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Plastic deformation of ferrite—pearlite structures in steel
Materials Science and Engineering, 17 (1975) 209--219 © Elsevier Sequoia S.A., Lausanne --Printed in The Netherlands
Plastic Deformation o f Ferrite -- Pearlite Structures in Steel B. KARLSSON and G. LINDI~N Department of Engineering Metals, Chalmers University of Technology, G6teborg (Sweden)
(Received May 7, 1974)
SUMMARY
different yield strength ratios between the phases has only recently been investigated
The yield and work hardening of ferrite pearlite aggregates with continuous ferrite matrix have been studied, special attention being paid to the relative deformation of the individual constituents, which was followed by TEM and measurements of microhardness and X-ray line broadening. FEM calculations of the deformation of inclusions of similar geometry in a softer matrix have been performed for varying hardness ratio between inclusion and matrix. Hard and soft pearlite inclusions plastify at different degrees of overall strain. The flow stress of the aggregate structure can be described in terms of load transfer from the matrix to the inclusions, using a mixture rule which divides both stress and strain among the constituents in the ratio of their volume fractions.
[7,81. In steel with ferrite - pearlite structure the ferrite is continuous at carbon contents of 0.4 - 0.5 wt.%. In the context of the plastic deformation of the aggregate, the pearlite inclusions can be considered as macroscopically homogeneous. Therefore, it should be possible to describe the properties of the aggregate in terms of the plastic properties of the constituents individually, if it were known how strain and stress are distributed between the constituents and how they interact during deformation. It is the purpose of this paper to study the behaviour of ferrite and pearlite during deformation of carbon steels. Particularly, the behaviour of the cementite phase within the pearlite is investigated.
EXPERIMENTAL INTRODUCTION Two-phase structures with grain sizes of the order of ten to hundred microns are c o m m o n in ordinary steels. It is well known that aggregates of two phases with different properties often combine good tensile strength with desired ductility. Despite the ample documentation in the literature on yield and tensile strength for materials with various volume fractions of harder phases, the individual behaviour of the phases during plastic deformation is rarely dealt with. Earlier experimental work [1 - 6] has primarily been concerned with the influence of volume fraction of the second phase on yield strength, tensile strength and fracture strain. The influence of
Four different carbon steels have been used in this study, Table 1. The main part of the investigation is done on the low carbon steel. Specimens of 2 mm thickness were taken in the rolling direction of the material which was supplied as hot rolled strip. Heat treatments included austenitization at l l 0 0 ° C for 1 h (1000°C for the eutectoid steel) followed by furnace cooling (7 deg C/ min) to A 1 and holding at this temperature for 1 h. Subsequent air cooling (15 deg C/s) or furnace cooling (2 d e g C / m i n ) yielded pearlite with different fineness in all materials. These cooling sequences gave, for the low carbon steel, rather equiaxed pearlite regions with a volume fraction of 0.25 and with
210 TABLE 1 C h e m i c a l analysis in wt. % Alloy
C
Si
Mn
Cr
1. 2. 3. 4.
0.023 0.20 0.55 0.85
~ 0.22 0.24 0.18
0.41 0.48 0.37
0.07 0.03
Pure ferritic steel L o w c a r b o n steel High c a r b o n steel E u t e c t o i d steel
Ni
Cu
all < 0 . 0 0 5 0.11 0.12 0.04 0.03
P
S
Remarks
0.006 0.025
-~ 0.008 0.030
* S a n d v i k 4LS Sandvik llL Swedish s t a n d a r d SIS 1 7 7 4
* T h e analysis is t a k e n o n t h e s u p p l i e d p o w d e r m a t e r i a l b e f o r e v a c u u m melting.
a mean free path in the ferrite between pearlire inclusions of 75 pm. In order to stabilize the ferrite phase against ageing all specimens were finally annealed at 400°C for 4 h. All heat t r e a t m e n t s were carried out in argon. Tensile tests were made with a Zwick universal testing machine at a strain rate of 2 X 10 - 4 s - 1 . The specimens were strained to different levels f r om 0.01 up t o fracture. Larger h o m o g e n e o u s strains were imposed by rolling. Microhardness measurements of prestrained specimens were p e r f o r m e d on a Reichert metallograph at a load of 20 g. Samples for transmission electron microsc o p y of the low carbon steel were thinned with a chromic - acetic acid electropolishing sOlution. F o r transmission studies, the cementite plates were e xt r act ed f r o m the material. The specimen surface was carefully polished mechanically and t he n electrolytically in order to avoid surface zones affected by earlier grinding and polishing. T he ferrite was t h e n heavily etched in a solution of 50% HNO3 in water. Carbon was evaporated o n t o the surface, which t he n was electrolytically etched in 10% Nital solution. By this technique it was possible to e xt r act large single cementite plates or even colonies from pearlite
regions. Plastic deformation of cementite has also been investigated by measurements of X-ray line broadening in diffractograms of cementite plates e x t r a c t e d f r om the high carbon steel. T h e e x t r a c t i o n was carried out by c om pl et e removal of the ferrite in an electrolyte consisting of 50 g C6HsNa3OT, 12 g KI and 6 g KBr in 1000 ml H 2 0 . T he strong peak at 2{) = 44.2 ° (containing (112) and (021)) and a t t h e one at 20 = 64 ° (containing (220), (004) and (023)) were measured. T he peak shapes were d e t e r m i n e d by scanning in steps of 0.05 ° and
counting the pulses for a fixed time of 60 s (corresponding to roughly 3000 counts). Cobalt radiation was used with suppression of the K~ c o m p o n e n t s of the radiation by an Fe filter.
RESULTS
Mechanical tests The low carbon steel has been heat treated to give two different hardness levels of the pearlite inclusions. T h r o u g h o u t this paper we will characterize the state of the constituents by their hardness ratio in the u n d e f o r m e d state, ~ = HV(pearlite)/HV(ferrite). Air and
A "I-
300
o
L
hogdness
!
j
"~
f
°
-i-I"
m m Iu
0
Ol
0
03
strain
Fig. 1. C o m p a r i s o n of t h e stress - s t r a i n b e h a v i o u r o f t h e ferrite - pearlite aggregate in the l o w - c a r b o n steel w i t h t h e m i c r o h a r d n e s s o f t h e individual c o n s t i t u e n t s . Hard a n d soft pearlite c o r r e s p o n d t o K = 2.4 a n d 1.9 respectively.
211 furnace cooling gave u-values of 2.4 and 1.9 respectively. Figure 1 shows the stress - strain relations of the material in these states. Apart from a slight difference in lower yield stress, it is seen that the higher u-value gives a somewhat higher work-hardening rate but no difference in true yield stress. (As shown elsewhere [ 7 ] , still higher K-values will increase the work hardening considerably at given volume fraction of inclusions.) Figure 1 also shows the microhardness in matrix and inclusions at different strains. T he dashed lines extend the curves to a higher strain p r o d u c e d by rolling. While the hardening of the ferritic matrix during d e f o r m a t i o n is fairly insensitive to the initial pearlite hardness, soft pearlite hardens mo r e rapidly than the harder pearlite. The work hardening of pearlite has been studied separately [9] and has been f o u n d to be i n d e p e n d e n t of the initial hardness or yield stress. Typical results for two pearlites with hardness levels similar to our inclusions are shown in Fig. 2. (Identical heat treatments give slightly harder pearlite in the eut ect oi d than in the low carbon steel, Figs. 1 and 2.) Comparing with Fig. 1, we conclude t hat the soft pearlite inclusions start deforming immediately when the aggregate is strained and that t h e y undergo greater strain (as indicated by their stronger work hardening) than the
, s ~
harder inclusions. Although there seems to be little or no strain hardening in the stronger inclusions up to a b o u t 0.07 total strain, t h e y t oo will eventually plastify. Thus in bot h states of the low carbon steel, the pearlite is subjected to stress levels well above that of the flow stress of the aggregate. Geometrical measurements
To ascertain the shape change of the pearlitic inclusions during tension, linear intercept measurements were perform ed bot h in the und e f o r m e d and in the d e f o r m e d condition. Table 2 shows the mean linear intercept length of pearlite regions for the " s o f t " material (K = 1.9) parallel and perpendicular to the tensile direction. In each case a minimum of 300 pearlite inclusions was recorded. If the change in intercept length was exclusively due to strain, the strain would be 0.24 parallel to the tensile direction and --0.19 perpendicular to it. However, the macroscopic mean strains are 0.23 and --0.12 respectively. The discrepancy can be at t ri but ed to rigid body rot at i on of the inclusions which, together with plastic deform at i on, will c o n t r i b u t e to the intercept ratio. Microscopic observations of identical surface regions before and after d e f o r m a t i o n d e m o n s t r a t e t hat especially " f i n s " of the pearlite regions are t urned towards the tensile axis at larger macroscopic strains. In order to separate the plastic d e f o r m a t i o n from rigid b o d y rotation, an out er surface of a tensile specimen of the " s o f t " material (~ = 1.9) was metallographically prepared, and the same area was successively p h o t o g r a p h e d during stepwise straining. The lengths of clearly recognizable lines across pearlite regions were measured at zero strain and after a mean (engineering) strain of 0.15, Fig. 3. The relative changes in length of these lines thus reflect
,, s
TABLE 2 Mean intercept length of pearlite regions in the low carbon material with K = 1.9
_o
! o
1
005 plastic strain
Macroscopic strain _OI
Fig. 2. Stress and microhardness at different strain
levels of soft and hard pearlite in t h e e u t e c t o i d steel cf. Fig. 1.
