NUCLEATION OF DEFORMATION TWINS IN ZSNC BIC RYSTALS Thesis hr the Degree of Ph. D. Ml-CMGAN STATE UNIVERSITY Chi Kwun Chyung 196.5 LIBRARY Michigan St!" University This is to certify that the thesis entitled NUCLEATION OF DEFORMATION TWINS IN ZINC BICRYSTAIS presented [)9 Chi Kwun Chyung has been accepted towards "fulfillment of the requirements for PhD degree in Metallurgy Date $31.44 J3 / '9 6-K“ \ 0-169 ABSTRACT NUCLEATION OF DEFORMATION TWINS IN ZINC BICRYSTALS by Chi Kwun Chyung The nucleation of twins has been investigated in zinc bi- crystals of various orientations subjected to tension. The resolved shear stress for twinning was found to vary from 25 to 4,900 g/mm2 depending upon the orientation of the bicrys- tals as well as the relative orientation of the adjacent crystal grains. The maximum possible stress concentration associated with the lowest observed r.s.s.t. has been esti- mated tx: be 12-15 kg/mm% which is one order of magnitude smaller than the theoretical value based on the homogeneous shear model but approximately equal to the stress required to form two coherent twin interfaces or a twin fault. A mechanism of twin nucleation based on the movement of existing twin dislocations in a twin embryo associated with a slip dislocation may be suggested. The necessary stress con— centration required to cause the twin dislocations to move until a twin lamella of critical size of approximately 700 X is formed can be provided by a piled-up group of dislocations against the grain boundary. It appears that the twin fault energy of 2.7 i 0.7 ergs/cm2 is the controlling factor in the nucleation of twins and a Chi Kwun Chyung shear stress of 14 i 5 kg/mm2 acting on the twin plane in the twinning direction can be taken as the critical r.s.s.t. in zinc crystals. NUCLEATION OF DEFORMATION TWINS IN ZINC BICRYSTALS BY Chi Kwun Chyung A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Material Science 1965 ACKNOWLEDGEMENTS The author wishes to express gratitude to Dr. C. T. Wei who suggested the problem and gave valuable guidance and counsel throughout the project. Sincere thanks are also due to the members of his guidance committee, Dr. A. J. Smith, Dr. L. E. Malvern and Dr. D. Montgomery. Finally, he would like to thank Mr° J. W. Hoffman and the Division of Engineering Research, College of Engineering, Michigan State UniVersity. This research has been made possible through a grant from the National Science Founda- tion G-l9652° ii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . . . CHAPTER I INTRODUCTION . . . . . . . . . . . . . . . CHAPTER II EXPERIMENTAL PROCEDURE . . . . . . . . . . 2.1 Growth of Zinc Bicrystals . . . . . . . . . . 2.2 Preparation of Tensile Specimens . . . . . . 2.3 Specimen Orientation . . . . . . . . . . . . 2.4 Tensile Testing . . . . . . . . . . . . . . . CHAPTER III RESULTS . . . . . . . . . . . . . . . . . 3.1 Photomicrographs . . . . . . . . . . . . . . 3.1.1 Bicrystals with orientation S—G . . . . 3.1.2 Bicrystals with orientation H . . . . . 3.1.3 Bicrystals with orientation S—H-G . . . 3.1.4 Nucleation of twins at a twin interface 3.2 Tensile Data . . . . . . . . . . . . . . . . 3.2.1 Stress—strain curves . . . . . . . . . 3.2.2 Tensile stress data . . . . . . . . . . CHAPTER IV DISCUSSION . . . . . . . . . . . . . . . . 4.1 Effects of Orientation . . . . . . . . . . . 4.1.1 Modes of deformation by slip . . . . . iii Page vi 10 10 11 12 13 21 21 21 27 3O 32 34 34 35 67 67 67 4.1.2 Accommodation of slips in adjacent grains at the grain boundary . . . . . , , ‘70 4.2 Effects of Prior Slip and Dislocation Pile—ups. . 75 4.2.1 Dislocation pile-ups on the basal slip plane . . . . . . . . . . . . . . . . 75 4.2.2 Dislocation pile-ups on the pyramidal slip plane . . . . . . . . . . . . . . . . 83 4.3 Effects of Relative Orientation . . . . . . . . . 84 4.4 Stress Concentration and the Energy to Form a Twin Fault o o o o o o o o o o o o o o o o _ 93' 4.5 Twin Nucleation at the Interface of an EXiSting Win 0 O O O O O O O C O O O O O O O O O 954' 4.6 A mechanism of Twin Nucleation . . . . . . . . . 98 4.6.1 Twin embryo . . . . . . . . . . . . . . . . 99 4.6.2 The size of the stable twin lamella . . . . 102 4.6.3 The stress required to form the critical twin lamella . . . . . . . . . . . 103 4.6.4 A model of twin nucleation . . . . . . . . 105 CHAPTER V CONCLUSION . . . . . . . . . . . . . . . . . . 114 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . 116 APPENDIX A STRESSES ASSOCIATED WITH A DISLOCATION PILE-UP . . . . . . . . . . . . . . . . . . . 121 APPENDIX B THE EFFECTS OF GRAIN BOUNDARY ON THE PLASTIC DEFORMATION BY SLIP . . . . . . . . . 124 iv LI ST OF TABLES Tables Page I Summary of Tensile Data . . . . . . . . . . . . . 65 II OT, TT and za- 0 o o o o o o o o o o o o o o o o o 66 III OT, 7% and 7; . . . . . . . . . . . . . . . . . . 66 LIST OF FIGURES Figures Page 1. Furnace setup . . . . . . . . . . . . . . . . . 16 2. a) Zinc blank (mm) . . . . . . . . . . . . . 17 b) Tensile specimen (mm) . . . . . . . . . . 17 c) A bicrystal specimen ready for tensile testing . . . . . . . . . . . . . . . . . 18 3. Specimen curing jig . . . . . . . . . . . . . . l9 4. Bicrystal orientations and their designations . . . . . . . . . . . . . . . . 20 5. Nucleation of twins in specimen No. 19 (S-lO) at a r.s.s.t. of 200 gr/mmz. 100 X . . . . . 37 6. Nucleation of twins.in specimen No. 19 (S-10) 80 X C C O O O O O O O O O O O O O O O O O O 37 7. Nucleation of a twin in specimen No. 20 (8-20) at a r.s.s.t. of 90 gr/mm2. 100 X . . . . . 38 8. Grain boundary distortion in specimen No. 20 (S-20) after 30 o/o elongation . . . . . . . 38 9. Formation of slip bands associated with the grain boundary distortion in specimen No. 20 (8-20) . 200 X C O O O O O C O O O O O C O 39 10. Nucleation of a twin in specimen No. 3 (8-30) at a r.s.s.t. of 25 gr/mmZ. 100 X . . . . . 40 ll. Nucleation of twins in specimen No. 3 (8—30) after 34.3 o/o elongation. 100 X . . . . . . 40 12. Nucleation of a twin and grain boundary distortion in specimen 35 (S—30). 250 X . . . 41 vi 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. A typical region near the grain boundary in specimen No. 21 (S-45) a) After 10 o/o elongation. 100 X b) After polishing. 100 X . . . . . . . Twins and grain boundary distortion in specimen No. a) after 21.4 o/o elongation. 100 X 3 (8-30) b) After polishing. A cleavage fracture formed in specimen 100 X No. 21 (S-45) after 35 o/o elongation . Accommodation of slip at the grain boundary after 11 o/o in specimen No. elongation Nucleation of twins in specimen No. 8 (Ha) 37 (8—60) at a r.s.s.t. of 2.56 kg/mmz. Specimen No. a) The pyramidal slip traces on the (0001) plane. b) Etch pip pattern revealed on the (0I10) plane. 26 (Ha) 100 X 100 X 100 X Nucleation of twins in specimen No. at a r.s.s.t. of 4.9 kg/mmz. Twins and the pyramidal slip traces on the 100 X 27 (Hb) (0001)B plane in specimen No. 27 (Hb). 100 X. The pyramidal slip traces on the (0001) plane 27 (Hb). in specimen No. Specimen No. a) Etch pip pattern revealed on the (I010) plane. 27 (Hb) 60 X vii 100 X 42 43 44 45 46 47 47 48 49 50 51 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. b) Same as (a). 200 X Specimen No. 22 (S—Ha-45) a) Nucleation of twins at a r.s.s.t. of 1.35 kg/mmZ. 100 x . . b) Same as (a). Polished. 100 X c) Growth of the twins shown in reloading . . . . (a) upon Nucleation of twins in specimen No. at a r.s.s.t. of 1.93 kg/mmz. 100 Nucleation of twins in specimen No. at r.s.s.t. of 1.70 kg/mmz. The basal slip in specimen No. after 1.8 o/o elongation. 70 X 70 X 31 (s-Hb—3o) x 33 (S—Hc-45) 33 (S—Hc-45) Nucleation of a twin at a twin interface in specimen No. 6 (H; single crystal). 100 X . Nucleation of twins along a twin interface in specimen No. 8 (Ha). 100 X . . Nucleation of twins along the interfaces of massive twins in specimen No. 100 X C O O O O O O O 31 (S-Hbi-BO) . Slip traces associated with the growth of a twin in specimen No. Stress-strain curves of orientation 8-9 0 o o o Stress-strain curves of at various strain rates Stress-strain curves of orientation Ha and Hb Stress-strain curves of orientation S-Hb-G . . viii 100 X 0 the bicrystals with specimen No. 3 (s-3o) the bicrystals with the bicrystals with 52 53 53 54 55 56 56 57 57 58 59 6O 61 62 63 Stress-strain curves of the bicrystals with orientation S—Ha—45, S-Hb-45, S-Hc-45 . . . . . 64 Schematic representation of slip bands in a symmetrical bicrystal of Zn and the formation of steps along the grain boundary . . . . . . . 71 The transferred shear stress due to the pile-up of dislocations in the adjacent grain . . . . . 73 Continuous distribution of edge dislocations in a pile-up and the coordinate system . . . . . 80 The stress field Ox in units of 75 for an edge dislocation piIe—up situated between x/a = -1 and 0 . . . O O O O O O O O O O O O O O 81 The stress field Ox in units of G/400fll1-V) of a single edge dislocation . . . . . . . . . . 82 Superposition of the shear of grain A onto grain B O O O O O O O O O O O O O O O O O O O O 86 (Sine cose) Nil vs 8 . . . . . . . . . . . . . . 89 Reference twin systems . . . . . . . . . . . . . 9O Schematic model of twin nucleation at the grain boundary . . . . . . . . . . . . . . . . . . . . 106 The coordinate systems for the transformation of the stress fields of the edge dislocation E to (10I2)[1OII] twin system in grain B in the bicrystal with orientation S-30 . . . . . . . . 110 The growth of the twin embryo in the bicrystal with orientation S-30 . . . . . . . . . . . . . 113 ix I INT RODUCT I ON Deformation twinning is one of the two fundamental modes of plastic deformation of crystalline solids. The ductility of metals, however, stems mainly from slip, and twinning plays only a minor role in all but a few special cases. Consequent- ly, the study of twinning has been somewhat neglected in favor of slip, and a gap has been developed in the understanding of the two deformation processes. It is therefore desirable that the knowledge of twinning be brought to the level that has been achieved in the area of deformation by slip. Deformation twinning is a process by which a portion of a crystal lattice becomes reoriented as a result of a coopera- tive shear of neighboring planes over one another a definite fraction of an interatomic spacing. The sheared portion of the lattice bears a definite orientation relationship with respect to the matrix; they can be mirror images of each other across the twin plane. Such a twinning process can occur only when a proper shear stress is acting on the twin plane in the twinning direction. Some important problems associated with the deformation twinning are i) the choice of the twinning elements, ii) the validity of a critical shear stress criterion for twinning, iii) the dynamics of twinning which consist of both the nu- cleation and the growth processes, iv) the relationship of twinning to brittle fracture, etc. In the present work efforts will be concentrated on the experimental investiga- tion of the stress conditions under which twins are nucleated in zinc in the hope that an explanation can be arrived at for the nucleation mechanism of twinning in such a hexagonal close-packed metal. It is known that deformation twinning occurs in all three most common metallic structures under proper loading condi— tions and at proper temperatures. A high strain rate and a low temperature promote the tendency of a crystal to undergo deformation twinning (l, 2, 3). Although there had been some evidence of deformation twinning in Al under impact loading (4), it was not until 1957 when a definite proof of deforma- tion twinning in f.c.c. metals was obtained (1, 5). The practical importance of the twinning process rests on the following facts: i) Twinning is capable of producing a lattice orientation such that further deformation by slip becomes possible. ii) Twinning can be the primary mode of plastic deforma- tion in some metals and alloys. iii) Under special loading conditions and at certain- temperatures the tendency for a metal to twin may in- crease. iv) The evidences of the interplay between twinning and brittle fracture are mounting. Such practical importance, together with the theoretical im- portance of twinning being one of the two deformation pro- cesses, might have led to the recent resurgence in the study of twinning. Various mechanisms have been proposed to account for the nucleation of twins in different crystal structures. A brief review of the important mechanisms follows. Pgle Mechanism Cottrell and Bilby (5), and Thompson and Millard (6), independently proposed the corresponding pole mechanisms for the b.c.c. and the h.c.p. structures. A twin can be formed in the b.c.c. structure by a shear of a/6 on successive {112} atomic planes. Consider a perfect dislocation line with a Burgers vector of a/2[lll] lying in the (112) plane. If sufficient energy is available from external sources, the a/2[lll] dislocationmay dissociate into a sessile dislocation and a twin partial on the (112) plane a a a - $1111.. §[112] + E—_)[111] Then the twin partial a/6[1lI] can glide onto (I21) plane and sweep around the node with the a/2[111] dislocation as a pole. Since the a/2[lll] dislocation has a component perpendicular to the (I21) plane equal to the spacing (= a/6[I21]) of these planes, the sweeping dislocation thus climbs one layer along the pole for each revolution, and thereby creates twinning on successive layers. The pole mechanism requires a screw dis- location (i.e. the pole dislocation) whose Burgers vector has a component perpendicular to the twin plane,which is the same as, or some multiple of, the spacing of the planes, and a twin partial which sweeps around the pole dislocation. Thus the dissociation of one suitable dislocation can account for the production of the successive layers of a twinned volume. Similarly, Thompson and Millard proposed that in h.c.p. structurezn1[0001] dislocation, which has a component ac/37(3a2 + c2)1/2 perpendicular to the twin plane equal to twice the spacing of the (10I2) twin plane, serves as the pole for two twin partials or a double twinning (zonal) dislocation. An [0001] dislocation, upon meeting with an existing twin interface, transforms into the double twinning dislocation which is a step in the interface. The residue of such a dis- location reaction appears to be two basal dislocations in the twin. The growth of a twin is explained by the repeated passages of such twin dislocations. Therefore, the