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The bauschinger effect, surface and interior stresses in cyclically strained 70-30 alpha brass single crystals
~cro mera//. Vol. 34. No. 4. pp. 721-733. 1986 Printed m Great Britain. All rights reserved
oool-6160’86
53.00 + 0.00 Ltd
Copyri8hc c 1986 Pergamon Press
THE BAUSCHINGER EFFECT, SURFACE AND INTERIOR STRESSES IN CYCLICALLY STRAINED 70-30 ALPHA BRASS SINGLE CRYSTALS Z. WANG and H. MARGOLIN Department
of Physical and Engineering Metallurgy, Polytechnic Institute of New York, Brooklyn, NY 11201, U.S.A. (Received
2 May 1985; in revised /arm 20 June 1985)
Abstract-Single crystals of 7@-30 alpha brass, oriented for easy glide, were cyclically strained under total strain control As,/2 = 0.15%. The peak stress, up, and the Bauschinger stress, ua. defined as the 0.00005 strain offset stress in the reverse direction, were determined as the number of cycles, N, increased. up increased and us decreased with increasing N. This was explained by assuming that the surface flow stress, us, was lower than the interior flow stress, u,. This difference in flow stress would produce residual surface and interior stresses on unloading to zero applied stress. These residual stresses increased during cycling because u, increased more rapidly than us. These assumptions were tested by longitudinally electropolishing half the gage length and observing the bending, due to residual stresses, while the specimen was in the top grip. Bending was independent of where along the gage length polishing took place. The direction of bending agreed with the assumption of a lower surface flow stress. From the magnitude of the deflection, the elastic modulus and the volume fraction of surface and interior material, it was possible to calculate the residual stresses on the surface and the interior and the flow stress of the surface and interior. As electropolishing of the entire gage length took place up decreased and us increased. No further changes occurred, when about 0.35 mm was removed from the diameter, and the values of up and us approached the values of the virgin single crystal after one cycle. The changes in up and us with surface removal along the entire gage length suggested that the surface was harder than the interior. To rationalize the two apparently contradictory behaviors it was assumed that the surface consisted of three different dislocation density areas: a high surface dislocation density region, followed by a larger low dislocation density region and at 150-175 pm below the surface the third region, a narrow highest dislocation density volume. This region would act as a barrier for dislocations to move out of or into the crystal. The entire surface region was assumed to have a lower average dislocation density than the interior, thus producing the lower flow stress in the surface region. Qualitative TEM observations revealed a higher dislocation density in the interior than in the surface. The presence of pile-ups near the surface was taken as partial support of the postulated high dislocation density region at 150-170 pm below the surface. R&m&-Nous avons dCforme cycliquement des monocristaux de laiton alpha 70-30 orient& pour le glissement facile jusqu’b une diformation totale contr6lte As,/2 = 0,15%. Nous avons dCterminC la contrainte au pit up et la contrainte de Bauschinger us en fonction du nombre de cycles N. up et us diminuaient lorsque N augmentait. Nous avons expliqut cela en supposant que la contrainte d’tcoulement i la surface us Ctait inftrieure a la contrainte d’tcoulement inteme u,. Cette diffirence de contrainte d’icoulement produirait des contraintes tisiduelles supeticielle et inteme lors de la d&harp. Ces contraintes rCsidue11esaugmentaient avec le nombre de cycles, car u, augmentait plus vite que us. Nous avons vtrifit ces hypoth&es en polissant Clectrolytiquement longitudinalement la moitiC de la longueur de l’ichantillon et en observant la flexion due aux contra&es rtsiduelles, alors que l’&chantillon Ctait dans la mordache sup&ieure. La flexion ne dtpendait pas de l’endroit de l’tchantillon qui avait CtCpoli. La direction de la flexion &it en accord avec l’hypoth&se d’une contrainte d’bcoulement en surface plus faible. A partir de la grandeur de la dCflexion, du module ilastique et des fractions volumiques de materiau supetficiel et inteme, nous avons pu calculer les contraintes rtsiduelles I la surface et dans la masse, et la contrainte d’ifoulement P la surface et dans la masse. Lorsqu’on polissait Clectrolytiquement toute la longueur de I’Cchantillon, up diminuait et us augmentait. I1 n’y avait pas d’autre changcment lorsqu’on enlevait environ 0,35 mm du diamitre et les valeurs de u et de ua tendaint vers les valeurs du monocristal vierge apr&s un cycle. Les variations de up et ua lorsq&n enlevait de la matitre i la surface de toute la longueur de I’Cchantillon donnent B penser que la surface &it plus dure que I’intCrieur. Afin de concilier ces deux comportements apparemment contradictoires, nous avons suppoti que la surface Ctait constituie par trois domaines de densitis de dislocations diffbrentes: une r&ion superficielle I forte densiti de dislocations, puis une plus grande r&ion P faible densitt de dislocations et enfin une troisitme r&ion itroite B 150-175 pm sous la surface avec la densitt de dislocations la plus forte. Cette r&ion agirait comme une barri& s’opposant P l’entrk ou P la sortie des dislocations. Nous avons supposC que toute la r&ion superlicielle avait une densiti moyenne de dislocations inftrieun B celle de l’inttrieur, ce qui entrainait une plus faible contrainte d’icoulement dans la r&ion superiicielle. Des observations qualitatives par MET ant montrC une densiti de dislocations P l’indrieur suptrieure d celle de la surface. Nous pensons que la prfsence d’empilements pris de la surface confirme partiellement l’existence de la r&ion de plus forte densitt de dislocations d 150-175 pm sous la surface. 721
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Zusammenfassung-Einkirsralle aus 70-30 ~-Messing. die fiir Einfachgleitung orientiert waren. wurden unter Kontrolle der Gesamtdehnung AE,!~ = 0,15% zyklisch verformt. Die Spitzenspannung op und die Bauschingerspannung us. definiert als die Offsetspannung bei Dehnung 0.00005 in der entgegengesetzten Richtung. wurden mit ansteigender Zyklenzahl N bestimmt. up nahm zu und us sank mit zunehmendem N. Dieser Befund IiiBt sich mit der Annahme erkliren, dal3 die FlieDspannung an der Oberfllche. 0s. niedriger war als die Spannung im Inneren da Kristalles, u,. Aus einer solchen Differenz ergebensich Restspannungen an der Oberfliiche und im Inneren, wenn auf eine aul3ere Spannung von Null entlastet wird. Diese Restspannungen nahmen mit der Zyklenzahl zu, da u, rascher als us anstieg. Diese Annahmen wurden nachgepriift, indem die Hiilfte der Probenlange poliert und dann die Durchbiegung gemessen wurde. Diese Durchbiegung hing nicht von dem Ort ab, an dem die Probe poliert wurde. Die Richtung der Durchbiegung stimmte mit der Annahme einer gegeniiber dem Inneren kleineren Oberflachenspannung iiberein. Aus Gr6Be der Durchbiegung, elastischem Modul und dem Verhiiltnis der Volumina an der Oberflache und im Inneren konnten die Restspannungen an OberflHche und im Inneren und die FheBspannungen der Oberfllche und des inneren berechnet werden. Nach dem Polieren der gesamten Probenlange sank up ab und ua nahm zu. Nach Abpolieren von etwa 0.35 mm gab es keine weiteren Anderungen und die Werte fiir up und ua erreichten die Werte des jungfriiulichen Kristalles nach einem Zykhts. Aus den Anderungen von up und ua bei Abpolieren der gesamten Oberflache konnte geschlossen werden, daB die ObertIlche barter als das innere war. Urn diese beiden augenscheinlich widerspriichlichen Verhaltensweisen zu erkliiren, wurde angenommen, daB die OberIIiiche aus drei Gebieten mit unterschiedlicher Versetzungsdichte zusammengesetzt ist: ein Gebiet hoher Dichte an der Oberflache. darunter ein gr6Beres Gebiet mit niedriger Versetzungsdichte und in einer Tiefe von 150-175 pm ein drittes Gebiet. ein enges Volumengebiet mit hiichster Versetzungsdichte. Dieses Gebiet wirkt als Hindernis fiir Versetzungen, die in den oder aus dem Kristall gleiten miichten. Die mittlere Versetzungsdichte dieser Oberfliiche wird als kleiner als im Inneren angenommen, damit eine kleinere Fliegspannung an der Oberfllche vorhegt. Qualitative Beobachtungen im Durchstrahlungselektronenmikroskop wiesen auf eine hdhere Versetzungsdichte an der Oberflache hin. AuBerdem fanden sich Aufstauungen in Oberfliichennihe, die als Teilbeweis fir die geforderte hohe Versetzungsdichte in einer Tiefe von 150-I 75 pm unterhalb der Oberfliiche gewertet werden.
