CT F CRY u—RE :mzm‘c; .. y 4. a... 2. it. #4.»...rn5 iaz. g .7914?! v1 1 :2! e . w, 2.. .xflz‘... “up. mérfl... I n: u - Y .53, A, :"r J. <,i L . ’51P}; HIGA-N 16 M T THESIS all-'1‘;- .5, LIBRARY Michigan Stat: :1 University This is to certify that the thesis entitled FRACTURE IN ZINC CRYSTALS presented by Bhupendra U . Shah has been accepted towards fulfillment of the requirements for Eh . D . degree in _M§i&ll_wgy /: / g, :71,» Major professor/l é. / I). ‘1 . .7‘ ". Date /L/1’j7 J": A} /? [f 7 / 0-169 t‘ a. 0'- FRACTURE IN ZINC CRYSTALS By Bhupendra Umedchand Shah A THESIS Submitted to Michigan State University in partial fuifiiiment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metaiiurgy, Mechanics and Materiais Science 1967 c Winn; 3-~11;«c‘:6 ABSTRACT FRACTURE IN ZINC CRYSTALS by Bhupendra Unedchand Shah A multiple-slip-band model for fracture in zinc crystals based on the stress concentration due to piled-up groups of dislocations on many nearby slip bands has been developed. The stress concentration factor for each of the various stress components of the stress field of the dislocation pile-ups near a grain boundary is calculated as a function of the geometric and defonnation variables. The results indicate that for the same number of dislocations per slip band the stress concentration ahead of the dislocation pile-ups is long-ranged and much higher than what is obtainable with the single-slip-band model, while the stress concentration on the side of the pile-ups is short- ranged and not as high. Experiments with asymmetric zinc bicrystals with controlled orientations show that fracture always takes place in the neighboring crystal grain ahead of the slip bands, and is caused mainly by the shear stress acting on the cleavage plane substantiating the present theory. A double pile-up model has also been developed for fracture in zinc single crystals based on the stress concentration produced by piled- up groups of dislocations on intersecting basal and pyramidal planes. Using the experimental results of Bell and Cahn (1958) the positions of the dislocations on the pyramidal plane are calculated as a function of the number of dislocations on the two planes. The results indicate that 2 Bhupendra Umedchand Shah the stress concentration at the tip of such a configuration of dislocations may also lead to the nucleation of a crack. The present investigation suggests that 0111 = K2, where 01 and 11 are respectively the internal tensile and shear stresses acting on the cleavage plane, and K, a material constant of the order of magnitude of the theoretical strength of the material, can be used as a criterion for the type of brittle fracture occurring in zinc. ACKNOWLEDGMENTS The author is very grateful to his advisor Professor C. T. Wei for his guidance and encouragement during the course of this study and throughout the graduate program. He is also indebted to the other members of his graduate committee, Professors D. J. Montgomery, A. J. Smith and J. G. Hocking. He wishes to extend his gratitude to Dr. J. w. Hoffman and the Division of Engineering Research for continuous financial support. Acknowledgment and thanks are due to the entire faculty and staff of the Department of Metallurgy, Mechanics and Material Science for their assistance and helpful discussions. Finally, special appreciation is expressed to Dr. E. S. Rowland, Director of Research of the Timken Roller Bearing Company, for the help which made it possible to carry this thesis through to completion. TABLE OF CONTENTS Page I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . 1 II. THEORY. . . . . . . . . . . . . . ..... . . . . . . . . ll 2.l. The stress field of a dislocation . . . . . . . . . . ll 2.2. Single— slip— band model. . . ....... l4 2. 2.l. Derivation of the equations for the stress concentration factors . . . . . . . . . . . . l4 2.3. Multiple- -slip— band model for crack nucleation . . . . l8 2. 3 l. Equations for the stress concentration factors . . .............. l8 2.3.2 Transformation of the coordinate system . . . l9 2.3.3 Effect of the orientation of slip bands . . . 22 2.3.4 Distribution of shear and normal stresses along the central slip band . . . . . 