pole mech— anism in h.c.p. metals as proposed by Thompson and Millard can only account for the growth of a twin, whereas the pole mech- anism can account for the nucleation as well as the growth of a twin in b.c.c. metals. In the latter case the dissociation of a/2[lll] dislocation can be regarded as the nucleation and the sweeping of the a/6[1lI] twin partial as the growth of a twin (7). Venable's Mechanism for F.C.C. Metals The pole mechanism proposed by Cottrell and Bilby could only account for the formation of a monolayer twin in f.c.c. metals. A.unit dislocation a/2[110] can dissociate on the (I11) or (lIl) planes in the following way a/2[110]—-a/3[lll] + a/6[112] where a/3[lll] is a sessile dislocation and a/6[ll2] a twin- ning dislocation. The rotation of the twinning dislocation about the node forms a one-layer twin, a stacking fault. After one revolution, however, the dislocation would reunite with the a/3[1ll] sessile dislocation. Repetition of this process on the same plane would not be possible, because a forbidden stacking sequence of either 2311231 or 2312231 would be produced otherwise. This picture was used for explaining the absence of twin- ning in f.c.c. metals,until copper was found to be capable of twinning at low temperatures (1). Venable (8, 9, 10) then rationalized these experimental results by allowing the re- united a/2[110] dislocation to glide from one (111) plane to the next between revolutions of the twinning dislocation. Homogeneous Shear Mechanism Orowan (13) has considered the formation of twin nuclei and concluded that they would form at a region of stress con— centration where the theoretical twinning stress might havelxxni reached. In their study of thin wires of Zn single crystals, Bell and Cahn (14) found that the nucleation of twins occurred only after both the pyramidal and the basal slips had taken place. They suggested that the necessary stress concentra- tion was caused by sessile dislocations (Lomer-Cottrel type) resultumgfrom the interaction of the basal and the pyramidal slip dislocations. They further reasoned that a twin is created as a whole by a locally homogeneous shear of the lat- tice and not by the progressive motion of a dislocation. The mean value of the resolved shear stress for twinning (r.s.s.t.) was found to be 3.5 kg/mm2 and hence it would require a stress concentration factor of approximately 35 in order to reach the theoretical shear stress (G/2x)y & 120 kg/mmz, where y is the twinning shear and G the shear modulus. The pile-up of dis- locations against a sessile dislocation on the pyramidal slip plane was regarded to be necessary to raise the local stress to the theoretical value. The necessity of dislocation pile-ups for the nucleation of twins was demonstrated by Wei (15). He deformed a small quasi-whisker crystal (20 u in diameter) of zinc of "hard" orientation at high stresses to 20 o/o strain, but found no sign of twinning. Wei then contended that in such a small crystal the dislocations are free to move under stress and hence attributed the result to a lack of dislocation pile-ups. Price (11) observed that the nucleation of twins in dis- location-free whiskers of zinc took place at an average shear stress Of 50 kg/mm2 without prior slip, and that the twins were nucleated in the crystals at re-entrant corners or at the specimen grips in the absence of any dislocations. He also found that the twins propagated at shear stresses ranging from 2 to 20 kg/mm2 without an operating pole mechanism, and suggested that the growth of the twins was a result of the repeated nucleation and movement of twinning dislocations. It may be seen from the above review that the twinning behaviors of thin sections or dislocation-free whiskers can be quite different from those of bulk crystals. In the former case an effective stress concentration by dislocation pile-ups and a dislocation configuration required by the pole mechanism are probably absent. Twins are nucleated only at high re- solved shear stresses close to the theoretical value. The homogeneous shear mechanism may be favored. In the latter case the lack of a consistent critical resolved shear stress for twinning may point to the possibility that a stress con- centration factor may vary from one experiment to another. Furthermore, the existing experimental observations do not seem to be able to establish either of the proposed mechan- ismscfiftwin nucleation and exclude the other. Additional ex- perimental work is necessary in order to shed some light on the mechanism of twin nucleation. Since a direct observation of twin nucleation on an atomic scale is only possible in thin sections with electron trans— mission microscopy, the nucleation of twins in bulk crystals can be investigated only indirectly by a systematic study of the twinning behavior of the bulk crystals as a function of such variables as temperature, orientation, stacking fault energy, and strain rate. By comparing the experimental results with the type of dependence of the twinning behavior on such variables based on theoretical considerations one may hope to find a clue to the solution of the problem of twin nucleation. In the present work the effect of stress concentration at the grain boundary on twinning in zinc bicrystals is investi- gated systematically as a function of the relative orientation of the individual crystal grains. II EXPERIMENTAL PROCEDURE 2&1_ Growth of Zinc Bicrystals Zinc (99.99+ pure) bicrystals were grown from the melt by using a modified Bridgman technique (26). As shown in Fig. 1, a graphite crucible is held stationary between two heating ele- ments and subjected to a constant temperature gradient. Heat— ing and cooling of the furnace are programmed and automatically controlled. A heat conduction sleeve surrounds the crucible and is thermally insulated to reduce the radial heat loss. Dry nitrogen gas is introduced to provide a non-oxidizing at- mosphere. Single crystals grown in this arrangement were found to have dislocation densities ranging from 4 to 9 x 104 cm-Z. Etch pip study (27) and Pseudo-Kossel patterns taken with the crystals showed that, except some local polygonization with a disorientation ranging from 5 to 25 seconds of arc, sub—boun- daries could be eliminated by using an off-set method (28). However, no attempts were made to eliminate sub-boundaries in the bicrystals used in this work. The total dislocation den- sity in the crystals was estimated to be of the order of 105 cm-2, In order to grow bicrystals in the desired orientations, 10 11 zinc blanks (2.5 x 13 x 100 mm), as shown in Fig. 2a, were machined from rolled slabs and two seed crystals were welded to their bottom. 2.2 Preparation of Tensile Specimens The grown bicrystals were then cut with an acid saw to an appropriate size. Tensile specimen holders (made of Al alloy 2017) were cemented to both ends of a bicrystal. The cement was prepared by mixing Armstrong Epoxy Resin C-4 with activa- tor D at a volume ratio of 4 parts to one. A special jig as shown in Fig. 3 was used for holding the crystal and the ten— sile specimen holder in alignment during a curing operation of one hour at 2000 F. Fig. 2b shows a prepared tensile specimen and its typical dimensions, and Fig. 2c shows a bicrystal specimen ready for tensile testing. Stainless steel pins were used for connect- ing the specimen to the loading fixture of an Instron testing machine. The cemented bond between the zinc and aluminum metals appeared to be strong enough for the subsequent tensile test- ings. Never once in the course of the tensile testing the bond failed. Prior to the tensile testing, back—reflection Laue patterns were taken with the bicrystal specimens to determine 12 their orientations. They were then chemically polished with the solution suggested by Vreeland et al. A special fixture was used to handle the tensile specimen during the polishing operation in order to avoid any accidental damage. 243_ Specimen Orientations Fig. 4 shows the three different groups of bicrystals hav— ing specific orientation relationship with respect to the ten- sile axis. The first group consists of symmetric bicrystals in which the two individual grains are mirror images of each other across the grain boundary. Both grains have the "soft" orientation, henceforth designated as orientation 8-8. The angle 8 between the tensile axis and the basal plane of each individual grain varies from 100 to 600. Bicrystals with five different 8 angles (10°, 20°, 30°, 450, 600) were grown and will be referred to in the future as S-lO, 8-20, ..., S—60. The broad surfaces of this group of bicrystals are parallel to the {1100} plane. The second group consists of two bicrystals in the "hard” orientations, henceforth designated as orientation H. The ori- entation relationship between the two grains in each of the two bicrystals is a rotation of 900 of one grain with respect to the other about the tensile axis. Both grains have their basal planes parallel to the tensile axis; i.e. 8 = 0 . Two 13 of these isoaxial type bicrystals, designated as Ha and Hb, have respectively their [2110] and [1010] directions parallel to the tensile axis. The last group consists of bicrystals each of which is a combination of two crystal grains respectively in the soft and hard orientations. The first two types of bicrystals of this group are designated as S-Ha-Q and S-Hb-G, depending on whether the broad surface of the grain A is parallel to the basal plane or to (0110) plane. The tensile axis of grain A in both types is parallel to the [2110] direction. The third type of bicrystal, designated as S-Hc-8, has the broad surface of grain A parallel to (1210) plane, and the tensile axis in the [I010] direction. When 6 = 45°, the bi— crystal (S~Hc-45) has a relative orientation of the two crys— tal grains such that the direction of the basal slip in grain B is nearly in the twinning direction of grain A. 2.4 Tensile Testinq An Instron machine with a typical cross head speed of 0.005 cm/min and a chart speed of 2 cm/min was used for the tensile testing. This cross head speed corresponds to a strain rate of approximately 3 x 10—5 sec-1 for a gauge length of the specimen of 25 mm. Occasionally strain rate of ten to one 14 hundred times greater than this value was used in order to investigate its effects on the deformation behavior of the bicrystals. During the tensile testing the specimens were at times unloaded for observations of the slip traces with an optical microscope. The specimens were reloaded following each examination until the onset of twinning. Such an interrupted loading method was also used for the study of the subsequent twin growth. In order to prevent the nucleation of twins at the ce- mented region of the specimen it was necessary to constrict the center portion of the specimens, especially the ones in the hard orientation. The cross sectional area of the con- stricted portion was made less than 60 o/o of that of the head portion to ensure the nucleation of twins to take place within the gauge length of the specimen. Further microscopic examinations were made immediately following the nucleation of twins. The subsequent growth of the twins was studied by reloading the specimen after pol— ishing. The polishing enhances the appearance of the original twin traces which can be clearly distinguished from any fresh ones whose movements are the direct results of the reloading. This technique was found to be very useful in determining 15 the stress level at which the growth of existing twins took place, and in detecting new twins formed upon reloading. It is also useful in studying slip traces whidh are associated with the twin growth, since the polishing removes all the previous slip traces. 16 Alumel wire Zinc specimen IOO mm long duction sleeve Graphite crucible T/C control T/C differential insulator :1 :2: ' T/C differential Alumina powder Seeds 25mm long \— Zircc tube 118T. Thermal blocks ill N2 Fig. l. Furnace setup (schematic) h u‘ .1 #— 5—HT”? t“. “l”— Jib—‘11 \ 01 e. ‘h T .4 u . I? i" I e— E fli— “’0 44' 2J: 8 ‘— “hf/co _1 12 '] _ _ 17 28 "” ‘. 7:_ l £3 8 - -. i L 45“" L i -.-] i-z.5 L—zz __ V I IO‘ Fig. 2 a) Zinc blank (mm) b) Tensile specimen (mm) 18 Fig. 2-c. A bicrystal specimen ready for tensile testing 19 Fig. 3. Specimen curing jig 20 o 0 6 5 A. O 3 0 2 m .. 9 e _ S \./ _ \ \Q " Ac. .4 +- ,x 6 _ / . lint. Ir \ a \O _ e . u: in W A B _ _ e llllll IL/ 3 H n K/ 81. II _ a \\Q _ 9 III II JIIII b ‘llll Ill 0%. I I ‘lcPIAI lam .. 6 _ H . e _ _ .. iIIIL/ xii -IIIF S B _ a \Q n I. IIIII - \ 9 e . . l, H Jl .. o _ . o . llllll If; .Illlil L/ S. Flg4 Bicrystal orientatations and their designations III RESULTS 3.1 Photomicrographs QLILI Bicrystals with orientation S-8 All bicrystals with orientation S-8 deformed easily under uniaxial tension by {0001}<2I10) type slip. Nucleation of {1012}<10II> type deformation twins took place exclusively at the grain boundary, and was characterized by low resolved shear stresses for twinning (25 gr/mm2 to about 200 gr/mm? depending on the angle 8), and by relatively large amount of elongation prior to the nucleation of twins (ranging from, several to tens of o/o elongation). Load relaxation upon twinning was in general less than 1/10 of the twinning load, and the twins formed were small and narrow. The bicrystals with 8 = 450 and 600 did not twin under the uniaxial tension, whereas the bicrystals with 0 = 10°, 20°, 300 twinned extensively. The stress and elongation were recorded in each case. Subsequent reloading produced a large number of twins at a higher load level. i) Bicrystals with orientation S-lO Fig. 5 shows the twins nucleated in specimen No. 19 which had the angle 8 = 12°, They were nucleated at a resolved 2 . shear stress for twinning (r.s.s.t.) of 200 gr/mm in the most 21 22 favorable twin system after 3 o/o elongation. The same crys— tal was polished and reloaded to approximately 10 o/o elonga- tion. More twins were nucleated at the grain boundary as shown in Fig. 6. A set of traces making an angle of approxi- mately 500 with the grain boundary can be seen in grain A. These traces on (OIlO) plane make an angle of 620-630 with the basal slip traces. Although it is not possible to deter— mine from one set of traces the slip system which is respon- sible for its formation, the most favorable pyramidal slip sys- tem is able to produce these traces. Upon careful examination a faint set of similar traces can also be seen in grain A near the grain boundary in Fig. 5. It is also observed in Fig. 6 that the grain boundary is considerably distorted after 10 o/o elongation. In grain B, fine slip traces are grouped into coarse slip bands which extend to the grain boundary. Slip bands formed in grain A are less extensive and do not extend to the grain boundary. The width of the slip bands ranges from a few to about 30 u, and the spacings between the slip bands are of the order of 10-60 M. The resolved shear stress on the basal slip system at the 2 point of the first twin nucleation was approximately 100 gr/mm . 