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Lüders deformation in polycrystalline iron
LijDERS
DEFORMATION
IN POLYCRYSTALLINE
IRON
HIROSHI FUJIT.4 and SHUICHI MIYAZAKI Department of Materials Science and Engineering. Facul_ty_of Engineering. Osaka UCersity. Yamada-Kami. Suita, Osaka 263. Japan
Abstract--The Liiders deformation in poiycrysta~line iron has been examined using specially shaped specimens under various conditions. The results obtained are summarized as follows: (a) The macroscopic deformation starts in a narrow region at the upper yield stress and rapidiy propagates across the cross-section of specimen. i\fter the band-structure is formed. a further deformation is necessary to cause the Liiders deformation. (b) When the ratio of specimen thickness to grain size is smaller than a critical value. both the upper and lower yield stresses markedly decrease with decreasing specimen thickness and the deformation occurs at random in a wide region. The critical value for the upper yield stress is higher than that for the lower one. (c) The upper yield stress becomes insensitive to the strain rate under a certain condition. while the lower yield stress always shows the strong strain rate dependence. (d) In the strain rate insensitive range, width of the band-structure is almost constant. Trace of the band-structure is retained in the deformed region of the Liiders deformation when the strain rate is small. (e) The Row stress of Liiders deformation reaches a constant value after the stress relaxation nevertheless the foregoing strain rate. Based on the results, the deformation process at each stage of deformation is discussed in terms of the mutual interaction among individual grains.
R&urn&On a 6tudi6 la deformation de Liiders d’~chanti~lons de fer poi>cristallin de forme sp&iale. dans drs conditions divcrses. Voici un r&urn6 drs rPsultats obtenus: (a) La deformation macroscopique dibute dans une rigion Ptroite B la limite kiastique superirure et se propage rapidement en travers de I’&hantillon. Lorsque la structure de bandes est form&, ii faut unc dtformation supplCmentaire pour provoquer la deformation de Liiders. (b) Lorsque le rapport de I’ipaisseur de l’&hantillon 6 la taille des grains est infirieur h une valeur critique. les limitcs Clastiques infirieure et superieure diminuent notablement lorsqu’on diminue I’Ppaisseur des ichantillons, et la deformation se produit alkatoirement dans une grandr region. La valeur critique pour la limite ilastique superieure est plus grande que pour la limite elastique inferieure. (c) La limite elastique suptrieure devient, dans certaines conditions. independante de la vitesse de dkformation. alors que la limite elastique infkrieure en dCpend toujours fortement. (d) Dans le domaine indkpendant de la vitesse de d&formation. la largeur de la structure de bandes est pratiquement constante. II restr des traces de la structure de bandes. apr& la deformation de Liiders. lorsque la vitesse de dPformation est faible. (e) La contrainte d’&coulement de la d&formation de Liiders atteint une valeur constante aprts la relaxation de la contrainte. quelle que soit la vitesse de dkformation antirieure. En s’appuyant sur ces r&&tats. on discuts, du mPcanisme de ia d~formatlon. a chacun de ses stadcs g partir de l’interaction entre grains. Zusammenfassung-Mittels besonders geformter Proben wurde die Verformung polykristailinen Eisens unter verschiedenen Bedingungen untersucht. Die erhaltenen Ergebnissc lassen sich folgendermaBen zusammenfassen: (a) Makroskopische Verformung setzt in einem eng begrenzten Gebiet bei der oberen FlieDspannung ein und breitet sich rasch durch den Querschnitt der Probe aus. Nach der Ausbildung der Bandstruktur ist weitere Verformung notwendig. urn die Liiders-Vcrformung einzuleiten. (b) Unterschreitet das Verhlltnis zwischen Probendicke und Korngr6De einen kritischen Wert. dann sinken obere und untere FlieRspannung mit sinkender Probendicke ab: die Verformung lluft zuftillig und in einem weiten Bereich ab. Das kritische Verhsltnis ist ftir die obere FlieBspannung hiiher als ftir die untere. (c) Unter einer bestimmten Bedingung hlngt die obere FlieDspannung nur noch wenig von der Dehngeschwindigkeit ab. wohingegen die untere FlieDspannung immer die starke Dehngeschwindigkeitsabh&gigkeit aufweist. (dt Im Bereich der Dehngeschwindigkeitsunempfindlichkeit ist die Weite dcr Bandstruktur nahezu konstant. Die Spur der Bandstruktur wird im Liiders-verformten Gebiet beibehalten. wenn die Dehngeschwindigkeit klein ist. (e) Unabh~ngig von der vorhergehenden Dehngeschwindigkeit erreicht die FlieBspannung der L~dersverformun~ nach Spannun~rel~ation einen konstanten Wert. Aufbauend auf den Ergebnissen werden die Verformungsprozesse eines jeden Verformungsbereiches anhand der Wechselwirkung zwischen einzelnen Karnern diskutiert.