0 0.23
Mean intercept length (pm) Parallel to tensile axis
Perpendicular to tensile axis
24.5 31.0
25.0 20.6
212
before straining
ofter straining
tensile
strong inhomogeneity of matrix strain, which has been shown to be characteristic for twophase aggregates [8].
direction
I - to st rQin
=
[ o
Fig. 3. Schematic representation of the length of lines in the pearlite regions.
the plastic strain and are recorded in Fig. 4 as a function of the angle with the tensile direction. The dashed line in Fig. 4 indicates the macroscopic mean strain in a tensile test defined in the same way, which from elementary continuum mechanics is equal to (e/4) (1 + 3 cos 20), where 0 is the angle with the tensile axis and e is the macroscopic strain (0.15). Assuming the analytical form of the curve to be the same, a least-square evaluation of the data gives the full line in Fig. 4, indicating a mean strain level of the inclusions of about 0.1. The scatter is partly a surface effect, but in the main it is certainly due to the
-
=tteln
Transmission electron microscopy Transmission electron microscopy of pearlite regions from the deformed low carbon steel showed considerable plastic deformation for both soft and hard pearlite. Figure 5 shows a high dislocation density in the ferritic phase of hard pearlite after a macroscopic strain of 0.10. Even within the cementite one frequently finds parallel series of contrast lines, often almost parallel to dislocations in adjacent ferrite, indicating plastic deformation of cementite. This is more evident in dark field micrographs of cementite, Fig. 6. These observations indicate that shear is transmitted on nearly parallel planes in ferrite and cementite. The systematic survey b y Andrews [10] on crystallographic relations between ferrite and cementite in pearlite shows that continuity in glide plane across the b o u n d a r y ferrite/cementite is to be expected and has indeed often been observed. Transmission studies of extracted cementite plates further elucidate the deformation of
et¢
inclusion S~rqln
O.Z
Q
o
0.1
•
-0.1 30
ongle
60
°
g0
[degrees]
Fig. 4. Engineering strain of lines in the pearlite regions (• = 1.9) of the low-carbon steel, cf. Fig. 3. For details, see the text.
Fig. 5. Transmission electron micrograph showing high dislocation density in the ferrite within pearlite regions in the low-carbon steel deformed 0.10. Bright field. 1000 kV.
213
Fig. 8. Transmission electron micrograph of a cementite plate extracted from the low-carbon steel deformed 0.30. For details, see text. Bright field. 100 kV.
Fig. 6. Transmission electron micrograph showing plastic deformation of the pearlitic cementite. Same region as Fig. 5. Dark field using a cementite reflection. 1000 kV.
this phase. In the u n d e f o r m e d c o n d i t i o n t h e c e m e n t i t e plates o f t e n e x h i b i t low-angle grain b o u n d a r i e s and stacking faults associated with partial dislocations, Fig. 7. T h e s e defects originate f r o m g r o w t h faults in the c e m e n t i t e . Direct observations and the absence o f distortion in wide area d i f f r a c t o g r a m s s h o w t h a t these d e f e c t s are n o t too c o m m o n . C e m e n t i t e plates e x t r a c t e d f r o m the " h a r d " low c a r b o n steel d e f o r m e d b y rolling t o e = 0.30 have high dislocation density, Fig. 8. T h e c o n t r a s t dots along distinguished parallel glide traces m a y be low angle b o u n d a r i e s o f twist t y p e ; a n e t o f screw dislocations can p r o d u c e such c o n t r a s t effects. As seen at the left o f Fig. 8, thickness fringes are o f f s e t at the glide traces, as e x p e c t e d f o r a r o w a screw dislocations. An alternative e x p l a n a t i o n of these c o n t r a s t lines might be m i c r o t w i n n i n g . T h e d i f f r a c t i o n spots s h o w n in Fig. 8 are fairly sharp, indicating a m o d e r a t e plastic strain; d i f f r a c t o g r a m s t a k e n f r o m o t h e r plates s h o w e d m o r e p r o n o u n c e d arcing of spots. F r e q u e n t l y there are t w o discrete glide trace o r i e n t a t i o n s in the c e m e n t i t e plates, Fig. 9.
X-Ray measurements
Fig. 7. Transmission electron micrograph of a cementite plate, extracted from the low-carbon steel in undeformed condition, showing low angle grain boundaries and stacking faults. Dark field. 1000 kV.
T h e high dislocation d e n s i t y in d e f o r m e d c e m e n t i t e plates, Figs. 8 and 9, s h o u l d also cause line b r o a d e n i n g in X-ray d i f f r a c t i o n . T h e X-ray line b r o a d e n i n g gives a m e a n value o f the dislocation d e n s i t y while the transmission micrographs necessarily give o n l y spot values. E x t r a c t e d carbide lamellae f r o m t h e high c a r b o n steel in u n d e f o r m e d and rolled
214
scatter it is evident t hat an increased macroscopic strain leads t o a broadening of the cementite reflections and thus to increased dislocation density.
Continuum mechanical analysis (FEM)
Fig. 9. Transmission electron micrograph of cementite plates extracted from the low-carbon steel deformed 0.30. Shear in two discrete planes (see text). Dark field. 1000 kV.
condition were studied. (Insufficient volume of cemen tite could be e x t r a c t e d f r om the 0.20-C steel to give good X-ray peaks.) Table 3 gives the half-height width of these lines as a f u n ctio n o f macroscopic plastic strain. All values are means of at least 3 measurements f r o m d i f f e r e n t specimens. In spite of some
C o n t i n u u m mechanical calculations (FEM) were made with the same m e t h o d as in a previous paper by Sundstr~m [ 1 2 ] . A twodimensional plate model was constructed by idealization of a ferrite - martensite microstructure from an optical micrograph, Fig. 10. This was taken from a com pani on project [ 8 ] , where the volume fraction of inclusions was slightly higher {0.30 compared with 0.25 in our case) b u t with virtually identical inclusion m o r p h o l o g y . T he slight difference in volume fraction is t h o u g h t to be of minor importance for the results. The model is loaded in plane strain with prescribed d e f o r m a t i o n corresponding to a tensile test. The elastic modulus is taken as 210 G N / m 2 and assumed equal for the two phases [ 1 3 ] . Linear work hardening is chosen as E / 1 0 0 for the matrix and E / 50 for t he inclusions. T he initial yield stress for the matrix is chosen as 150 MN/m 2 and that for the inclusions as ~ times this value with ~ = 2, 5 and 10. Calculated stress - strain curves for different ~-values are shown in Fig. 11. The mean stress in the tensile direction in each phase is also indicated. Already at small strains the stress level in the inclusions lies above the mean stress. This is particularly accent uat ed for higher K-values. Even the matrix stress
TABLE 3 Half-height width of the (220, 004, 023) and (112, 021) lines after different macroscopic strains Line
Macroscopic strain
Half-height width, A (20), degrees
(220)vw (004)m (023)m*
0 0.12 0.22 0.34 0 0.12 0.22 0.34
0.34 0.38 0.43 0.38 0.32 0.35 0.41 0.40
( l l 2 ) v s , (021)vs*
vw = very weak, m -- medium, vs = very strong ( according to Koch [ 11 ]).
Fig. 10. FEM model constructed by idealization of an optical micrograph of ferrite - pearlite aggregate in the low-carbon steel.
215 1.0
Flow
i
i = inclusions
, /
m= matrix /
o.8
t
/
.~
5i/
j~ / ///
./
/ /
!
t"
//
~
stress of the matrix
The initial yielding of the aggregate material is determined by the continuous soft ferrite matrix [9]. Its work hardening rate is governed by the accumulation of dislocations. The flow stress can be calculated using the concept of geometrically necessary dislocations proposed by Ashby [15]. The flow stress of the matrix is determined by the dislocation density p via the well-documented empirical relation
J/
/
m
/
10
/
o = o o + aGbx/p t-
O.l~ I
~
,
~
It
~
O.~'i 2o o
~
o.o,
~
....
--
"
..................
2.7................ .: _ o.o2
o.o3
(1)
10m
o o~
o.os
strain
Fig. 11. C a l c u l a t e d s t r e s s - s t r a i n c u r v e s o f t h e s t r u c t u r e in Fig. 10. K c o r r e s p o n d s t o t h e y i e l d s t r e n g t h ratio between inclusion and matrix. Full lines indicate macroscopic stress - strain behaviour.
rises above that for single phase matrix material, partly owing to hydrostatic stress components set up during straining, and partly owing to the creation of geometrically necessary dislocations. The calculation of load distribution in the two phases is rather sensitive to the local stress resolution in the FEMmodel (dependent on element net size). Details of these calculations are reported elsewhere [8].