23 ii) Bicrystals with orientation S-20 The nucleation of a twin occurred in specimen No. 20 at a r.s.s.t. of 90 gr/mm2 after 12.3 o/o elongation. At this point the resolved shear stress on the basal slip systems was approximately 100 gr/mmz, The specimen was then polished and reloaded to a total strain of about 20 o/o. Fig. 7 shows the twin formed after the first loading (dark area) and how it widened during the second loading. Two sets of slip traces can be seen inside the widened region of the twin. Further deformation of the specimen to 30 o/o elongation increased the intensity of the basal slip traces and the grain boundary distortion as shown in Fig. 8. At this point the resolved shear stress in the basal slip system is approxi- mately 200 gr/mmz. A second set of slip traces in addition to the basal slip traces appears in grain B near the grain boun- dary. These traces make an angle of approximately 620 with the basal slip traces and are in agreement with the most favorable pyramidal slip system. The traces extend to approxi— mately 0.2-0.3 mm away from the grain boundary. The spacing between these traces is approximately 5-10 u. Such a set of traces is also visible above the twin near the grain boundary in Fig. 7. Similar traces are found in different regions of the same 24 crystal. Fig. 9 shows a set of traces making an angle of approximately 640 with the basal slip traces at a higher magnification of 200 x. It also shows more clearly the grain boundary distortion due to the basal slips in the adjacent grains. The steps along the grain boundary have heights of the order of 20-30 u and appear to be directly related to the formation of the slip bands. This is clearly seen at the left hand side of the grain boundary near the top of Fig. 9. The originally straight grain boundary is distorted severely by the protrusion of regions of 20—30 u in dimension into the neighboring grain. Such a formation of steps is par- ticularly pronounced in the bicrystals of orientation S-8 with the angle 8 ranging from 200 to 35°. iii) Bicrystals with orientation S-30 In these crystals, twins are nucleated at the lowest r.s.s.t. among all the bicrystals tested in this work. A twin nucle- ated at the grain boundary of specimen no. 2 is shown in Fig. 10. It was nucleated at an r.s.s.t. of 25 gr/mm2 after 1.54 o/o elongation. The amount of deformation prior to twin nu- cleation in the crystals with similar orientation varied con— siderably. Twins in specimens No. 3 and No. 35 were nucleated after 6.3 and 4.4 o/o elongations respectively. 25 Twins nucleated at such a low r.s.s.t. are usually Of the order of 100 u long, and considerably smaller than the twins formed at a higher stress level. They are also ac- companied by small load drops of less than 10 o/o of the twinning load. An interrupted loading method with a step- wise increase in the strain rate was used for specimen No. 3. After a total of 34.3 o/o elongation, the last 8 o/o of which was loaded with a strain rate of 3 x 10_3 sec-l, a large number of twins were nucleated at the grain boundary. A typ- ical region near the grain boundary is shown in Fig. 11. Sev— eral twins of different sizes can be seen. The r.s.s.t. var- ied from 25 to 70 gr/mm2 as the strain rate was varied from 3 x 10.5 to 3 x 10.3 sec-l. A twin formed in specimen No. 35 is shown in Fig. 12. The specimen was loaded to 28 o/o elongation in two stages; the first 12 o/o at a strain rate of 3 x 10-5 sec-1 and the rest at 3 x 10- sec-1. The twin was nucleated at an r.s.s.t. of approximately 70 gr/mm2 after 23 o/o elongation. The figure also shows a set of faint traces making an angle of approxi- mately 650 with the basal slip traces, and steps along the grain boundary. 26 iv) Bicrystals with orientation S-45 No twinning occurred in the bicrystal of this orientation after 35cvt>elongation at a strain rate of 3 x 10.5 sec-l. Neither formed are the steps along the grain boundary. Fig. 13a shows a region near the grain boundary of specimen No. 21 after 10 o/o elongation. The basal slip traces are fine and uniformly distributed throughout the specimen. Coarse slip bands developed in the bicrystals described previously are absbnt in this bicrystal. Fig. 13b is a similar region of the bicrystal after polishing. The grain boundary remains sharp and straight. This is compared with specimen No. 3 with the orientation S-30. After 21.4 o/o elongation, the grain boun- dary has been distorted considerably as shown in Fig. 14a. The same area after being polished shows the distortion of the grain boundary as shown in Fig. 14b. Fig. 15 shows a cleavage formed along the basal slip trace and the accompanying kink bands in specimen No. 21 at a tensile stress of 405 gr/mmz. The corresponding resolved shear stress on the most favorable twin system is approximately 100 gr/mmz. Twinning did not occur despite the fact that the r.s.s.t. is considerably higher than that of the bicrystals with the 27 orientation 8-30 It is noted that the width of the grain boundary has in- creased substantially after 35 070 elongation. There are no traces corresponding to the most favorable pyramidal slip system in this crystal even after 35 o/o elongation. v) Bicrystals with orientation S-60 Deformation twinning did not take place in these crystals. The tensile behavior is in fact similar to that of the bicrys— tal with orientation 8-45. The only difference is that the formation of a few coarse slip bands in these crystals as shown in Fig. 16. 3.1.2 Bicrystals with orientation H In all bicrystals with orientation H basal slip was con— strained to a great extent. The angle 6 was in most cases less than 20 and hence the resolved shear stress in the basal i‘ slip system was small. However, at a high tensile stress level of the order of 2 kg/mm2 a slight deviation from 9 = 00 would produce enough resolved shear stress in the basal slip system to activate fine basal slips in the specimens particu— larly along the grain boundary. The r.s.s.t. has been found to be very sensitive to the angle 8. The closer the angle 8 was to zero, the higher was the stress level at which the 28 nucleation of twins took place. These crystals were charac- terized by the formation of massive twins in an avalanche after relatively small amount of plastic deformation by slip. The twins were also formed at a considerably higher stress level (r.s.s.t. greater than 1 kg/mmz). A large number of twins might be formed simultaneously accompanied by a large load drop ranging from 30 to as much as 90 o/o of the twin- ning load. The subsequent twin growth upon reloading took place at a much reduced stress level. Fig. 17 show twins nucleated in specimen No. 8 with ori- entation Ha. The twins were nucleated at a r.s.s.t. of 2.56 kg/mm2 after 2.9 o/o elongation. A great burst of noise was heard upon twinning accompanied by a load drop of 69 o/o. On the (0001) plane there aretsets of slip traces; one set is perpendicular to the tensile axis, the other sets make 300 angle with the tensile axis. The first set of slip traces is faint and uniformly distributed in the background. The other two sets have relatively strong contrast and are local— ized at the neighborhood of the twins. The resolved shear stress on the most favorable pyramidal slip system (2II2) [2113] and its conjugate is 2.68 kg/mm2 at the point of twin nucleation. The slip traces on the (0001) plane (broad sur- face of the specimen) and an etch pip pattern on the (OIlO) 29 plane (side surface) of specimen No. 26 with orientation Ha are shown in Fig. 18a and 18b respectively. The etch pips are aligned in the pyramidal slip direction [21I3] and [2113] in Fig. 18b. The slip traces on the (0001) plane and the etch pip patttern on the (0110) plane are consistent with the most favorable pyramidal slip systems (2112) [2113] and (2112) [2113]. Deformation twins nucleated at a r.s.s.t. of 4.9 kg/mm after 2 o/o elongation in specimen No. 27 with orientation Hb are shown in Fig. 19. In this bicrystal almost all the twins were those of the most favorable conjugate twin systems (10I2) [101T] and (1012) [$011]. This stress of 4.9 kg/mm2 was the highest r.s.s.t. observed in all specimens. The re- solved shear stress on the most favorable pyramidal slip sys- tem was 2.58 kg/mmz. Thus specimens No. 8 and No. 27 have approximately the same value of resolved shear stresses on the most favorable pyramidal slip systems. The twins in specimen No. 27 were observed to start widen- ing at a tensile stress of about 800 gr/mm2 which was less than 10 o/o of the twinning load. Fig. 20 and 21 show slip traces on the basal plane of the side surface of grain B in specimen No. 27. The traces make an angle of about 620 with the tensile axis, the [1010] direction. 30 The picture was taken after loading to a tensile stress of 9.95 kg/mmz. An etch pip pattern of the cross section per- pendicular to the tensile axis of the specimen is shown in Fig. 22a and 22b. In the grain B, one set of the etch pips is aligned in the direction making an angle of about 420 with the basal plane and is consistent with the pyramidal slip sys- tem (1122) [I123]. The two intersecting sets of etch pips are consistent with the conjugate pyramidal slip traces (I212) [1213] and (1212) [1213] within 20 of accuracy. These con- jugate slip traces correspond to the vertical slip traces observed in Fig. 19. It is interesting to note that the ap- pearance of the pyramidal slip traces on the (0001) plane is ”wavy" and discontinuous as shown in Fig. 21. 3.1.3 Bicrystals with orientation S-H—8 Some difficulties were encountered in growing these bi- crystals with straight and vertical grain boundaries. The grain boundaries have a tendency to follow the contour of the constriction. The grains with hard orientation tend to grow larger at the expense of the soft grains in such a way that grain boundaries are inclined toward the soft grains. Upon loading, the grains with hard orientation support almost the entire tensile load beyond the yield point of the grain 31 B with soft orientation. Thus the stress-strain curves of these bicrystals show the typical characteristics ofaa crys- tal with hard orientation. Because of this uneven response of the two grains to the applied load these bicrystals often exhibit a tendency to bend about the horizontal axis per— pendicular to the broad surface so that the grain boundary becomes convex toward the grains with soft orientation. In— variably the basal slips were initiated in both grains near the curved region of the grain boundary upon tensile loading. Twins were nucleated most frequently at this region in these bicrystals. Consequently, it has been found difficult to nucleate twins within the gauge length of these specimens. However, in specimen No. 22 with orientation S—Ha-45 twins were nucleated near the inside edge of the gauge length where the cross sectional area of the hard grain was the smallest. Fig. 23a shows such twins formed at the grain boundary at a r.s.s.t. of 1.35 kg/mm2 after 1.9 o/o elongation. Fig. 23b shows the same twins after polishing. The specimen was then reloaded to a tensile stress OT = 2.5 kg/mm2 (930 gr/mm2 re- solved on the most favorable twin system) at which the twins started to grow extensively as shown in Fig. 23c. Two sets of pyramidal slip traces can be seen in this figure; one set makes an angle of approximately 900 with the twin traces, the 32 other set about 300. Specimen No. 31 with orientation S-Hb-30 twinned at a r.s.s.t. of 1.93 kg/mm2 after 2.1 o/o elongation. Near the bent portion of the grain boundary extensive basal slips took place in both grains. A large number of twins were nucleated in this region as shown in Fig. 24. Essentially the same be- havior was observed in specimen No. 34 with orientation S—Hb—60. Specimen No. 34 twinned at a r.s.s.t. of 1.36 kg/mm2 after 1.8 o/o elongation. In specimen No. 33 with orientation S-Hc-45, twins were also formed near the curved portion of the grain boundary where most extensive basal slips took place in both grains as shown in Fig. 25. A large number of twins were formed in grain A with their traces in grain A parallel to the basal slip traces in grain B. Fig. 26 shows a region near the grain boundary within the gauge length of the specimen No. 33. In grain A the vertical slip traces corre- spond to the basal slip. The basal slip traces in grain B are not extended to the grain boundary. 3.1.4 Nucleation of twins at a twin interface A twin formed at one of the twin interfaces in specimen No. 6 is shown in Fig. 27. It was nucleated at a tensile stress of CT = 2 kg/mm2 upon reloading after being polished, at the intersection of two basal slip bands respectively in the 33 matrix and the twin. Fig. 28 shows similar twins formed along a twin interface in specimen No. 8 during a large load drop in the tensile stress from 6.8 to 2.6 kg/mmz. It is to be noted that the basal slip bands in the matrix and/or in the twin are associated with the twin nucleation. Fig. 29 shows another example of the twins formed between the massive primary twins. They were formed in grain A of specimen No. 31 during a tensile stress drop from 7.27 to 1.7 kg/mmz. All but one of the secondary twins were formed at the intersection of the slip bands in the matrix and the twins similar to specimen No. 6 and No. 8. Sudh behavior of twin nucleation along an existing twin interface has been observed also in crystals with soft orienta— tion at a much lower stress level. It is not always true that twins are nucleated along a twin interface whenever the basal traces in the matrix and in the twin intersect each other. Sometimes an existing twin may continue to grow as shown in Fig. 30. The (1012) [IOlI] twin was observed to grow at a tensile stress of approximately 2 kg/mmz. In fact, such nucleation of new twins and the growth of existing twins are often observed to take place simultaneously. Fig. 30 also shows two intersecting sets of pyramidal traces in the twinned region. 