1. INTRODUCTION Mechanical properties of metals, such as the yield phenomenon, the flow stress, the fracture mechanism and so on, are closely related to the heterogeneous deformation which occurs in various scaies. There-
fore, in order to discuss details of the mechanical properties, it is necessary to kpqw why the deformation is localized. Many theories [l-4] have heen proposed on the deformation of polycrystalline metals. Some [I, 21 of them, however, based on an assump tion that the polycrystal is homogeneously deformed, 1273
127-t
FUJIT.4
ASD
MIYUAKI:
LCDERS
DEFUR~~ATI~~
and others [3.-t] assumed that the interaction occurs only between adjacent two grains. In the former case, it is diRicuIt to discuss the localized mode of deformation. The slip mode. which is closely related to the mechanical properties. is too much simplified in the latter. as will be mentioned later. These theories, however. suggest that the interaction among indivi&al grains must be taken into account to discuss the flaw stress of polycrystals. ?vlany experiments have been carried out to investigate the interaction using bycrystals witk interesting results [S. 61. Each grain of a bicrystal. however, is constrained by only one n~igkbor, so that the slip .mode in each grain becomes simple compared with those in polycrystals with many neighbors. Since the flow stress is closely related to the slip mode in individual grains. it is difficult to estimate the flow stress of polycrystals from those of bicrystals. In fact. the flow stress of polycrystals markedly decreases with decreasing specimen thickness when the number of grains involving along the thickness direction becomes smailer than the critical value[7-t I]. The critical value is larger than 5 and increases with decreasing grain size [7]. This suggests that the interaction among individual grains occurs in a long-range passing across the first nearest neighbors. On the other hand, in a previous paper [It]. it was shown that the serrated yielding in polycrystatline AI-ICZg alloys is quite different from that in single crystals. At the later stage of deformation in these polycrystals, one step yield is caused by rapid formation of a band-structure which hardly widens further. Another band-structure, i.e. the LUders-band, is also formed at the beginning of deformation in polycrystals of these alloys and followed by successive expansion of the deformed region. These facts were dis-
1N P~LYCRYSTALLI~E
IRON
Table I. Chemical composition of the specimens Elements wt. 0;
C
Si
Mn
P
S
Al
0.009 0.006 0.001 0.003 0.002 0.001
cussed in terms of the long-range interaction among individual grains of polycrystafs [12]. As mentioned above, formation of the band-structure in the potycrystals is closely related to the Iongrange interaction among grains. This is a reason why, the Liiders deformation has been investigated in the present experiment, especialty on initiation of the Liiders-band and transition from the band-structure to the Liiders deformation. Many experiments and discussions have been made on the Lilders deformation [13-151. Detailed investigation of initiation of the Ltiders-band, however, has not been made SO many, because it is very sensitive to the specimen and deformation conditions. Petck [16] applied a new method to accurate measurement of the upper yield stress and obtained many valuable results. In his experiment. however, the strain rate dependence of the flow stress was obtained using polycrystaIs with considerably large grain sizes. In such polycrystals, the Liiders-band was not formed, as he mentioned. This means that the Liiders deformation is sensitive to the gram size as well as the specimen thickness. Furthermore, his method is difficult to apply to any polycrystals obtained by various treatments. From this point of view, in the present experiment, the specimens were specially shaped so that the deformation always started in the center region of the specimen at a constant stress Ievel. Specimens used were ~lycrystalline iron, and the Liiders deformation have been examined as a function of the grain size, specimen thickness, strain rate and deformation temperature. The results have been discussed in terms of the fang-range interaction among individual grains. 2. SPECIMENS AND EXPERLMENTAL PROCEDURES
d ?
,”
Specimens used were potycrystalline iron plates with a chemical composition shown in Tabte 1. These specimens were heavify cross-roiled at room temperature and followed by annealing in order to avoid the rolling texture. Specimen thickness was mainly I mm and the grain size was controIled in the range 17.6-313 jrm. For accurate measurement of the upper yield stress (ae). the deformation was initiated in the center part of the specimen using two types of the specimens shown in Figs. I(a) and (b). respectively. Here, a parameter (K) expressed by K a 2(A,J&i”)
Fig. 1, Two tgpes of specimens revealing a constanf upper yield stress.
(1)
was used, where r is the stress concentration factor, and A,,, and z&in are the maximum and the minimum cross-section of the specimen. A suitable value
FUJ1T.A
10)
0.045
(bf
n-n-n thick
I.200
mm thick
oOf0 Goo0
> Z v) 200: 5
/ &J, (_/J
0
/
l
/.-0
e-
.
0 : Upper yield stress l : Lower yield stress
100-i
Grain size : 25 pm 0
o-2 (8)
0.4 (16)
(0;:)
0.8 (32)
I.0 (40)
I.2 (48)
t (mm) (t/d)
Fig. 2. Thickness eff:ct on the upper and lower yield stresses in the specimens of 3pn-1 in pram size. of K was determined by changing the radius of curvature at the center part of the specimen using the specimens of type (a). In the specimens of type la). the value of eL. corrected by K becomes always constant when the value of K in the center part of the specimen is larger than about 1.6. and the deformation always occurs in the center part of the specimen. The value of Go, however, fluctuates when the value of K is smaller than about 1.5. and the deformation preferentially occurs near the specimen grip. Therefore. the value of K at the shoulders of parallel part of the specimens of type (b), which were used for investigating Liiders deformation has been fixed at 2.17 in the following experiments. The specimens were stretched with a Shimadzu Autograph IS-5000 tensile machine, whose hardness is 18.48 >lN,m under the operating condition, by crosshead changing velocity from 0.005 to 500 mm,‘min. 3. EXPERIMESTAL 3.1 Deformation fo grain size
RESULTS
and the ratio of sprcirnen
thickness
The flow stress of polycrystals is very sensitive not only to the grain size (6) but to the specimen thickness (f). Figure 2 shows the effect of t.d on the values of ur and Go by changing specimen thickness in the specimens of 25 pm in grain size. It is noted in Fig. 2 that both values oi cc and CJ~ abruptly decreases with decreasing specimen thickness when the value of t,‘d is smaller than about 20 and 10, respectively. Micrographs in Fig. 2 show the deformed regions in the specimens whose values of t d are 48 and 1.8, respectively. It is noted in Fig. Z that the deformation occurs in a narrow band-region u-hen the value of
rid is sufficiently large. and it is follow-ed by the Liiders deformation. In this case, the ratio of ur to oL is quite large. When the value of r,d is small. however, the deformation occurs at random in a wide region so that no band-structure is formed. Figure 3 is another example showing the effect of I d on the deformation by changing grain size in the specimens of 1 mm thick. The corresponding micrographs were taken just after the upper yield point as indicated with arrows. Here. the critical value of r/d, above which the value of gG becomes almost constant, decreases with increasing grain size, i.e. it is about -20 for 30pm in grain size and about 5 for 250pm. With the exception of the critical value, the effect of t/d on the deformation is quite similar to the case of the specimens with a fixed grain size, Fig. 2. 3.2 Deformation and the lower yield stress Figure 4 is an example showing the effect of prestrain on the yield phenomenon. At fust. doublenotched specimens were stretched to various stress levels, as shown in micrographs (a), (b) and (c). These specimens were gotten into the shape of the specimens of type (a) as shown with (a’), (b’) and (c’). and then they were further stretched. It is noted in Fig. 4 that the yield drop appears even after the deformed region just expands across the cross-section of specimen, as seen on a stress-strain curve of reshaped specimen (b’). When the deformed region expands to a certain width, the yield drop does not appear, as seen on a stress-strain curve of specimen (c’). The results shows that the flow stress does not reach the lower vieid stress even after the macroscopic deformation just expands across the cross-section of Specimen.