INTRODUCTION
This work was undertaken as a result of previous observations on fi brass single crystals used for Bauschinger effect studies [I]. It is of interest to review this work briefly. After prior compression or tension the reverse loading revealed new slip bands in the region of Bauschinger reverse strain at strains as low as 0.005%. It was reasoned that, if Bauschinger behavior were due to dislocations piled up during forward loading, then on reverse loading unslipping would occur and new slip should not appear, especially at the small strains used. However, new slip did indeed appear at these low strains. It was considered that a flow stress difference between surface and interior would conceivably account for such behavior. Such a flow stress difference had been reported by Fourie to develop as a result of prior deformation [2] and had been confirmed in subsequent investigations [3-51. If the flow stress of the surface is below that of the interior, after tensile strain, then, on unloading the average stress to zero, the surface would be in compression and the interior in tension [l]. On subsequent reloading in compression, a smaller applied stress would be required to cause the surface to yield at the stress, in compression, corresponding to the unloading flow stress in tension. New slip would occur for the same reason that it occurs in unidirectional testing, i.e. strain hardening, but at an applied stress below the average flow stress of the specimen prior to unloading [l]. No attempt was made [l] to determine whether the surface hardened more rapidly than the interior. Such behavior might perhaps be anticipated from the results of excess dislocation density at the surface
reported by Panghom er al. [6] and partially confirmed in a note added in proof in a paper by Ungar er al. [7]. However, as will later be shown, the evidence suggested that the interior hardens more rapidly than the surface. If the surface layer were initially soft, as proposed [I], the presence of this soft surface layer would indicate that slip was taking place in a flow stress gradient. The present investigation undertook to study how slip and persistent slip band formation would be affected by an imposed stress gradient in x brass alloys. This paper reports initial results on uniform cross section, 70-30 alpha brass, easy glide single crystals. EXPERIMENTAL (A) Material and single crystal growrh
Alpha brass, containing 70.03% Cu, 0.04% Pb, 0.03% Fe, 0.01% Sn and 29.89% Zn (by difference), was obtained from the Olin Corporation as 1.8 cm thick plate. This plate was used to grow unseeded single crystals 10 cm long and I .2 cm in diameter in a Bridgman furnace with a temperature gradient of 32”C/cm. The orientations were checked by Laue back reflection, and the stress axes were found to lie within the central region of the stereographic triangle, Fig. 1. Twenty crystals were grown and ten crystals provided the data subsequently reported. (B) Specimen preparation
The as-grown single crystals were initially annealed at 35O’C for two hours and subsequently machined to produce button head specimens of total length
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until a load of 22 N (5 lb) was applied on the specimen. The screws on the bottom grip were then tightened so that no movement of the dial gage occurred. Both the top grip, with a single hemisphere, and the bottom grip, with a double hemisphere, were lubricated with a teflon spray. A 1.27cm long strain gage, with an error of 0.004% in strain. was used to control total strain, BE,*, of 0.15%. 2. Electropolishing experiments. Two types of electropolishing experiments were carried out. For both experiments the bottom grip was removed and an electrolytic polishing apparatus was placed on the movable crosshead and moved into position with the specimen still attached to the top grip. In the first experiment material was removed all around the specimen. In the second series of experiments half the gage length was masked along the length and electropolishing occurred along the other half. Electropolishing was carried out at 15’C or less. It was anticipated that bending would take place, and a mirror was placed on the bottom part of the specimen. A laser beam passed through the solution to the mirror and was reflected backward as electropolishing took place. As the specimen bent the light was deflected either upward or downward. Since the laser beam was located 1.72m away from the mirror, it was possible to detect a deflection of 0.025 deg. A schematic diagram of the laser beam set-up is shown in Fig. 2. moved upward
Fig. 1. Orientations of single crystals tested. Specimens 12 and 13 with the same orientation.
9.3cm and 5%5.7mn-1 in diameter in the gage length. Considerable care was taken to make the machined ends perpendicular to the specimen axis. This perpendicularity permitted the careful alignment to be carried out as indicated below. Following machining, the specimens, still in the lathe, were hand polished through 600 x grit emery paper to remove 0.054075 mm from the diameter. This was sufficient to remove the machining marks. The crystals were then annealed at 350°C for 2 h again. Following the anneal, 0.5475 mm was removed from the diameter by electropolishing. Laue photograms, after electropolishing, revealed no asterism. (C) Testing 1. Alignment. An MTS hydraulic machine was used to obtain a tension-compression strain rate of 3 x 10-‘/s. A double hemisphere grip was used to permit careful alignment. To align the specimen it was first inserted in the top grip and a dial gage, with individual divisions of 0.0127mm was mounted on the movable bottom head. The head was moved up and down and the top grip screws were adjusted so that the misalignment was less than 0.005 mm/cm. The dial gage was then moved to one of the supporting rods and adjusted so that it just touched the specimen. The bottom grip was then mounted and
(D) Slip observations 1. Optical microscopy. To obtain photomicrographs of slip bands a microscope and camera were mounted on the fatigue machine together with appropriate lighting. Photomicrographs were taken without halting the testing. This procedure limited the magnifications which could be used to x 150.
Fig. 2. Schematic diagram of apparatus to measure specimen deflection during electropolishing. 1. Laser generator; 2. Recording graph; 3. Specimen--shadowed area on specimen is covered by acid-resistant lacquer; 4. Mirror (polished single crystal silicon chip); 5. Magnetic stirrer; 6. Stainless steel plate; 7. Upper grip; 8. MT’S machine;-9. Electropolishing solution = 50% phosphorus acid + 50% methanol.
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2. Transmission electron microscopy. In order to observe the dislocation structure near the surface, the cyclically strained specimens were electroplated with Cu to a thickness of 4-5 mm. The plated specimens were spark cut into slices 0.2 mm thick. These slices were then either spark cut into smaller disks or simply punctured to produce the required circular disks. Some pits appeared during jetting to produce the dish form in the disk. This pitting became more severe during final polishing. Various techniques were tried to eliminate pitting but they were not successful. Examination was carried out between pitted areas. RESULTS (A) Change of peak average resolved shear stress, up, and acerage Bauschinger resolved shear stress, ag, with number of cycles
The peak stress, up, increased and reached a plateau between 1000 and 1500 cycles. Typical hysteresis loops at the 2nd, 150th and 2500th cycles are shown in Fig. 3. Serrated yield behavior, seen in the 2nd cycle, continued for several cycles. Strain bursts were observed in one specimen at 65, 387 and 500 cycles and were manifested by sudden increases in width of the hysteresis loop and a decrease in up. The increase in width of the hysteresis loop existed for about 10-20 cycles, during which the width gradually decreased, and ep gradually increased. This strain burst behaviour is presented in Fig. 4 and serrations are also evident in several of the strain burst loops. After the strain burst disappeared at 514 cycles no further strain bursts were observed. Similar behavior was observed in all specimens similarly tested. The Bauschinger stress, ea. was measured at an offset strain of 0.00005. The maximum error in this
measurement is estimated to be f0.45 MPa. The magnitude of ug decreased with increasing number of cycles, as trp increased. This is revealed in Fig. 5. Results for both tension and compression are plotted in a positive direction. It can be seen that when no further increase in op occurred, no further decrease in ug took place. Values of ca, measured in compression show negative values of ua. This indicates that Bauschinger behavior occurred during unloading of the forward cycle. (B) Change of crPand ug with surface removal
It was planned to examine changes of up and us as surface material was removed by electropolishing at no load. In order to separate out the effects of surface removal from possible recovery effects taking place during polishing. measurements of up and ug were made during continuous testing, after 24 h and after electropolishing, all at room temperature. Holding time and electropolishing occurred at no load. Results are presented in Fig. 6(a,b). Figure 6(a) reveals the behavior of up. Curve 1 corresponds to the virgin crystal and the initial value of up is lowest. up increased continuously until a plateau was reached which continued up to 3000 cycles. After 3000 cycles, testing was halted, and the specimen was held for 24 h before testing was resumed, curve 2. Cycling was continued for curve 2 for another 3000 cycles. The initial value of or, in curve 2, was below tr, after 3000 cycles in curve 1 and indicated that recovery had occurred. However, this value of or, was followed by a decline in magnitude, as strain aging effects were removed. Subsequently, the stress increased and the original plateau was reached, albeit at a smaller number of cycles. After the second test of 3000 cycles, the specimen
Fig. 3. Hysteresis loops showing hardening during cycling at a total strain, A&,/2. of 0.15%.
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Fig. 4. Strain burst behavior between cycles 499 to 514. was electropohshed. The entire time for electropolishing and reassembty was Z$h. After electropofishing to remove 0.4 mm from the diameter, curve 3, the initial value of at, was below that for the initial value of crPin curve 2 but above that of curve I. in a different series of experiments in which material was removed in a series of tests, it was found that the
magnitude of the drop in crpdepended on the amount of material removed. The initial value of 6, decreased continuously as material was pohshed off. Removal of more than 0.4 mm did not show very much change in up The relatively large drop in epr following electropolishing to remove 0.4mm. curve 3 Fig 6(a), and
Fig. 5. Average resolved shear (RS) peak stress. up, and average Bauschinger RS stress, a,, as a function of number of cycles.
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902 No. of Cyh (a)
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ND. of Cyekr (b) Fig. 6. (a) Average RS peak stress, up, in the tension cycle, as a function of numbers of cycles. Curve 1: original crystal tested continuously to 3000 cycks. Curve 2: after 3000 q&s, specimen held at room temperature for 24 h, then tested to another 3UOOcycles. Curve 3: after 6ooo cycles, specimen electropolished to remove 0.4 mm &fore cycIing. @) Average ~~~~r RS stress, us, for the tests shown in 6(a).
WANG anr: MARGOLIN: Tension
BAL’SCHINGER Compression
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Compression
(b)
Fig. 7. Decreasing Bauschinger stress produced as a result of more rapid strain hardening of mtcrlor thdn surface. Both tension and compression cycles are plotted in the same direction. It is assumed that the flt~t stress of the surface material is below the stress of the interior material which has a larger volume iracrion The reverse stress-strain curve (compression) of the surface and the interior are drawn. FIN. 7(b). so that flow in each xglon begins at the ..:me stress magnitude at Hhich flow ended in the forward dIrectIon When the mtc:lc)r and surface flo\r stress curbes are combined. 11can be seen that the average flnu stress begins to delldIe from linearit! at a stress considerably below the average flow stress just belo\\ unloading. The Bauschinger stress. at an offset of 0.005 %. ug. is marked in Fig. 12(b) and (c). In Fig. ]?,(a~ CSA~ line) and Fig. 12(b), the surface and interior strain harden at the same rate. However. when the lnlerlor hardens more rapidly than the surface. uB decreases. see Fig l?(a) (dashed line) and Fig I?(c).
the general decrease in up with increased surface removal suggested that the surface was behaving like a hard material. This behavior is in contradiction to the assumption of a soft surface layer and will be considered in the discussion. The behaviour of ua. corresponding to the three tests shown in Fig. 6(a) is revealed in Fig. 6(b). With the exception of curve 2 there is a continuous decrease in ~a. accompanying the increase in up, until a plateau is reached. For curve 2 there is an initial increase in ua in the first few cycles following the 24 h recovery period. This increase has been observed in the four specimens where the entire gage length was electropolished and must be considered genuine. Beyond about 6-10 cycles the data for the three tests scatter around a single curve, and the scatter is within the error of measurement. Figure 6(a) reveals that a plateau in crPis reached after about 1000 cycles in original and electropolished states but the plateau is reached after 100-300 cycles in the recovered specimen. The plateau in ua, Fig. 6(b), is reached after about 1000 cycles for all three conditions. In other tests of original specimens, the two plateaus are reached at about the same number of cycles, Fig. 5, and other recovered specimen show the relationship of Fig. 6(a. b). It should be noted Specimen 17 curve 1, Fig. 8. was subjected to tests l-3 of Fig. 6(a). before the tests of Fig. 8 were carried out.
that the plateau in ug for the recovered specimen !a@~ the plateau in up.