34 3.2 Tensile Data 3.2.1 Stress—strain curves Fig. 31 shows the typical stress-strain curves for the bicrystals with orientations 8-10, 8—20, 8—30. The tensile stress is calculated from the applied load and the original cross sectional area and is plotted against the strain as measured from the recorded chart. At each point of twin nucleation the values of the tensile stress and the r.s.s.t. are given. The r.s.s.t. are calculated from the true tensile stresses when the specimens have been deformed more than 3 o/o. When the strain rate is increased successively from approxi- mately 3 x 10-5 to 3 x 10-3 sec-1, the resolved shear stresses for the most favorable twin systems are increased from 25 to 70 gr/mmz. The rate of work hardening is also increased somewhat.) The stress-strain curves of the bicrystals with orienta- tion Ha and Hb are shown in Fig. 33. Both the specimens No. 8 (Ha) and No. 27 (Hb) had the angle 8 less than 10 and showed the highest r.s.s.t. among the bicrystals with the respective orientations. As shown in Fig. 32, the load relaxations upon twinning are rather small compared with those shown in Fig. 33. In the latter cases the load relaxations of specimen No. 8 and No. 27 are 69 and 86 o/o respectively. 35 Fig. 34 shows the stress—strain curves of the bicrystals with orientations S-Hb-30, S-Hb-45, and S-Hb-60. The stress- strain curves of the bicrystals with orientations S—Ha-45, S-Hb-45, S-Hc-45 are shown in Fig. 35. It is observed that, although the r.s.s.t. for those bicrystals with orientation S-H-e vary from 1.36 to 2.12 kg/mmz, they are within i 25 o/o 2 . . . of an average value of 1.7 kg/mm . This variation does not appearto be systematic, but random. 3.2.2 Tensile stress data Table I summarizes the data obtained from the tensile test- ings. True stress and o/o elongation are measured at the point of the first twin nucleation at a strain rate of approximately 3 x 10-5 sec-1. The r.s.s.t. are calculated for the most favorable twin systems. The most important features observed are: l. The r.s.s.t.‘7& in the bicrystals with orientation 8—8 vary systematically with the angle 8, ranging from 25 gr/mm2 for 8 = 300 to 200 gr/mm2 for 8 = 100, 2. The values of‘7& for the bicrystals with orientation H are much higher than those for the single crystals of similar orientation. The maximum value of IT ob— 2 served was 4.9 kg/mm , 36 3. The r.s.s.t. is a very sensitive function of the angle 8 for the single crystal and bicrystals with hard orientation. As the angle 8 becomes larger, 7% de- creasesconsiderably, and twins tend to be formed at the grip even when the basal slip is entirely con- strained. 4. The r.s.s.t. for the bicrystals with orientation S-H-e are of approximately the same order of magni- tude. They are within 1 25 o/o of 1.7 kg/mmz, except specimen No. 23 in which twins are nucleated at the grip. Table II shows approximate values of the tensile stresses 0 the resolved shear stress‘j; on the basal slip system, and the T’ r.s.s.t. 7% on the most favorable twin system for the bicrys— tals with orientation 8-8. In spite of the considerable vari— ations of 6T and z} with the angle 8,‘7; remains substantially constant at approximately 100 gr/mmz. Table III shows the resolved shear stress 7; on the most favorable pyramidal slip system for specimen No. 8 and No. 27 with orientation Ha and Hb respectively. The specimens have the angle 8 within 1 lo, and the values of‘YE are approxi- mately the same for both bicrystals. A B (0110)B Fig. 5. Nucleation of twins in specimen No. 19 (S—lO) at a r.s.s.t. of 200 gr/mmz. 100 x A B (OJ—.10)B Fig. 6. Nucleation of twins in specimen No. 19 (S-lO) after 10% elongation. 80 x 38 [51101A A B (0I10)B Fig. 7. Nucleation of a twin in specimen No. 20 (S—20) at a r.s.s.t. of 90 gr/mmz. 100 x A B (0110) B Fig. 8. Grain boundary distortion in specimen No. 20 after 30% elongation. 100 x 39 [5110] “20 u Fig. 9. Formation of slip bands associated with the grain boundary distortion in specimen No. 20 (S-20). 200 x 40 [illolA A B (0I10)B Fig. 10. Nucleation of a twin in specimen No. 3 (S-30) at a r.s.s.t. of 25 gr/mmz. 100 x U‘."I I" :II [I , A B (0110) B Fig. 11. Nucleation of twins in Specimen No. 3 (S-30) after 34.3% elongation. 100 x 41 [5110]B Fig. 12. Nucleation of a twin and grain boundary distortion in specimen 35 (S-30). 250 x A B (oilo) B (a) (0110) (b) Fig. 13. A typical region near the grain boundary in specimen No. 21 (S—45) (a) After 10% elongation. 100 x (b) After polishing. 100 x 43 (0110)B (a) (0110) B (b) Fig. 14. Twins and g.b. distortion in_specimen No. 3. (a) After 21.4% elongation. 100 x (b) After polishing. 100 x 44 [5110]B T.A. Fig. 15. A cleavage fracture formed in specimen No. 21 (S—45) after 35% elongation. 100 x 45 (0I10) B Fig. 16. Accommodation of slip at the grain boundary in specimen No. 37 (S-60). 100 x 46 (0001)A B (oi10)B Fig. 17. Nucleation of twins in specimen No. 8 (Ha) at a r.s.s.t. of 2.56 kg/mmz. 100 x 47 T.A. [5110] (oi10)A (b) Fig. 18. Specimen No. 26 (Ha) (a) The pyramidal slip traces on the (0001) plane. 100 x (b) Etch pip pattern on the (OIlO) plane. 100 x 48 ‘- ’st . IL;,, (0001)A (1510) B Fig. 19. Nucleation of twins in specimen No. 27 (Hb) at a r.s.s.t. of 4.9 kg/mmz. 100 x 49 (0001)B Fig. 20. Twins and the pyramidal slip traces on (0001)B plane in specimen No. 27 (Hb). 100 x 50 (0001)B Fig. 21. The pyramidal slip traces on the (0001) plane in . B spec1men No. 27. (Hb) 51 Fig. 22a. Etch pip pattern on (I010) plane of specimen No. 27. (Hb) 60 x h 52 22b. Same as Fig. 22a. Enlarged to 200 x. 53 (Oilols (0001)A Fig. 23a) Nucleation of twins in specimen No. 22 (S-Ha-45) at a r.s.s.t. of 1.35 kg/mmz. 100 x Fig. 23b. Same as Fig. 23a. Polished. 100 x 54 B A (0001)A Fig. 23c. Growth of twins in specimen No. 22 upon reload- ing. 100 x 55 [5110]B Fig. 24. Nucleation of twins in specimen No. 31 (S-Hb-30) at a r.s.s.t. of 1.93 kg/mmZ. 100 x - 56 [i010]A Fig. 25. Nucleation of twins in specimen No. 33 (S-Hc-45) at a r.s.s.t. 1.70 kg/mm2. 70 x [I0101A A B Fig. 26. The basal slip in specimen No. 33 (S-Hc-45) after 1.8% elongation. 70 x (0110) Fig. 27. Nucleation of a twin at a twin interface in specimen No. 6'(H, single crystal).. 100 x T.A. [5110]A (oilo) Fig. 28. Nucleation of twins along a twin interface in specimen No. 8 (Ha). 100 x 58 T.A. [[51101A (oilo)A Fig. 29. Nucleation of twins along the interfaces of massive twins in specimen No. 31 (S-Hb—30) . 100 x 59 Growing interface T M (1210)A [i0101A Orig. interface Fig. 30. Slip traces associated with the growth of a twin in specimen No. 34 (S—Hb—60). 100 x 60 600‘ 135% load drop No. I9 , S-IO 2 500. 0"},3570 Of/lllllz gflmm 0:.- {-7.200 (gr/nmzl 400 300 9% load drop ' No.24, s-zo o- .305 gr/IIIII T 2 T7: 35 gr/mm 200’. 6% load 'drop 100‘ Tues, s-so 2 0}=l22 dr/mm T1530 ar/mmz 0 i f i 4i is is i 6 3 F0 % elongtlon F.c. 71. Stress-strait cur/es o? the ‘zcrvstsls WLth orientation S—G. 61 sauna swnuum unawua> an Aomlmv m 102 seafloomn mo nabhso sflnhunleuauum .mm .mwm 6:60:20 oxo . - m.» cm _- aw o.~ n. o. o m e m o .Iooeln: an“ ..W bk To: vuo. xmuw 7000 ”to. unuw It fit 18. on? oeumo 8.qu .00. .OON a. £5.58 Gnu-N .—. Wu .0 now. .onN Nessa Ohumw s Nessa Ohm a& can so a n 62 IO‘ No.27, "b 03:: 9-94 kg/mm2 9 13:43.9 kg/mmz 84 0'1- ,86 as load drop (kg/mmz) 1rNo.8, Ho ‘B-BKgImmz 6‘ TTs2°56 liq/mm2 5‘ A V6996lood dron 44 . . 3 ML 24 h )1 o I Y 3 7 95 elongation Fig. 33. Stress-strain curves of the bicrystals with orientations He and Hb 63 No.54, S'HS'SO 8 . a}: 8'34 TT: 3'10 7] No.3l, 344.," 30' 0': 6'2? T1,: 2'32 6 1 01 N0. 3|, 8-11-30 ( kgnnmzl . a}: 5-20 _\ 01"" 73 5 . T s 2' '2 T 8 I '93 T T 4 , "0.3‘. S-H-GO 8 3'67 P33 3 . i 2 i i I 1 o i 3. i 5 5 95 elongation Fig. 34. Stress—strain curves of the .icrystals With orientation S-I'C‘-9 35] 3.4. 33‘ 6'04 3_2-4 No.38, $41545 3" ‘ O}:s-73 T1,: Z'IZ 5'0- 3'0 I'6 T (kg/mi; 60 36 load drop No.3 3, S-Hc-45\ l 0} =3-5o ) 3-04 Tr = l-TO No.22,s-Hg45 . 0;=3'65 TT=|-35 2'0. I-Ot °/o elongation Fig. 35. stress-strain curves of the bicrystals with orientations 5-Ha'—45, S-mx—45, and S-Hc-45 Orig. Cross Sec. Gage Twinning Relaxed % Deform. Spec. Area length load load before No. Orientation (mmz) (mm) (gr) (gr) Twinning l S-30 32.5 30.0 7.100 200 2.17 2 S-30 27.0 32.5 3,600 250 1.54 3 S-30 26.4 31.3 3,200 200 6.27 35 S—30 15.4 25.0 1.500 100 4.4 20 S-20 18.0 29.0 6,100 80 2.3 24 S—20 10.5 25.0 3,200 270 8.4 19 S-10 12.6 25.0 7,200 400 3.0 21 S-45 20.0 29.0 --- --- --- 37 S-60 216.0 25.0 --- --- --- 8 Ha 24.0 34.0 163,000 100,000 2.93 26 Ha* 13.6 29.0 53,700 1,070 1.9 27 Hb 6.5 25.0 64,600 55,300 2.0 40 Hb** 15.0 25.0 53,000 41,000 2.8 23 S-Ha-30 10+ 25.0 24,300 1,800 0.86 22 S-Ha-45 10+ 29.0 36,500 9,000 1.9 31 S-Hb-30 12.6+ 25.0 65,600 8,000 2.1 32 S-Hb-45 10.0+ 29.0 55,000 7,000 2.5 38 S-Hb-45 13.0+ 25.0 74,500 44,500 3.5 34 S-Hb-60 9.0+ 25.0 33,000 2,800 1.8 33 S-Hc-45 11.0+ 25.0 38,000 2,600 1.8 6 H(Single) 22.0 35.0 42,000 3,500 0.75 39 H(Single) 6.0 29.0 21,000 2,000 1.1 Table I. Summary 65 Re- laxed Ten- St. R.S.S.T. Frac- sile (Ten- 1' ture Stress sile)2 T Load Remarks (g/mmz) (g/mm ) (g/mm2) (gr) 218 6.2 40 10,800 Cross Head Speed = 0.01 cm/min 133 9.3 25 7,500 126 7.6 30 9,700 105 7.3 25 --- 340 4.7 90 12,300 328 28.0 85 --- 588 32.0 200 --- --- --- --- 3,200 No twinning occurred 6,800 4,700 2,560++ -—— 0 3,950 787 l,465++ --— * Grain A had 8 = 4 9,950 8,520 4,900 ~-- 0 3,530 2,740 l,730++ -—— ** Grain B had 8 = 5 2,430 180 900++ 26,000 +Min. Cross Sec. Area of Grain 3 ’ 650 900 l I 350 ___ + II II II II II II 5 ’ 200 635 1’ 930 ___ + II II II II II II 5,500 700 2,040 55,000 + " " " " " " 5,730 3,420 2,120 --- 3,670 310 1,360 --- 3,450 _236 1,700 --- 1,900 160 710++ --- Orient. same as Grain A of Ha. 3,500 335 1,720 Orient. same as Grain B of Hb. ++ Twins formed at the grip. of Tensile Data. 66 2 -- 2 ‘ 2 9 OT(gr/mm ) TT(gr/mm ) 7;(cgr/mm ) o 10 600 200 100 20° 340 90 100~110 o 30 130 130 55~6O Table II. 0T, 7T' and 7; for the bicrystals with orientations S-lO, S-ZO, and S-30. 2 t 2 2 Spec. No. 0T(kg/mm ) TTikg/mm ) 'Z'p(gr/mm ) 8 (Ha) 6.8 2.56 2.68 27 (Hb) 9.95 4.90 2.58 Table III. 0T, 7%, and 7% for the bicrystals with orientations Ha and Hb. IV DISCUSSION $11. Effects of Orientation 4,1,1l Modes of deformation by slip Zinc crystals deform by two different modes of slip at room temperature; the basal slip {0001} <2IIO> and the pyra- midal slip [2112) (2113). The existence of the second order pyramidal slip in Zn was first established by Bell and Cahn (14) who found the critical resolved shear stress for this slip to be approximately 1.0 — 1.5 kg/mmz. The c.r.s.s. for the basal slip is about 35 gr/mm2 (14, 29). Because of this large difference in the c.r.s.s., the basal slip is the pre- dominant mode of deformation, unless it is constrained to such an extent that the resolved shear stress on the pyra— midal slip system exceeds the c.r.s.s. for the pyramidal slip. Hence, the predominant modes of deformation in the soft and the hard bicrystals are the basal and the pyramidal slip re- spectively. The bicrystals with orientation S-H-8 still de- form primarily by the pyramidal slip, since the soft grain can only deform to the extent the hard grain does. In order to confirm the existence of the pyramidal slip in the hard crystals, the etching technique suggested by Vreeland et al. was used to reveal the dislocation etch pips 67 68 in a {lIOO} plane (27), after many attempts to observe the pyramidal slip traces on this plane without etching failed. Fig. 18 and 22a prove the existence of the pyramidal slip. In the right hand side of Fig. 18b where the pyramidal slip has taken place uniformly, it is difficult to distinguish the pyramidal slip bands due to the uniform distribution of the etch pips. The pyramidal slip traces appear to be wavy and dis- continuous on the basal planes as shown in Fig. 21. Such wavy slip traces were observed in iron and were attributed to the fact that many slip systems with a common slip direc- tion operate simultaneously by cross slip (30). However, it is not established as yet whether cross slip takes place in zinc crystals.g The exact cause of the wavy slip bands re- mains to be investigated. In the bicrystals with orientation S-8, grains A and B deform extensively by the corresponding basal slips which must be accommodated with one another at the grain boundary in order to maintain a continuity of the strains across the boundary.. The macroscopic compatibility (see appendix B)is satisfied in this case. On the other hand, the bicrystals with orientation H and S-H-8 are macroscopically incompatible, since the primary slip systems in each single crystal grains 69 do not satisfy the continuity conditions. When an incompatible bicrystal deforms on its two primary slip systems, high elastic stresses are built up at the grain boundary. In order that these stresses can be relieved, secondary slip systems are activated (20, 23). Even if a bicrystal is macroscopically compatible, it can still be harder than its component single crystals because of microscopic incompatibility. In other words, macroscopic compatibility is not satisfied everywhere along the grain boundary. This leads to the formation of piled up groups of dislocations against the grain boundary and build- ing up of internal stresses. These in turn may activate the secondary slip systems and mark the beginning of the turbulent flow at least near the grain boundary, and as a result the rate of work hardening increases. It has been clearly shown in the photomicrographs through- out the text that twins are nucleated at the grain boundary in the bicrystals with soft as well as hard orientations. This in itself confirms the hypothesis that the nucleation of ‘twins takes place at the point of high stress concentration. If twins are not nucleated at the grain boundary as observed occasionally in the bicrystals of hard orientation, they are nucleated near the specimen grip where there is a known stress concentration. 70 However, as shown in Table I, the stress levels at which twins are nucleated at the site of high stress concentration differ considerably for the various specimens. This will be discussed later in this chapter. 