DISCUSSION
The macroscopic stress - strain curve of a two-phase material with sizeable volume fraction of a hard, second phase is governed by the stresses and strains in the phases, the geo m e t r y of the microstructure and the volume fraction of the hard phase. Stresses and strains vary within each phase [8], but only mean values are considered in the following discussion. For the volume fraction (0.25 of the hard phase) and geometry studied, the effects of mutual constraint will be small [14].
where ao is a grain size independent friction stress [16], a a constant approximately equal to 1 [17], G shear modulus and b Burgers vector. The dislocation density p consists of statistical dislocations and geometrically necessary dislocations. The density of statistical dislocations corresponds to that which appears in single crystals upon straining. Grain boundaries and second phase particles will severely reduce the slip length, necessitating an increased dislocation density. This geometrical contribution to the total dislocation density is the d o m i n a n t part for ordinary grain sizes at low strain [15]. Therefore, the statistical dislocations are omitted in the following treatment. Furthermore, it is assumed that the dislocation densities caused by the grain boundaries in the matrix and by the inclusions are simply additive. Annihilation of dislocations is also neglected; the possible effect of this will be discussed below. A schematic two-phase structure is shown in Fig. 12. X~ denotes the mean free path in the fi-phase, whose volume fraction is f~. The mean dislocation slip length in the a-phase is X~; this is determined by both a / a and a/~ boundaries. At a total plastic strain ~ the matrix and inclusion strains are e~ and e~ respectively. Consider first a typical a grain in a hypothetical structure where the /3 regions have the same deformation properties as a. The dislocation density is then given by 1 PG1 = C1 b--~ " g
(2)
where X~ 1 = slip distance in the typical grain. Now let fi have different deformation properties. Then, a volume fraction f~ will deviate from the overall strain by Ae = e -- e~.
216
the number of ~ particles and ~/~ grain boundaries per unit length along a random test line is approximately equal, which means that ~c~2 ~ ~ . Further, X~I = X~. In this case, the flow stress o~ in the s-phase is given by eqns. (1) and (5):
°,x = ° o + C " G .-
Fig. 12. S c h e m a t i c two-phase structure. Test line and m e a n free path in each phase are indicated.
This requires the generation of additional dislocation loops of total length N per unit volume. If the mean slip length o f these dislocations is X~2, then N" ~'~2" b = C 2 • f~(~- - - ~/~)
i
.
l/1
+
f#
e--e~e
(6)
where C is a constant. The strains e~, e~ and ~ above are plastic strains, but at the strain levels considered, it is justified to neglect the elastic strain. Annihilation of dislocations will not change the contribution PGz in eqn. (5) compared with that in single phase ~, but PG 2 would decrease. However, it is believed that this is of minor importance. The strain in the pearlite regions can be evaluated from their microhardness, using the
(3)
where Ce is the constant of proportionality between shear and tensile strains. (The quantity X~ 2 is introduced because it is n o t certain that the slip length of the dislocations generated by the two mechanisms is the same.) The resulting length of dislocations per unit volume of s-phase is N 1 f~ PG2 - l--f% - C2" bX,~2 " 1--f----~" (~- --e~). (4)
hard
pearltte
1.0
e z
The total density of geometrically necessary dislocations in the s-phase is given by
Vl ferrite/hard
pearilte
Po
0.5
= PG1 + ,0G2
=1__ C~ b
~
+C2~
1--
ferrite/soft pearhte
(5)
The quantities C1/X~I and C2/X,~2 could in principle be determined by comparative studies of single phase and two-phase materials with low-volume fraction of inclusions. For the purpose of this discussion, we will assume statistically homogeneous distribution of the dislocations in the available space and equal propagation modes for the dislocations involved in eqns. (2) and (4). In our structures
i
I experir~enls
I
equo~ strain
I
modet
i
0o
Eqs. ? o~1 8
0.O5
011
ors
strain
Fig. 13. Stress - strain behaviour of ferrite, pearlite and the aggregate in the low-carbon steel. E x p e r i m e n tal values and m o d e l results.
217 h a r d n e s s - s t r a i n - r e l a t i o n s h i p o f t h e corres p o n d i n g fully pearlitic m a t e r i a l s (Fig. 2) as a c a l i b r a t i o n curve. ( T h e F E M c a l c u l a t i o n s indicate t h a t the strain levels in t h e inclusions are generally highest n e a r t h e i n t e r f a c e . As t h e h a r d n e s s i n d e n t a t i o n s c o u l d o n l y be m a d e a w a y f r o m the i n t e r f a c e s , the strains t h u s det e r m i n e d are p r o b a b l y s o m e w h a t low.) F o r instance, a t a m e a n strain of 0.1, t h e strain in t h e s o f t a n d h a r d pearlite is a p p r o x i m a t e l y 0.01 a n d 0 . 0 6 r e s p e c t i v e l y (Figs. 1 a n d 2). T h e m a t e r i a l w i t h h a r d pearlite is t h u s v e r y n e a r the case w i t h rigid inclusions, while t h e m a t e r i a l w i t h s o f t pearlite u n d e r g o e s a plastic d e f o r m a t i o n of a p p r o x i m a t e l y h a l f t h e m a c r o scopic strain. This is in r e a s o n a b l e a c c o r d a n c e w i t h the results f r o m t h e g e o m e t r i c a l m e a s u r e m e n t s , Fig. 4. I n s e r t i n g t h e inclusion strains of 0.01 a n d 0.06 estimated from the microhardness measu r e m e n t s in eqn. (6), w e o b t a i n a f l o w stress i n c r e m e n t for the s - p h a s e ( a s - - Oo ) o f 6% f o r t h e m a t e r i a l w i t h s o f t pearlite a n d 14% f o r t h e h a r d pearlite. C o m p a r i s o n w i t h experim e n t is possible using t h e m i c r o h a r d n e s s m e a s u r e m e n t s on t h e ferrite p h a s e in t h e aggregate m a t e r i a l , Fig. 1. T h e p r o c e d u r e is as follows: F r o m t h e e x p e r i m e n t a l f l o w c u r v e f o r single p h a s e f e r r i t e * , Fig. 13, we find o~(e = 0.1) = 0.35 G N / m 2. T h e value o f ao f o r ferrite is 0 . 0 4 G N / m 2 [ 2 0 ] . T h e r e f o r e (o~ - Oo)c=0.1 = 0.31 G N / m 2. A t this level, t h e 6 or 14% d i f f e r e n c e in ( a s - - ao ) b e t w e e n m a t e rial w i t h s o f t a n d h a r d pearlite t r a n s l a t e s to a d i f f e r e n c e in f l o w stress of 23 M N / m 2. Acc o r d i n g to t h e r e l a t i o n o ~ H V / 3 [ 2 1 ] , this s h o u l d c o r r e s p o n d t o a d i f f e r e n c e in m i c r o h a r d n e s s o f 8 units. Figure 1 s h o w s t h a t at e = 0.1, t h e ferrite m i c r o h a r d n e s s of m a t e r i a l w i t h h a r d p e a r l i t e is ca. 10 units higher t h a n t h a t o f m a t e r i a l w i t h s o f t pearlite. T h e a g r e e m e n t is g o o d e n o u g h to a l l o w t h e c o n c l u s i o n t h a t the i n f l u e n c e o f h a r d a n d s o f t pearlite inclu-
* The grain size and alloy content of the pure ferrite material deviate somewhat from those of the ferrite in the low carbon steel, Table 1. As the work hardening is fairly insensitive to grain size [17 ] and substitutional alloying [18,19], the measured stress -strain curve for the pure ferrite was uniformly raised by 30 MN/m 2 to correctly correspond to the alloyed ferrite in the low alloy steel. For both the ferrite and low carbon steels, the flow curves in Fig. 13 have been extrapolated back to zero plastic strain assuming a linear log a - log e relation.