23 2.3.5. Transferred shear stress of grain A onto grain B . . . ........... . . . . . 24 2.4. Dislocation pile— —ups in the multiple— slip- band model. 26 2. 4. l. Equilibrium positions of dislocations in the multiple- -slip- -band model. . . ....... 27 2.4.2 Effect of a group of slip bands . . . . . . . 28 2.4.3 Effect of N, n, s, and k. . . . . 29 2.4.4 Effect of bicrystal orientation and slip° modes . . ............... 3l 2.4.5 Process ofo fracture . . . . . . . . . . . . . 32 2.4.6 Application . . . . . . . . . . . . . . . . . 34 2.5. Crack nucleation in single crystals . . . . . . . . . 36 2.5. l. Basal- pyramidal interaction . . . . 36 2.5.2. Equilibrium positions of dislocations inc a0 double pile- up of dislocations. ....... 37 2.5.3. Transformation of the coordinate system . . 42 2.5.4. Stress components at the tip of the double“ pile- up of dislocations ..... . . . . . 43 2.5.5. Calculation of p(k) and the s. c. f.. . 45 2.5.6. Stress concentration due to a doublea pile- -up of dislocations ............... 46 III. EXPERIMENTAL PROCEDURES AND RESULTS .......... 79 3.l. Preparation of the specimens. . . . . . ..... 79 Table of Contents -- (continued) 3.2. e lts ....................... l. Asymmetric bicrystals ............ 2. Symmetric bicrystals with orientation (S-A) . 3. Single crystals ............... su .2. .2. .2. 00000030 IV. DISCUSSION ......................... 4.1. Location of cracks .................. 4.l.l. Bicrystals .................. 4.l.2. Single crystals ............... 4.2. Crack nucleation ................... 4.3. River pattern . . .................. 4.4. A criterion for the fracture process in zinc ..... V. CONCLUSIONS ........................ BIBLIOGRAPHY .......................... APPENDICES ........................... 99 99 99 TOT l02 103 107 lO9 112 Table 1. LIST OF TABLES s.c.f. components on the plane (X = 0) normal to the plane of the pile—up. . . . . . . . . . . . . . ........ (a) s.c.f. as a function of N ............... (b) s.c.f. as a function of n. . . ............ (c) Summary of the results obtained by applying the isotropic and anisotropic elasticity theories. . Positions of pyramidal dislocations in a double pile- up of dislocations. . . . . . . . . . . . . . . . . . (s 5°C°f')rs at the tip Q on the pyramidal plane as a function of Z for various values of n ........... Summary of tensile data .......... K values for bicrystals . . . . . . . . . . ..... Equilibrium positions of dislocations in the single- -slip- band model. . . . . . . . . . .............. Page 74 75 75 76 77 78 98 106 Figure 10. 11. 12. 13. 14. 15° 16. 17. 18. LIST OF FIGURES Page Five possible positions of a crack with respect to a barrier ...................... . . . . 9 Slip systems and twin system in zinc ............ 10 Single-slip-band model ............... . . . . 50 (s.c.f.)xx distribution along the negative Y axis . . . . . 51 (s.c.f.)XX and (s.c.f.)xy as a function of n. . ...... 52 Multiple-slip-band model ........ . . . . . . . . . . 53 (a) Transformation of the coordinate system for the multiple- slip-band model, A = 45° ................ 54 (b) Symmetric bicrystal .................. 23 (c) Coordinate systems for the superposition of stresses of grain A onto grain B .................. 55 Interactions between dislocations in different slip bands . 56 s.c.f. as a function of N ................. 57 (a) (s.c.f.) as a function of k . . ......... . . 58 (b) (s c.f.):i and (s.c.f.)yy as a function of k. . . . . . 59 s.c.f. as a function of 10910 5 .............. 6O s.c.f. as a function of n .......... . . . . . . . 61 (s.c.f.)Xy as a function of A ............... 62 (s.c.f.)xx and (s.c.f.)yy as a function of A. . . . . . . . 63 (s.c f )xx as a function of 10910 s .......... . . 64 (s.c.f.)xy as a function of 10910 s . . . . . . . . . . . . 65 ( s.c.f.) as a function of log 5 . . . . . . . . . . . . 66 yy 10 Cancellation of stress fields of dislocations in various regions .......................... 