The assumption that the surface had a loner flo\\ stress than the interior. as noted earlier. imphed that after a tension half cycle there should be residual stresses, with the surface in compression and the interior in tension. After a compression halfqclc. the stresses would be reversed. This residual stress behaviour is depicted in Fig. 7. To test this assumption half the surface was electropolished. after 3500-4OOO cycles. Four tests were conducted and the results are given in Fig. 8. After a tension half cycle bending uould be expected to take place in a downward or negative direction, as demonstrated in Fig. 8. curves 1 and 3. and anticipated from Fig. 9(b). For a compression half cycle the bending would be positive. curves 2 and 4, Fig. 8 and Fig. 9(b). Of particular concern n-as ths possibility that the bending stresses due to possible misalignment might have caused the observed bending. To this end the polishing was carried out on the surface where the positive edge dislocations emerged. curves It and 2. where the negative edge dislocations emerged. curve 3, and where the screw dislocations emerged, curve 4. In each case bending occurred according to the prediction of Fig. 9. and nas mdependent of the position at which polishing took place.
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Fig. 8. Bending occurring when half the surface of the gage length was electropohshed.
Fig. 10. Dislocation structure in the interior.
The results of Fig. 8 are. thus. in accord with the assumption of a soft exterior.
system occurred. A typical slip appearance after 3500 cycles is revealed in Fig. IO.
SLIP OBSERVATIONS (.4 ) Opticol
Pnmary slip bands formed in clusters and the number of slip bands along the gage length increased with number of cycles to about 2000 cycles. With a larger number of cycles some cross slip appeared and occasionally isolated slip on a non-primary slip
Stop at Tension
(B) TEM
After 35004000 cycles. which corresponds to the plateau in 6,. Figs. 5 and 6a. the dislocation structure is not complex. However. a difference in dislocation structure was noted between surface and interior. Near the surface structures reminiscent of pile-ups were observed. Fig. 11. Within the interior at least two slip systems were operating. This can be seen in
Stop at Comprrsion
Fig. 9. Schematic illustration showing positive or upward and negative or downward bending during surface removal after a compression half cycle or tension half cycle. respectively. T = tension and C = compression residual stress. Shadowed region was removed from the specimen surface during polishing.
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BACSCHI5GER EFFECT IN BRASS
Fig. 1I. Dislocation pile-up near specimen surface
Fig. I? in which the different orientations of slip systems are present at A and B. In addition. within the interior jogs and kinks frequent]! appeared as did two-dimensional dislocation networks. A general survey indicated that dislocation densit! mcreased from surface to interior. However. no attempt at quantitative measurement uas undertaken. DISCUSSION
It had been proposed. as noted earlier. that the Bauschinger effect in single crystal beta brass [I] was due to a difference in flow stress betueen surface and interior. the surface having a loner floti stress. The same mechanism can be used to explain the decrease in CT~accompan!,in.g the increase in uP. Fig. 7. If it is assumed that the interior strarn hardens more rapid11 than the surface. then on unloading from a tensile cycle. the softer surface would be placed further into compression than when the surface and interior strain harden at the same rate. The larger residual com-
A
Fig. 12. Primar! slip band and cross slip in specimen 15 after 3500 cycles ar AC:2 = 0.15’ O.
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pressive stress on the surface would produce the smaller value of uB. compare Fig. 7tb. c). According to Fig. 7 the value of Go depends onl! on the residual stress developed between the surface and the interior. Houever. if this were true. there should be an exact match between the number of cycles at which the plateaus in gP and CT~are reached. Since this matching is true onI\ for the original and electropolished states. but not the reco\,ered state. several other factors must be considered. Figure 3 suggests that dislocation motton occurs at different applied stress. For example. cycle 2 shoas deviation from linearity at loads whtch are considerably below the peak stress. where serrations take place. This indicates that dislocation motion occurs below the peak stress. as would be expected with a soft surface. The serrations occur because dtslocations are freed from their binding atmospheres. While dislocations are being freed of their atmospheres. other dislocations which have been freed to move undergo some hardening. Thus dislocatton release. strain hardening and recapture b) atmospheres occur in the same cycle at the surface and within the interior and the local stress whtsh the dislocations “see” varies according to the stress required for their movement. Local differences m flnu stresses develop. and these local stress differences also contribute to the Bauschinger beha\,ior. after WIloadrnp. These local stress t:ifferences occur at both surface and interior. and therefore. both regtons sdn contribute to Bauschinper behavior in addttion to the overall Bauschmger effect. caused by the difference in stress level between surface and interior. When the release of dislocations take place. during straining after the first half cycle after recovery. the dislocations. free of atmospheres. are available to produce the Bauschinger flow in reverse flow. and the residual stress on them is relatively low because the local difference in flow stress is small. This low residual stress raises the applied stress required to cause Bauschinper flow. This applied stress continues to increase after the first cycle as the number of dislocations which are freed- of their atmospheres increases. When the number of freed dislocations decreases. relative to those vvhich have been freed but have undergone hardening or recapture. the maximum in the Bauschinger stress is passed. and the process of decreasing Bauschinper stress with increasing number of cycles continues. Fig. 6(b). From the discussion above one would anticipate that serrated behaviour would be possible during the rise to approximately constant stress. curve 2. Frg. 3. Dislocations in the softer surface would also undergo capture by and release from atmosphere. Such serrations are seen in the hystereses curves but they are sufficiently small that they can not. with confidence. be Separated from the fluctations produced by the hydraulic system. Such suspicion 1s not applicable to the flat portion of the curve of cycle 2. in Fig 3. Another possible source for the difference tn num-
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ber of cycles at which bp and err,reach their respective plateaus may be related to the friction stress. It has been proposed that the friction stress increases during cycling [IO]. This would mean that dislocations which have experienced different amounts of flow on a given slip system would have different friction stresses and, therefore, different flow stresses. If this occurs during the present work, the difference in number of cycles at which the plateaus in up and ergare reached implies that a relatively uniform friction stress has not been reached until the plateau in ua is reached. (B) Harder surface than interior
As noted with respect to Fig. 6(a) the surface appears to behave as if it were hard, and yet the Bauschinger behavior suggests that the surface was soft. If the initially soft surface had subsequently become hard as a result of repeated cycling, then, instead of decreasing continuously with cycling, there would have had to be a reversal, an upward movement of ug. That the surface continued to be softer than the interior was manifested by the bending experiments, Fig. 8. However. the apparent pile-ups of Fig. 1I suggest a way of rationalizing the two apparently contradictory behaviors. This would require the existence of two conditions: (1) The presence just below the surface, approximately at 150 to 175 pm, of a narrow high dislocation density region which would inhibit the ingress of dislocations from the surface or exit from the interior. The choice of 150-175 pm as the position of the dislocation density peak is made because no further appreciable changes in up and no further bending occur when this much material is removed from the radius. (2) An average dislocation density of the surface which was below the average dislocation density of the interior. Pangbom et al. [6] have indicated that the surface has a higher dislocation density than the interior. The present qualitative results provide support only for dislocation density increasing away from the surface. However, a higher surface dislocation density can be accommodated in the present explanation as long as the average dislocation density of the surface region is below that of the interior. Thus, a schematic dislocation density would appear as shown in Fig. 13(a).
EFFECT IN BRASS
;6_ &_ _ a*
YJn
Fig. 13. (a) Schematic illustration of the dislocation density, p, change from surface to interior. (b) Dislocation density. p, rearrangement as the surface is being polished.
It is suggested that the dislocation arrangement is not fixed, and that when material is polished off the surface, the dislocations rearrange themselves as shown by the dashed line in Fig. 13(b). As polishing takes place dislocations from the peak region move inward and to the surface tending to raise the dislocation density in both regions. At the same time the reduction in the peak dislocation density would reduce the resistance of this area to dislocation passage and the peak stress would decrease. With increase in polishing the peak internal dislocation density disappears. With respect to bending during polishing Fig. 13 would be in accord with observations. The two peak dislocation densities of the surface and interior. as well as the interior region, would be in tension after a tension half cycle, while the region between the two peaks would be in compression. The volume average stress of the three surface regions would produce a net compression stress. Removal of the initial surface region would still leave the surface region in compression and the bending would continue until the low density dislocation region was almost entirely eliminated. Fabiniak and Kuhlmann-Wilsdord [B] have also reported a dual behaviour of a surface region in Al single crystals. They concluded that regions just below the surface acted as dislocation sources, while the surface acted as a barrier to dislocation motion. This behavior is not strictly parallel to the postulated behavior but is an indication that dual surface behavior is possible. (C) Calculation of surface and interior residual stress and surface and interior jaw stress
The surface region would thus consist of two high dislocation regions separated by a low dislocation area. It is assumed as noted above that the volume average of dislocation density of the three regions comprising the surface is below the dislocation density of the interior. Figure 13 also differs from the measurements of Panghorn er al. (61 by the assump-
It is possible to calculate the residual stresses in the surface and the interior from the measured deflection, the elastic modulus and the moment of inertia of specimen cross section. The calculations are illustrated in the Appendix for Specimen 20, Fig. 8, which was stressed finally in compression. The calculated residual surface stresses, uRs, and the calculated residual interior stresses, uRl, for specimens 17-20 are given in Table 1. The residual stresses were developed after 3000-4000 cycles but, according to Fig. 6(a).
tion of the internal
these stresses
peak in dislocation
density.
may have developed
after about
1000
WANG and MARGOLIN:
BAUSCHINGER
cycles. It can be seen from Table 1 that the absolute value of the residual surface stress is approximately 6.5 times the absolute value of the residual interior stress. From Fig. 7 and the determination of uRs and uR, it is possible to determine the peak values of internal flow stress, CT,.and surface flow stress, us. The average peak stress, u,,, can be expressed at constant strain as e*v =u,V,+usVs
(I)
where V, and V, are the volume fractions of internal and surface material. Since V, + V, = 1 U ,,=U,+(%-U,>VS
IUS
-
u1 1 =
(2) 1 URS -
uRl
(3)
1.