sions on the w o r k h a r d e n i n g of t h e m a t r i x is s a t i s f a c t o r i l y d e s c r i b e d in t e r m s o f g e o m e t r i cally n e c e s s a r y dislocations. E q u a t i o n (6) allows us to e s t i m a t e the t o t a l f l o w stress i n c r e m e n t of the ferrite p h a s e caused b y t h e pearlite inclusions. T h e result is 43 M N / m 2 f o r the h a r d pearlite, a n d 20 M N / m 2 f o r the s o f t pearlite, b o t h at a m a c r o scopic strain o f 0.1. Figure 13 shows t h a t at a strain of 0.1, t h e f l o w stress o f t h e aggregate o f ferrite w i t h h a r d pearlite is 150 M N / m 2 higher t h a t t h a t of single p h a s e ferrite. This is f o u r t i m e s as m u c h as the f l o w stress increm e n t in the m a t r i x caused b y g e o m e t r i c a l l y n e c e s s a r y dislocations. A similar result is a c h i e v e d f o r the case o f s o f t pearlite. T h e r e f o r e , a m a j o r p a r t of t h e w o r k h a r d e n i n g o f the aggregate m u s t be a t t r i b u t e d to o t h e r effects. F l o w stress o f the ferrite - pearlite aggregate Figure 13 s h o w s t h a t t h e yield stress o f t h e aggregate is quite insensitive to the h a r d n e s s o f the inclusions. It is, in f a c t , equal to t h a t o f single-phase ferrite. T h e p r e s e n c e o f inclusions leads to m o r e r a p i d w o r k h a r d e n i n g , w h i c h m u s t be a t t r i b u t e d to t h e higher stresses supported by them. I f the strain d i s t r i b u t i o n in t h e aggregate w e r e k n o w n , t h e n t h e stress - strain b e h a v i o u r o f the individual c o n s t i t u e n t s c o u l d be used to e s t i m a t e t h e f l o w stress o f t h e aggregate. A s c h e m a t i c i n t e r p r e t a t i o n o f the stress - strain b e h a v i o u r o f t h e ferrite - pearlite s t r u c t u r e s can be based on t h e a s s u m p t i o n o f equal strain in b o t h c o n s t i t u e n t s n e g l e c t i n g interact i o n effects. A resulting stress - strain c u r v e for t h e aggregate is t h e n easily c o n s t r u c t e d f r o m t h e stress - strain d a t a f o r each c o n s t i t u e n t a n d t h e v o l u m e f r a c t i o n s . S u c h curves f o r t h e " h a r d " a n d " s o f t " m a t e r i a l are i n c l u d e d in Fig. 13. T h e y r e s e m b l e the e x p e r i m e n t a l o n e s in s h a p e b u t lie at a higher stress level. This is to be e x p e c t e d , since the pearlite in reality u n d e r g o e s less plastic strain t h a n t h e m a t r i x (Figs. 1 a n d 2) so t h a t its stress will be l o w e r t h a n a s s u m e d in the equal-strain m o d e l . On the o t h e r h a n d , b e c a u s e o f t h e a b s e n c e o f m e c h a n i c a l i n t e r a c t i o n b e t w e e n the c o n s t i t u ents, the m o d e l d o e s n o t t a k e i n t o a c c o u n t t h e h y d r o s t a t i c stress c o m p o n e n t s in t h e m a trix, w h i c h will raise t h e m a t r i x stress s o m e what. A b e t t e r a p p r o x i m a t i o n of t h e stress -
218
strain behaviour of the aggregate would have to include the partl~fon of both stresses and strains between the constituents according to some sort of mixture rule. In an earlier paper [9] a mixture rule of the following type was used: a = (1 -
f,).
= (1 -
+ Cc, +
(7) %
•
(8)
This gives a less extreme partitioning of stresses than the equal-strain model. The strains used in both of these models are total strains. To test the validity of eqns. (7) and (8), we turn to the FEM results (Figs. 11 and 14) as a model case in which the stresses and strains for both constituents are known. The stresses given in Fig. 11 are found to add up to the overall stress when weighted together by eqn. (7) within 15%. In the same way, eqn. (8) is satisfied by the strains in Fig. 14 to within 25% of the total strain. The discrepancies are largest for ~ = 10, while the figures are about half as large for ~ = 2. We note in passing that the strains in Fig. 14 are effective plastic strains, whereas eqn. (8) envisages total unidirectional strains. In the estimation, the total unidirectional strain has been considered equal to the sum of effective plastic strain and corresponding elastic strain. In the present case, the strain of the pearlite constituent can be obtained from the microhardness measurements (Fig. 1) with the aid of the hardness - strain relationship of pure pearlite (Fig. 2). (This procedure is applicable only to the regime where both constitu0.08
I
I,t v ¢--
0.06 -
o
i
2
o
S
•
10-
o
,P/"
matrix
t.. u') o
&O&
U3
~-
0.02
, ° ,,,,,,4
0
0 01
o.o2
0.03
0.o4
o.os
strain Fig. 14. Calculated strain d i s t r i b u t i o n in t h e s t r u c t u r e in Fig. 10.
ents have in fact plastified; the region of very low overall strains, where the inclusions remain elastic, cannot be treated in this way.) The matrix strain can be deduced from the inclusion strain and the overall strain by eqn. (8). Combining the flow stresses for the constituents (Fig. 13) at their respective strains in the manner of eqn. (8), the flow curve for the aggregate material is obtained (Fig. 13). It is in good agreement with experiment. The high initial stress level estimated for the ferrite/soft pearlite aggregate is attributed to an overestimation of the pearlite strain because of uncertainty in the hardness measurements (Fig. 1). As an alternative to the use of eqn. (8), the matrix strain could be determined from the microhardness of the s-phase. However, the low work hardening of ferrite in combination with the strong inhomogeneity of matrix strain [8] would make this determination rather uncertain. In contrast with ferrite, the hardness of pearlite is a sensitive indicator of strain (Fig. 13). To summarize, the mixture law stated in eqns. (7) and (8) allows one to predict with reasonable accuracy the stress- strain behaviour of the aggregate.
CONCLUSIONS
The yield stress of ferrite - pearlite aggregates with continuous ferrite matrix is about the same as that extrapolated from singlephase ferrite of equal grain size and independent of the hardness of the pearlite inclusions. The flow stress is considerably higher than t h a t of single-phase ferrite and rises with increasing inclusion hardness. During plastic deformation of the aggregate the pearlite inclusions also deform, at first elastically and then plastically. Softer inclusions deform more than do harder ones. At higher overall strains, the cementite within the pearlite deforms plastically. The flow stress of the matrix can be satisfactorily described in terms of geometrically necessary dislocations. The flow stress of the aggregate can be described in terms of load transfer from the matrix to the inclusions, using a mixture law which divides both stress and strain among the constituents in the ratio of their volume fractions.
219 ACKNOWLEDGEMENTS T h e a u t h o r s wish t o t h a n k Dr. B.O. S u n d str~im f o r c a r r y i n g o u t t h e F E M c a l c u l a t i o n s . H e l p f u l d i s c u s s i o n s w i t h Prof. H. F i s c h m e i s t e r are g r a t e f u l l y a c k n o w l e d g e d . Dr. B. J a e n s s o n has a s s i s t e d in t h e X - r a y w o r k a n d Prof. K . E . E a s t e r l i n g has t a k e n p a r t in t h e H V E M s t u d i e s . T h e m a t e r i a l was s u p p l i e d b y S a n d v i k AB. The project has been financially s u p p o r t e d by the Swedish Board for Technical Development.
REFERENCES 1 I.L. Mogford, Met. Rev., No. 114, 12 (1967). 2 F.B. Pickering and T. Gladman, Iron Steel Inst., Spec. Rept. 81, 1963, p. 10. 3 T. Gladman, I.D. McIvor and F.B. Pickering, J. Iron Steel Inst. (London), 210 (1972) 916. 4 I. Tamura et al., Trans. Iron Steel Inst. Japan, 13 (1973) 283. 5 K.W. Burns and F.B. Pickering, J. Iron Steel Inst. (London), 185 (1964) 899. 6 H.J. Kouwenhoven, Am. Soc. Metals, Trans. Quart., 62 (1969) 437.
7 H. Fischmeister, J.-O. Hj~ilmered, B. Karlsson, G. Linddn and B. SundstrSm, Proc. 3rd Int. Conf. on Strength of Metals and Alloys, 1973, p. 621. 8 B. Karlsson and B.O. Sundstr~m, Mater. Sci. Eng., 16 (1974} 161. 9 B. Karlsson and G. Linden, Mater Sci. Eng., 17 (1975) 153. 10 K.W. Andrews, Acta Met., 11 (1963) 939. 11 W. Koch, Metallkundliche Analyse, Verlag Stahleisen, DUsseldorf, 1965, p. 426. 12 B.O. Sundstr(im, Mater. Sci. Eng., 12 (1973) 265. 13 G.R. Speich and W.C. Leslie, Met. Trans., 4 (1973) 1873. 14 D.C. Drucker, J. Mater., 1 (1966) 873. 15 M.F. Ashby, in A. Kelly and R.B. Nicholson (eds.), Strengthening Methods in Crystals, Elsevier, London, 1971, p. 137. 16 J.-P. Bailon, A. Loyer and J.-M. Dorlot, Mater. Sci. Eng., 8 (1971) 288. 17 W. Roberts and Y. BergstriSm, Z. Metallk., 62 (1971) 752. 18 W.C. Leslie, Met. Trans., 3 (1972) 5. 19 T. Gladman, B. Holmes and F.B. Pickering, J. Iron Steel Inst. (London), 208 (1970) 172. 20 C.T. Liu, R.W. Armstrong and J. Gurland, J. Iron Steel Inst. (London), 209 (1971) 142. 21 F.A. McClintock and A.S. Argon, Mechanical Behavior of Materials, Addision-Wesley, Reading, Mass., 1966, p. 453.