4:1;2. Accommodation of slips in adjacent grains at the grain boundary If a bicrystal is compatible everywhere along the grain boundary, there will be a one to one correspondence of the slip lines at the grain boundary and hence the boundary would remain undistorted upon deformation. However, such a uni- formity in slip does not exist in any of the bicrystals tested. As 'shown in Fig. 9, severe distortion of the grain boundary is a typical example of the inhomogeneity of deformation by slip in the bicrystals with orientation S-10, S-20, and S-30. Even when a bicrystal is symmetrical, slip starts in- variably in one of the grains, say A, earlier than in the other grain B. Because of this non-uniform slip, slip bands are formed earlier in grain A than grain B. The grain boun- dary thus must accommodate the slip bands to relieve the stress build-up due to the dislocations in the slip bands. Fig.4-1 shows the formation of steps along the grain boundary. Such steps are the largest in the bicrystals with orientation S-20. 71 Fig. 4—1. Schematic representation of slip bands in a sym- metric bicrystal of zinc and the formation of steps along the grain boundary. 72 In the bicrystals with orientation S-45, the steps along the grain boundaryareeso small that the boundary appears undistorted as shown in Fig. 13. The slip traces in the bicrystals with orientation S-8 do not in general meet with each other at the grain boundary because of the following reasons. First, compatibility does not prevail everywhere along the grain boundary and hence there is no one to one correspondence between the slip traces in the adjacent crystal grains. Second, there is a strong repulsion between the leading dislocations of the pile-ups associated with each of the slip bands. Therefore, once slip takes place in grain A earlier than in grain B due to a slight difference in the orientations of the two grains,it is diffi- cult for the dislocations in the slip bands in grain B to approach the leading dislocations of the piled-up array in grain A. Instead, slip bands in grain B, which would meet the grain boundary midway between the slip bands in grain A, are attracted toward the grain boundary (31, 33). - The spacing of the slip lines in zinc single crystals has been observed to be SOC-3,000 A at room temperature (34, 35). If this is taken to be also true in zinc bicrystals, it is reasonable to assume that the Slip lines of grain A and. B meet the grain boundary alternatively with a spacing of about 250-1,500 X. 73 It is often observed that the slip bands in one of the grains of the bicrystals with orientation_S-8 do not extend to the grain boundary, but terminate at a certain distance xO (taken perpendicular to the boundary) away from it. In gen- eral, the distance x0 increases with decreasing 8 as shown in Fig. 5, 9, and 12. The variation in the distance x0, which ranges up to about 200 u, can be understood if one considers the accommodation of the transferred shear stresses of grain A onto grain B. In Fig. 4-2, at point P the transferred shear 25 due to the pile-up array in grain A is (see equation A.6 in appendix 1/2 A)'7; = [1 + (g) ]'7;, where s = xo/sine. Such a shear TOT stress must be accommodated by grain B. For the distance 5 = 200-400 u, the length L of the pile-up needs to be only equal to s to have 75 equal to 27;. The length of the pile—up of the order of several hundred microns can easily exist in the bicrystals as will be discussed later in this chapter. In the bicrystal with orientation S-45, the transferred shear stress is compatible with the resolved shear stress in the slip Fig. 4-2. plane of grain B as shown in Fig. 4-2. 74 Therefore, the transferred shear stresses assist the slip in grain B. The distance 5 vanishes in this bicrystal as shown in Fig. 13a. When 8 is less than 22.50 the transferred shear stress will have a component opposing the resolved shear stress in grain B and resists the slip in the later grain. In the bicrystals of cubic structures, the internal stresses can always be relieved by suitably oriented secondary slip systems, and a region of multiple slip appears along the grain boundary. Hauser and Chalmers (23) have shown that such a region of multiple slip extended up to 2-3 mm. In zinc bicrystals, however, the critical resolved shear stress for the pyramidal slip is so high compared to that of the basal slip that the transferred shear stresses could not be relieved by the pyramidal slip, except perhaps in the region extremely close to the grain boundary. Because of this, it is possible that even in the macrosc0pica11y compatible bi- crystals of zinc deformation twinning is invoked to relieve the microscopic strains near the grain boundary. Pugh (37) and Hauser et a1. (38) have suggested that twinning in Mg provides additional modes of deformation when there are an insufficient number of independent slip systems for the strain continuity. Recently, Tegart (39) refuted this suggestion on the grounds that twinning could not 75 produce sufficient deformation to account for the ductility of Mg crystals. However, it is possible that twinning could. provide sufficient deformation to relieve the microscopically incompatible strains in a macroscopically compatible bicrystal. 4.2 Effects of Prior Slip and Piled-up Group of Dislocations Previous workers (14, 41, 42) have shown that a certain amount of plastic deformation always preceded twinning in bulk crystals. Bell and Cahn attributed the prior deformation to the necessity of building up the sufficiently high stresses to nucleate a twin. They asserted that the actual stresses in the vicinity of dislocation pile-ups reached the theoreti- cal shear stress of approximately 100 kg/mmz. This led them to propose the homogenous mechanism for the nucleation of twins. In the following, an attempt will be made to find out whether or not such a stress concentration has reached the theoretical value according to the experimental data obtained in the present work. 4.2.1 Dislocation pile-ups on the basal slip plane Consider the case in which the bicrystals with orienta- tion S-30 twinned at the smallest observed r.s.s.t. of 30 2 gr/mm . The corresponding resolved shear stress on the basal 76 plane is approximately 60 gr/mm2 at which dislocations (pre— sumably the edge type) on the basal plane pile-up against the grain boundary. According to equation (A.2), the number of dislocations n that can be piled up in a length L of the slip plane is n = (1(K/Gb) L7;l (A.2) From equation (A.4), the maximum shear stress concentration due to the n dislocations is n7;' Therefore, when'Y; = 60 gr/mmz, n must be about 1,500-2,000 in order that the shear stress is to reach the theoretical value. This means that the length of the slip band would have to be approximately 2 cm according to equation (A.2). ,However, L.= 2 cm is much larger than the entire slip band length which is only 4-6 mm. Hence, it must be concluded that the theoretical shear stress has not been reached at 7; = 60 gr/mm2° (See appendix A.) The choice of the slip band length over which dislocations pile up becomes a primary importance in the calculation of the stress concentration. A reasonable choice of L can be made from the following arguments. 1) Seeger and Trfiuble (35), and Boéek et al. (43) have shown that the thermally activated climb of edge dislocations can cause an extensive fanning of the basal slip bands in zinc crystals at room temperature, and hence the higher the temperature, the shorter become the 77 active slip lines. At ~300C., the typical length of the ac- tive slip lines is several millimeters long (35). Therefore, the value L is expected to be shorter than several milli- meters at room temperature. 2) when specimens of the same orientation but of different size (see specimens No. 2, 3 and 35 in Table I) are tested, the r.s.s.t. remained more or less constant. Table II shows that twins in the bicrystals with orientation S—lO, S-20, and S—30 are formed at the resolved shear stress of approximately 100 gr/mm2 on the basal plane, despite the large variation in the slip band length. This implies that the length of the dislocation pile-up is much shorter than the length of the entire slip band. 3) if the dislocations in a pile-up are generated from a Frank-Read source, the dislocations of the same sign occupy on the ave- rage only one half of the slip band. Therefore, it is reason- able to choose L = 2-3 mm. Using L = 2-3 mm, G-= 5.5 x 1011 dynes/cmz, and 2/= 1/3 in equation (A.2), one obtains n = 200-300. Thus the maxi- mum number of dislocations that can be piled up against the grain boundary is approximately 200-300. They are capable of magnifying at the most 200-300 times the resolved shear stress near the leading dislocation of the pile-up. The distance x1 is given by equation (A.3) and is 78 estimated to be about 500 A. The distance rO of the closest approach to the grain boundary of the leading dislocation has been estimated by Jaswbn and Foreman (44) to be approximately 5 atomic Spacings at a typical yield stress in an elastically isotrOpic material. Although such an isotropy is not realized in Zn bicrystals, the shear moduli nevertheless do not change greatly in passing from one grain to the other. Therefore, it is reasonable to adopt the estimation of Jawson and Foreman in the present case. The maximum stress concentration can be calculated by using equation (A.4) in the region r<<.xl. Hence'Zi = n7; = (200)(60 gr/mmZ) = 12 kg/mmz. Thus the maximum shear stress ahead of the dislocation pile-up is approximately 12-15 kg/mm2° This value is one order of magnitude smaller than the theoreti- cal shear stress for the homogeneous nucleation of a twin. A stable twin nucleus of 250 A would probably not be formed as a whole in such a manner as proposed by Bell and Cahn. The stress fields around an edge dislocation array at a distance ¥%'Q:r «:Lhave been calculated by using the approxi- mation of Leibfried (45) that the discrete dislocations are replaced by a continuous dislocation density. They evaluated such a stress distribution via the stress function obtained from the superposition of the stress fields of the hypothetically 79 infinitesimal dislocations making up the distribution. A group of positive edge dislocations piled up against a bar- rier under a shear stress 7; is shown in Fig. 4-3. The com- ponents of the stresses at P(x,y) are (45, 46): Oxx = % Ta (§)1/2 [% cos 9%.? - 21:— cos 353—9 + 4 sin 352] OYY = 27a (91/2 [—% cos 9¥+ ¥ cos :39] Oxz = Oyz = 0 The stresses at near the leading dislocation can be found by putting FL=IM ¢==0, r «Z L, and y = r sin 8. These equa— tions then become the approximate expressions based on the model of a pile-up of discrete dislocations (48). Mitchell (48) has plotted the stress fields for f.c.c. metals as a function of r and 8. Fig. 4-4 shows the stress components Oxy in units of 7; for an edge dislocation pile-up situated between x/a = —1 and 8. The back stresses due to the pile—up are shown on the left hand side of the diagram and are always smaller than the applied shear stress. It can be seen that the maximum shear stress exists in front of the pile-up. By comparing the stress field in Fig. 4-4 with that of a single edge dislocation in Fig. 4-5, it can be shown that the 80 P(x,y) dislocation density Fig. 4-3. Continuous distribution of edge dislocations in a pile-up and the coordinate system (46) 81 wk 0 u A O Fig. 4-4. The stress field: Oxy in units of 7; for an edge dislocation pile-up situated between x/a — —1 and 0 (48) 82 I WT W STIISS: p I.” I l l- '1 W? U 0137‘“). Fig. 4-5. The stress field Ox in units of G/4007C(1 -L/) of a single edge dislocation (31) 83 stress fields of a pile-up at large distances approach that of a single dislocation of strength nb situated at the center of gravity (-a/4,0). At smaller distances x1/15 << r << L, it is expected that the stress fields may have a functional dependence on r and 8 similar to that shown in Fig. 4-4. The existence of such high stress fields due to the pile-up of dislocations against the grain boundary has been verified in MgO bicrystals by the birefringence patterns observed in polarized light (49). géng. Dislocation pile-ups on the pyramidal slip plane When the basal slip is constrained, a zinc bicrystal will deform by pyramidal slip. The dislocations in the pyramidal slip plane can pile up against the grain boundary. The stress concentration due to the piled up array of the pyramidal slip dislocations can be estimated in a similar manner as described previously. For the bicrystals with orientation H, twins are nucle- ated at the resolved shear stresses on the pyramidal slip planes of approximately 1.5-2.5 kg/mmz. If a slip band length L = 0.2 cm is used for the calculation of the number of dislocations in the pile-up, it would result in a stress concentration which is much greater than the theoretical 84 r.s.s.t. Hence, the stress concentration factor in these bicrystals must be much less than that in the soft crystals. At the average applied StreSS‘Yb = 2 kg/mmz, only about 6-8 dislocations are needed to magnify the applied shear stress to 12-15 kg/mmz. From equation (A.2), the corresponding slip band length L is about 2-3 microns. Such a stress concentra- tion can be easily achieved at many locations along the grain boundary in the macroscopically incompatible bicrystals with hard orientation. Perhaps, this is the reason why the twins are formed in large numbers over considerable portions (~l cm long) of the specimens. 4.3 Effects of Relative Orientation The wide variations in the resolved shear stresses for twinning With respect to the relative orientation of the neighboring grains in the bicrystals require an explanation. In Table I, it is shown that the resolved shear stresses for twinning for the bicrystals with hard orientation are much higher than those for the bicrystals with soft orientation and the single crystals with hard orientation. This implies that the mere presence of a grain boundary does not necessarily insure a high stress concentration factor. The relative orientation of the neighboring grains and the modes of 85 deformation in these crystal grains dictate the stress con- centration behavior. Even among the bicrystals with orientation S—8, the r.s.s.t. varies considerably ranging from 25 gr/mm2 for 8 = 30 to 200 gr/mm2 for 8 = 10°. In order to account for these variations, consider a bicrystal such as shown in Fig. 4-6. The crystal grains A and B are assumed to have the same resolved shear stresses on their corresponding slip systems due to a certain tensile stress as they would have if they were pulled separately to the same tensile stress. Consider the effect of the primary slip taking place in grain A on grain B, or vice versa. When the basal slip takes place in grain A, it will superimpose on grain B the shear taking place on its basal plane. This means that the extension of the basal slip planes of grain A in grain B is subjected to a shear stress which is the continuation of the shear stress on the basal slip plane in grain A. This superimposed shear stress on grain B can be resolved on all the slip or the twin 'systems in grain B. Let [n] be the unit normal of the slip plane and [s] the unit vector in the slip direction. If 0 is the shear stress 12 on the basal slip system (n 31) of grain A, the resolved ll shear stress 012 on system (ni,s ) in grain B is l . 