Equation (3) follows from the fact that the difference between u, and us is maintained when the applied stress is reduced to zero, Fig. 7(a). The calculated values of us and u, for specimens 17-20 are given in Table 1. For specimens 18-20 al/us = 1.41-1.55. However. for specimen 17 with the largest deflection ul/as = 2.01. The presence of a flow stress gradient is similar to the results obtained by Mughrabi [9] and Fourie [2]. The volume fraction of surface material was 0.46, corresponding to a surface depth of 0.49mm for a specimen 3.68 mm in diameter in single crystal Cu [9). The volume fraction and depth are much larger than were encountered in the present case. (D) Strain bursts
As noted earlier strain bursts in all but one case occurred below 1000 cycles after which no strain bursts took place. However. if 0.4 mm was removed from the diameter, the strain bursts would reoccur. This suggested the possibility that the strain bursts were associated with a breaching of the high dislocation density region postulated to exist at about 15S175 pm below the surface. The general absence of the strain bursts beyond 1000 cycles may have two possible explanations. It may be that the postulated high dislocation density below the surface becomes more impenetrable or that the dislocation structure presents a sufficiently strong pile-up to develop which could penetrate the postulated high dislocation density layer. No persistent slip bands or persistent Luders bands
731
EFFECT IN BRASS
[lo] were found and this may be due either to insufficient cycling to reach the platau where persistent slip bands are found [lo] or that the strain was too low. However, the strain bursts do not involve persistent slip bands. Some serrated yielding is found during the strain bursts and this behaviour, as seen in the hysteresis loops (Fig. 4). is quite similar to the serrated yielding seen when cycling begins on a fresh specimen or after recovery. The onset of strain aging during the first few initial cycles at the outset syggests that it is occurring during the testing. The serrations disappear. when strain hardening raises the flow stress above that required for serrated flow. In effect the higher strain hardened flow stress governs flow rather than the unpinning stress. (E) Duplex slip
Optical microscopy and TEM observation indicated that slip at the outer surface occurred essentially on a single slip system, whereas duplex slip was found in the interior. Table I shows the resolved shear stresses T, and ~1 for the primary and secondary interior slip systems. The resolved shear stresses are. of course, higher for the primary slip system. If strain hardening during the fatigue test were the same as in undirectional straining, then latent hardening would be expected to require a stress on the secondary slip which would be above the resolved shear stress on the primary slip system. The fact that the secondary slip system operates at a lower resolved shear stress implies that the friction stress on the primary slip system is higher than it is on the secondary slip system and that generally there is no interaction between the primary and secondary slip systems. Figure 12 supports this non-interaction suggestion.
SUMMARY 1. A series of 70-30 alpha brass single crystals was cyclically strained under strain control, with a total strain of de,/2 = 0.15%. The peak stress, up, and Bauschinger stress, bg, defined as the 0.00005 offset in reverse flow, were determined as a function of number of cycles, N. up increased and uB decreased as N increased, both stresses reaching a plateau after 1000-l 500 cycles.
Table 1. Calculated surface and interior residual stress and flow stress Specimen No.
No. of cycles
17 18 :‘o
4000 z3500
(d&c)
E (MPa)
0.27 0.18 0.16 0.19
5567 4100 4500 4452
=as
ras
-22.6 -8.9 11.6 5.6 -10.5 12.7 -4.7 5.9
%I
rat
3.5 -1.8 -2.0 1.6
-0.9 -0.9 0.7
1.4
@*v (MPa?
rs
01
rl
i,
48.5 25.9 10.2 52.0 20.5 18.4 -35.8 -24.2 -11.9 -37.6 -18.4 -18.0 -42.3 39.8 -29.6 29.3 -13.9 13.2 -44.3 41.4 -20.8 18.6 -18.6 15.9
NOW: Positive and negative values represent tension and comptassion stress, respectively. 6 = Deflection angle measured. E - Elastic constant along specimen aais. ass. oat = Residual stress in surface and interior. mspcctivcly. r asr rat = resolved shear stress on primary slip system in surface and interior, raspcctively. calculated from residual stress. o,,v = Measured average peak flow stress. 0s. V, = Plow stress acting in surface and interior. respectively. Q, I, = Resolved shear stress on primary slip system in surface and interior, respectively, calculated from flow stress cs and 0,. i, = Resolved shear stress on secondary slip system in interior calculated from flow stress 0,.
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WANG and MARGOLIN:
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2. After cyclically straining 3OOwOOO cycles and electropolishing to remove 0.34-0.4mm from the
diameter, up decreased and crs increased. Both values approached the original values reached after straining the virgin crystal one cycle. When lesser amounts of surface were removed, the changes in oP and ~a were reduced. This behavior was consistent with a harder surface. 3. It was proposed that the flow stress of the surface was below the flow stress of the interior. On unloading from the forward strain this flow stress difference would produce residual stresses which would be larger on the surface than the interior because of the smaller volume fraction of the surface. It was also proposed that, as N increased, the flow stress of the interior increased more rapidly than that of the surface. To check these assumptions half the gage length surface was electropolished along the length and the accompanying deflection were measured. The deflections were independent of where along the gage length electropolishing took place and agreed with the assumptions, thus indicating that the surface material was softer than the interior. It was possible to calculate residual surface and interior stresses and surface and interior flow stresses from the deflections which occurred during polishing, the volume fraction of surface and interior and the elastic modulus. 4. The apparent contradition in items 2 and 3 was rationalized by assuming the presence of a surface high dislocation density layer and a subsurface layer 150-l 75 pm below the surface. This distance corresponded to the point of maximum deflection during bending tests and to the amount of material required to be electropolished from the radius to restore the specimen properties almost to the original values. It was also assumed that the average dislocation density of the surface, including outer surface and subsurface high density region, was below that of the interior. This lower average dislocation density would permit the surface to have a lower average flow stress than the interior, and the sub-surface high dislocation density region served as an obstacle to the outward movement of interior dislocations, thus maintaining the flow stress at the interior at a higher level than the surface. TEM observations indicated that the dislocation density of the interior was higher than that of the surface. Dislocation pile-ups near the surface were taken as evidence of the postulated high dislocation subsurface area. No direct TEM evidence of this high dislocation density area was obtained. 5. Strain bursts were observed below 1000 cycles, but not above this value. It was suggested that strain bursts occurred when the postulated high dislocation density area was breached.
EFFECT IN BRASS REFERENCES
1. H. Yaguchi and H. Margolin, Scripra merall. 17, 1213 (1983). 2. 1. T. Fourie, Phil. Mag. 17, 735 (1968). 3. J. T. Fourie, Phil. Mag. 21, 977 (1970). 4. H. Mughrabi, Physica starus solidi 39, 317 (1970). 5. H. Himstadt and N. Neuhiiuser, Scripra melaN. 6, I 151 (1972). 6. R. N. Pangborn, S. Weissman and M. Wilken, Me/a//. Trans. A 12A, 109 (1981). 7. T. Ungar, H. Mughrabi and M. Wilkens, Acfa merall. 30, 1861 (1982). 8. R. C. Fabiniak and D. Kuhlmann-Wilsdorf, Enrironmeni-Sensitive Mechanical Behavior (edited by A. R. C. Westwood and N. S. Stoloff), p. 147. Gordon & Breach, New York (1966). 9. H. Mughrabi Surface E&cu in f&to/ Plasriciry (edited by R. M. Latanison and J. T. Fourie), p. 533. Series E Appl. Sci. NO. 17. Nordhoff-Lender, Reading, Mass. (1977). IO. C. Laird and L. Buchinger. Proc. Symp. Comm JO Anniv. Inrroduction of Dislocations, Detroit, Mich. (1984). Metal/. Trans. A. To be published. APPENDIX I. CALCULATION OF SURFACE AND INTERIOR STRESSES (A) Estimate of volume fraction of surface and interior
An estimate of the volume fraction of the surface region can be made from the observation of the amount of material which must be removed by electropolishing to obtain no further changes in behavior. From Fig. 7, Specimen 20, this is taken to be approximately 0.35 mm from a 5 mm diameter specimen. This corresponds to a volume fraction of 13.5%. (B) Assumptions
It is assumed that the stresses in the surface and interior are uniform, that is, average stresses. It is also assumed that bending begins at the end of the gage length nearest the left grip, as shown in Fig. Al(a). The moment of inertia is
b) ,&knowledgemenrs-The authors wish to acknowledge help ful discussions with Professors S. Nourbakhsh, J. P. Hirth, R. Asaro and C. Laird. This work was supported under National Science Foundation Grant NO. DMR8312963.
Fig. Al:Specimen configuration used to calculate bending stress. Cs-shape center of surface semicircle layer; C-shape center of interior section; Ccsshape center of interior section; Cca-shape center of whole cross section after half surface is removed.
WANG and MARGOLIN:
BAUSCHINGER
calculated from the cross Section A-A shown in Fig. Al(b). The shape center lies 0.11 mm above the center of the figure. The force couple is also shown in Fig. Al(c). The shape center of the surface shown is indicated at C,, and the shape center of the interior at C, Fig. Al(c). (C) Culculution procedure The deflection, Y, is given by the following expression YZg
(Al)
where E = elastic constant along axis of Specimen 20, 4452 MPa, x = the length of specimen over which bending takes place, Fig. Al, 15 mm, I = moment of inertia, M = moment of force couple, Y = x .tan 0, where 8 is the measured deflection angle, Fig. 7, for Specimen 20, 6 =0.095’. In the present case, I, the moment of inertia for the whole crossection about the horizontal axis passing through the shape center, can be expressed as I = I’ - ; (I: + r:,c;.
642)
Here rL = the radius of the original cross section [see Fig.
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EFFECT IN BRASS
Al(b)], rs = the radius of the cross section after polishing [see Fig. Al(b)]. C, = distance from the specimen center to the shape center which is 0.11 mm [see also Fig. Al(b)]. I’ = the moment of inertia of the whole cross section about the horizontal axis passing the specimen center. I’ = I, + I,. I, and I, are the moments of inertia for the semi-circles. one with the radius rL, the other with rs. respectively, both are about the horizontal axis passing the specimen center. And I’ was determined to be 26.6mm’. M is determined from equations (Al) and (A2) and is 516N.mm. This moment of force is illustrated in Fig. Al(c). M = o,,~A,~I
= o,,,A,,I
(A3)
where cks and ua, are the surface and internal residual stresses, respectively; A, and A, are the areas over which uRS and ~a, are distributed, respectively, I = C,-
C, = lS4mm
[see Fig. Al(c)].
(A4)
From equations (A3) and (A4) uRS and uR, can be determined oRs = 12.7 MPa,
rra, = - 2.0 MPa.