Plastic Deformation o f Ferrite -- Pearlite Structures in Steel B. KARLSSON and G. LINDI~N Department of Engineering Metals, Chalmers University of Technology, G6teborg (Sweden)
(Received May 7, 1974)
SUMMARY
different yield strength ratios between the phases has only recently been investigated
The yield and work hardening of ferrite pearlite aggregates with continuous ferrite matrix have been studied, special attention being paid to the relative deformation of the individual constituents, which was followed by TEM and measurements of microhardness and X-ray line broadening. FEM calculations of the deformation of inclusions of similar geometry in a softer matrix have been performed for varying hardness ratio between inclusion and matrix. Hard and soft pearlite inclusions plastify at different degrees of overall strain. The flow stress of the aggregate structure can be described in terms of load transfer from the matrix to the inclusions, using a mixture rule which divides both stress and strain among the constituents in the ratio of their volume fractions.
[7,81. In steel with ferrite - pearlite structure the ferrite is continuous at carbon contents of 0.4 - 0.5 wt.%. In the context of the plastic deformation of the aggregate, the pearlite inclusions can be considered as macroscopically homogeneous. Therefore, it should be possible to describe the properties of the aggregate in terms of the plastic properties of the constituents individually, if it were known how strain and stress are distributed between the constituents and how they interact during deformation. It is the purpose of this paper to study the behaviour of ferrite and pearlite during deformation of carbon steels. Particularly, the behaviour of the cementite phase within the pearlite is investigated.
EXPERIMENTAL INTRODUCTION Two-phase structures with grain sizes of the order of ten to hundred microns are c o m m o n in ordinary steels. It is well known that aggregates of two phases with different properties often combine good tensile strength with desired ductility. Despite the ample documentation in the literature on yield and tensile strength for materials with various volume fractions of harder phases, the individual behaviour of the phases during plastic deformation is rarely dealt with. Earlier experimental work [1 - 6] has primarily been concerned with the influence of volume fraction of the second phase on yield strength, tensile strength and fracture strain. The influence of
Four different carbon steels have been used in this study, Table 1. The main part of the investigation is done on the low carbon steel. Specimens of 2 mm thickness were taken in the rolling direction of the material which was supplied as hot rolled strip. Heat treatments included austenitization at l l 0 0 ° C for 1 h (1000°C for the eutectoid steel) followed by furnace cooling (7 deg C/ min) to A 1 and holding at this temperature for 1 h. Subsequent air cooling (15 deg C/s) or furnace cooling (2 d e g C / m i n ) yielded pearlite with different fineness in all materials. These cooling sequences gave, for the low carbon steel, rather equiaxed pearlite regions with a volume fraction of 0.25 and with
210 TABLE 1 C h e m i c a l analysis in wt. % Alloy
C
Si
Mn
Cr
1. 2. 3. 4.
0.023 0.20 0.55 0.85
~ 0.22 0.24 0.18
0.41 0.48 0.37
0.07 0.03
Pure ferritic steel L o w c a r b o n steel High c a r b o n steel E u t e c t o i d steel
Ni
Cu
all < 0 . 0 0 5 0.11 0.12 0.04 0.03
P
S
Remarks
0.006 0.025
-~ 0.008 0.030
* S a n d v i k 4LS Sandvik llL Swedish s t a n d a r d SIS 1 7 7 4
* T h e analysis is t a k e n o n t h e s u p p l i e d p o w d e r m a t e r i a l b e f o r e v a c u u m melting.
a mean free path in the ferrite between pearlire inclusions of 75 pm. In order to stabilize the ferrite phase against ageing all specimens were finally annealed at 400°C for 4 h. All heat t r e a t m e n t s were carried out in argon. Tensile tests were made with a Zwick universal testing machine at a strain rate of 2 X 10 - 4 s - 1 . The specimens were strained to different levels f r om 0.01 up t o fracture. Larger h o m o g e n e o u s strains were imposed by rolling. Microhardness measurements of prestrained specimens were p e r f o r m e d on a Reichert metallograph at a load of 20 g. Samples for transmission electron microsc o p y of the low carbon steel were thinned with a chromic - acetic acid electropolishing sOlution. F o r transmission studies, the cementite plates were e xt r act ed f r o m the material. The specimen surface was carefully polished mechanically and t he n electrolytically in order to avoid surface zones affected by earlier grinding and polishing. T he ferrite was t h e n heavily etched in a solution of 50% HNO3 in water. Carbon was evaporated o n t o the surface, which t he n was electrolytically etched in 10% Nital solution. By this technique it was possible to e xt r act large single cementite plates or even colonies from pearlite
regions. Plastic deformation of cementite has also been investigated by measurements of X-ray line broadening in diffractograms of cementite plates e x t r a c t e d f r om the high carbon steel. T h e e x t r a c t i o n was carried out by c om pl et e removal of the ferrite in an electrolyte consisting of 50 g C6HsNa3OT, 12 g KI and 6 g KBr in 1000 ml H 2 0 . T he strong peak at 2{) = 44.2 ° (containing (112) and (021)) and a t t h e one at 20 = 64 ° (containing (220), (004) and (023)) were measured. T he peak shapes were d e t e r m i n e d by scanning in steps of 0.05 ° and
counting the pulses for a fixed time of 60 s (corresponding to roughly 3000 counts). Cobalt radiation was used with suppression of the K~ c o m p o n e n t s of the radiation by an Fe filter.
RESULTS
Mechanical tests The low carbon steel has been heat treated to give two different hardness levels of the pearlite inclusions. T h r o u g h o u t this paper we will characterize the state of the constituents by their hardness ratio in the u n d e f o r m e d state, ~ = HV(pearlite)/HV(ferrite). Air and
A "I-
300
o
L
hogdness
!
j
"~
f
°
-i-I"
m m Iu
0
Ol
0
03
strain
Fig. 1. C o m p a r i s o n of t h e stress - s t r a i n b e h a v i o u r o f t h e ferrite - pearlite aggregate in the l o w - c a r b o n steel w i t h t h e m i c r o h a r d n e s s o f t h e individual c o n s t i t u e n t s . Hard a n d soft pearlite c o r r e s p o n d t o K = 2.4 a n d 1.9 respectively.
211 furnace cooling gave u-values of 2.4 and 1.9 respectively. Figure 1 shows the stress - strain relations of the material in these states. Apart from a slight difference in lower yield stress, it is seen that the higher u-value gives a somewhat higher work-hardening rate but no difference in true yield stress. (As shown elsewhere [ 7 ] , still higher K-values will increase the work hardening considerably at given volume fraction of inclusions.) Figure 1 also shows the microhardness in matrix and inclusions at different strains. T he dashed lines extend the curves to a higher strain p r o d u c e d by rolling. While the hardening of the ferritic matrix during d e f o r m a t i o n is fairly insensitive to the initial pearlite hardness, soft pearlite hardens mo r e rapidly than the harder pearlite. The work hardening of pearlite has been studied separately [9] and has been f o u n d to be i n d e p e n d e n t of the initial hardness or yield stress. Typical results for two pearlites with hardness levels similar to our inclusions are shown in Fig. 2. (Identical heat treatments give slightly harder pearlite in the eut ect oi d than in the low carbon steel, Figs. 1 and 2.) Comparing with Fig. 1, we conclude t hat the soft pearlite inclusions start deforming immediately when the aggregate is strained and that t h e y undergo greater strain (as indicated by their stronger work hardening) than the
, s ~
harder inclusions. Although there seems to be little or no strain hardening in the stronger inclusions up to a b o u t 0.07 total strain, t h e y t oo will eventually plastify. Thus in bot h states of the low carbon steel, the pearlite is subjected to stress levels well above that of the flow stress of the aggregate. Geometrical measurements
To ascertain the shape change of the pearlitic inclusions during tension, linear intercept measurements were perform ed bot h in the und e f o r m e d and in the d e f o r m e d condition. Table 2 shows the mean linear intercept length of pearlite regions for the " s o f t " material (K = 1.9) parallel and perpendicular to the tensile direction. In each case a minimum of 300 pearlite inclusions was recorded. If the change in intercept length was exclusively due to strain, the strain would be 0.24 parallel to the tensile direction and --0.19 perpendicular to it. However, the macroscopic mean strains are 0.23 and --0.12 respectively. The discrepancy can be at t ri but ed to rigid body rot at i on of the inclusions which, together with plastic deform at i on, will c o n t r i b u t e to the intercept ratio. Microscopic observations of identical surface regions before and after d e f o r m a t i o n d e m o n s t r a t e t hat especially " f i n s " of the pearlite regions are t urned towards the tensile axis at larger macroscopic strains. In order to separate the plastic d e f o r m a t i o n from rigid b o d y rotation, an out er surface of a tensile specimen of the " s o f t " material (~ = 1.9) was metallographically prepared, and the same area was successively p h o t o g r a p h e d during stepwise straining. The lengths of clearly recognizable lines across pearlite regions were measured at zero strain and after a mean (engineering) strain of 0.15, Fig. 3. The relative changes in length of these lines thus reflect
,, s
TABLE 2 Mean intercept length of pearlite regions in the low carbon material with K = 1.9
_o
! o
1
005 plastic strain
Macroscopic strain _OI
Fig. 2. Stress and microhardness at different strain
levels of soft and hard pearlite in t h e e u t e c t o i d steel cf. Fig. 1.