j/
Ox
Cfosshea~.~~~~in
Crosshead 5ispi~~ement Fig. 3. Grain size effwt on stress-strain
curves and the corresponding deformed regions saused by
the yteld drop. 3.3 The effect of srrain rate on thr deformation
Both values of oti and CT,_ are sensitive to the strain rate in general as well as other factors such as the grain size, shape of the specimen. the deformation temperature and the hardness of machine. Under a
0
0
certain condition. however, there is a range in which the value of o”~ is insensitive to the strain rate. .&n example in the specimens of 3Opm in grain size is shown in Fig. 5. In 300 K deformation, the value of CT~,does not change when the crosshead velocity is smaller than about 5mmrmin, while the value of 6t
0
Crosshead Displacement Fig. 4. Effect or prestrain on the yield drop.
0: 0005
005
05
5
50
01 500
Crossnead
0005
Velocity
005
05
5
50
. 500
i mm/m:n)
Fig 6. Initiation of the band-structure in grain
across the cross-section of specimen. The specimens of 30,~m velocities of (a) 0.5. (b) 5 and (c) 50 mm ‘min. respectively. Each micrograph was picked up from a high speed tine film.
size were deformed
with crosshead
increases with increasing crosshead velocity in the same range. The same phenomenon also occurs in 196 K deformation. though the critical crosshead velocity decreases to about 0.5 mm min. The strain rate insensitive range is closely related to both va!ues of pi and r. It is noted in Fig. 5 that the strain rate insensitive range oi Go is hardly observed in the specimen whose grain size is larger than about 100~1m. In such specimens the deformation at cc occurs in a wide repion. as seen in Fig.
3. The strain rate dependence of the flow stress also decreases with decreasing the value of t,d. and the deformation at (rl. occurs in a wide region when the talus of t d is small. as seen in Fig. 3. The results show that the strain rate insensitive range is closely related to the width of deformed region at cc. Furthermore, the ratio of Go to bt in Fig. 5 decreases with increasing grain size. and both values of oC and gL. show almost the same strain rate dependence in ihe *(i@Y single crystnIs+ with many active slip systems. In the strain rate insensitirc range. the effect of strain rate on the deformation process has been exam-
Fig. 7. Stress-strain curves and the corresponding micrographs of the specimens deformed wirh various crosshead velocities in the strain rate insensnrve range. The specimens were deformed at 3C_x+l K.
curve. The same band-structure is form& in all sp&imens. Width and the amount of strain of these bandstructures are not sensitive to the strain rate in this range Hhich is lower than 5 mm,‘min in crosshead
ined at room temperature using the specimens of 30pm in grain size, as shown in Figs. 6. 7 and 8.
Each micrograph in Fig. 7 was taken at a strain indicated with an arrow on.each individual stress-strain
250
I
L
5
Crossrtsad
vsiociry
(of 5-O mm/min
inmm/min
f
t
I
20
40
60
Time
(see>
Fig. 8. Propagation of the Liiders-band front during the stress relaxation and the corresponding stresstime curves in the specimens shown in Fig. 7.
FUJ!T.I
.~\n
Vi\-.‘.Z.‘X::
Li:DERS
DEFOR\I.\TION
velocity. as seen in Fig. 6. It is noted in these micrographs that trace of the band-structure formed at low strain rates is retained even after the Ltiders deformation proceeds, as seen in Figs. 7tc) and td). Strain rate dependence of (To is considered to be due to the strain rate dependence of the deformation at the Liiders-band front. In order to clarify this phenomenon. the relaxation test has heen carried out in the midst of the Liiders deformation. as seen in Fig. S. In micrographs in Fig. 8. the crosshead is abruptly stopped when the Liiders deformation propagates to a position indicated with a lower arrow. Then the deformed region further expands during the relaxation to a position indicated with an upper arrow along the forward direction until the stress level reaches the final one which corresponds to an extremely small strain rate. Amount of deformation during the relaxation increases with increasing strain rate. but the final stress level is exactly the same in all cases. Here it is noted that the Liiders-band front sharply appears even when the stress reaches the final one, and that the ratio of Go to the final stress is about 2.5.
1% P@LYCRfST-\LLI~E
IRO?;
1279
is increased by the deformation of the trigger grain. Radius of the affected zone: is determined by the value of Q, which is a function of the grain size. the slip mode of individual grains, the anchoring force of dislocations and the deformation temperature. Here the slip mode is closely related to the stacking fault energy. In room temperature deformation of polycrystalline iron, the atfected zone expands to the fourth or fiith nearest neighboring grains when the grain size is ?O,~rn or less. When the gain size increases to about 1OOltm or more, the large internal stress accumulates on the grain boundary of the trigger grain. but it can almost be relieved by the distortion of the first nearest neighbors. Therefore. the afIected zone in such cases expands to only the first nearest neighboring grains, and finally the deformation of the trigger grain propagates into only one or
1. DISCI_SSIOS 4.1 Formsion
of the hand-structure
The values of oL. and CJ‘ markedly decrease with decreasing specimen thickness when the value of t/d is smaller than the critical one, as seen in Fig. 2. This fact is closely related to the long-range interaction among individual grains [7]. Each trigger grain in thick polycrystals is constrained three-dimensionally with its surrounding grains, and thus the macroscopic deformation hardly occurs until the applied stress (li,) reaches a certain value (a,) [7] at which all surrounding grains are deformed at the same time nevertheless their orientation factors [5] against the trigger grain. Here, a region involving these surrounding grains is named ‘the atfected zone’ in which the local stress 1 Details will be published
elsewhere.