All the other calculated results are given in Table 1.
oool-6160’86
53.00 + 0.00 Ltd
Copyri8hc c 1986 Pergamon Press
THE BAUSCHINGER EFFECT, SURFACE AND INTERIOR STRESSES IN CYCLICALLY STRAINED 70-30 ALPHA BRASS SINGLE CRYSTALS Z. WANG and H. MARGOLIN Department
of Physical and Engineering Metallurgy, Polytechnic Institute of New York, Brooklyn, NY 11201, U.S.A. (Received
2 May 1985; in revised /arm 20 June 1985)
Abstract-Single crystals of 7@-30 alpha brass, oriented for easy glide, were cyclically strained under total strain control As,/2 = 0.15%. The peak stress, up, and the Bauschinger stress, ua. defined as the 0.00005 strain offset stress in the reverse direction, were determined as the number of cycles, N, increased. up increased and us decreased with increasing N. This was explained by assuming that the surface flow stress, us, was lower than the interior flow stress, u,. This difference in flow stress would produce residual surface and interior stresses on unloading to zero applied stress. These residual stresses increased during cycling because u, increased more rapidly than us. These assumptions were tested by longitudinally electropolishing half the gage length and observing the bending, due to residual stresses, while the specimen was in the top grip. Bending was independent of where along the gage length polishing took place. The direction of bending agreed with the assumption of a lower surface flow stress. From the magnitude of the deflection, the elastic modulus and the volume fraction of surface and interior material, it was possible to calculate the residual stresses on the surface and the interior and the flow stress of the surface and interior. As electropolishing of the entire gage length took place up decreased and us increased. No further changes occurred, when about 0.35 mm was removed from the diameter, and the values of up and us approached the values of the virgin single crystal after one cycle. The changes in up and us with surface removal along the entire gage length suggested that the surface was harder than the interior. To rationalize the two apparently contradictory behaviors it was assumed that the surface consisted of three different dislocation density areas: a high surface dislocation density region, followed by a larger low dislocation density region and at 150-175 pm below the surface the third region, a narrow highest dislocation density volume. This region would act as a barrier for dislocations to move out of or into the crystal. The entire surface region was assumed to have a lower average dislocation density than the interior, thus producing the lower flow stress in the surface region. Qualitative TEM observations revealed a higher dislocation density in the interior than in the surface. The presence of pile-ups near the surface was taken as partial support of the postulated high dislocation density region at 150-170 pm below the surface. R&m&-Nous avons dCforme cycliquement des monocristaux de laiton alpha 70-30 orient& pour le glissement facile jusqu’b une diformation totale contr6lte As,/2 = 0,15%. Nous avons dCterminC la contrainte au pit up et la contrainte de Bauschinger us en fonction du nombre de cycles N. up et us diminuaient lorsque N augmentait. Nous avons expliqut cela en supposant que la contrainte d’tcoulement i la surface us Ctait inftrieure a la contrainte d’tcoulement inteme u,. Cette diffirence de contrainte d’icoulement produirait des contraintes tisiduelles supeticielle et inteme lors de la d&harp. Ces contraintes rCsidue11esaugmentaient avec le nombre de cycles, car u, augmentait plus vite que us. Nous avons vtrifit ces hypoth&es en polissant Clectrolytiquement longitudinalement la moitiC de la longueur de l’ichantillon et en observant la flexion due aux contra&es rtsiduelles, alors que l’&chantillon Ctait dans la mordache sup&ieure. La flexion ne dtpendait pas de l’endroit de l’tchantillon qui avait CtCpoli. La direction de la flexion &it en accord avec l’hypoth&se d’une contrainte d’bcoulement en surface plus faible. A partir de la grandeur de la dCflexion, du module ilastique et des fractions volumiques de materiau supetficiel et inteme, nous avons pu calculer les contraintes rtsiduelles I la surface et dans la masse, et la contrainte d’ifoulement P la surface et dans la masse. Lorsqu’on polissait Clectrolytiquement toute la longueur de I’Cchantillon, up diminuait et us augmentait. I1 n’y avait pas d’autre changcment lorsqu’on enlevait environ 0,35 mm du diamitre et les valeurs de u et de ua tendaint vers les valeurs du monocristal vierge apr&s un cycle. Les variations de up et ua lorsq&n enlevait de la matitre i la surface de toute la longueur de I’Cchantillon donnent B penser que la surface &it plus dure que I’intCrieur. Afin de concilier ces deux comportements apparemment contradictoires, nous avons suppoti que la surface Ctait constituie par trois domaines de densitis de dislocations diffbrentes: une r&ion superficielle I forte densiti de dislocations, puis une plus grande r&ion P faible densitt de dislocations et enfin une troisitme r&ion itroite B 150-175 pm sous la surface avec la densitt de dislocations la plus forte. Cette r&ion agirait comme une barri& s’opposant P l’entrk ou P la sortie des dislocations. Nous avons supposC que toute la r&ion superlicielle avait une densiti moyenne de dislocations inftrieun B celle de l’inttrieur, ce qui entrainait une plus faible contrainte d’icoulement dans la r&ion superiicielle. Des observations qualitatives par MET ant montrC une densiti de dislocations P l’indrieur suptrieure d celle de la surface. Nous pensons que la prfsence d’empilements pris de la surface confirme partiellement l’existence de la r&ion de plus forte densitt de dislocations d 150-175 pm sous la surface. 721
722
WANG and MARGOLIN:
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EFFECT IN BRASS
Zusammenfassung-Einkirsralle aus 70-30 ~-Messing. die fiir Einfachgleitung orientiert waren. wurden unter Kontrolle der Gesamtdehnung AE,!~ = 0,15% zyklisch verformt. Die Spitzenspannung op und die Bauschingerspannung us. definiert als die Offsetspannung bei Dehnung 0.00005 in der entgegengesetzten Richtung. wurden mit ansteigender Zyklenzahl N bestimmt. up nahm zu und us sank mit zunehmendem N. Dieser Befund IiiBt sich mit der Annahme erkliren, dal3 die FlieDspannung an der Oberfllche. 0s. niedriger war als die Spannung im Inneren da Kristalles, u,. Aus einer solchen Differenz ergebensich Restspannungen an der Oberfliiche und im Inneren, wenn auf eine aul3ere Spannung von Null entlastet wird. Diese Restspannungen nahmen mit der Zyklenzahl zu, da u, rascher als us anstieg. Diese Annahmen wurden nachgepriift, indem die Hiilfte der Probenlange poliert und dann die Durchbiegung gemessen wurde. Diese Durchbiegung hing nicht von dem Ort ab, an dem die Probe poliert wurde. Die Richtung der Durchbiegung stimmte mit der Annahme einer gegeniiber dem Inneren kleineren Oberflachenspannung iiberein. Aus Gr6Be der Durchbiegung, elastischem Modul und dem Verhiiltnis der Volumina an der Oberflache und im Inneren konnten die Restspannungen an OberflHche und im Inneren und die FheBspannungen der Oberfllche und des inneren berechnet werden. Nach dem Polieren der gesamten Probenlange sank up ab und ua nahm zu. Nach Abpolieren von etwa 0.35 mm gab es keine weiteren Anderungen und die Werte fiir up und ua erreichten die Werte des jungfriiulichen Kristalles nach einem Zykhts. Aus den Anderungen von up und ua bei Abpolieren der gesamten Oberflache konnte geschlossen werden, daB die ObertIlche barter als das innere war. Urn diese beiden augenscheinlich widerspriichlichen Verhaltensweisen zu erkliiren, wurde angenommen, daB die OberIIiiche aus drei Gebieten mit unterschiedlicher Versetzungsdichte zusammengesetzt ist: ein Gebiet hoher Dichte an der Oberflache. darunter ein gr6Beres Gebiet mit niedriger Versetzungsdichte und in einer Tiefe von 150-175 pm ein drittes Gebiet. ein enges Volumengebiet mit hiichster Versetzungsdichte. Dieses Gebiet wirkt als Hindernis fiir Versetzungen, die in den oder aus dem Kristall gleiten miichten. Die mittlere Versetzungsdichte dieser Oberfliiche wird als kleiner als im Inneren angenommen, damit eine kleinere Fliegspannung an der Oberfllche vorhegt. Qualitative Beobachtungen im Durchstrahlungselektronenmikroskop wiesen auf eine hdhere Versetzungsdichte an der Oberflache hin. AuBerdem fanden sich Aufstauungen in Oberfliichennihe, die als Teilbeweis fir die geforderte hohe Versetzungsdichte in einer Tiefe von 150-I 75 pm unterhalb der Oberfliiche gewertet werden.
INTRODUCTION
This work was undertaken as a result of previous observations on fi brass single crystals used for Bauschinger effect studies [I]. It is of interest to review this work briefly. After prior compression or tension the reverse loading revealed new slip bands in the region of Bauschinger reverse strain at strains as low as 0.005%. It was reasoned that, if Bauschinger behavior were due to dislocations piled up during forward loading, then on reverse loading unslipping would occur and new slip should not appear, especially at the small strains used. However, new slip did indeed appear at these low strains. It was considered that a flow stress difference between surface and interior would conceivably account for such behavior. Such a flow stress difference had been reported by Fourie to develop as a result of prior deformation [2] and had been confirmed in subsequent investigations [3-51. If the flow stress of the surface is below that of the interior, after tensile strain, then, on unloading the average stress to zero, the surface would be in compression and the interior in tension [l]. On subsequent reloading in compression, a smaller applied stress would be required to cause the surface to yield at the stress, in compression, corresponding to the unloading flow stress in tension. New slip would occur for the same reason that it occurs in unidirectional testing, i.e. strain hardening, but at an applied stress below the average flow stress of the specimen prior to unloading [l]. No attempt was made [l] to determine whether the surface hardened more rapidly than the interior. Such behavior might perhaps be anticipated from the results of excess dislocation density at the surface
reported by Panghom er al. [6] and partially confirmed in a note added in proof in a paper by Ungar er al. [7]. However, as will later be shown, the evidence suggested that the interior hardens more rapidly than the surface. If the surface layer were initially soft, as proposed [I], the presence of this soft surface layer would indicate that slip was taking place in a flow stress gradient. The present investigation undertook to study how slip and persistent slip band formation would be affected by an imposed stress gradient in x brass alloys. This paper reports initial results on uniform cross section, 70-30 alpha brass, easy glide single crystals. EXPERIMENTAL (A) Material and single crystal growrh
Alpha brass, containing 70.03% Cu, 0.04% Pb, 0.03% Fe, 0.01% Sn and 29.89% Zn (by difference), was obtained from the Olin Corporation as 1.8 cm thick plate. This plate was used to grow unseeded single crystals 10 cm long and I .2 cm in diameter in a Bridgman furnace with a temperature gradient of 32”C/cm. The orientations were checked by Laue back reflection, and the stress axes were found to lie within the central region of the stereographic triangle, Fig. 1. Twenty crystals were grown and ten crystals provided the data subsequently reported. (B) Specimen preparation
The as-grown single crystals were initially annealed at 35O’C for two hours and subsequently machined to produce button head specimens of total length
WANG and MARGOLIN:
BAUSCHINGER
EFFECT
IN BRASS
723
until a load of 22 N (5 lb) was applied on the specimen. The screws on the bottom grip were then tightened so that no movement of the dial gage occurred. Both the top grip, with a single hemisphere, and the bottom grip, with a double hemisphere, were lubricated with a teflon spray. A 1.27cm long strain gage, with an error of 0.004% in strain. was used to control total strain, BE,*, of 0.15%. 2. Electropolishing experiments. Two types of electropolishing experiments were carried out. For both experiments the bottom grip was removed and an electrolytic polishing apparatus was placed on the movable crosshead and moved into position with the specimen still attached to the top grip. In the first experiment material was removed all around the specimen. In the second series of experiments half the gage length was masked along the length and electropolishing occurred along the other half. Electropolishing was carried out at 15’C or less. It was anticipated that bending would take place, and a mirror was placed on the bottom part of the specimen. A laser beam passed through the solution to the mirror and was reflected backward as electropolishing took place. As the specimen bent the light was deflected either upward or downward. Since the laser beam was located 1.72m away from the mirror, it was possible to detect a deflection of 0.025 deg. A schematic diagram of the laser beam set-up is shown in Fig. 2. moved upward
Fig. 1. Orientations of single crystals tested. Specimens 12 and 13 with the same orientation.