67 vi Figure 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. List of Figures -- (continued) Dimensions and orientations of the specimens. Crack nucleation in specimen No. l (46 Crack nucleation in specimen No. 2 (55 Crack nucleation in specimen No. 3 (59 Crack nucleation in specimen No. 4 (49 (a) Upper half of the fractured specimen Crack nucleation in specimen No. 6 (5 Crack nucleation in specimen No. 7 (5 Crack nucleation in specimen No. 8 (A Basal and pyramidal slip traces in specimen No. (A = 26°) ......................... ) Crack nucleation in specimen No.11 (A (a Eb) Basal and pyramidal slip traces . c - 63) . Double pile-up of dislocations in zinc. . Transformation of the coordinate system for the double pile-up of dislocations in zinc ............ 31). . . . (41 - Ha). (b) Cleavage face of grain B ................ 40) ....... 53) . . . . ) Stress - strain curve of the crystal No. .f. )x distribution along the central slip band . .f. )W distribution along the central slip band. .f. )g distribution along the central slip band . . fe rred shear stress of grain A onto grain B. . 0000000 0000000 10 11 {1° ° Crack nucleation along (000])twin in specimen No. Crack nucleation along (OOOl)tw1-n in specimen No. vii Page 68 69 7O 71 72 73 84 85 86 87 88 89 89 9O 91 92 LIST OF APPENDICES Page Lattice parameters and elastic constants of zinc ....... 112 Equilibrium positions of dislocations in the single-slip- band model .......................... 113 Decrease in strain energy due to the basal-pyramidal interaction ....................... . . 115 viii I. INTRODUCTION The fracture of crystalline solids is a phenomenon of considerable importance from both the theoretical and the application points of view. It is a phenomenon usually associated with plastic deformation. The tendency for many materials to undergo brittle fracture under high stresses severely limits their applications as engineering materials. Furthermore, brittle fracture also limits the extent to which materials can be strengthened by alloying, cold working, or other means. Brittle fracture in iron and steel has been, to a large extent, responsible for stimulating the research activities in the field of fracture in the last two decades. In zinc, a similar type of fracture occurs. Although the detailed mechanism of fracture in zinc may differ from that in steel, there are certain features common to both steel and zinc with regard to the conditions for crack initiation and propagation (l) (2). The present work is devoted to the study of fracture in zinc as a step toward the understanding of the fracture process in crystalline solids in general. Since industrial metals and alloys are generally polycrystalline, it is essential that the fracture behavior of the individual grains in a polycrystalline metal and the role played by the grain boundary in the fracture process be understood first. Therefore, the fracture process in single crystals or bicrystals is usually the subject of theoretical and experimental studies. Fracture may be treated as a nucleation and a growth or propagation process. Using an approach similar to that for the slip process (3a), one may estimate the stress concentration factor (abbreviated henceforth as s.c.f.) required for the homogeneous nucleation of a crack in a crystal subjected to an applied stress. In order to break the atomic bonds across a plane, the stress should, in theory, reach approximately G/Zn for a perfect crystal, where G is the shear modulus of the material. Using a refined analysis based on realistic laws of atomic forces, Dieter (4) arrived at a theoretical shear strength of approximately G/30. For zinc crystals the reported shear stress at fracture If may be as low as 25 g/mm2 to 200 g/mm2 depending on the crystal orientation (5) (6). Using an average value of T of 100 g/mm2 and G = 3.72 x 1011 dynes/cm2 f (Appendix A), one obtains s.c.f. = = 6000 ZflTf Thus the applied stress alone is not sufficient for the homogeneous nucleation of a crack, and some form of an internal stress concentration is necessary for fracture to occur. It is generally observed that plastic yielding precedes fracture in crystalline solids, suggesting that fracture is induced by plastic deformation (5). Slip and mechanical twinning are the only fundamental processes by which crystalline solids can be plastically deformed. When a slip band or a twin ends within a crystal, a