0 0.23
Mean intercept length (pm) Parallel to tensile axis
Perpendicular to tensile axis
24.5 31.0
25.0 20.6
212
before straining
ofter straining
tensile
strong inhomogeneity of matrix strain, which has been shown to be characteristic for twophase aggregates [8].
direction
I - to st rQin
=
[ o
Fig. 3. Schematic representation of the length of lines in the pearlite regions.
the plastic strain and are recorded in Fig. 4 as a function of the angle with the tensile direction. The dashed line in Fig. 4 indicates the macroscopic mean strain in a tensile test defined in the same way, which from elementary continuum mechanics is equal to (e/4) (1 + 3 cos 20), where 0 is the angle with the tensile axis and e is the macroscopic strain (0.15). Assuming the analytical form of the curve to be the same, a least-square evaluation of the data gives the full line in Fig. 4, indicating a mean strain level of the inclusions of about 0.1. The scatter is partly a surface effect, but in the main it is certainly due to the
-
=tteln
Transmission electron microscopy Transmission electron microscopy of pearlite regions from the deformed low carbon steel showed considerable plastic deformation for both soft and hard pearlite. Figure 5 shows a high dislocation density in the ferritic phase of hard pearlite after a macroscopic strain of 0.10. Even within the cementite one frequently finds parallel series of contrast lines, often almost parallel to dislocations in adjacent ferrite, indicating plastic deformation of cementite. This is more evident in dark field micrographs of cementite, Fig. 6. These observations indicate that shear is transmitted on nearly parallel planes in ferrite and cementite. The systematic survey b y Andrews [10] on crystallographic relations between ferrite and cementite in pearlite shows that continuity in glide plane across the b o u n d a r y ferrite/cementite is to be expected and has indeed often been observed. Transmission studies of extracted cementite plates further elucidate the deformation of
et¢
inclusion S~rqln
O.Z
Q
o
0.1
•
-0.1 30
ongle
60
°
g0
[degrees]
Fig. 4. Engineering strain of lines in the pearlite regions (• = 1.9) of the low-carbon steel, cf. Fig. 3. For details, see the text.
Fig. 5. Transmission electron micrograph showing high dislocation density in the ferrite within pearlite regions in the low-carbon steel deformed 0.10. Bright field. 1000 kV.
213
Fig. 8. Transmission electron micrograph of a cementite plate extracted from the low-carbon steel deformed 0.30. For details, see text. Bright field. 100 kV.
Fig. 6. Transmission electron micrograph showing plastic deformation of the pearlitic cementite. Same region as Fig. 5. Dark field using a cementite reflection. 1000 kV.
this phase. In the u n d e f o r m e d c o n d i t i o n t h e c e m e n t i t e plates o f t e n e x h i b i t low-angle grain b o u n d a r i e s and stacking faults associated with partial dislocations, Fig. 7. T h e s e defects originate f r o m g r o w t h faults in the c e m e n t i t e . Direct observations and the absence o f distortion in wide area d i f f r a c t o g r a m s s h o w t h a t these d e f e c t s are n o t too c o m m o n . C e m e n t i t e plates e x t r a c t e d f r o m the " h a r d " low c a r b o n steel d e f o r m e d b y rolling t o e = 0.30 have high dislocation density, Fig. 8. T h e c o n t r a s t dots along distinguished parallel glide traces m a y be low angle b o u n d a r i e s o f twist t y p e ; a n e t o f screw dislocations can p r o d u c e such c o n t r a s t effects. As seen at the left o f Fig. 8, thickness fringes are o f f s e t at the glide traces, as e x p e c t e d f o r a r o w a screw dislocations. An alternative e x p l a n a t i o n of these c o n t r a s t lines might be m i c r o t w i n n i n g . T h e d i f f r a c t i o n spots s h o w n in Fig. 8 are fairly sharp, indicating a m o d e r a t e plastic strain; d i f f r a c t o g r a m s t a k e n f r o m o t h e r plates s h o w e d m o r e p r o n o u n c e d arcing of spots. F r e q u e n t l y there are t w o discrete glide trace o r i e n t a t i o n s in the c e m e n t i t e plates, Fig. 9.
X-Ray measurements
Fig. 7. Transmission electron micrograph of a cementite plate, extracted from the low-carbon steel in undeformed condition, showing low angle grain boundaries and stacking faults. Dark field. 1000 kV.
T h e high dislocation d e n s i t y in d e f o r m e d c e m e n t i t e plates, Figs. 8 and 9, s h o u l d also cause line b r o a d e n i n g in X-ray d i f f r a c t i o n . T h e X-ray line b r o a d e n i n g gives a m e a n value o f the dislocation d e n s i t y while the transmission micrographs necessarily give o n l y spot values. E x t r a c t e d carbide lamellae f r o m t h e high c a r b o n steel in u n d e f o r m e d and rolled
214
scatter it is evident t hat an increased macroscopic strain leads t o a broadening of the cementite reflections and thus to increased dislocation density.
Continuum mechanical analysis (FEM)
Fig. 9. Transmission electron micrograph of cementite plates extracted from the low-carbon steel deformed 0.30. Shear in two discrete planes (see text). Dark field. 1000 kV.
condition were studied. (Insufficient volume of cemen tite could be e x t r a c t e d f r om the 0.20-C steel to give good X-ray peaks.) Table 3 gives the half-height width of these lines as a f u n ctio n o f macroscopic plastic strain. All values are means of at least 3 measurements f r o m d i f f e r e n t specimens. In spite of some
C o n t i n u u m mechanical calculations (FEM) were made with the same m e t h o d as in a previous paper by Sundstr~m [ 1 2 ] . A twodimensional plate model was constructed by idealization of a ferrite - martensite microstructure from an optical micrograph, Fig. 10. This was taken from a com pani on project [ 8 ] , where the volume fraction of inclusions was slightly higher {0.30 compared with 0.25 in our case) b u t with virtually identical inclusion m o r p h o l o g y . T he slight difference in volume fraction is t h o u g h t to be of minor importance for the results. The model is loaded in plane strain with prescribed d e f o r m a t i o n corresponding to a tensile test. The elastic modulus is taken as 210 G N / m 2 and assumed equal for the two phases [ 1 3 ] . Linear work hardening is chosen as E / 1 0 0 for the matrix and E / 50 for t he inclusions. T he initial yield stress for the matrix is chosen as 150 MN/m 2 and that for the inclusions as ~ times this value with ~ = 2, 5 and 10. Calculated stress - strain curves for different ~-values are shown in Fig. 11. The mean stress in the tensile direction in each phase is also indicated. Already at small strains the stress level in the inclusions lies above the mean stress. This is particularly accent uat ed for higher K-values. Even the matrix stress
TABLE 3 Half-height width of the (220, 004, 023) and (112, 021) lines after different macroscopic strains Line
Macroscopic strain
Half-height width, A (20), degrees
(220)vw (004)m (023)m*
0 0.12 0.22 0.34 0 0.12 0.22 0.34
0.34 0.38 0.43 0.38 0.32 0.35 0.41 0.40
( l l 2 ) v s , (021)vs*
vw = very weak, m -- medium, vs = very strong ( according to Koch [ 11 ]).
Fig. 10. FEM model constructed by idealization of an optical micrograph of ferrite - pearlite aggregate in the low-carbon steel.
215 1.0
Flow
i
i = inclusions
, /
m= matrix /
o.8
t
/
.~
5i/
j~ / ///
./
/ /
!
t"
//
~
stress of the matrix
The initial yielding of the aggregate material is determined by the continuous soft ferrite matrix [9]. Its work hardening rate is governed by the accumulation of dislocations. The flow stress can be calculated using the concept of geometrically necessary dislocations proposed by Ashby [15]. The flow stress of the matrix is determined by the dislocation density p via the well-documented empirical relation
J/
/
m
/
10
/
o = o o + aGbx/p t-
O.l~ I
~
,
~
It
~
O.~'i 2o o
~
o.o,
~
....
--
"
..................
2.7................ .: _ o.o2
o.o3
(1)
10m
o o~
o.os
strain
Fig. 11. C a l c u l a t e d s t r e s s - s t r a i n c u r v e s o f t h e s t r u c t u r e in Fig. 10. K c o r r e s p o n d s t o t h e y i e l d s t r e n g t h ratio between inclusion and matrix. Full lines indicate macroscopic stress - strain behaviour.
rises above that for single phase matrix material, partly owing to hydrostatic stress components set up during straining, and partly owing to the creation of geometrically necessary dislocations. The calculation of load distribution in the two phases is rather sensitive to the local stress resolution in the FEMmodel (dependent on element net size). Details of these calculations are reported elsewhere [8].