86 /_(ni' Si) \sgl Fig. 4-6. Superposition of the shear of grain A onto grain B 87 012 NilOlZ = [a11a22 + a12a211°12 (4‘1) where a c08(‘#[sl] and [si]) 11 a22 = cos(4[n1] and [ni]) a12 = cos(‘4[si] and [n1]) a21 = cos(.4[ni] and [31]). The factor Ni1 is derived from a transformation of the shear stress in [$1] direction on the basal slip system (n1,sl) in grain A to that in [Si] direction of the system (ni,si) in grain B. Hence, the total shear stress acting on the (ni,si) system in grain B is 7total =‘75 + MO12 (4‘2) where 7% is the applied resolved shear stress on (ni,si) system in grain B, and M is the stress concentration factor due to the pile-up group of dislocation on the primary slip system (n $1) in grain A. ll If the system (ni,si) in grain B is a twin system, equa— tion (4.2) gives the total resolved shear stress acting on this twin system. For the symmetric bicrystals with orien- tation S-8, 0 can be evaluated as a function of 8. Since 12 012 = OT c038 sine, where CT is the applied tensile stress, equation (4.2) becomes 88 = T + M(Ni1) (sin8 cose)o (4.3) 7total T T For a high stress concentration, the first term is negligible compared with the second, and hence -Ztotal:z M(Ni1)(31n8 cose)oT (4.4) Therefore, Ytotal is determined by the product (Nil)(sin8 , cose) for a particular value of M. The factor (sin8 cose) is the Schmidt factor for the basal (primary) slip system (n1,sl). The factor Nil can be calcu- lated with the help of a stereographic projection as the angle 8 varies. Fig. 4-7 shows N sin8 c058, and their product, 11' together with N21 s1n8 c038 and N31 Sine cose, where N11, N21 and N31 are the corresponding factors for the itwin systems (T1,tl),(T2,t2) and (T3,t3) as shown in Fig. 4—8. It can be clearly seen in Fig. 4-7 that the product Nil sin8 cose reaches its maximum valuewhen 8 is approxi— mately 250-350. This is in agreement with the experimental data that twins are nucleated at the smallest r.s.s.t. (and hence at the maximum stress concentration) in the bicrystals with orientation S-30. Fig. 4-7 also explains the facts that the bicrystals with smaller 8 angle twinned at progressively higher resolved shear stresses, and the bicrystals with 8 = 450 did not twin (Sine 0008) N" LOW 0.9] 0.8‘ 0.7 r 0.5‘ 04* 0.3 . 0.2+ O.I« 89 SinOCoss (SinOCosO) N3| (SInOCOIO) N” ‘ (Sine C008) "2| / 6 lb I3 2'0 2'5 3'0 33 fa 4o Fig. 4-7. (SInOCosG) H" vs 9 90 0012) J) l I) J [I .H (I ‘/a fp/ / 1 'g)i L'I I I ,f l f } ”i/J—ee—«rnfiW x\ 1 //o/ [2qu \\\ ~. I ' EQ \—-fl|02) \L‘(OITZ) ((Tl.tl); (1012) [10111 L 2,.t4); (1102) [1101] ’1: [(32,ti); (0112) [0111] L(T5,t.); (0112) [0111] —(u .t ); (1102) [1101] . 3 3 L-(T6,t6); (1012) [1011] Fig. 4—f. Reference twin systems 91 at all. The Ni sin8 cos8 curve decreases very fast as 8 1 approaches 450 and eventually becomes negative beyond 8 = 45°. This explains the absence of deformation twinning in the bicrystal specimens No. 21 and No. 37 with the respective orientations S-45 and S—60. The above analysis can be extended to a calculation of the transferred shear stress on the twin systems in one grain due to the pile—up of dislocations on the pyramidal slip plane in the other grain. However, the critical resolved shear stress for the pyramidal slip of 1.0-1.5 kg/mm2 is high, and shear stresses higher yet than this value must be applied in order to nucleate twins. The bicrystals with hard orien- tation have been found to twin at a resolved shear stresses of 1.5-2.5 kg/mm2 on the pyramidal slip systems. Therefore, the wide-spectrum of the r.s.s.t. in the bicrystals of vari- ous orientations can be accounted for by using this trans— ferred shear stress criterion. For a given 8, the bicrystals with orientation S-8 show a consistency in the r.s.s.t., implying that the stress con- V// centration factor is more or less constant. Table-I shows the r.s.s.t. of 25—30 gr/mm2 for the three bicrystals with orientation S—30, and 80-90 gr/mm2 for the bicrystals with orientation S-20. 92 The consistency in the r.s.s.t. for a given orientation» for the bicrystals with hard orientation is not as good due to the fact that the r.s.s.t. depends sensitively on the angle 8. When 8 is larger than 20 considerable basal slip takes place prior to twinning, and twins are formed at a much lower stress level than in the case of ~19< 8 < 10. In the former case twins are formed exclusively at the grip due to the stress concentration resulthngrom the basal slip, where- as in the later case twins are nucleated at the grain boundary. For the bicrystals with orientation S-H-8, the data in Table I do not show any syStematic variations in the r.s.s.t. with respect to the angle 8. The r.s.s.t. are found to vary within i.25C%3 of an average value of 1.7 kg/mmz. In these cases it is hard to draw a conclusion about the effects of the relative orientation on the twinning stress. The soft grain B can only deform to the extent limited by the hard grain A, otherwise some bending would result as mentioned in the previous chapter. The effects of the basal dislocation pile-up in grain B on the r.s.s.t. in grain A become insig- nificant because the applied stress in grain A is so high that it is sufficient to nucleate twins in grain A before there can be any significant stress concentration at the grain boun- dary due to the dislocation pile-ups in grain B. 93 In most cases in the bicrystals with orientation S-H-8, twins were nucleated near the curved region of the grain boundary where extensive basal slip took place. In this region, bunched basal slip bands are often observed and per- haps sessile dislocations resulting from the interaction of the pyramidal with the basal slip dislocations act as effec— tive barriers for dislocation pile-ups as discussed by Bell and Cahn (14). The basal/pyramidal interaction can be writ- ten as follows (14, 53, 54): 1/3 <§110> + 1/3 <2II3>-<0001> (4.5) It is interesting to note that specimen no. 33 with orien- tation S—Hc—45 exhibits a behavior of "smooth" load relaxation upon twinning as shown in Fig. 35. In this bicrystal the direction of the basal slip in the soft grain is nearly par- allel to the most favorable twin system in the hard grain. A large number of narrow twins can be seen in grain A near the grain boundary in Fig. 25. Such a “smooth" relaxation can be interpreted as a superposition of a large number of small load relaxations over a time interval. This behavior was never observed in any other orientation. 4.4 Stress Concentration and the Energy to Form a Twin Fault The energy associated with the coherent twin interface for the {lOlZ} <10li> twin systems in Zn has been estimated to be 94 approximately 1.4 I 0.4 ergs/cm2 (51). In the same work the interaction force Fd between two parallel edge type twin par- tials has also been calculated. The calculation took into account the anisotropy of zinc crystals. The twin interface energy was found;by applying the principle of virtual work. If the leading twin dislocation is displaced by a small amount 6x from the equilibrium position, the area of the co- herent twin boundaries will be increased by 25x per unit length of the twin dislocation line. Hence under equilibrium conditions 2vt6x = Fd5x or F6: 2yt, where Vt is the coherent twin interface energy. In order to form a twin fault which is bounded by two 2 Vt. bt is the Burgers vector of the twin partial. Using bt = 2.35 x twin interfaces, a shear stress 0 = is required, where bt 10- cm (6), one obtains o 2:10‘~ 15 kg/mmz. It seems that the stress required for the formation of a twin fault is ap- proximately equal to the stress built up at the grain boundary by dislocation pile—ups. Thus the formation of a twin fault may be regarded as the first stage of twin nucleation, and the shear stress required to overcome the drag of the twin fault may possibly be the controlling factor of the twin nucleation. It should be noted that the energy of a twin fault is equivalent to that of two coherent twin interfaces, and hence 95 it does not require any more energy to overcome the drag of the twin interfaces of a thick twin than a single layer twin. A mechanism of twin nucleation based on the movement of existing twin dislocations will be discussed later. I g;§_ Twin Nucleation at the Interfaces of an Existing Twin The fundamental hypothesis of the present work is that twins are nucleated at the site of high stress concentration, regardless whether the stress concentration occurs at a grain boundary or at a twin boundary. In fact each of the bicrystals with orientation S-8 has a grain boundary across which the neighboring grains are mir- ror images of each other. However, there exists an important distinction between a grain boundary and a twin boundary as far as dislocation interactions are concerned. Across the twin boundary, specific crystallographic relationship exists between the twin and the matrix. Nabarro (19, 24) pointed out that a dislocation would not pass from one grain to another when the slip directions in these two grains are different. .If it were to continue in the next grain without changing its Burgers vector, the dislocation would be trailed behind by a fault in whiCh atoms are badly dis- arranged. This would require a stress 0 = E-, where € is the 96 energy of a large angle boundary, and hence 0 would be of the order of the theoretical shear strength of the material. Al— ternatively, the original Burgers vector b may dissociate 1 into 62 and b , where b is a suitable slip vector in the 3 2 neighboring grain, and 53 remains at the grain boundary. In general, El and 32 are equal and the dissociation increases the total elastic energy in proportion to b2 A high stress 3. will be needed to enforce the dissociation. Therefore, it is expected that slip will not spread from one grain into its neighbor in this manner but rather by the process of inducing sources of dislocations in the neighbor- ing grain to become active. The most probable way to activate the dislocation sources in the neighboring grains is through stress concentration at the grain boundary as described pre- viously. On the other hand at a twin interface dislocations may pass through the boundary in accordance with the specific crystallographic arrangement between the matrix and the twin. Dislocation reactions such as described by Thompson and Mil- lard (6) may take place at the twin interface if they are energetically favorable. As shown in Fig. 27 and 28, twins are nucleated at the twin interface presumably due to the stress concentration 97 resulthu;from the basal slip in either the matrix or the twin where the basal slip has taken place. The fact that the heavy basal slip bands are associated with the twin nu- cleation may provide a clue to the mechanism of the twin nucleation. Fig. 27, 28 and 29 show that heavy basal slip bands in both the matrix and the twin meet at the existing twin inter— face where new twins are nucleated. Especially at points P, Q, R, S in Fig. 27, there appears to be a definite one to one correspondence between the slip bands in the matrix and those in the twin along the twin trace. It seems most probable that the points P, Q, R, S are the potential nucleation sites for twins. This is seen to be true in Fig. 28 as well as in Fig. 29. The basal slip plane in a twin in Fig. 27 has its normal making an angle of 300 with the tensile axis, and that in the matrix, an angle of 900. Hence the basal slip would take place easier in the twin than in the matrix. The basal slip dis- locations in the twin would pile up at the interface. If the resultimgstress concentration is high enough twins will be formed as in the case of near a grain boundary. Price ob- served such a piled up group of basal dislocations against the twin interface in cadmium platelets (43). 98 It has been observed that deformation twins can be formed at the existing interfaces inside as well as outside of the twin. Whether or not basal slip takes place in either the twin or the matrix (or both) is usually determined by the orienta- tion of the specimen and the way the load is applied. When it takes place in the twin, the c.r.s.s. is much higher than the ordinary value. .Bell and Cahn's experimental value for the c.r.s.s. for the basal slip inside the twin is approximately 185 gr/mm2. It is also known that such basal slip is accom- panied by a work softening. This general behavior is con- firmed by the present work. The nature of such work-soften- ing remains to be investigated.< The twins nucleated at the existing twin interfaces appear to be the conjugate of the existing ones, as shown in Fig. 27, 28, and 29. This is perhaps due to the fact that the trans- ferred shear stress due to the pile-up of the basal disloca- tions in the twin is a maximum for the conjugate twin system. 4.6 A Mechanism of Twin Nucleation A nucleation mechanism should be based on the following observed facts as well as the reasonable interpretation of them: i) Twins were formed at the sites of stress concentration 99 such as grain boundary, twin boundary, and specimen grip, etc. . ii) The streSs concentration at the grain boundary was not sufficient to create a stable twin nucleus in the bicrystals with orientation S-30, but approximately equal to the stress required to form two coherent twin interfaces. iii) In the bicrystal specimens, twins were formed at one side of the grain boundary under the influence of dis- location pile-ups on the opposite side. iv) In general, twins formed in the bicrystals with hard orientations were larger, more numerous, and grow to a greater extent upon reloading than those in the bi- crystals with soft orientations. In the case the stress concentration did not reach the theoretical twinning stress, a stable nucleus could not have been created as a whole as proposed by the homogeneous nucleation mechanism. Hence the nucleation process of a twin must involve the motion of pre-existing twin partials. 