Fig. 10. Formation process of the Liiders-band
front.
twe grains of the first nearest neighbors with large orientation factors when the grain size is sufficiently large. Now, the a&ted zone is simply assumed to cover the first nearest neighbors as shown in Fig. 9. Since the shape of affected zone also depends on the shear component of applied stress. the affected zone is elongated along a direction of the maximum resolved shear stress which is determined by the constraining force of the trigger grain. as shown with an elongated circle in Fig. 9 in which the trigger grain is denoted by &. When the applied stress reaches the vaiue of c;,, aii surrounding grains in the &ected zone are deformed at the same time, and then many affected zones are newly formed around individual deformed grains AZ, B2. Cr and so on. These affected zones overlap each other in some grains. When the affected zones overlap. the local stress increases depending on the number of overlapping. because the maximum shear displacement of individual grains occurs along the same direction in the affected zone. Now, the deformation is assumed hereafter to simply propagate two-dimensionally. Then, in Fig. 9, four affected zones overlap in grains AS. BJ. C3. D: and E,, three in grains DL and E2, and so on. When the local stress [a*) in the overlapped aKected zones increases :L’times as much as that in a single affected zone (ai), G* can be expressed by g* = d It’.4. ,Yfl. !I
(2)
The applied stress rather increases at the tip of deformed region during the propagation of deformation. Thus. once the avalanche of deformation occurs at cr, in a single affected zone of grain do. the deformed region rapidly propagates under stress G* along the direction of the maximum resolved shear stress (1). On the other hand, the applied stress is relieved considerably, depending on the hardness of machine in regions where the avalanche of deformation occurs. Thus the sidewise propagation of deformation is decelerated markedly so that the width of deformed zone at the beginning of deformation will be a few times as large as that of a single affected zone. This is the formation process of the bandstructure. The avalanche of deformation generally starts at the specimen surface where the constraining force is small. and the volume fraction of trigger grains is extremely small until the stress reaches F&.ahen the grain size is small. Therefore, the value of cc becomes nearly equal to o, and insensitive to the strain rate when the strain rate is lower than a certain value. as shown in Fig. 5
the periphery of affected zone. a large amount of deimmion ot’curj only in grains whose value of .V is sufficiently large, ai expect& from equation (2). As a result of this fact, the periphery of band-structure becomes zig-zag in a scale of grain size. as shown in Fig. 10(a). In Fig. t0, the deformed grains are shown by hatching and each affected zone caused by the deformed grain is shown with a circle. Here, each affected zone is assumed to simply expand isotropic&y to the first nearest neighboring grains. and the number of overlapping of the affected zones is denoted by indites !. 2. 3 and 4. Grains of index I are hardly deformed until the applied stress reackes the value bf ti,. Grains of indices 2. 3 and i. however, are deformed at small applied stresses depending on the value of 5’ in the affected zones. When the deformation propagates into the neighboring grains whose index is larger than 1 at this stage, the propagation of deformation occurs only along a direction indicated with an arrow in Fig. iqb). Finally the deformed grains are arranged on a plane along the maximum resolved shear stress. as shown in Fig. 10(c). The criticai value of c if for Go will be reduced by this fact, as shown in Fig. 2, because half of the affected zone is already relieved by the deformation. This is the formation process of the Liiders-band front. After that. once a grain is deformed in the first nearest neighbors of these deformed grains, the deformation quickly propagates toward the direction indicated with arrows along arrays of grains of index ? which ,ire parallel to the Liiders-band front. According to this process, the specimen is bent around an axis perpendicular to the shear direction in the tramition zone between the deformed and undeformed regions. as shown in Fig. 11. The tensile stress in-
1.1 Fortnarion of‘ the Liiders-hand frotzt After the band-structure is formed. some incuvation period is necessary to make the Li.iders deformation. as expected from Fig. 4. The stress level in both sides of the band-structure is markedly decreased when the avalanche of deformation pass through. Thus. near
Fig. i I. Additional Stress2s cuusrd by focltl bznding of the specimen accompanying the Liiders dciurmation. Thickness and grain six of the specimen are 3 mm and 5Oym. respecti~2l~.
Ft’JfT,\
creases creases
-4%~ MIk’lZrtf(I:
Lt;DERS
DEFORMATION
at the tension (23 side of bending and deat the compression tCI side. as same as in
the case of single crystals of Cu-Al alloy [lg]. The deformation always starts at the T-side and propagates to the C-side. This process is repeated layer by layer in a scale of grain size. Since the propagation Liiders-band front occurs
of
deformation
at
the
with a considerably high speed. the local stress at the Liiders-band front is determined by the deformation speed as weil as the stress expressed by equation 12). as expected from Fig. 8.
As mentioned above. the deformation of each individual grain propagates into the affected zone. The number of trigger grains at the beginning of deformation increases with the decrease in the value of cr,. Therefore. the avalanche of deformation in soft polycrystafs occurs in various places at random. Since the directions of shear displacement in these deformed zones are different from each other, each individual deformed zone does not expand into a wide region because of the interaction among different deformed zones. Namely, the deformation mode and distribution of the local stress in polycrystats are ctosely related to the number of the trigger grams at the beginning of deformation.
IN
POLYCRYSThLLINE
IRON
izst
.~c~nuwledgemenr-me authors wish to express their hearty thanks to &Ir. I(. Shibata for his help in the present experiment
REFEREWCES t. G. Sacks. 2. Fer. deer. fng. 72, 734 f 1928. 2. G. i. Taylor. 1. inst. Merais 62. 30-i (19383. 3. E. 0. &all. PFUC. php. .%c. Lond. 64, 747 ffQ51). 4. %. J. Pet&, J. fFOn .%d fnSf. 174, 35 ti%% 5. For example. J. D. Livingston and B. Chalmers. Acra ~F~~f~~~. 5, 322 (19371. 6. For example, H. Fujita, K. Toyoda and Y. Kanetsuki, Trm. Japan fnsr. Xetafs I6, t5t (197% 7. H. Fujita. &ii. Joprrn inst. Merats iin Japanese) I+ 537 (197% 8. W. T. Peil-Waipole. J. Insr. ?iierais 69, 131 (1943). 9. R. L. Fleischer and W, F. Hosford. Trans. TN.S.A.I..Cf.E. 221, 2J.l (1961). 10. R. W. Armstrong. J. Tech. Php. Solids 9, 196 (1961). 11. A. W. Thompson. Scripts metalt. 8, 145 (1974). 12. H. Fujita and T. Tabata, .+tcta metall. 25, 793 (t977). 13. A. Nadai. Theary of Flow and Fracrurr 0fS01id.s. Vol. i. McGraw-Hill. New York (1950). 14. E. 0. Hail, yield P&r Pke~o~e~~ in .tlerds utui Moys. Plenum Press. New York (1970). 15. E. Sudo, Srrerc~ler-Srrai~ (in Japanese). Japan Inst. of Metals (i9701. 16. N. .I. Fetch. Itcrcz metall. 12, 59 (1964). t?. T. Takeuchi, J, phf.s. Sot. Japan 26, 351 (1969). t8. T. Mori and H. Fuyta, Japan ffisr. &ferals (in Japanese) 39, 872 (1974).
