated temperatures to lock dislocations effectively. As shown in Table 2 by the unusually large ratio of pEGBDslxtead to pEGBDfozllmedfor ultrafine-grained steel, the grain boundaries in ultrafine material seem to be more resistant to the loss in strength than are those in mediumand large-grained structures. Tentatively, this finding might be explained on the assumption
L9 t h a t in p a r t d e f o r m a t i o n o c c u r r e d b y grain b o u n d a r y sliding. As t h e u l t r a f i n e - g r a i n e d material c o n t a i n s a high d e n s i t y o f i n t r a g r a n u l a r carbides [ 25 ], t h e y m i g h t h i n d e r grain b o u n d a r y sliding, t h e r e b y increasing t h e grain b o u n d ary strength. In t h e i n t e r e s t o f c o m p l e t i o n it m a y also be m e n t i o n e d t h a t t h e grain b o u n d a r y sources, w h i c h p l a y a m a j o r role in yielding (low def o r m a t i o n s ) , are a t h e r m a l in n a t u r e a n d t h a t , u n d e r t h e c o n d i t i o n pEGBDformed~ pEGBDsPread, ay is n o t e x p e c t e d t o v a r y w i t h t e m p e r a t u r e , a n d a p l a t e a u m a y be o b s e r v e d w i t h i n a certain t e m p e r a t u r e range. In view o f t h e a b o v e discussion it m a y be seen t h a t a p r o p e r c o n s i d e r a t i o n o f t h e relative rates of f o r m a t i o n and spreading of EGBDs d u r i n g t h e yielding o f f.c.c, p o l y c r y s t a i s a n d specifically an a u s t e n i t i c stainless steel o f f e r s a m o r e s a t i s f a c t o r y e x p l a n a t i o n f o r t h e prese n c e o f a p l a t e a u and t h e s u d d e n d r o p in oy w i t h increasing t e m p e r a t u r e .
ACKNOWLEDGMENT This w o r k was s u p p o r t e d b y grants f r o m t h e N a t u r a l Sciences a n d E n g i n e e r i n g Research C o u n c i l o f C a n a d a w h i c h are g r a t e f u l l y acknowledged.
REFERENCES 1 C. J. Ball, Philos. Mag., 2 (1957) 1011;J. Iron Steel Inst., London, 191 (1959) 232. 2 P. B. Hirsch and D. H. Warrington, Philos. Mag., 6 (1961)735. 3 P. C. J. Gallagher, Philos. Mag., 15 (1967) 51. 4 T. H. Alden, Philos. Mag., 25 (1972) 785. 5 D. H. Sastry and K. Tangri, Philos. Mag., 32 (1975) 513.
6 S. L. Mannan, K. G. Samuel and P. Rodriguez, in R. C. Gifkins (ed.), Proc. 6th Int. Conf. on the Strength o f Metals and Alloys, Melbourne, August 16-20, 1982, Pergamon, Oxford, 1982, p. 637. 7 Y. Ishida, T. Hasegawa and F. Nagata, Trans. Jpn. Inst. Met., Suppl., 9 (1968) 504. 8 P. H. Pumphrey and H. Gleiter, Philos. Mag., 30
(1974) 593. 9 P. H. Pumphrey, J. Phys. (Paris), Colloq. C4, 36 (10, suppl.) (1975) 23. 10 R. A. Varin, Phys. Status Solidi A, 52 (1979) 347. 11 R. A. Varin, J. Mater. Sci., 14 (1979) 811. 12 R. A. Varin, Mater. Sci. Eng., 66 (1984) 97. 13 R. A. Varin and E. Romanowska-Haftek, Proc. 4th Japan Institute o f Metals Int. Symp. on Grain Boundary Structure and Related Phenomena, November 25-29, 1985, Minakami-Spa, in the
press. 14 T. Johannesson and A. ThSlen, Met. Sci. J., 6 (1972) 189. 15 W. Lojkowski, Ph.D. Thesis, Warsaw Technical University, 1980. 16 W. Lojkowski and M. W. Grabski, in N. Hansen, A. Horsewell, T. Leffers and H. Lilholt (eds.), Deformation o f Polycrystals: Mechanisms and Microstructures, Proc. 2nd Ris~ Int. Syrup. on Metallurgy and Materials Science, September 1418, 1981, Ris6 National Laboratory, Ris~b, 1981,
p. 329. 17 A. ThSlen, Proc. 8th Int. Congr. on Electron Microscopy, Canberra, 1974, Vol. 1, p. 606. 18 R. V. Patil and B. D. Sharma, Met. Sci., 16 (1982) 389. 19 L. E. Murr, in Electron and Ion Microscopy and Microanalysis, Dekker, New York, 1982, p. 740, Table D.4. 20 B. Weiss and R. Stickler, MetaU. Trans., 3 (1972) 851. 21 H. M. Ledbetter, Phys. Status Solidi A, 84 (1984) 89. 22 C. F. Jenkins and C. V. Smith, Trans. AIME, 245 (1969) 2149. 23 D. J. Michel, J. Moteff and A. J. Lovell, Acta Metall., 21 (1973) 1269. 24 J. T. Barnby, J. Iron Steel Inst., London, 203 (1965) 392. 25 R. A. Varin and K. Tangri, Mater. Sci. Eng., 72 (1985) 177.