DISCUSSION
The macroscopic stress - strain curve of a two-phase material with sizeable volume fraction of a hard, second phase is governed by the stresses and strains in the phases, the geo m e t r y of the microstructure and the volume fraction of the hard phase. Stresses and strains vary within each phase [8], but only mean values are considered in the following discussion. For the volume fraction (0.25 of the hard phase) and geometry studied, the effects of mutual constraint will be small [14].
where ao is a grain size independent friction stress [16], a a constant approximately equal to 1 [17], G shear modulus and b Burgers vector. The dislocation density p consists of statistical dislocations and geometrically necessary dislocations. The density of statistical dislocations corresponds to that which appears in single crystals upon straining. Grain boundaries and second phase particles will severely reduce the slip length, necessitating an increased dislocation density. This geometrical contribution to the total dislocation density is the d o m i n a n t part for ordinary grain sizes at low strain [15]. Therefore, the statistical dislocations are omitted in the following treatment. Furthermore, it is assumed that the dislocation densities caused by the grain boundaries in the matrix and by the inclusions are simply additive. Annihilation of dislocations is also neglected; the possible effect of this will be discussed below. A schematic two-phase structure is shown in Fig. 12. X~ denotes the mean free path in the fi-phase, whose volume fraction is f~. The mean dislocation slip length in the a-phase is X~; this is determined by both a / a and a/~ boundaries. At a total plastic strain ~ the matrix and inclusion strains are e~ and e~ respectively. Consider first a typical a grain in a hypothetical structure where the /3 regions have the same deformation properties as a. The dislocation density is then given by 1 PG1 = C1 b--~ " g
(2)
where X~ 1 = slip distance in the typical grain. Now let fi have different deformation properties. Then, a volume fraction f~ will deviate from the overall strain by Ae = e -- e~.
216
the number of ~ particles and ~/~ grain boundaries per unit length along a random test line is approximately equal, which means that ~c~2 ~ ~ . Further, X~I = X~. In this case, the flow stress o~ in the s-phase is given by eqns. (1) and (5):
°,x = ° o + C " G .-
Fig. 12. S c h e m a t i c two-phase structure. Test line and m e a n free path in each phase are indicated.
This requires the generation of additional dislocation loops of total length N per unit volume. If the mean slip length o f these dislocations is X~2, then N" ~'~2" b = C 2 • f~(~- - - ~/~)
i
.
l/1
+
f#
e--e~e
(6)
where C is a constant. The strains e~, e~ and ~ above are plastic strains, but at the strain levels considered, it is justified to neglect the elastic strain. Annihilation of dislocations will not change the contribution PGz in eqn. (5) compared with that in single phase ~, but PG 2 would decrease. However, it is believed that this is of minor importance. The strain in the pearlite regions can be evaluated from their microhardness, using the
(3)
where Ce is the constant of proportionality between shear and tensile strains. (The quantity X~ 2 is introduced because it is n o t certain that the slip length of the dislocations generated by the two mechanisms is the same.) The resulting length of dislocations per unit volume of s-phase is N 1 f~ PG2 - l--f% - C2" bX,~2 " 1--f----~" (~- --e~). (4)
hard
pearltte
1.0
e z
The total density of geometrically necessary dislocations in the s-phase is given by
Vl ferrite/hard
pearilte
Po
0.5
= PG1 + ,0G2
=1__ C~ b
~
+C2~
1--
ferrite/soft pearhte
(5)
The quantities C1/X~I and C2/X,~2 could in principle be determined by comparative studies of single phase and two-phase materials with low-volume fraction of inclusions. For the purpose of this discussion, we will assume statistically homogeneous distribution of the dislocations in the available space and equal propagation modes for the dislocations involved in eqns. (2) and (4). In our structures
i
I experir~enls
I
equo~ strain
I
modet
i
0o
Eqs. ? o~1 8
0.O5
011
ors
strain
Fig. 13. Stress - strain behaviour of ferrite, pearlite and the aggregate in the low-carbon steel. E x p e r i m e n tal values and m o d e l results.
217 h a r d n e s s - s t r a i n - r e l a t i o n s h i p o f t h e corres p o n d i n g fully pearlitic m a t e r i a l s (Fig. 2) as a c a l i b r a t i o n curve. ( T h e F E M c a l c u l a t i o n s indicate t h a t the strain levels in t h e inclusions are generally highest n e a r t h e i n t e r f a c e . As t h e h a r d n e s s i n d e n t a t i o n s c o u l d o n l y be m a d e a w a y f r o m the i n t e r f a c e s , the strains t h u s det e r m i n e d are p r o b a b l y s o m e w h a t low.) F o r instance, a t a m e a n strain of 0.1, t h e strain in t h e s o f t a n d h a r d pearlite is a p p r o x i m a t e l y 0.01 a n d 0 . 0 6 r e s p e c t i v e l y (Figs. 1 a n d 2). T h e m a t e r i a l w i t h h a r d pearlite is t h u s v e r y n e a r the case w i t h rigid inclusions, while t h e m a t e r i a l w i t h s o f t pearlite u n d e r g o e s a plastic d e f o r m a t i o n of a p p r o x i m a t e l y h a l f t h e m a c r o scopic strain. This is in r e a s o n a b l e a c c o r d a n c e w i t h the results f r o m t h e g e o m e t r i c a l m e a s u r e m e n t s , Fig. 4. I n s e r t i n g t h e inclusion strains of 0.01 a n d 0.06 estimated from the microhardness measu r e m e n t s in eqn. (6), w e o b t a i n a f l o w stress i n c r e m e n t for the s - p h a s e ( a s - - Oo ) o f 6% f o r t h e m a t e r i a l w i t h s o f t pearlite a n d 14% f o r t h e h a r d pearlite. C o m p a r i s o n w i t h experim e n t is possible using t h e m i c r o h a r d n e s s m e a s u r e m e n t s on t h e ferrite p h a s e in t h e aggregate m a t e r i a l , Fig. 1. T h e p r o c e d u r e is as follows: F r o m t h e e x p e r i m e n t a l f l o w c u r v e f o r single p h a s e f e r r i t e * , Fig. 13, we find o~(e = 0.1) = 0.35 G N / m 2. T h e value o f ao f o r ferrite is 0 . 0 4 G N / m 2 [ 2 0 ] . T h e r e f o r e (o~ - Oo)c=0.1 = 0.31 G N / m 2. A t this level, t h e 6 or 14% d i f f e r e n c e in ( a s - - ao ) b e t w e e n m a t e rial w i t h s o f t a n d h a r d pearlite t r a n s l a t e s to a d i f f e r e n c e in f l o w stress of 23 M N / m 2. Acc o r d i n g to t h e r e l a t i o n o ~ H V / 3 [ 2 1 ] , this s h o u l d c o r r e s p o n d t o a d i f f e r e n c e in m i c r o h a r d n e s s o f 8 units. Figure 1 s h o w s t h a t at e = 0.1, t h e ferrite m i c r o h a r d n e s s of m a t e r i a l w i t h h a r d p e a r l i t e is ca. 10 units higher t h a n t h a t o f m a t e r i a l w i t h s o f t pearlite. T h e a g r e e m e n t is g o o d e n o u g h to a l l o w t h e c o n c l u s i o n t h a t the i n f l u e n c e o f h a r d a n d s o f t pearlite inclu-
* The grain size and alloy content of the pure ferrite material deviate somewhat from those of the ferrite in the low carbon steel, Table 1. As the work hardening is fairly insensitive to grain size [17 ] and substitutional alloying [18,19], the measured stress -strain curve for the pure ferrite was uniformly raised by 30 MN/m 2 to correctly correspond to the alloyed ferrite in the low alloy steel. For both the ferrite and low carbon steels, the flow curves in Fig. 13 have been extrapolated back to zero plastic strain assuming a linear log a - log e relation.