9.3cm and 5%5.7mn-1 in diameter in the gage length. Considerable care was taken to make the machined ends perpendicular to the specimen axis. This perpendicularity permitted the careful alignment to be carried out as indicated below. Following machining, the specimens, still in the lathe, were hand polished through 600 x grit emery paper to remove 0.054075 mm from the diameter. This was sufficient to remove the machining marks. The crystals were then annealed at 350°C for 2 h again. Following the anneal, 0.5475 mm was removed from the diameter by electropolishing. Laue photograms, after electropolishing, revealed no asterism. (C) Testing 1. Alignment. An MTS hydraulic machine was used to obtain a tension-compression strain rate of 3 x 10-‘/s. A double hemisphere grip was used to permit careful alignment. To align the specimen it was first inserted in the top grip and a dial gage, with individual divisions of 0.0127mm was mounted on the movable bottom head. The head was moved up and down and the top grip screws were adjusted so that the misalignment was less than 0.005 mm/cm. The dial gage was then moved to one of the supporting rods and adjusted so that it just touched the specimen. The bottom grip was then mounted and
(D) Slip observations 1. Optical microscopy. To obtain photomicrographs of slip bands a microscope and camera were mounted on the fatigue machine together with appropriate lighting. Photomicrographs were taken without halting the testing. This procedure limited the magnifications which could be used to x 150.
Fig. 2. Schematic diagram of apparatus to measure specimen deflection during electropolishing. 1. Laser generator; 2. Recording graph; 3. Specimen--shadowed area on specimen is covered by acid-resistant lacquer; 4. Mirror (polished single crystal silicon chip); 5. Magnetic stirrer; 6. Stainless steel plate; 7. Upper grip; 8. MT’S machine;-9. Electropolishing solution = 50% phosphorus acid + 50% methanol.
724
WANG and MARGOLIN:
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2. Transmission electron microscopy. In order to observe the dislocation structure near the surface, the cyclically strained specimens were electroplated with Cu to a thickness of 4-5 mm. The plated specimens were spark cut into slices 0.2 mm thick. These slices were then either spark cut into smaller disks or simply punctured to produce the required circular disks. Some pits appeared during jetting to produce the dish form in the disk. This pitting became more severe during final polishing. Various techniques were tried to eliminate pitting but they were not successful. Examination was carried out between pitted areas. RESULTS (A) Change of peak average resolved shear stress, up, and acerage Bauschinger resolved shear stress, ag, with number of cycles
The peak stress, up, increased and reached a plateau between 1000 and 1500 cycles. Typical hysteresis loops at the 2nd, 150th and 2500th cycles are shown in Fig. 3. Serrated yield behavior, seen in the 2nd cycle, continued for several cycles. Strain bursts were observed in one specimen at 65, 387 and 500 cycles and were manifested by sudden increases in width of the hysteresis loop and a decrease in up. The increase in width of the hysteresis loop existed for about 10-20 cycles, during which the width gradually decreased, and ep gradually increased. This strain burst behaviour is presented in Fig. 4 and serrations are also evident in several of the strain burst loops. After the strain burst disappeared at 514 cycles no further strain bursts were observed. Similar behavior was observed in all specimens similarly tested. The Bauschinger stress, ea. was measured at an offset strain of 0.00005. The maximum error in this
measurement is estimated to be f0.45 MPa. The magnitude of ug decreased with increasing number of cycles, as trp increased. This is revealed in Fig. 5. Results for both tension and compression are plotted in a positive direction. It can be seen that when no further increase in op occurred, no further decrease in ug took place. Values of ca, measured in compression show negative values of ua. This indicates that Bauschinger behavior occurred during unloading of the forward cycle. (B) Change of crPand ug with surface removal
It was planned to examine changes of up and us as surface material was removed by electropolishing at no load. In order to separate out the effects of surface removal from possible recovery effects taking place during polishing. measurements of up and ug were made during continuous testing, after 24 h and after electropolishing, all at room temperature. Holding time and electropolishing occurred at no load. Results are presented in Fig. 6(a,b). Figure 6(a) reveals the behavior of up. Curve 1 corresponds to the virgin crystal and the initial value of up is lowest. up increased continuously until a plateau was reached which continued up to 3000 cycles. After 3000 cycles, testing was halted, and the specimen was held for 24 h before testing was resumed, curve 2. Cycling was continued for curve 2 for another 3000 cycles. The initial value of or, in curve 2, was below tr, after 3000 cycles in curve 1 and indicated that recovery had occurred. However, this value of or, was followed by a decline in magnitude, as strain aging effects were removed. Subsequently, the stress increased and the original plateau was reached, albeit at a smaller number of cycles. After the second test of 3000 cycles, the specimen
Fig. 3. Hysteresis loops showing hardening during cycling at a total strain, A&,/2. of 0.15%.
WANG
and MARGOLIN:
BAUSCHINGER
EFFECT
IN BRASS
715
Fig. 4. Strain burst behavior between cycles 499 to 514. was electropohshed. The entire time for electropolishing and reassembty was Z$h. After electropofishing to remove 0.4 mm from the diameter, curve 3, the initial value of at, was below that for the initial value of crPin curve 2 but above that of curve I. in a different series of experiments in which material was removed in a series of tests, it was found that the
magnitude of the drop in crpdepended on the amount of material removed. The initial value of 6, decreased continuously as material was pohshed off. Removal of more than 0.4 mm did not show very much change in up The relatively large drop in epr following electropolishing to remove 0.4mm. curve 3 Fig 6(a), and
Fig. 5. Average resolved shear (RS) peak stress. up, and average Bauschinger RS stress, a,, as a function of number of cycles.
WANG and MARGOMN:
726
BAUSCHlNGER EFFECT IN BRASS
902 No. of Cyh (a)
I
I
lo
I
I
#
loJ
1V
ND. of Cyekr (b) Fig. 6. (a) Average RS peak stress, up, in the tension cycle, as a function of numbers of cycles. Curve 1: original crystal tested continuously to 3000 cycks. Curve 2: after 3000 q&s, specimen held at room temperature for 24 h, then tested to another 3UOOcycles. Curve 3: after 6ooo cycles, specimen electropolished to remove 0.4 mm &fore cycIing. @) Average ~~~~r RS stress, us, for the tests shown in 6(a).
WANG anr: MARGOLIN: Tension
BAL’SCHINGER Compression
EFFECT IS BRASS
121
Compression
(b)
Fig. 7. Decreasing Bauschinger stress produced as a result of more rapid strain hardening of mtcrlor thdn surface. Both tension and compression cycles are plotted in the same direction. It is assumed that the flt~t stress of the surface material is below the stress of the interior material which has a larger volume iracrion The reverse stress-strain curve (compression) of the surface and the interior are drawn. FIN. 7(b). so that flow in each xglon begins at the ..:me stress magnitude at Hhich flow ended in the forward dIrectIon When the mtc:lc)r and surface flo\r stress curbes are combined. 11can be seen that the average flnu stress begins to delldIe from linearit! at a stress considerably below the average flow stress just belo\\ unloading. The Bauschinger stress. at an offset of 0.005 %. ug. is marked in Fig. 12(b) and (c). In Fig. ]?,(a~ CSA~ line) and Fig. 12(b), the surface and interior strain harden at the same rate. However. when the lnlerlor hardens more rapidly than the surface. uB decreases. see Fig l?(a) (dashed line) and Fig I?(c).
the general decrease in up with increased surface removal suggested that the surface was behaving like a hard material. This behavior is in contradiction to the assumption of a soft surface layer and will be considered in the discussion. The behaviour of ua. corresponding to the three tests shown in Fig. 6(a) is revealed in Fig. 6(b). With the exception of curve 2 there is a continuous decrease in ~a. accompanying the increase in up, until a plateau is reached. For curve 2 there is an initial increase in ua in the first few cycles following the 24 h recovery period. This increase has been observed in the four specimens where the entire gage length was electropolished and must be considered genuine. Beyond about 6-10 cycles the data for the three tests scatter around a single curve, and the scatter is within the error of measurement. Figure 6(a) reveals that a plateau in crPis reached after about 1000 cycles in original and electropolished states but the plateau is reached after 100-300 cycles in the recovered specimen. The plateau in ua, Fig. 6(b), is reached after about 1000 cycles for all three conditions. In other tests of original specimens, the two plateaus are reached at about the same number of cycles, Fig. 5, and other recovered specimen show the relationship of Fig. 6(a. b). It should be noted Specimen 17 curve 1, Fig. 8. was subjected to tests l-3 of Fig. 6(a). before the tests of Fig. 8 were carried out.
that the plateau in ug for the recovered specimen !a@~ the plateau in up.