L5
Letter
A n e w approach to the temperature dependence of the yield stress in polycrystalline austenitic stainless steels
R. A. VARIN Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada)
K. KURZYDLOWSKI Institute of Materials Science and Engineering, Warsaw Technical University, 02-524 Warsaw, Narbutta 85 (Poland) K. TANGRI Metallurgical Sciences Laboratories, University of Manitoba, Winnipeg, Manitoba R3T 2N2 (Canada)
(Received July 25, 1985; in revised form December 24, 1985)
ABSTRACT The appearance o f a plateau followed by an abrupt decrease in the yield stress as a function o f test temperature observed in type 316 austenitic stainless steel is re-interpreted on the basis o f the relationship between the rate o f formation o f extrinsic grain boundary dislocations and the rate o f their spreading by core widening. The mathematical model derived gives good agreement with the experimental data.
1. INTRODUCTION It is n o w well recognized that both the yield and the flow stresses in single and polycrystals of metals and alloys drop relatively rapidly above approximately 0.5Tin where Tm is the melting temperature. The first suggestion that in polycrystals this p h e n o m e n o n was to some extent related to the processes occurring at the grain boundaries was put forward by Ball [ 1 ]. This idea was later challenged by Hirsch and Warrington [ 2 ] who performed experiments on aluminum and copper polycrystals 0025-5416/86/$3.50
and claimed to observe basically the same behaviour as for single crystals. Ever since then the models developed by other researchers to explain the above p h e n o m e n o n were focused solely on the behaviour of dislocations within the grains [ 3 - 5 ] although Gallagher [3] admitted in his paper that "grain boundary effects are liable to be important at elevated temperatures". Unfortunately, until quite recently there was no systematic study on the dependence of the elevated temperature deformation behaviour of polycrystals on grain size. The experiments by the above-mentioned researchers were limited to polycrystals with a singular grain size only. Very recently, Mannan et al. [6] clearly showed a dependence of the yield stress (measured at 0.2% p r o o f stress) on the grain size in a type 316 austenitic stainless steel deformed up to 1200 K. The grain sizes in the specimens were changed from 25 to 650 #m. Figure 1 shows the dependence of the yield stress (0.2% p r o o f stress normalized for Young's modulus) on the temperature for 25 and 650/~m grain sizes, replotted from the original data of Mannan et al. [6]. A plot for ultrafine-grained type 316 steel (grain size, about 2 pm) investigated in the present work is also included in Fig. 1. Three striking features are visible in Fig. 1. Firstly, there is a strong dependence of the "plateau" stress on grain size. Secondly, there is a substantial yield stress drop for a very small grain size and almost none for the 650 pm grain size, which as a matter of fact can be approximated to the behaviour of a single crystal. As seen, the yield stress curve is almost flat up to about 1220 K. Thirdly, the yield stress drop begins at a lower temperature with decreasing grain size. These three factors clearly show that the yielding, at least in an austenitic stainless steel, is strongly controlled by the grain boundaries especially for very small and medium grain sizes. Therefore, it is postulated here that the major mechanism responsible for the observed © Elsevier Sequoia/Printed in The Netherlands
L6
3.C
0
~\\\
200
400 600 800 TEMPERATURE(K)
10(30
1200
Fig. 1. Experimentally observed variation in the yield stress (0.2% proof stress) with temperature: ®, o, present work, d ~ 2 pro, e = 2.2 X 10 -4 s-l; [], data of Mannanetal. [6], d = 25 pro, ~ = 3 x 10 -4 s - 1 ; ~ , data of Mannan et al. [6], d = 650 pro, e = 3 X 10 -4 s-1. As ultrafine-grained steel (about 2 / l m grain size) exhibited inhomogeneous yielding behaviour (Liiders deformations at lower temperatures, the broken line also shows the 0.2% p r o o f stress values obtained by extrapolating the work-hardening region of the flow curve back to the elastic line. The yield stress curves for 25 and 650 pm grain sizes were replotted from the data of Mannan et al. [6].