4.6.1 Twin embryo A twin embryo can be defined as a region of high stresses, which is bounded by twin partials but its size is smaller than 100 a stable twin nucleus. It can be considered as a potential twin nucleus which can grow to the stable size when sufficient shear stress is applied. Consider the self-stress fields of an edge dislocation. The shear stress‘oXy around the edge dislocation is shown in Fig. 4-5, where the z-axis is taken parallel to the disloca— tion line with the extra half plane of atoms taken to be the half of yz plane with positive y. For an elastically iso- tropic medium the stress fields in analytical form are (19, 33): = _ y(3X2 + y2) _ _ sin8 (2 + c0328) Oxx 70b (X2 + YZ) Zj 21b r y(X2 + y?) sin8 cos 28 O = b ~ = b 4.6 YY 7% (x2 + y2)2 7% r ( ) O = -2V_('b _,__y_ : “2770b Sine 2 _ 2 cos8 c0528 0 = 7Tb X(X y g = 7"b where “Z3: G/ZWKl-V). with G being the shear modulus, b the Burgers vector, and y Poisson's ratio. If these stress fields are taken to be approximately true for zinc crystals, one can find the boun- dary for a region where Oxy >‘7th ( = [G/2K]y=0.02.G). The contour of oxyé=0.02 G extends as far as x==12.5b=30 3. Along this equi-stress contour the shear stress is sufficient to displace the atoms in the direction of Oxy by an amount lOl equal to the twinning shear. Therefore, this contour can be considered as the boundary of a region bounded by twin par- tials whose Burgers vector is in the direction of Oxy as shown in Fig. 4-9. The width of this region is approximately one—half of its length. Since the spacing of the twin planes is equal to 0.633b in zinc such a region contains about basal 10 twin partials and can be considered as a twin boundary. In general, the twinning direction is not parallel to the Burgers vector of the dislocation. In such a case the shear stress in the twinning direction can be found by a suitable transformation of the components of the stresses in accordance with the particular twin system. The shape and size of the equi—stress contour thus vary according to the particular twin system under consideration. For the sake of convenience, the twin system whose twinning direction is parallel to the Burgers vector of the edge dislocation will be considered in the following. Once the equi—stress contour of the theoretical twinning shear stress is found, the nucleation event can be explained in terms of the motion of the twin partials until a stable twin nucleus is formed. It should be noted, however, that the stress field of a dislocation cannot be a potential twin nucleus unless the dislocation is locked strongly, otherwise 102 the dislocation itself will move before the total shear stress acting on it becomes sufficiently high to activate the twin partials in the embryo, since the shear stress required to move the dislocation is much smaller than that to activate the twin partials. Therefore, strongly locked dislocations such as the leading dislocation of a pile-up against a barrier may become a potential twin nucleus. fi;§;§_ The size of the stable twin lamella From the calculation of the activation energy for the for- mation of a twin lamella, Orowan (13) obtained the critical size of a twin lamella at which the twin becomes stable. He considered circular loOpS of twin partials lying parallel to the twin plane. The critical diameter of the loops is given by dC = BFVbTy, where F is the specific energy of the twin dislocations, b is the spacing of the twin planes, and y is -5 the twinning shear. Using F”: 0.018 1n(r/r6) X 10 ergs/cm -6 -7 (51) or F“: 3 x 10 ergs/cm for r = 1 cm and r0 = 10 cm, - 2 b = 0.633 b = 1.7 X 10 8 cm, W = 0.14, and 77: 15 kg/mm , basal 0 one obtains the critical diameter dé=700 A. Here, the shear 2 . . . . stress 7T= 15 kg/mm is used, Since it is the stress required to form two coherent twin interfaces. This means that a twin 0 u lamella of 700 A long become stable. The stress concentration, 103 therefore, must be sufficient to push the twin partials beyond this critical dimension for the nucleation of a twin. The above analysis is based on the assumption that the twin partials are circular loops lying parallel to the twin plane. However, the radius of the loops would be determined by the spacing of the dislocations cutting through the twin plane. If the dislocation density F>is taken to be of the order of 1010 cm-z, the spacing I of the dislocations is equal to 1A¢5 = 1000 3. Therefore, at the critical stage the radius of curvature of the twin partials is about 500 2. Prior to the critical stage, the loops are probably elon- gated in the direction perpendicular to the twinning direction. QLQL; The stress required to form the critical twin lamella The critical stage for the nucleation of a twin is the formation of circular loops of twin partials with a radius of approximately 500 g. The twinning stress at the critical stage 7; is given by (10) ZVt th where Vt is the twin interface energy and a0 is the critical radius of the twin partials. The second term is the stress 104 required to overcome the curvature of the loop. Using a0 = 500 A, the second term is approximately 1 kg/mmz, and hence it does not contribute significantly to 7;. The first term is the stress required to form two coherent twin interfaces. Hence, one obtains 7; = 14 i 5 kg/mmz. In order for a twin to be nucleated the shear stress exerted on the leading twin partial should be at least equal to 14 t 5 kg/mmz. In the calculation of the maximum stress concentration in the bicrystals with orientation S—30, the value of 2-3 mm was used for the length L of the dislocation pile-up° For this value of L, the stress concentration at the distance r = 700 A ahead of the leading dislocation of the lipe-up is given by L 1/2 Tzz('f) 75210-13 kg/mmz. Although this stress is somewhat lower than that required for the nucleation of a twin, the actual stress exerted on the leading twin partials should be larger than this value due to the forces acting on it by the trailing twin partials in the group. When the spacings of the twin partials are very large compared to the spacing of the twin planes, the array of twin partials approximates a linear array of piled-up dislocations lying on a slip plane (56). At the critical stage, the array of twin partials does not approximate the linear piled—up array. Therefore, the stress exerted on the leading twin partial is less than ntYL, 105 where'j; is the total shear stress (applied plus internal) exerted on the twin system. However, it is reasonable to assume that the stress concentration factor is larger than one. Hence the stress exerted on the leading twin partial can be larger than 10 - 13 kg/mmz. Because of this stress concentration factor, the twin lamella may grow beyond the critical size. So far, the stress fields of the grain boundary have not been considered. The stress fields of a grain boundary de- crease exponentially with the distance r from it, and becomes negligible when r is a few times larger than h, the spacing of dislocations in the grain boundary (19, 23, 33). For a large angle boundary, h is of the order of one atomic spac- ing. Since the stress fields due to the dislocation pile—ups are long ranged, the stress fields of the grain boundary can be neglected without introducing much error. 4&§;4_ A model of twin nucleation Based on the analysis in sections 4.6.1 through 4.6.3 a model of twin nucleation is proposed as shown in Fig. 4-9. The slip dislocations in grain A pile up at the grain boun- dary. The twin embryo, associated with the self-stress fields of the leading dislocation of the pile-up, consists of about Grain A Fig. 4-9. Schematic model of twm nucleation at the grain boundary 107 10 twin partials lying along the equi-stress contour of the theoretical shear stress for twinning (0.02 GelOO kg/mmz). When the shear stress due to the pile-up of dislocations ex— ceeds the stress required to form two coherent twin inter- , faces and to overcome the curvature of the twin partials, the twin embryo will grow. The formation of the critical twin lamella requires a shear stress of 14 I 5 kg/mmz. The present nucleation model requires only the existence of the twin embryo associated with the self-stress fields of a locked edge dislocation and the stress concentration suffi- cient to form two coherent twin interfaces. The stress con- centration can be developed at a grain boundary, twin boun- dary or at the specimen grip, etc. The shear stress 14 i 5 kg/mm2 can be taken as the critical r.s.s.t. for zinc crystals. This means that zinc crystals will twin at this shear stress in the absence of any stress concentration at room temperature. However, ordinarily it is difficult to avoid completely the stress concentration, and consequently twinning occurs at much lower stress level than the critical value. It is also of significance that the c.r.s.s.t. of 14 t 5 kg/mm2 is approximately equal to the stress required to form two coherent twin interfaces (or a twin fault bounded by two twin partials). The twin interface 108 energy is the controlling factor of the nucleation of twins. Recently, Burr and Thompson (55) obtained a twinning stress of approximately 14-16 kg/mm2 in zinc single crystals with a gauge length of 4 mm and a diameter of about 1.5 mm. The basal slip planes of the crystals were parallel to the tensile axis and the basal slip in the gauge length was never observed prior to twinning. The authors attribute the high twinning stress to the absence of the [0001] sessile dislocations which require the interaction of the pyramidal and the basal slip dislocations (14, 43, 54). This is in agreement with the c.r.s.s.t. deduced from the present work. In his electron-microscopy study of Mo - 35 o/o Re alloy, Hull (56) observed the nucleation of twins next to a grain boundary at the opposite side of which there was a piled-up array of dislocations in the adjacent grain. This is in agreement with the nucleation model shown in Fig. 4—9. In the nucleation model shown in Fig. 4—9 a grain boun- dary was taken as the barrier against which dislocations pile up. The grain boundary can be replaced by a twin boundary or by a sessile dislocation line lying in the slip plane. It should be emphasized that the fundamental features of the nucleation model can be applied to the nucleation of twins in general. These fundamental features are: 109 l) Pre-existing twin partials in a twin embryo associated with the stress fields of a locked dislocation, 2) The presence of a stress concentration sufficient to form a twin lamella of a critical size of about 700 g, 3) A shear stress of 14 i 5 kg/mm2 acting on the twin plane in the twinning direction. It was mentioned earlier that, when the twinning direction in grain B is not parallel to the Burgers vector of the lead- ing dislocation of the pile-up in grain A, the equi-stress contour of the theoretical shear stress for twinning must be calculated for the particular twin system by a transformation of the components of the self-stress fields of the leading dislocation. In the following, the equi-stress contour will be calculated for the (1012) [1011] twin system in grain B in the bicrystals with orientation S-30. The self-stress fields of an edge dislocation are given by equation (4.6). The stress fields referred to the co— ordinate system shown in Fig. 4—5 can be transformed to the new coordinate system shown in Fig. 4-10. Let the x, y, z axes be parallel to the [5110]A, [0001]A, and [011'0]A direc- tions respectively. The x', y', z' axes are taken to be par— allel to the twinning direction [bt]B = [ioII]B, the twin 110 [5110]A [0001]A [01I01A [b 1 = [10iI1B Fig. 4-10. The coordinate systems for the transformation of the stress fields of the edge dislocation E to the (1012) [1011] twin system in grain B in the bi- crystal' with orientation S-30. 111 plane normal [nt]B, and the [1210]B direction respectively as shown in Fig. 4—10. The subscripts A and B refer to the grains A and B of the bicrystal respectively. Then Oxy in the new coordinate system is given by the transformation (59) O 12 = alka2£ok1 (4.9) ' = + + or O 12 all a21 011 (all a22 a12 321) 012 + a12 a22 + 022 a13 a23 033' (4‘10) where the subscripts l, 2, 3, refer to the x, y, z axes re- spectively and 013 = 023 = 0. For the twin system (1012) [1011] in grain B, the direction cosines are: all = cos (¢x[bt]B and [2110]A) = cos 22O a12 = cos (2¥[bt]B and [0001]A) = cos 82o al3 = cos (2¥[bt]B and [0110]A) = cos 70o a21 = cos (¢5[nt]B and [2110]A) = cos 1060 = - sinl6O a22 = cos (¢$[nt]B and [2110]A) = cos 27O a23 = cos (4i[nt]B and [0110]A) = cos 690 where the angles are found by using the stereographic projec- tion. 0'12 is then plotted over a range of the angle 8 and the equi-stress contour of the theoretical shear stress for twin— ning is found as shown schematically in Fig. 4-11. It may be noted that the size and shape of the twin embryo are altered somewhat, but the general appearance remains substantially 112 similar to that shown in Fig. 4-9. Along the equi-stress con- tour the twin partials lie with their Burgers vectors parallel to the twin plane. When a shear stress of approximately 14 i 5 kg/mm2 is exerted on the (1012) [1011] twin system the twin partials move, and the twin embryo grows to the stable twin lamella as described previously. Fig. 10 and 12 show the (1012) [1011] twins nucleated at the grain boundary in the bicrystals with orientation S-30. Essentially similar analysis for the nucleation of twins in the bicrystals with different orientations can be made. Such an analysis can also be applied to the case in which twins are formed at the existing twin interface as shown in Fig. 27 and 28. It is further contended that the nucleation of twins in single crystals of zinc can be explained also in terms of the nucleation model shown in Fig. 4-9. In this case it is most likely that the pyramidal slip dislocation locked by the [0001] sessile dislocation has the twin embryo associated with its stress fields and may become a potential twin nucleus. 