DEFORMATION
IN POLYCRYSTALLINE
IRON
HIROSHI FUJIT.4 and SHUICHI MIYAZAKI Department of Materials Science and Engineering. Facul_ty_of Engineering. Osaka UCersity. Yamada-Kami. Suita, Osaka 263. Japan
Abstract--The Liiders deformation in poiycrysta~line iron has been examined using specially shaped specimens under various conditions. The results obtained are summarized as follows: (a) The macroscopic deformation starts in a narrow region at the upper yield stress and rapidiy propagates across the cross-section of specimen. i\fter the band-structure is formed. a further deformation is necessary to cause the Liiders deformation. (b) When the ratio of specimen thickness to grain size is smaller than a critical value. both the upper and lower yield stresses markedly decrease with decreasing specimen thickness and the deformation occurs at random in a wide region. The critical value for the upper yield stress is higher than that for the lower one. (c) The upper yield stress becomes insensitive to the strain rate under a certain condition. while the lower yield stress always shows the strong strain rate dependence. (d) In the strain rate insensitive range, width of the band-structure is almost constant. Trace of the band-structure is retained in the deformed region of the Liiders deformation when the strain rate is small. (e) The Row stress of Liiders deformation reaches a constant value after the stress relaxation nevertheless the foregoing strain rate. Based on the results, the deformation process at each stage of deformation is discussed in terms of the mutual interaction among individual grains.
R&urn&On a 6tudi6 la deformation de Liiders d’~chanti~lons de fer poi>cristallin de forme sp&iale. dans drs conditions divcrses. Voici un r&urn6 drs rPsultats obtenus: (a) La deformation macroscopique dibute dans une rigion Ptroite B la limite kiastique superirure et se propage rapidement en travers de I’&hantillon. Lorsque la structure de bandes est form&, ii faut unc dtformation supplCmentaire pour provoquer la deformation de Liiders. (b) Lorsque le rapport de I’ipaisseur de l’&hantillon 6 la taille des grains est infirieur h une valeur critique. les limitcs Clastiques infirieure et superieure diminuent notablement lorsqu’on diminue I’Ppaisseur des ichantillons, et la deformation se produit alkatoirement dans une grandr region. La valeur critique pour la limite ilastique superieure est plus grande que pour la limite elastique inferieure. (c) La limite elastique suptrieure devient, dans certaines conditions. independante de la vitesse de dkformation. alors que la limite elastique infkrieure en dCpend toujours fortement. (d) Dans le domaine indkpendant de la vitesse de d&formation. la largeur de la structure de bandes est pratiquement constante. II restr des traces de la structure de bandes. apr& la deformation de Liiders. lorsque la vitesse de dPformation est faible. (e) La contrainte d’&coulement de la d&formation de Liiders atteint une valeur constante aprts la relaxation de la contrainte. quelle que soit la vitesse de dkformation antirieure. En s’appuyant sur ces r&&tats. on discuts, du mPcanisme de ia d~formatlon. a chacun de ses stadcs g partir de l’interaction entre grains. Zusammenfassung-Mittels besonders geformter Proben wurde die Verformung polykristailinen Eisens unter verschiedenen Bedingungen untersucht. Die erhaltenen Ergebnissc lassen sich folgendermaBen zusammenfassen: (a) Makroskopische Verformung setzt in einem eng begrenzten Gebiet bei der oberen FlieDspannung ein und breitet sich rasch durch den Querschnitt der Probe aus. Nach der Ausbildung der Bandstruktur ist weitere Verformung notwendig. urn die Liiders-Vcrformung einzuleiten. (b) Unterschreitet das Verhlltnis zwischen Probendicke und Korngr6De einen kritischen Wert. dann sinken obere und untere FlieRspannung mit sinkender Probendicke ab: die Verformung lluft zuftillig und in einem weiten Bereich ab. Das kritische Verhsltnis ist ftir die obere FlieBspannung hiiher als ftir die untere. (c) Unter einer bestimmten Bedingung hlngt die obere FlieDspannung nur noch wenig von der Dehngeschwindigkeit ab. wohingegen die untere FlieDspannung immer die starke Dehngeschwindigkeitsabh&gigkeit aufweist. (dt Im Bereich der Dehngeschwindigkeitsunempfindlichkeit ist die Weite dcr Bandstruktur nahezu konstant. Die Spur der Bandstruktur wird im Liiders-verformten Gebiet beibehalten. wenn die Dehngeschwindigkeit klein ist. (e) Unabh~ngig von der vorhergehenden Dehngeschwindigkeit erreicht die FlieBspannung der L~dersverformun~ nach Spannun~rel~ation einen konstanten Wert. Aufbauend auf den Ergebnissen werden die Verformungsprozesse eines jeden Verformungsbereiches anhand der Wechselwirkung zwischen einzelnen Karnern diskutiert.