behaviour is the annihilation of lattice dislocations entering grain boundaries (and possibly also nucleated at the boundaries) by spreading. An individual dislocation impinging on a random (general) grain boundary forms a socalled extrinsic grain b o u n d a r y dislocation (EGBD) which is thermally Unstable, however, and as such it may be easily incorporated into the grain b o u n d a r y when the temperature increases. This process is manifested in a gradual disappearance of the contrast of an EGBD when heating in situ in a transmission electron microscope [ 7 - 1 1 ] and it is c o m m o n l y called "the spreading of an EGBD". Now, considering the deformation process dynamically, we have on the one hand a continuous supply of lattice dislocations interacting with grain boundaries and forming EGBDs and on the other hand, if the temperature is high enough, a continuous spreading of newly formed EGBDs. The mechanical behaviour of polycrystalline material is therefore governed by the relative rates of these processes, i.e. the rate pEGBDfo~medof formation of EGBDs and the rate ~S6BDs~ea d of their spreading. It should be pointed o u t that b y "spreading" we assume here a complete relaxation of the EGBD's stress field. It is also assumed here that EGBD strain energy is completely dissi-
pated when its core width approaches the value of the order of two extinction distances in the transmission electron microscope which, in turn, corresponds to complete disappearance of EGBD contrast in the transmission electron microscope. The aim of the present work is to propose a formal mathematical description of the model presented and to establish the dislocation conditions at the grain boundaries leading to a drop in the yield stress at elevated temperatures.
2. FORMAL DESCRIPTION OF EXTRINSIC G RA I N BOUNDARY DISLOCATION BEHAVIOUR DURING D EF O RMA TI O N AT ELEVATED TEMPERATURES
Recent experiments on austenitic stainless steel (type 316 [12] and 0.01wt.%C-19wt.%Cr18wt.%Ni [ 13 ] ) have shown that the kinetics of the spreading of EGBDs pre-existing in rand o m grain boundaries is best described by an equation of the general form
ta=ATaexp(AHb~
(1)
\RTd ]
where ta is the spreading time, Td the spreading temperature, A H b the activation enthalpy for grain boundary diffusion and A a constant. Equation (1) was derived theoretically by Johannesson and ThSlen [ 14 ], Lojkowski [15] and Lojkowski and Grabski [16]. The constant A in the Lojkowski and Grabski model has the following form: A -
11kSm a
G6~2Dob
(2)
where k is Boltzmann's constant, Sm the core width of an EGBD at which its image becomes invisible using transmission electron microscopy (equal to about t w o extinction distances [ 17 ] ), G the shear modulus, 5 the grain boundary width for diffusion (normally assumed to be 5 × 10 -l° m), $2 the atomic volume and D0 b the pre-exponential term in the expression for the grain boundary diffusivity (Db = D 0 b e x p ( - - z~r-Ib/RT)).
Let us assume for simplicity that the grains are cubes with sides equal to d (where d is the average grain size) and they deform with the strain rate ~g. In time dt the deformation is
L7 TABLE 1
d% = ~g dt. This d e f o r m a t i o n level m u s t be achieved b y the m o v e m e n t o f s o m e d e n s i t y Pm o f lattice d i s l o c a t i o n s such as
deg ~ pmbYc
(3)
w h e r e ~ ~ d in view o f t h e fact that, at yielding, grain b o u n d a r i e s act also as sources and g e n e r a t e d d i s l o c a t i o n s will travel across t h e grains. T h e t o t a l l e n g t h o f m o b i l e dislocations is o b v i o u s l y prod 3. T h u s t h e d e n s i t y o f E G B D s at t h e grain b o u n d a r y w h i c h is shared b y t w o grains can be expressed as pEGBD -- Prod3 3d 2
(4)
S u b s t i t u t i n g Pm f r o m eqn. (4) f o r ~ = d, we obtain pEGBD __ deg 3b
(5)
Experimentally observed temperatures of the yield stress drop (taken at 95% of the plateau stress value) and the ratios of the rate of spreading of extrinsic grain boundary dislocations to the rate of formation of extrinsic grain boundary dislocations Grain
Tdrop
size
(a t O. 95Gplateau)
(Urn)
(K)
2 25 40-60 125 270 650
850 960 1025 1040 1040 1030
AEGBD t P spread/ pEGBDformed
0.11 0.10 0.23-0.16 0.10 0.05 0.02
The following parameters were used for the calculations: ~ ' ~ b = 177 kJ mo1-1 [22];D0 b = 6.1 x 10-4 m 2 s - 1 [22];S m ~ 6 x 10-8 m [19]; ~ = 11.64 X 10-30 m 3 (zero on the assumption of a lattice parametera of 3.598 X 10-1° m [20]);G = 7.74 X 10 l° N m-2 [21].