113 onlm aoflumusowuo :ue3 Hmumhuown asp QM omnhfia afl3u may no spzoum 0:8 .aala .mah «coco. .6 ”crap; 5:: e~o.onw.b 52:8 33.» Eco V CONCLUSIONS Deformation twins are always nucleated in zinc bicrystals subjected to tension at the sites of stress concentration such as the grain boundary, the twin boundary, or the specimen grip. The resolved shear stress for twinning is a function of the orientation of the bicrystal as well as the relative orientation of the neighboring grains. Twins are nucleated at the grain boundary under the in- fluence of the pile-ups of dislocations on the opposite side of the boundary. The maximum shear stress at the grain boundary due to the piled up group of dislocations has been found to be 12-15 kg/mmz. This value is one order of magnitude smaller than the theoretical shear stress for twinning and hence it does not support the homoge- neous shear mechanism, according to which the critical twin lamella is assumed to be created as a whole at a stress level corresponding to the theoretical shear stress for twinning. The nucleation of a twin is explained by the motion of the pre-existing twin partials in a twin embryo until a twin lamella of critical size of approximately 700 A is formed. The stress required to form such a critical twin 114 115 lamella has been estimated to be 14 I 5 kg/mm2 which can be taken as the critical r.s.s.t. in zinc crystals. The twin interface energy or the energy associated with the twin fault is the controlling factor in the nuclei— tion of twin in zinc crystals. Slip always precedes the nucleation of twins in zinc bi- crystals. However the activation of one, not necessarily both, of the basal and the pyramidal slips seems to be sufficient to nucleate twins. 10. 11. 12. BIBLIOGRAPHY Blewitt, T. H., Coltman, R. R. and Redman, J. K., "Low- Temperature Deformation of Copper Single Crystals," J. Appl. Phys., vol. 28, 651 (1957). Smith, C. S., "Metallographic Studies of Metals after Ex- plosive Shock," Trans. AIME, vol. 212, 574 (1958). Tartif, H. P., Claisse, F. and Chollet, P., Response of Metals to High Velocity Deformation, edited by P. G. Shewmon and V. F. Zackay, Interscience Publishers, New York, 1961, p. 389. Banerjee, B. R., "A Study of Strain Markings in Aluminum," Trans. AIME, vol. 188, 1126 (1950). Cottrell, A. H. and Bilby, B. A., “A Mechanism of the Growth of Deformation Twins in Crystals," Phil. Mag., vol. 42, 573 (1951). Thompson, N. and Millard, D. J., "Twin Formation in Cd,“ Phil. Mag., vol. 43, 421 (1952). Ogawa, K. and Maddin, R., "Transmission ElectroneMicro- scopic Studies of Twinning in Mo — Re Alloys," Acta Met., vol. 12, 712 (1964). Venable, J. A., "Deformation Twinning In F.C.C. Metals," Phil. Mag., vol. 6, 379 (1961). Venable, J. A., "The Electron Microscopy of Deformation Twinning," J. Phys. Chem. Solids, vol. 25, 685 (1964). Venable, J. A., "Nucleation and Propagation of Deforma— tion Twins," ibid., vol. 25, 693 (1964). Price, P. B., "Nucleation and Growth of Twins in Disloca- tion—Free Zinc Crystals," Proc. Roy. Soc. London, vol. A 260, 251 (1960). Fourie, J. T., Weinberg, F., and Boswell, F. C., "The Growth of Twins in Sn Single Crystals as Observed by 116 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 117 Transmission Electron Microscopy," Acta Met., vol. 8, 851 (1960). Orowan, B., Dislocations in Metals, AIME, New York, 1954, p. 116. Bell, R. L.*and Cahn, R. W., "The Dynamics of Twinning and the Interrelation of Slip and Twinning in Zinc Crystals," Proc. Roy..Soc. London, vol- A 239, 494 (1957). Wei, C. T., "Bending of a Small Zinc Single Crystal," J. Appl. Phys., vol. 30, 1457 (1959). Friedel, J., Dislocations, Pergamon Press, London, 1964, p. 262. Eshelby, J. D., Frank, F. C., and Nabarro, R. N., "The Equilibrium Arrays of Dislocations," Phil. Mag., vol. 42, 351 (1951). Cottrell, A. H., Progress in Metal Physics I, Ch. 2, edited by B. Charlmers, Butterworths Scientific Pub., Inc., London, 1949. Cottrell, A. H., Dislocations and Plastic Flow in Crys- tals, Oxford University Press, 1953. Livingston, J. D. and Charlmers, B., "Multiple Slip in Bicrystal Deformation," Acta Met., vol. 5, 322 (1957). Livingston, J. D., ibid., vol. 6, 216 (1958). Elbaum, C., "Relation Between the Plastic Deformation of Single Crystals and of Polycrystals," Trans. AIME, vol. 218, 444 (1960). Hauser, J. J. and Chalmers . B., "The Plastic Deformation of Bicrystals of F.C.C. Metals," Acta Met., vol. 9, 802 (1961). Nabarro, R. N., "Some Recent Developments in Rheology," British Rheologist's Club, London (1950). Frank, F. C., Report of Pittsburg, Conference on Plastic Deformation of Crystals, Washington (1950). P. 100. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 118 Chyung, C. K., "On the Improvement of Lattice Perfection in High Purity Zinc Crystals Grown From the Melt," M. S. Thesis, Michigan State University, 1962. Brandt, R. C., Adams, K. H., and Vreeland, T. Jr., "Etch— ing of High Purity Zinc," J. Appl. Phys., vol. 34, 587 (1963). Chyung, C. K. and Taylor, W., “Elimination of Small- Angle Boundary in High Purity Zinc Crystals Grown From the Melt," J. Appl. Phys., vol. 35, 731 (1964). Harper, S. and Cottrell, A. H., Proc. Phys. Soc. Vol. B63, 331 (1950). Barrett, C. 8., Structure of Metals, McGraw-Hill, 1952, p. 340. Li, J. C. M., “The Interaction of Parallel Edge Disloca- tions with a Simple Tilt Dislecation Wall," Acta Met., vol. 8, 296 (1960). Li, J. C. M., "Some Elastic Properties of an Edge Dislo- cation Wall," Acta Met., vol. 8, 563 (1960). Li, J. C. M., "Theory of Strengthening by Dislocation Groupings," Electron Microscopy and Strength of Crys- tals, ch. 15, edited by G Thomas and J. Washburn, In- terscience Pub., 1963. Madar, S., ibid., "Surface and Thin-Foil observations of the Substructures in Deformed F.C.C. and H.C.P° Metal Single Crystals," ch. 4. Seeger, V. A. and Trauble, H., "Die Plastische Verformung von Zinkkristallen," Z. Metall., vol. 51, 435 (1960). Read, W. T. Jr., Dislocations in Crystals, McGraw—Hill, New York, 1953, p. 118. Pugh, S. F., Rev. Metall., vol. 51, 683 (1964). Hauser, F. E., Starr, C. D., Tietz, L., and Dorn, J. E., "Deformation Mechanisms in Polycrystalline Aggregates of Mg," Trans. ASM, vol. 47, 102 (1955). 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 119 Tegart, W. T. McG., "Independent Slip Systems and Duc- tility of Hexagonal Polycrystals," Phil. Mag., Ser. 8, vol. 9, 339 (1964). Gilman, J. J., ”Fracture of Zn - Mono and Bicrystals," Trans. AIME, vol. 212, 783 (1958). Priestner, R., "The Relationship between Brittle Cleav- age and Deformation Twinning," AIME-Univ. Florida Conf. on Deformation Twinning (Mar. 21-22, 1963), ed. Reed-Hill et a1., Gordon and Breach Pub., N.Y., 1964, p. 321. Priestner, R. and Leslie, W. C., "Nucleation of Deforma- tion Twins at Slip Plane Intersections in B.C.C. Lat- tice," Phil. Mag., vol. 11, 895 (1964). Kratochvil, P. and Bocék, M., "Slip Lines in Deformed Zinc Crystals," Phys. Stat. 801., vol. 6, K69 (1964). Jaswon, M. A. and Foreman, J. E., "Non-Hookean Interac- tion of Dislocation with a Lattice Inhomogeneity," Phil. Mag., vol. 43, 201 (1952). Hausen, P. and Leibfried, G., Nachr. Akad. Wiss. G6ttingen, No. 2, 31 (1954). Head, A. K., "The Stress Fields Around Some Dislocation Arrays," Aust. J. Phys., vol. 12, 613 (1960). Stroh, A. N., "The Formation of Cracks as a Result of Plastic Flow," Proc. Roy. Soc., vol. A 223, 404 (1954). Mitchell, T. E., "The Stress-Fields Around Group of Dis- locations in F.C.C. Metals," Phil. Mag., Ser. 8, vol. 10, 301 (1964). Ku, R. C. and Johnston, T. L., "Fracture Strength of MgO Crystals," Phil. Mag., vol. 9, 98 (1964). Schmidt, E. and Boas, W., Plasticity of Crystals, F. A. Hughes Pub., 1935. Yoo, M. H. and Wei, C. T., to be published in Phil. Mag. 52. 53. 54. 55. 56. 57. 58. 59. 120 Price, P. B., "Direct Observations of Glide, Climb, and Twinning in Hexagonal Metal Crystals," Electron Mi- croscopy and Strength of Crystals, ch. 2, ed. G. Thomas and J. washburn, Interscience Pub., 1963. Bocék, M. und Kaska, V., "Die Orientierungs- und Tem- peratur-Abhangigkeit der Verformungkurven von Zink- kristallen," Phy. Stat. 801., vol. 4, 325 (1964). Burr, D. J. and Thompson, N., "Dislocations and Cracks in Zinc," Phil. Mag., vol. 7, 1773 (1962). Burr, D. J. and Thompson, N., "Twinning and Fracture in Zn Single Crystals," Phil. Mag., vol. 12, 229 (1965). Hull, D., "Growth of Twins and Associated Dislocation Phenomena," AIME- Univ. Florida Conf. on Deformation Twinning (Mar. 21-22, 1963), ed. Reed-Hill et a1., Gordon and Breach Pub., New York, 1964, p. 121. Stiegler, J. D. and McHargue, C. J., ibid., "The Effect of Impurities on Mechanical Twinning and Dislocation Behavior in B.C.C. Metals,“ p. 209. Berghezan, A. and Foudeux, A., "Transmission Electron Microscopy Studies of Mechanism of Plastic Deforma- tion," J. Appl. Phys., vol. 30, 1913 (1959). Nye, J. F.,.Physical Properties of Crystals, Clarendon Press, Oxford (1957). ‘ APPENDIX .A STRESSES ASSOCIATED WITH A DISLOCATION PILE—UP Eshelby et a1. studied the characteristics of a pile-up group of dislocations (17). .When dislocations emitted from a source encounter a barrier, they will pile up against the bar- rier and form a linear array. The positions of the disloca- tions of xi(i = l, 2, ..., n) can be defined as the values of x at which the polynomial f(X) = (x - x1) (x - x2).......(x - xn) (A.1) vanishes. The equilibrium condition of the dislocations can be expressed in terms of a differential equation, the solu- tion of which is the polynomial f(x). The equation f(x) = 0 has n distinct and real solutions of x which are the equilib- rium positions of the dislocations in the pile—up array. Eshelby et al. showed that the distribution of the n disloca— tions along the slip plane is the same as the radial distribu- tion of nodes in the ns state of a hydrogen atom except a constant factor. Other results of the solution of Equation (A.1) can be summarized as follows (16, 18) 1. The number of dislocations that can be piled up in a length L of the slip plane is 121 122 n = (1rk/Gb) LTa (A. 2) where k = (1 -2/) for an edge dislocation l for a screw dislocation, G = shear modulus, applied shear stress on the slip plane, Dfl u U‘ ll magnitude of the Burgers vector of the dislocation. 2. The distance between the leading dislocation and its nearest neighbor is x1 = (1.84) (ch/zink-(a) (A.3) 3. The stresses in the neighborhood of the leading dis- location: a. Near the leading dislocation (r «:xl) the piled up group exerts in its glide plane a shear stress 71,3117; (A.4) b. At large distance (rcL) the piled up group exerts the same stress as a single dislocation of Burgers vector nb located at its centre of gravity would, T3z(L/2r)7; (A.5) c. At intermediate distance (XI/15 «:r «:L) the stress is “[241 + (L/r)1/2]'(:_;1 (A.6) 123 It may be noted that Equation (A.4) can be derived by ap- plying the principle of virtual work (18, 19). The piled up group of dislocations are in equilibrium and none of them can move unless the leading dislocation moves. If the Leading one moves by a small distance 5x, all the dislocations will move forward by this amount. Thus the work done by the leading dis— location against the internal stress 21 is‘Zlbdx. The work done by the applied stress is nzébdx. At equilibrium these are equal so that Tianjg (A.4) Thus the stress concentration factor at near the leading dis- location is the same as the number of dislocations in the pile—up group. In the immediate neighborhood of the leading dislocation (r < X1/15), the stress field of its own is predominant. As the distance r increases beyond the core of the leading dis— location, the self stress field will be successively replaced by ZIInYa’ zéz[l + (%-1/2]2;, and 73e§%-7fa. The shear stress is expected to be a smooth and continuous function of the distance r. APPENDIX B THE EFFECTS OF GRAIN BOUNDARY ON THE PLASTIC DEFORMATION BY SLIP It is known that the grain boundary does not contribute any strength per se (20, 21). The fact that the strains must be continuous across the grain boundary dictates the mechanism of deformation in the adjacent grains which may in turn con- tribute to the strength of the bicrystal. The contribution of the grain boundary, therefore, depends on the mechanism of deformation that prevails in each grain (22). The term "com- patible" is usedtto describe specimens in which, on the assump— tion of homogeneous shear on the most favorable (primary) slip system only, each crystal grain deforms as if it were indepen- dent of the other crystal grains. The condition for compati- bility is satisfied if the strains {GT | arising from deformation on one slip B system in each crystal is identical, I or in a mirror image relationship, Y across the grain boundary. I I I I I I l I I --~3}--»- . Consider a bicrystal shown in , .I Fig. B—l. The grain boundary is de- Fig. B-l fined as xz plane and the tensile 124 125 axis is along the z axis. The three strain components which describe the deformation of the grain boundary are gxx' 6X2, 222' Then the condition for compatibility of the bicrystal is satisfied when the strain components of the corresponding primary systems of grains A and B are equalsat the grain boundary, i.e.. 5A =£B . EA =53. 6A = EB (B.7) xx xx xz xz 22- 22 Thus if Equation (B.7) is not satisfied by the primary slip systems, the bicrystal is said to be incompatible. In the most incompatible bicrystals, at least four in- dependent slip systems must be activated in order to maintain the strain continuity across the grain boundary (22, 23). The three continuity conditions in equation (B.7) and the elonga- tion along the tensile axis (fizz) constitute the four require- ments which must be satisfied by slip on at least four indepen- dent slip systems. So far the concept of compatibility has been defined on the basis of macroscopically homogeneous strains. Deforma— tion by slip, however, is highly heterogeneous, and slip takes place only on a small number of all the available slip planes. This leads to a microscopic incompatibility in which micro- scopic strains in the neighborhood of a grain boundary are not equal (22, 23). In general there is no one to one correspondence 126 between slip lines in the neighboring grains along the grain boundary. Because of the microscopic incompatibility the secondary slip systems are activated near the grain boundary in order that the internal stresses built up by the microscopic strains can be relieved. The most likely process by which secondary slip systems are activated is by means of disloca- tion pile-up which causes the slip sources in the adjacent grain to be activated (24, 25). The plastic deformation in bicrystals of cubic metals has been investigated extensively (21, 22, 23). Similar study of h.c.p. metals is, however, rarely found in the literature. W I! {IHIHIIIHIIIIIHI 93 03046 4162 Inimiiiiiui