The assumption that the surface had a loner flo\\ stress than the interior. as noted earlier. imphed that after a tension half cycle there should be residual stresses, with the surface in compression and the interior in tension. After a compression halfqclc. the stresses would be reversed. This residual stress behaviour is depicted in Fig. 7. To test this assumption half the surface was electropolished. after 3500-4OOO cycles. Four tests were conducted and the results are given in Fig. 8. After a tension half cycle bending uould be expected to take place in a downward or negative direction, as demonstrated in Fig. 8. curves 1 and 3. and anticipated from Fig. 9(b). For a compression half cycle the bending would be positive. curves 2 and 4, Fig. 8 and Fig. 9(b). Of particular concern n-as ths possibility that the bending stresses due to possible misalignment might have caused the observed bending. To this end the polishing was carried out on the surface where the positive edge dislocations emerged. curves It and 2. where the negative edge dislocations emerged. curve 3, and where the screw dislocations emerged, curve 4. In each case bending occurred according to the prediction of Fig. 9. and nas mdependent of the position at which polishing took place.
728
WANG and MARGOLIN:
BAUSCHINGER
EFFECT IN BRASS
Fig. 8. Bending occurring when half the surface of the gage length was electropohshed.
Fig. 10. Dislocation structure in the interior.
The results of Fig. 8 are. thus. in accord with the assumption of a soft exterior.
system occurred. A typical slip appearance after 3500 cycles is revealed in Fig. IO.
SLIP OBSERVATIONS (.4 ) Opticol
Pnmary slip bands formed in clusters and the number of slip bands along the gage length increased with number of cycles to about 2000 cycles. With a larger number of cycles some cross slip appeared and occasionally isolated slip on a non-primary slip
Stop at Tension
(B) TEM
After 35004000 cycles. which corresponds to the plateau in 6,. Figs. 5 and 6a. the dislocation structure is not complex. However. a difference in dislocation structure was noted between surface and interior. Near the surface structures reminiscent of pile-ups were observed. Fig. 11. Within the interior at least two slip systems were operating. This can be seen in
Stop at Comprrsion
Fig. 9. Schematic illustration showing positive or upward and negative or downward bending during surface removal after a compression half cycle or tension half cycle. respectively. T = tension and C = compression residual stress. Shadowed region was removed from the specimen surface during polishing.
\+‘.4NG ind \1,4RG@LIS
BACSCHI5GER EFFECT IN BRASS
Fig. 1I. Dislocation pile-up near specimen surface
Fig. I? in which the different orientations of slip systems are present at A and B. In addition. within the interior jogs and kinks frequent]! appeared as did two-dimensional dislocation networks. A general survey indicated that dislocation densit! mcreased from surface to interior. However. no attempt at quantitative measurement uas undertaken. DISCUSSION
It had been proposed. as noted earlier. that the Bauschinger effect in single crystal beta brass [I] was due to a difference in flow stress betueen surface and interior. the surface having a loner floti stress. The same mechanism can be used to explain the decrease in CT~accompan!,in.g the increase in uP. Fig. 7. If it is assumed that the interior strarn hardens more rapid11 than the surface. then on unloading from a tensile cycle. the softer surface would be placed further into compression than when the surface and interior strain harden at the same rate. The larger residual com-
A
Fig. 12. Primar! slip band and cross slip in specimen 15 after 3500 cycles ar AC:2 = 0.15’ O.
‘29
pressive stress on the surface would produce the smaller value of uB. compare Fig. 7tb. c). According to Fig. 7 the value of Go depends onl! on the residual stress developed between the surface and the interior. Houever. if this were true. there should be an exact match between the number of cycles at which the plateaus in gP and CT~are reached. Since this matching is true onI\ for the original and electropolished states. but not the reco\,ered state. several other factors must be considered. Figure 3 suggests that dislocation motton occurs at different applied stress. For example. cycle 2 shoas deviation from linearity at loads whtch are considerably below the peak stress. where serrations take place. This indicates that dislocation motion occurs below the peak stress. as would be expected with a soft surface. The serrations occur because dtslocations are freed from their binding atmospheres. While dislocations are being freed of their atmospheres. other dislocations which have been freed to move undergo some hardening. Thus dislocatton release. strain hardening and recapture b) atmospheres occur in the same cycle at the surface and within the interior and the local stress whtsh the dislocations “see” varies according to the stress required for their movement. Local differences m flnu stresses develop. and these local stress differences also contribute to the Bauschinger beha\,ior. after WIloadrnp. These local stress t:ifferences occur at both surface and interior. and therefore. both regtons sdn contribute to Bauschinper behavior in addttion to the overall Bauschmger effect. caused by the difference in stress level between surface and interior. When the release of dislocations take place. during straining after the first half cycle after recovery. the dislocations. free of atmospheres. are available to produce the Bauschinger flow in reverse flow. and the residual stress on them is relatively low because the local difference in flow stress is small. This low residual stress raises the applied stress required to cause Bauschinper flow. This applied stress continues to increase after the first cycle as the number of dislocations which are freed- of their atmospheres increases. When the number of freed dislocations decreases. relative to those vvhich have been freed but have undergone hardening or recapture. the maximum in the Bauschinger stress is passed. and the process of decreasing Bauschinper stress with increasing number of cycles continues. Fig. 6(b). From the discussion above one would anticipate that serrated behaviour would be possible during the rise to approximately constant stress. curve 2. Frg. 3. Dislocations in the softer surface would also undergo capture by and release from atmosphere. Such serrations are seen in the hystereses curves but they are sufficiently small that they can not. with confidence. be Separated from the fluctations produced by the hydraulic system. Such suspicion 1s not applicable to the flat portion of the curve of cycle 2. in Fig 3. Another possible source for the difference tn num-
730
WANG and MARGOLIN:
BAUSCHINGER
ber of cycles at which bp and err,reach their respective plateaus may be related to the friction stress. It has been proposed that the friction stress increases during cycling [IO]. This would mean that dislocations which have experienced different amounts of flow on a given slip system would have different friction stresses and, therefore, different flow stresses. If this occurs during the present work, the difference in number of cycles at which the plateaus in up and ergare reached implies that a relatively uniform friction stress has not been reached until the plateau in ua is reached. (B) Harder surface than interior
As noted with respect to Fig. 6(a) the surface appears to behave as if it were hard, and yet the Bauschinger behavior suggests that the surface was soft. If the initially soft surface had subsequently become hard as a result of repeated cycling, then, instead of decreasing continuously with cycling, there would have had to be a reversal, an upward movement of ug. That the surface continued to be softer than the interior was manifested by the bending experiments, Fig. 8. However. the apparent pile-ups of Fig. 1I suggest a way of rationalizing the two apparently contradictory behaviors. This would require the existence of two conditions: (1) The presence just below the surface, approximately at 150 to 175 pm, of a narrow high dislocation density region which would inhibit the ingress of dislocations from the surface or exit from the interior. The choice of 150-175 pm as the position of the dislocation density peak is made because no further appreciable changes in up and no further bending occur when this much material is removed from the radius. (2) An average dislocation density of the surface which was below the average dislocation density of the interior. Pangbom et al. [6] have indicated that the surface has a higher dislocation density than the interior. The present qualitative results provide support only for dislocation density increasing away from the surface. However, a higher surface dislocation density can be accommodated in the present explanation as long as the average dislocation density of the surface region is below that of the interior. Thus, a schematic dislocation density would appear as shown in Fig. 13(a).
EFFECT IN BRASS
;6_ &_ _ a*
YJn
Fig. 13. (a) Schematic illustration of the dislocation density, p, change from surface to interior. (b) Dislocation density. p, rearrangement as the surface is being polished.
It is suggested that the dislocation arrangement is not fixed, and that when material is polished off the surface, the dislocations rearrange themselves as shown by the dashed line in Fig. 13(b). As polishing takes place dislocations from the peak region move inward and to the surface tending to raise the dislocation density in both regions. At the same time the reduction in the peak dislocation density would reduce the resistance of this area to dislocation passage and the peak stress would decrease. With increase in polishing the peak internal dislocation density disappears. With respect to bending during polishing Fig. 13 would be in accord with observations. The two peak dislocation densities of the surface and interior. as well as the interior region, would be in tension after a tension half cycle, while the region between the two peaks would be in compression. The volume average stress of the three surface regions would produce a net compression stress. Removal of the initial surface region would still leave the surface region in compression and the bending would continue until the low density dislocation region was almost entirely eliminated. Fabiniak and Kuhlmann-Wilsdord [B] have also reported a dual behaviour of a surface region in Al single crystals. They concluded that regions just below the surface acted as dislocation sources, while the surface acted as a barrier to dislocation motion. This behavior is not strictly parallel to the postulated behavior but is an indication that dual surface behavior is possible. (C) Calculation of surface and interior residual stress and surface and interior jaw stress
The surface region would thus consist of two high dislocation regions separated by a low dislocation area. It is assumed as noted above that the volume average of dislocation density of the three regions comprising the surface is below the dislocation density of the interior. Figure 13 also differs from the measurements of Panghorn er al. (61 by the assump-
It is possible to calculate the residual stresses in the surface and the interior from the measured deflection, the elastic modulus and the moment of inertia of specimen cross section. The calculations are illustrated in the Appendix for Specimen 20, Fig. 8, which was stressed finally in compression. The calculated residual surface stresses, uRs, and the calculated residual interior stresses, uRl, for specimens 17-20 are given in Table 1. The residual stresses were developed after 3000-4000 cycles but, according to Fig. 6(a).
tion of the internal
these stresses
peak in dislocation
density.
may have developed
after about
1000
WANG and MARGOLIN:
BAUSCHINGER
cycles. It can be seen from Table 1 that the absolute value of the residual surface stress is approximately 6.5 times the absolute value of the residual interior stress. From Fig. 7 and the determination of uRs and uR, it is possible to determine the peak values of internal flow stress, CT,.and surface flow stress, us. The average peak stress, u,,, can be expressed at constant strain as e*v =u,V,+usVs
(I)
where V, and V, are the volume fractions of internal and surface material. Since V, + V, = 1 U ,,=U,+(%-U,>VS
IUS
-
u1 1 =
(2) 1 URS -
uRl
(3)
1.