1. INTRODUCTION Mechanical properties of metals, such as the yield phenomenon, the flow stress, the fracture mechanism and so on, are closely related to the heterogeneous deformation which occurs in various scaies. There-
fore, in order to discuss details of the mechanical properties, it is necessary to kpqw why the deformation is localized. Many theories [l-4] have heen proposed on the deformation of polycrystalline metals. Some [I, 21 of them, however, based on an assump tion that the polycrystal is homogeneously deformed, 1273
127-t
FUJIT.4
ASD
MIYUAKI:
LCDERS
DEFUR~~ATI~~
and others [3.-t] assumed that the interaction occurs only between adjacent two grains. In the former case, it is diRicuIt to discuss the localized mode of deformation. The slip mode. which is closely related to the mechanical properties. is too much simplified in the latter. as will be mentioned later. These theories, however. suggest that the interaction among indivi&al grains must be taken into account to discuss the flaw stress of polycrystals. ?vlany experiments have been carried out to investigate the interaction using bycrystals witk interesting results [S. 61. Each grain of a bicrystal. however, is constrained by only one n~igkbor, so that the slip .mode in each grain becomes simple compared with those in polycrystals with many neighbors. Since the flow stress is closely related to the slip mode in individual grains. it is difficult to estimate the flow stress of polycrystals from those of bicrystals. In fact. the flow stress of polycrystals markedly decreases with decreasing specimen thickness when the number of grains involving along the thickness direction becomes smailer than the critical value[7-t I]. The critical value is larger than 5 and increases with decreasing grain size [7]. This suggests that the interaction among individual grains occurs in a long-range passing across the first nearest neighbors. On the other hand, in a previous paper [It]. it was shown that the serrated yielding in polycrystatline AI-ICZg alloys is quite different from that in single crystals. At the later stage of deformation in these polycrystals, one step yield is caused by rapid formation of a band-structure which hardly widens further. Another band-structure, i.e. the LUders-band, is also formed at the beginning of deformation in polycrystals of these alloys and followed by successive expansion of the deformed region. These facts were dis-
1N P~LYCRYSTALLI~E
IRON
Table I. Chemical composition of the specimens Elements wt. 0;
C
Si
Mn
P
S
Al
0.009 0.006 0.001 0.003 0.002 0.001
cussed in terms of the long-range interaction among individual grains of polycrystafs [12]. As mentioned above, formation of the band-structure in the potycrystals is closely related to the Iongrange interaction among grains. This is a reason why, the Liiders deformation has been investigated in the present experiment, especialty on initiation of the Liiders-band and transition from the band-structure to the Liiders deformation. Many experiments and discussions have been made on the Lilders deformation [13-151. Detailed investigation of initiation of the Ltiders-band, however, has not been made SO many, because it is very sensitive to the specimen and deformation conditions. Petck [16] applied a new method to accurate measurement of the upper yield stress and obtained many valuable results. In his experiment. however, the strain rate dependence of the flow stress was obtained using polycrystaIs with considerably large grain sizes. In such polycrystals, the Liiders-band was not formed, as he mentioned. This means that the Liiders deformation is sensitive to the gram size as well as the specimen thickness. Furthermore, his method is difficult to apply to any polycrystals obtained by various treatments. From this point of view, in the present experiment, the specimens were specially shaped so that the deformation always started in the center region of the specimen at a constant stress Ievel. Specimens used were ~lycrystalline iron, and the Liiders deformation have been examined as a function of the grain size, specimen thickness, strain rate and deformation temperature. The results have been discussed in terms of the fang-range interaction among individual grains. 2. SPECIMENS AND EXPERLMENTAL PROCEDURES
d ?
,”
Specimens used were potycrystalline iron plates with a chemical composition shown in Tabte 1. These specimens were heavify cross-roiled at room temperature and followed by annealing in order to avoid the rolling texture. Specimen thickness was mainly I mm and the grain size was controIled in the range 17.6-313 jrm. For accurate measurement of the upper yield stress (ae). the deformation was initiated in the center part of the specimen using two types of the specimens shown in Figs. I(a) and (b). respectively. Here, a parameter (K) expressed by K a 2(A,J&i”)
Fig. 1, Two tgpes of specimens revealing a constanf upper yield stress.
(1)
was used, where r is the stress concentration factor, and A,,, and z&in are the maximum and the minimum cross-section of the specimen. A suitable value
FUJ1T.A
10)
0.045
(bf
n-n-n thick
I.200
mm thick
oOf0 Goo0
> Z v) 200: 5
/ &J, (_/J
0
/
l
/.-0
e-
.
0 : Upper yield stress l : Lower yield stress
100-i
Grain size : 25 pm 0
o-2 (8)
0.4 (16)
(0;:)
0.8 (32)
I.0 (40)
I.2 (48)
t (mm) (t/d)
Fig. 2. Thickness eff:ct on the upper and lower yield stresses in the specimens of 3pn-1 in pram size. of K was determined by changing the radius of curvature at the center part of the specimen using the specimens of type (a). In the specimens of type la). the value of eL. corrected by K becomes always constant when the value of K in the center part of the specimen is larger than about 1.6. and the deformation always occurs in the center part of the specimen. The value of Go, however, fluctuates when the value of K is smaller than about 1.5. and the deformation preferentially occurs near the specimen grip. Therefore. the value of K at the shoulders of parallel part of the specimens of type (b), which were used for investigating Liiders deformation has been fixed at 2.17 in the following experiments. The specimens were stretched with a Shimadzu Autograph IS-5000 tensile machine, whose hardness is 18.48 >lN,m under the operating condition, by crosshead changing velocity from 0.005 to 500 mm,‘min. 3. EXPERIMESTAL 3.1 Deformation fo grain size
RESULTS
and the ratio of sprcirnen
thickness
The flow stress of polycrystals is very sensitive not only to the grain size (6) but to the specimen thickness (f). Figure 2 shows the effect of t.d on the values of ur and Go by changing specimen thickness in the specimens of 25 pm in grain size. It is noted in Fig. 2 that both values oi cc and CJ~ abruptly decreases with decreasing specimen thickness when the value of t,‘d is smaller than about 20 and 10, respectively. Micrographs in Fig. 2 show the deformed regions in the specimens whose values of t d are 48 and 1.8, respectively. It is noted in Fig. Z that the deformation occurs in a narrow band-region u-hen the value of
rid is sufficiently large. and it is follow-ed by the Liiders deformation. In this case, the ratio of ur to oL is quite large. When the value of r,d is small. however, the deformation occurs at random in a wide region so that no band-structure is formed. Figure 3 is another example showing the effect of I d on the deformation by changing grain size in the specimens of 1 mm thick. The corresponding micrographs were taken just after the upper yield point as indicated with arrows. Here. the critical value of r/d, above which the value of gG becomes almost constant, decreases with increasing grain size, i.e. it is about -20 for 30pm in grain size and about 5 for 250pm. With the exception of the critical value, the effect of t/d on the deformation is quite similar to the case of the specimens with a fixed grain size, Fig. 2. 3.2 Deformation and the lower yield stress Figure 4 is an example showing the effect of prestrain on the yield phenomenon. At fust. doublenotched specimens were stretched to various stress levels, as shown in micrographs (a), (b) and (c). These specimens were gotten into the shape of the specimens of type (a) as shown with (a’), (b’) and (c’). and then they were further stretched. It is noted in Fig. 4 that the yield drop appears even after the deformed region just expands across the cross-section of specimen, as seen on a stress-strain curve of reshaped specimen (b’). When the deformed region expands to a certain width, the yield drop does not appear, as seen on a stress-strain curve of specimen (c’). The results shows that the flow stress does not reach the lower vieid stress even after the macroscopic deformation just expands across the cross-section of Specimen.