sions on the w o r k h a r d e n i n g of t h e m a t r i x is s a t i s f a c t o r i l y d e s c r i b e d in t e r m s o f g e o m e t r i cally n e c e s s a r y dislocations. E q u a t i o n (6) allows us to e s t i m a t e the t o t a l f l o w stress i n c r e m e n t of the ferrite p h a s e caused b y t h e pearlite inclusions. T h e result is 43 M N / m 2 f o r the h a r d pearlite, a n d 20 M N / m 2 f o r the s o f t pearlite, b o t h at a m a c r o scopic strain o f 0.1. Figure 13 shows t h a t at a strain of 0.1, t h e f l o w stress o f t h e aggregate o f ferrite w i t h h a r d pearlite is 150 M N / m 2 higher t h a t t h a t of single p h a s e ferrite. This is f o u r t i m e s as m u c h as the f l o w stress increm e n t in the m a t r i x caused b y g e o m e t r i c a l l y n e c e s s a r y dislocations. A similar result is a c h i e v e d f o r the case o f s o f t pearlite. T h e r e f o r e , a m a j o r p a r t of t h e w o r k h a r d e n i n g o f the aggregate m u s t be a t t r i b u t e d to o t h e r effects. F l o w stress o f the ferrite - pearlite aggregate Figure 13 s h o w s t h a t t h e yield stress o f t h e aggregate is quite insensitive to the h a r d n e s s o f the inclusions. It is, in f a c t , equal to t h a t o f single-phase ferrite. T h e p r e s e n c e o f inclusions leads to m o r e r a p i d w o r k h a r d e n i n g , w h i c h m u s t be a t t r i b u t e d to t h e higher stresses supported by them. I f the strain d i s t r i b u t i o n in t h e aggregate w e r e k n o w n , t h e n t h e stress - strain b e h a v i o u r o f the individual c o n s t i t u e n t s c o u l d be used to e s t i m a t e t h e f l o w stress o f t h e aggregate. A s c h e m a t i c i n t e r p r e t a t i o n o f the stress - strain b e h a v i o u r o f t h e ferrite - pearlite s t r u c t u r e s can be based on t h e a s s u m p t i o n o f equal strain in b o t h c o n s t i t u e n t s n e g l e c t i n g interact i o n effects. A resulting stress - strain c u r v e for t h e aggregate is t h e n easily c o n s t r u c t e d f r o m t h e stress - strain d a t a f o r each c o n s t i t u e n t a n d t h e v o l u m e f r a c t i o n s . S u c h curves f o r t h e " h a r d " a n d " s o f t " m a t e r i a l are i n c l u d e d in Fig. 13. T h e y r e s e m b l e the e x p e r i m e n t a l o n e s in s h a p e b u t lie at a higher stress level. This is to be e x p e c t e d , since the pearlite in reality u n d e r g o e s less plastic strain t h a n t h e m a t r i x (Figs. 1 a n d 2) so t h a t its stress will be l o w e r t h a n a s s u m e d in the equal-strain m o d e l . On the o t h e r h a n d , b e c a u s e o f t h e a b s e n c e o f m e c h a n i c a l i n t e r a c t i o n b e t w e e n the c o n s t i t u ents, the m o d e l d o e s n o t t a k e i n t o a c c o u n t t h e h y d r o s t a t i c stress c o m p o n e n t s in t h e m a trix, w h i c h will raise t h e m a t r i x stress s o m e what. A b e t t e r a p p r o x i m a t i o n of t h e stress -
218
strain behaviour of the aggregate would have to include the partl~fon of both stresses and strains between the constituents according to some sort of mixture rule. In an earlier paper [9] a mixture rule of the following type was used: a = (1 -
f,).
= (1 -
+ Cc, +
(7) %
•
(8)
This gives a less extreme partitioning of stresses than the equal-strain model. The strains used in both of these models are total strains. To test the validity of eqns. (7) and (8), we turn to the FEM results (Figs. 11 and 14) as a model case in which the stresses and strains for both constituents are known. The stresses given in Fig. 11 are found to add up to the overall stress when weighted together by eqn. (7) within 15%. In the same way, eqn. (8) is satisfied by the strains in Fig. 14 to within 25% of the total strain. The discrepancies are largest for ~ = 10, while the figures are about half as large for ~ = 2. We note in passing that the strains in Fig. 14 are effective plastic strains, whereas eqn. (8) envisages total unidirectional strains. In the estimation, the total unidirectional strain has been considered equal to the sum of effective plastic strain and corresponding elastic strain. In the present case, the strain of the pearlite constituent can be obtained from the microhardness measurements (Fig. 1) with the aid of the hardness - strain relationship of pure pearlite (Fig. 2). (This procedure is applicable only to the regime where both constitu0.08
I
I,t v ¢--
0.06 -
o
i
2
o
S
•
10-
o
,P/"
matrix
t.. u') o
&O&
U3
~-
0.02
, ° ,,,,,,4
0
0 01
o.o2
0.03
0.o4
o.os
strain Fig. 14. Calculated strain d i s t r i b u t i o n in t h e s t r u c t u r e in Fig. 10.
ents have in fact plastified; the region of very low overall strains, where the inclusions remain elastic, cannot be treated in this way.) The matrix strain can be deduced from the inclusion strain and the overall strain by eqn. (8). Combining the flow stresses for the constituents (Fig. 13) at their respective strains in the manner of eqn. (8), the flow curve for the aggregate material is obtained (Fig. 13). It is in good agreement with experiment. The high initial stress level estimated for the ferrite/soft pearlite aggregate is attributed to an overestimation of the pearlite strain because of uncertainty in the hardness measurements (Fig. 1). As an alternative to the use of eqn. (8), the matrix strain could be determined from the microhardness of the s-phase. However, the low work hardening of ferrite in combination with the strong inhomogeneity of matrix strain [8] would make this determination rather uncertain. In contrast with ferrite, the hardness of pearlite is a sensitive indicator of strain (Fig. 13). To summarize, the mixture law stated in eqns. (7) and (8) allows one to predict with reasonable accuracy the stress- strain behaviour of the aggregate.
CONCLUSIONS
The yield stress of ferrite - pearlite aggregates with continuous ferrite matrix is about the same as that extrapolated from singlephase ferrite of equal grain size and independent of the hardness of the pearlite inclusions. The flow stress is considerably higher than t h a t of single-phase ferrite and rises with increasing inclusion hardness. During plastic deformation of the aggregate the pearlite inclusions also deform, at first elastically and then plastically. Softer inclusions deform more than do harder ones. At higher overall strains, the cementite within the pearlite deforms plastically. The flow stress of the matrix can be satisfactorily described in terms of geometrically necessary dislocations. The flow stress of the aggregate can be described in terms of load transfer from the matrix to the inclusions, using a mixture law which divides both stress and strain among the constituents in the ratio of their volume fractions.
219 ACKNOWLEDGEMENTS T h e a u t h o r s wish t o t h a n k Dr. B.O. S u n d str~im f o r c a r r y i n g o u t t h e F E M c a l c u l a t i o n s . H e l p f u l d i s c u s s i o n s w i t h Prof. H. F i s c h m e i s t e r are g r a t e f u l l y a c k n o w l e d g e d . Dr. B. J a e n s s o n has a s s i s t e d in t h e X - r a y w o r k a n d Prof. K . E . E a s t e r l i n g has t a k e n p a r t in t h e H V E M s t u d i e s . T h e m a t e r i a l was s u p p l i e d b y S a n d v i k AB. The project has been financially s u p p o r t e d by the Swedish Board for Technical Development.
REFERENCES 1 I.L. Mogford, Met. Rev., No. 114, 12 (1967). 2 F.B. Pickering and T. Gladman, Iron Steel Inst., Spec. Rept. 81, 1963, p. 10. 3 T. Gladman, I.D. McIvor and F.B. Pickering, J. Iron Steel Inst. (London), 210 (1972) 916. 4 I. Tamura et al., Trans. Iron Steel Inst. Japan, 13 (1973) 283. 5 K.W. Burns and F.B. Pickering, J. Iron Steel Inst. (London), 185 (1964) 899. 6 H.J. Kouwenhoven, Am. Soc. Metals, Trans. Quart., 62 (1969) 437.
7 H. Fischmeister, J.-O. Hj~ilmered, B. Karlsson, G. Linddn and B. SundstrSm, Proc. 3rd Int. Conf. on Strength of Metals and Alloys, 1973, p. 621. 8 B. Karlsson and B.O. Sundstr~m, Mater. Sci. Eng., 16 (1974} 161. 9 B. Karlsson and G. Linden, Mater Sci. Eng., 17 (1975) 153. 10 K.W. Andrews, Acta Met., 11 (1963) 939. 11 W. Koch, Metallkundliche Analyse, Verlag Stahleisen, DUsseldorf, 1965, p. 426. 12 B.O. Sundstr(im, Mater. Sci. Eng., 12 (1973) 265. 13 G.R. Speich and W.C. Leslie, Met. Trans., 4 (1973) 1873. 14 D.C. Drucker, J. Mater., 1 (1966) 873. 15 M.F. Ashby, in A. Kelly and R.B. Nicholson (eds.), Strengthening Methods in Crystals, Elsevier, London, 1971, p. 137. 16 J.-P. Bailon, A. Loyer and J.-M. Dorlot, Mater. Sci. Eng., 8 (1971) 288. 17 W. Roberts and Y. BergstriSm, Z. Metallk., 62 (1971) 752. 18 W.C. Leslie, Met. Trans., 3 (1972) 5. 19 T. Gladman, B. Holmes and F.B. Pickering, J. Iron Steel Inst. (London), 208 (1970) 172. 20 C.T. Liu, R.W. Armstrong and J. Gurland, J. Iron Steel Inst. (London), 209 (1971) 142. 21 F.A. McClintock and A.S. Argon, Mechanical Behavior of Materials, Addision-Wesley, Reading, Mass., 1966, p. 453.