Equation (3) follows from the fact that the difference between u, and us is maintained when the applied stress is reduced to zero, Fig. 7(a). The calculated values of us and u, for specimens 17-20 are given in Table 1. For specimens 18-20 al/us = 1.41-1.55. However. for specimen 17 with the largest deflection ul/as = 2.01. The presence of a flow stress gradient is similar to the results obtained by Mughrabi [9] and Fourie [2]. The volume fraction of surface material was 0.46, corresponding to a surface depth of 0.49mm for a specimen 3.68 mm in diameter in single crystal Cu [9). The volume fraction and depth are much larger than were encountered in the present case. (D) Strain bursts
As noted earlier strain bursts in all but one case occurred below 1000 cycles after which no strain bursts took place. However. if 0.4 mm was removed from the diameter, the strain bursts would reoccur. This suggested the possibility that the strain bursts were associated with a breaching of the high dislocation density region postulated to exist at about 15S175 pm below the surface. The general absence of the strain bursts beyond 1000 cycles may have two possible explanations. It may be that the postulated high dislocation density below the surface becomes more impenetrable or that the dislocation structure presents a sufficiently strong pile-up to develop which could penetrate the postulated high dislocation density layer. No persistent slip bands or persistent Luders bands
731
EFFECT IN BRASS
[lo] were found and this may be due either to insufficient cycling to reach the platau where persistent slip bands are found [lo] or that the strain was too low. However, the strain bursts do not involve persistent slip bands. Some serrated yielding is found during the strain bursts and this behaviour, as seen in the hysteresis loops (Fig. 4). is quite similar to the serrated yielding seen when cycling begins on a fresh specimen or after recovery. The onset of strain aging during the first few initial cycles at the outset syggests that it is occurring during the testing. The serrations disappear. when strain hardening raises the flow stress above that required for serrated flow. In effect the higher strain hardened flow stress governs flow rather than the unpinning stress. (E) Duplex slip
Optical microscopy and TEM observation indicated that slip at the outer surface occurred essentially on a single slip system, whereas duplex slip was found in the interior. Table I shows the resolved shear stresses T, and ~1 for the primary and secondary interior slip systems. The resolved shear stresses are. of course, higher for the primary slip system. If strain hardening during the fatigue test were the same as in undirectional straining, then latent hardening would be expected to require a stress on the secondary slip which would be above the resolved shear stress on the primary slip system. The fact that the secondary slip system operates at a lower resolved shear stress implies that the friction stress on the primary slip system is higher than it is on the secondary slip system and that generally there is no interaction between the primary and secondary slip systems. Figure 12 supports this non-interaction suggestion.
SUMMARY 1. A series of 70-30 alpha brass single crystals was cyclically strained under strain control, with a total strain of de,/2 = 0.15%. The peak stress, up, and Bauschinger stress, bg, defined as the 0.00005 offset in reverse flow, were determined as a function of number of cycles, N. up increased and uB decreased as N increased, both stresses reaching a plateau after 1000-l 500 cycles.
Table 1. Calculated surface and interior residual stress and flow stress Specimen No.
No. of cycles
17 18 :‘o
4000 z3500
(d&c)
E (MPa)
0.27 0.18 0.16 0.19
5567 4100 4500 4452
=as
ras
-22.6 -8.9 11.6 5.6 -10.5 12.7 -4.7 5.9
%I
rat
3.5 -1.8 -2.0 1.6
-0.9 -0.9 0.7
1.4
@*v (MPa?
rs
01
rl
i,
48.5 25.9 10.2 52.0 20.5 18.4 -35.8 -24.2 -11.9 -37.6 -18.4 -18.0 -42.3 39.8 -29.6 29.3 -13.9 13.2 -44.3 41.4 -20.8 18.6 -18.6 15.9
NOW: Positive and negative values represent tension and comptassion stress, respectively. 6 = Deflection angle measured. E - Elastic constant along specimen aais. ass. oat = Residual stress in surface and interior. mspcctivcly. r asr rat = resolved shear stress on primary slip system in surface and interior, raspcctively. calculated from residual stress. o,,v = Measured average peak flow stress. 0s. V, = Plow stress acting in surface and interior. respectively. Q, I, = Resolved shear stress on primary slip system in surface and interior, respectively, calculated from flow stress cs and 0,. i, = Resolved shear stress on secondary slip system in interior calculated from flow stress 0,.
132
WANG and MARGOLIN:
BAUSCHINGER
2. After cyclically straining 3OOwOOO cycles and electropolishing to remove 0.34-0.4mm from the
diameter, up decreased and crs increased. Both values approached the original values reached after straining the virgin crystal one cycle. When lesser amounts of surface were removed, the changes in oP and ~a were reduced. This behavior was consistent with a harder surface. 3. It was proposed that the flow stress of the surface was below the flow stress of the interior. On unloading from the forward strain this flow stress difference would produce residual stresses which would be larger on the surface than the interior because of the smaller volume fraction of the surface. It was also proposed that, as N increased, the flow stress of the interior increased more rapidly than that of the surface. To check these assumptions half the gage length surface was electropolished along the length and the accompanying deflection were measured. The deflections were independent of where along the gage length electropolishing took place and agreed with the assumptions, thus indicating that the surface material was softer than the interior. It was possible to calculate residual surface and interior stresses and surface and interior flow stresses from the deflections which occurred during polishing, the volume fraction of surface and interior and the elastic modulus. 4. The apparent contradition in items 2 and 3 was rationalized by assuming the presence of a surface high dislocation density layer and a subsurface layer 150-l 75 pm below the surface. This distance corresponded to the point of maximum deflection during bending tests and to the amount of material required to be electropolished from the radius to restore the specimen properties almost to the original values. It was also assumed that the average dislocation density of the surface, including outer surface and subsurface high density region, was below that of the interior. This lower average dislocation density would permit the surface to have a lower average flow stress than the interior, and the sub-surface high dislocation density region served as an obstacle to the outward movement of interior dislocations, thus maintaining the flow stress at the interior at a higher level than the surface. TEM observations indicated that the dislocation density of the interior was higher than that of the surface. Dislocation pile-ups near the surface were taken as evidence of the postulated high dislocation subsurface area. No direct TEM evidence of this high dislocation density area was obtained. 5. Strain bursts were observed below 1000 cycles, but not above this value. It was suggested that strain bursts occurred when the postulated high dislocation density area was breached.
EFFECT IN BRASS REFERENCES
1. H. Yaguchi and H. Margolin, Scripra merall. 17, 1213 (1983). 2. 1. T. Fourie, Phil. Mag. 17, 735 (1968). 3. J. T. Fourie, Phil. Mag. 21, 977 (1970). 4. H. Mughrabi, Physica starus solidi 39, 317 (1970). 5. H. Himstadt and N. Neuhiiuser, Scripra melaN. 6, I 151 (1972). 6. R. N. Pangborn, S. Weissman and M. Wilken, Me/a//. Trans. A 12A, 109 (1981). 7. T. Ungar, H. Mughrabi and M. Wilkens, Acfa merall. 30, 1861 (1982). 8. R. C. Fabiniak and D. Kuhlmann-Wilsdorf, Enrironmeni-Sensitive Mechanical Behavior (edited by A. R. C. Westwood and N. S. Stoloff), p. 147. Gordon & Breach, New York (1966). 9. H. Mughrabi Surface E&cu in f&to/ Plasriciry (edited by R. M. Latanison and J. T. Fourie), p. 533. Series E Appl. Sci. NO. 17. Nordhoff-Lender, Reading, Mass. (1977). IO. C. Laird and L. Buchinger. Proc. Symp. Comm JO Anniv. Inrroduction of Dislocations, Detroit, Mich. (1984). Metal/. Trans. A. To be published. APPENDIX I. CALCULATION OF SURFACE AND INTERIOR STRESSES (A) Estimate of volume fraction of surface and interior
An estimate of the volume fraction of the surface region can be made from the observation of the amount of material which must be removed by electropolishing to obtain no further changes in behavior. From Fig. 7, Specimen 20, this is taken to be approximately 0.35 mm from a 5 mm diameter specimen. This corresponds to a volume fraction of 13.5%. (B) Assumptions
It is assumed that the stresses in the surface and interior are uniform, that is, average stresses. It is also assumed that bending begins at the end of the gage length nearest the left grip, as shown in Fig. Al(a). The moment of inertia is
b) ,&knowledgemenrs-The authors wish to acknowledge help ful discussions with Professors S. Nourbakhsh, J. P. Hirth, R. Asaro and C. Laird. This work was supported under National Science Foundation Grant NO. DMR8312963.
Fig. Al:Specimen configuration used to calculate bending stress. Cs-shape center of surface semicircle layer; C-shape center of interior section; Ccsshape center of interior section; Cca-shape center of whole cross section after half surface is removed.
WANG and MARGOLIN:
BAUSCHINGER
calculated from the cross Section A-A shown in Fig. Al(b). The shape center lies 0.11 mm above the center of the figure. The force couple is also shown in Fig. Al(c). The shape center of the surface shown is indicated at C,, and the shape center of the interior at C, Fig. Al(c). (C) Culculution procedure The deflection, Y, is given by the following expression YZg
(Al)
where E = elastic constant along axis of Specimen 20, 4452 MPa, x = the length of specimen over which bending takes place, Fig. Al, 15 mm, I = moment of inertia, M = moment of force couple, Y = x .tan 0, where 8 is the measured deflection angle, Fig. 7, for Specimen 20, 6 =0.095’. In the present case, I, the moment of inertia for the whole crossection about the horizontal axis passing through the shape center, can be expressed as I = I’ - ; (I: + r:,c;.
642)
Here rL = the radius of the original cross section [see Fig.
733
EFFECT IN BRASS
Al(b)], rs = the radius of the cross section after polishing [see Fig. Al(b)]. C, = distance from the specimen center to the shape center which is 0.11 mm [see also Fig. Al(b)]. I’ = the moment of inertia of the whole cross section about the horizontal axis passing the specimen center. I’ = I, + I,. I, and I, are the moments of inertia for the semi-circles. one with the radius rL, the other with rs. respectively, both are about the horizontal axis passing the specimen center. And I’ was determined to be 26.6mm’. M is determined from equations (Al) and (A2) and is 516N.mm. This moment of force is illustrated in Fig. Al(c). M = o,,~A,~I
= o,,,A,,I
(A3)
where cks and ua, are the surface and internal residual stresses, respectively; A, and A, are the areas over which uRS and ~a, are distributed, respectively, I = C,-
C, = lS4mm
[see Fig. Al(c)].
(A4)
From equations (A3) and (A4) uRS and uR, can be determined oRs = 12.7 MPa,
rra, = - 2.0 MPa.
All the other calculated results are given in Table 1.