j/
Ox
Cfosshea~.~~~~in
Crosshead 5ispi~~ement Fig. 3. Grain size effwt on stress-strain
curves and the corresponding deformed regions saused by
the yteld drop. 3.3 The effect of srrain rate on thr deformation
Both values of oti and CT,_ are sensitive to the strain rate in general as well as other factors such as the grain size, shape of the specimen. the deformation temperature and the hardness of machine. Under a
0
0
certain condition. however, there is a range in which the value of o”~ is insensitive to the strain rate. .&n example in the specimens of 3Opm in grain size is shown in Fig. 5. In 300 K deformation, the value of CT~,does not change when the crosshead velocity is smaller than about 5mmrmin, while the value of 6t
0
Crosshead Displacement Fig. 4. Effect or prestrain on the yield drop.
0: 0005
005
05
5
50
01 500
Crossnead
0005
Velocity
005
05
5
50
. 500
i mm/m:n)
Fig 6. Initiation of the band-structure in grain
across the cross-section of specimen. The specimens of 30,~m velocities of (a) 0.5. (b) 5 and (c) 50 mm ‘min. respectively. Each micrograph was picked up from a high speed tine film.
size were deformed
with crosshead
increases with increasing crosshead velocity in the same range. The same phenomenon also occurs in 196 K deformation. though the critical crosshead velocity decreases to about 0.5 mm min. The strain rate insensitive range is closely related to both va!ues of pi and r. It is noted in Fig. 5 that the strain rate insensitive range oi Go is hardly observed in the specimen whose grain size is larger than about 100~1m. In such specimens the deformation at cc occurs in a wide repion. as seen in Fig.
3. The strain rate dependence of the flow stress also decreases with decreasing the value of t,d. and the deformation at (rl. occurs in a wide region when the talus of t d is small. as seen in Fig. 3. The results show that the strain rate insensitive range is closely related to the width of deformed region at cc. Furthermore, the ratio of Go to bt in Fig. 5 decreases with increasing grain size. and both values of oC and gL. show almost the same strain rate dependence in ihe *(i@Y single crystnIs+ with many active slip systems. In the strain rate insensitirc range. the effect of strain rate on the deformation process has been exam-
Fig. 7. Stress-strain curves and the corresponding micrographs of the specimens deformed wirh various crosshead velocities in the strain rate insensnrve range. The specimens were deformed at 3C_x+l K.
curve. The same band-structure is form& in all sp&imens. Width and the amount of strain of these bandstructures are not sensitive to the strain rate in this range Hhich is lower than 5 mm,‘min in crosshead
ined at room temperature using the specimens of 30pm in grain size, as shown in Figs. 6. 7 and 8.
Each micrograph in Fig. 7 was taken at a strain indicated with an arrow on.each individual stress-strain
250
I
L
5
Crossrtsad
vsiociry
(of 5-O mm/min
inmm/min
f
t
I
20
40
60
Time
(see>
Fig. 8. Propagation of the Liiders-band front during the stress relaxation and the corresponding stresstime curves in the specimens shown in Fig. 7.
FUJ!T.I
.~\n
Vi\-.‘.Z.‘X::
Li:DERS
DEFOR\I.\TION
velocity. as seen in Fig. 6. It is noted in these micrographs that trace of the band-structure formed at low strain rates is retained even after the Ltiders deformation proceeds, as seen in Figs. 7tc) and td). Strain rate dependence of (To is considered to be due to the strain rate dependence of the deformation at the Liiders-band front. In order to clarify this phenomenon. the relaxation test has heen carried out in the midst of the Liiders deformation. as seen in Fig. S. In micrographs in Fig. 8. the crosshead is abruptly stopped when the Liiders deformation propagates to a position indicated with a lower arrow. Then the deformed region further expands during the relaxation to a position indicated with an upper arrow along the forward direction until the stress level reaches the final one which corresponds to an extremely small strain rate. Amount of deformation during the relaxation increases with increasing strain rate. but the final stress level is exactly the same in all cases. Here it is noted that the Liiders-band front sharply appears even when the stress reaches the final one, and that the ratio of Go to the final stress is about 2.5.