Thus, EGBD ~EGBDformed
--
_
~
_
dt
I lOC
N
~g
(6)
-
3b
Similarly, the rate o f spreading o f E G B D s can be expressed as dS
pEGBDspread = pEGBD v --
(7)
•~
I
IO00 /
~
g
90C
! !
/
/
2
bc~ 80C
dt GRAIN
w h e r e pEGBDv (m m -3) is n o w expressed per u n i t v o l u m e and is n u m e r i c a l l y equal to p ~ . T h e f a c t o r dS/dt is t h e rate o f c o r e spreading o f an E G B D and is equal t o Sm/ta w h e r e Sm is t h e final core w i d t h and td (the spreading time f o r S~) can be easily c a l c u l a t e d f r o m eqns. (1) and (2).
3. RESULTS T h e ratios o f pEGBDslrtead/pEGBDformed were calculated e m p l o y i n g eqns. (6) and (7) f o r t h e various grain sizes investigated. T h e results are listed in Table 1 t o g e t h e r w i t h t h e t e m p e r a tures o f t h e yield stress d r o p . I n o r d e r to ensure u n i f o r m i t y , these t e m p e r a t u r e s were read f r o m t h e o0.0o2 = f(T) p l o t s at t h e values o f yield stress c o r r e s p o n d i n g t o 0.95 o f t h e plat e a u stress (5% d r o p f r o m t h e p l a t e a u stress).
SIZE
[Izrn)
Fig. 2. Variation in the yield stress drop temperature (both calculated employing eqns. (6) and (7) and experimental data) with grain size: 8, this work; o, data of Mannan et al. [6 ]; - - -, theoretical, p E G B D s p r e a d ---0.1pEGBDormed, ~ = 3 X 10-4 s-1.
I t is clearly seen t h a t the d r o p o c c u r s w h e n the rate o f spreading is a p p r o x i m a t e l y equal t o 0.1 o f t h e rate o f f o r m a t i o n : pEGBDsl~ead
~
0,1~5EGBDfo~a
(8)
Figure 2 shows t h e d e p e n d e n c e o f t h e yield stress d r o p t e m p e r a t u r e o n t h e grain size calc u l a t e d on t h e a s s u m p t i o n t h a t eqn. (8) is fulfilled t o g e t h e r with e x p e r i m e n t a l p o i n t s obt a i n e d f r o m t h e plots o0.0o2 = f(T). It is clearly seen t h a t t h e t h e o r e t i c a l curve fits t h e exp e r i m e n t a l p o i n t s very well f o r grain sizes up t o a b o u t 150 p m . A t larger grain sizes t h e
L8 TABLE 2 Experimentally observed temperatures corresponding t o 0.1Oplatea u a n d t h e r a t i o s o f t h e r a t e o f s p r e a d i n g o f e x t r i n s i c grain b o u n d a r y d i s l o c a t i o n s t o t h e r a t e o f f o r m a t i o n o f e x t r i n s i c grain b o u n d a r y d i s l o c a t i o n s
Grain size
Temperature at O.10plateau
(urn)
(K)
2 25 40-60 125 270 650
~ ~ ~ ~
1220 1250 1250 1280 1280 1280
AEGBD t ~.EGBDSPread/ P formed
148 13 8-5.4 4 2 1
drop temperature becomes almost independent of grain size. Now, another important question arises: what is the ratio of the rate o f spreading to the rate of formation of EGBDs when the strengthening effect of grain boundaries disappears at higher temperatures? Table 2 shows the values of the temperature for which the yield stress is equal to merely 10% of the plateau stress (90% drop from the plateau stress) and the calculated values of the rate of spreading to the rate of formation of EGBDs. It is seen that the strengthening by the grain boundaries becomes negligible when the rate of spreading becomes slightly higher than the rate of formation of EGBDs. In the ultrafinegrained structure this ratio is unusually high. This result will be discussed later.