1% P@LYCRfST-\LLI~E
IRO?;
1279
is increased by the deformation of the trigger grain. Radius of the affected zone: is determined by the value of Q, which is a function of the grain size. the slip mode of individual grains, the anchoring force of dislocations and the deformation temperature. Here the slip mode is closely related to the stacking fault energy. In room temperature deformation of polycrystalline iron, the atfected zone expands to the fourth or fiith nearest neighboring grains when the grain size is ?O,~rn or less. When the gain size increases to about 1OOltm or more, the large internal stress accumulates on the grain boundary of the trigger grain. but it can almost be relieved by the distortion of the first nearest neighbors. Therefore. the afIected zone in such cases expands to only the first nearest neighboring grains, and finally the deformation of the trigger grain propagates into only one or
1. DISCI_SSIOS 4.1 Formsion
of the hand-structure
The values of oL. and CJ‘ markedly decrease with decreasing specimen thickness when the value of t/d is smaller than the critical one, as seen in Fig. 2. This fact is closely related to the long-range interaction among individual grains [7]. Each trigger grain in thick polycrystals is constrained three-dimensionally with its surrounding grains, and thus the macroscopic deformation hardly occurs until the applied stress (li,) reaches a certain value (a,) [7] at which all surrounding grains are deformed at the same time nevertheless their orientation factors [5] against the trigger grain. Here, a region involving these surrounding grains is named ‘the atfected zone’ in which the local stress 1 Details will be published
elsewhere.
Fig. 10. Formation process of the Liiders-band
front.
twe grains of the first nearest neighbors with large orientation factors when the grain size is sufficiently large. Now, the a&ted zone is simply assumed to cover the first nearest neighbors as shown in Fig. 9. Since the shape of affected zone also depends on the shear component of applied stress. the affected zone is elongated along a direction of the maximum resolved shear stress which is determined by the constraining force of the trigger grain. as shown with an elongated circle in Fig. 9 in which the trigger grain is denoted by &. When the applied stress reaches the vaiue of c;,, aii surrounding grains in the &ected zone are deformed at the same time, and then many affected zones are newly formed around individual deformed grains AZ, B2. Cr and so on. These affected zones overlap each other in some grains. When the affected zones overlap. the local stress increases depending on the number of overlapping. because the maximum shear displacement of individual grains occurs along the same direction in the affected zone. Now, the deformation is assumed hereafter to simply propagate two-dimensionally. Then, in Fig. 9, four affected zones overlap in grains AS. BJ. C3. D: and E,, three in grains DL and E2, and so on. When the local stress [a*) in the overlapped aKected zones increases :L’times as much as that in a single affected zone (ai), G* can be expressed by g* = d It’.4. ,Yfl. !I
(2)
The applied stress rather increases at the tip of deformed region during the propagation of deformation. Thus. once the avalanche of deformation occurs at cr, in a single affected zone of grain do. the deformed region rapidly propagates under stress G* along the direction of the maximum resolved shear stress (1). On the other hand, the applied stress is relieved considerably, depending on the hardness of machine in regions where the avalanche of deformation occurs. Thus the sidewise propagation of deformation is decelerated markedly so that the width of deformed zone at the beginning of deformation will be a few times as large as that of a single affected zone. This is the formation process of the bandstructure. The avalanche of deformation generally starts at the specimen surface where the constraining force is small. and the volume fraction of trigger grains is extremely small until the stress reaches F&.ahen the grain size is small. Therefore, the value of cc becomes nearly equal to o, and insensitive to the strain rate when the strain rate is lower than a certain value. as shown in Fig. 5
the periphery of affected zone. a large amount of deimmion ot’curj only in grains whose value of .V is sufficiently large, ai expect& from equation (2). As a result of this fact, the periphery of band-structure becomes zig-zag in a scale of grain size. as shown in Fig. 10(a). In Fig. t0, the deformed grains are shown by hatching and each affected zone caused by the deformed grain is shown with a circle. Here, each affected zone is assumed to simply expand isotropic&y to the first nearest neighboring grains. and the number of overlapping of the affected zones is denoted by indites !. 2. 3 and 4. Grains of index I are hardly deformed until the applied stress reackes the value bf ti,. Grains of indices 2. 3 and i. however, are deformed at small applied stresses depending on the value of 5’ in the affected zones. When the deformation propagates into the neighboring grains whose index is larger than 1 at this stage, the propagation of deformation occurs only along a direction indicated with an arrow in Fig. iqb). Finally the deformed grains are arranged on a plane along the maximum resolved shear stress. as shown in Fig. 10(c). The criticai value of c if for Go will be reduced by this fact, as shown in Fig. 2, because half of the affected zone is already relieved by the deformation. This is the formation process of the Liiders-band front. After that. once a grain is deformed in the first nearest neighbors of these deformed grains, the deformation quickly propagates toward the direction indicated with arrows along arrays of grains of index ? which ,ire parallel to the Liiders-band front. According to this process, the specimen is bent around an axis perpendicular to the shear direction in the tramition zone between the deformed and undeformed regions. as shown in Fig. 11. The tensile stress in-
1.1 Fortnarion of‘ the Liiders-hand frotzt After the band-structure is formed. some incuvation period is necessary to make the Li.iders deformation. as expected from Fig. 4. The stress level in both sides of the band-structure is markedly decreased when the avalanche of deformation pass through. Thus. near
Fig. i I. Additional Stress2s cuusrd by focltl bznding of the specimen accompanying the Liiders dciurmation. Thickness and grain six of the specimen are 3 mm and 5Oym. respecti~2l~.
Ft’JfT,\
creases creases
-4%~ MIk’lZrtf(I:
Lt;DERS
DEFORMATION
at the tension (23 side of bending and deat the compression tCI side. as same as in
the case of single crystals of Cu-Al alloy [lg]. The deformation always starts at the T-side and propagates to the C-side. This process is repeated layer by layer in a scale of grain size. Since the propagation Liiders-band front occurs
of
deformation
at
the
with a considerably high speed. the local stress at the Liiders-band front is determined by the deformation speed as weil as the stress expressed by equation 12). as expected from Fig. 8.
As mentioned above. the deformation of each individual grain propagates into the affected zone. The number of trigger grains at the beginning of deformation increases with the decrease in the value of cr,. Therefore. the avalanche of deformation in soft polycrystafs occurs in various places at random. Since the directions of shear displacement in these deformed zones are different from each other, each individual deformed zone does not expand into a wide region because of the interaction among different deformed zones. Namely, the deformation mode and distribution of the local stress in polycrystats are ctosely related to the number of the trigger grams at the beginning of deformation.
IN
POLYCRYSThLLINE
IRON
izst
.~c~nuwledgemenr-me authors wish to express their hearty thanks to &Ir. I(. Shibata for his help in the present experiment
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