4. D I S C U S S I O N
First o f all, it must be pointed o u t that evidence is provided in the present work which clearly indicates that grain b o u n d a r y processes are largely responsible for the drop in the yield stress at elevated temperatures. However, this effect seems to decrease for very large grain sizes (say, above a b o u t 150/~m for an austenitic stainless steel), and the yield stress drop becomes independent of grain size. It is therefore suggested that, because Hirsch and Warrington [2] used polycrystalline samples o f aluminum with a very large grain size (400 #m), they were not able to find any substantial difference between the behaviour of single crystals and so-called "polycrystals".
However, as shown by Gallagher [3] (see Fig. 4 in his paper) there is a considerable difference between the drop temperature for polycrystals (about 700 K - 0.52Tin) and that for single crystals (about 1220 K - 0.9Tin) of copper. The presence of plateaus followed by sudden drops in yield stress as a function of deformation temperature in austenitic stainless steel is not interpreted clearly in the literature. Jenkins and Smith [22] attributed these phenomena to a dynamic-strain-aging effect (manifested by serrations in the stress-strain curve) due to carbon-vacancy pairs between 473 and 673 K and due to chromium a t o m dislocation interactions for effects above 673 K. However, they observed the same plateau and sudden drop in yield stress (0.2% offset) in pure Fe-35wt.%Ni alloy which revealed no sign of dynamic strain aging. Michel e t al. [23] investigated the high temperature deformation (up to 1089 K) of t y p e 316 austenitic stainless steel and concluded that the observed plateau and sudden drop in ultimate tensile strength at about 0.5Tin were a result of the change in the character of the dislocation substructure from a cell structure to a subgrain structure. This explanation which might be reasonable for deformation levels larger than a few per cent is completely inadequate to explain plateaus and drops in the yield stress (0.2% offset) where the formation of any significant a m o u n t of substructure is hardly expected. Also, any dynamic-strainaging effects seem to be quite improbable for deformations e as low as 0.002. For example, at the strain rate ~ of 3 × 10 -4 s -1 employed by Mannan e t al. [6] the time required to attain e = 0.2% is only a b o u t 7 s. However, Barnby's results [24] indicate that a time of a b o u t 16 h at 623 K is required to obtain such a locking which results in only a small yield point in type 316 stainless steel. The whole effect seems to be rather weak. Thus, it is unlikely that any substantial diffusion can occur within about 7 s at elevated temperatures to lock dislocations effectively. As shown in Table 2 by the unusually large ratio of pEGBDslxtead to pEGBDfozllmedfor ultrafine-grained steel, the grain boundaries in ultrafine material seem to be more resistant to the loss in strength than are those in mediumand large-grained structures. Tentatively, this finding might be explained on the assumption
L9 t h a t in p a r t d e f o r m a t i o n o c c u r r e d b y grain b o u n d a r y sliding. As t h e u l t r a f i n e - g r a i n e d material c o n t a i n s a high d e n s i t y o f i n t r a g r a n u l a r carbides [ 25 ], t h e y m i g h t h i n d e r grain b o u n d a r y sliding, t h e r e b y increasing t h e grain b o u n d ary strength. In t h e i n t e r e s t o f c o m p l e t i o n it m a y also be m e n t i o n e d t h a t t h e grain b o u n d a r y sources, w h i c h p l a y a m a j o r role in yielding (low def o r m a t i o n s ) , are a t h e r m a l in n a t u r e a n d t h a t , u n d e r t h e c o n d i t i o n pEGBDformed~ pEGBDsPread, ay is n o t e x p e c t e d t o v a r y w i t h t e m p e r a t u r e , a n d a p l a t e a u m a y be o b s e r v e d w i t h i n a certain t e m p e r a t u r e range. In view o f t h e a b o v e discussion it m a y be seen t h a t a p r o p e r c o n s i d e r a t i o n o f t h e relative rates of f o r m a t i o n and spreading of EGBDs d u r i n g t h e yielding o f f.c.c, p o l y c r y s t a i s a n d specifically an a u s t e n i t i c stainless steel o f f e r s a m o r e s a t i s f a c t o r y e x p l a n a t i o n f o r t h e prese n c e o f a p l a t e a u and t h e s u d d e n d r o p in oy w i t h increasing t e m p e r a t u r e .
ACKNOWLEDGMENT This w o r k was s u p p o r t e d b y grants f r o m t h e N a t u r a l Sciences a n d E n g i n e e r i n g Research C o u n c i l o f C a n a d a w h i c h are g r a t e f u l l y acknowledged.
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