local stress concentration results. Consequently, the stress necessary to initiate fracture is generally found to be lower. than the theoretical shear strength of the material. The estimation of the stress concentration at the point where a slip band ends within a crystal was first attempted by Zener (7) using the elasticity theory of a crack. By considering the slip band as a freely slipping crack across which the applied shear stress is completely relieved, Zener obtained the s.c.f. as L q = (lg->2 where L is the length of the slip band and b is the radius of curvature at its end. In Zener's treatment the cohesion between the faces of the slip band was neglected. However, in a later refinement of the theory, Zener (8) proposes that a slip band can be represented by a sequence of edge dislocations extending from the source to an obstacle such as a grain boundary, sub-boundary, twin boundary, or a precipitated particle. Such a piled-up group of dislocations has the effect of magnifying the applied stress in the vicinity of the obstacle and may produce a sufficient stress concentration to initiate a crack. Koehler (9) estimated the tensile stress acting on a plane normal to the plane of the pile-up and in the vicinity of the tip of the pile-up to be of the order of hi, where n is the number of dislocations in the pile-up and r is the resolved shear stress acting on the slip plane. This concept, valid only when n is large, has been developed into a quantitative theory of fracture by Stroh (10) (ll), represented by the expression 1 /2 n 0n = 3 (F) 1 “‘< 2 'i “”11”“ < > s.c.f. = 22 yy j=l (Xj2 + yj2)2 n X- (X-2 - y-Z) J J J (s.c.f.) = 2 Z (23) The component at of the stress acting at a point is obtained from the sum of the internal stress (xx, yy, or xy) and the particular component of the applied stress 6a. Thus, 9t = 0a + (s.c.f.) T . (24) For a positive shear stress, the nature of a normal stress component is determined from the sign of the stress concentration factor; that is, it is tensile if the sign if positive, and it is compressive if the sign is negative, Let E(j) be the equilibrium positions of the dislocations. Transfonning E(j) according to the equation 17 5(3) = e(.i)'2% . (25) the s.c.f. as a function of n can be obtained by assigning pr0per values of n. The s.c.f. for the various stress components at the tip of the pile-up, the s.c.f. for the points on the nonnal to the slip plane, and the maximum values of the normal and shear components are investigated in the following by applying both the isotropic and anisotropic elasticity theories for comparison. By the substitution of xj = e(j) and yj = O in the equations (18) through (23), the stress concentration factors at the tip for xx and yy components are found to be equal to zero, and that for xy component, equal to n. Unless otherwise specified each plot of the stress concentration factor as a function of the indicated variable will hereafter represent the results obtained by applying isotropic elasticity along the negative y axis theory. The (s.c.f.)X as a function of y. J for n = 49, 69, and 99 is shown in Fig. 4. The maximum value onthe X (s.c.f.)XX obtained at a point on the y axis, and the (s.c.f.)Xy at that point as a function of n are shown in Fig. 5. The results obtained by using each of the isotrOpic and anisotrOpic elasticity theories are listed in Table l for comparison. The maximum value of the (s.c.f.)Xy is found to be equal to n, and occurs at the tip of the pile-up on the slip plane. It follows that the maximum values of the.(s.c.f.)XX and (s.c.f.) occur on the planes at +45° and -45° to the Slip plane yy respectively, and both are equal to n. 18 2.3. Multiple-slip-band model for crack nucleation A model for crack nucleation may be developed by considering a bicrystal as the first approximation to a polycrystalline aggregate, since a bicrystal has both the elements (grains and grain boundary) of a polycrystal. If an unstrained bicrystal of zinc suitably oriented for basal slipping in both grains is deformed, deformation by slip appears as many parallel slip lines across the crystal grains. Each slip line can be regarded as being associated with a pile-up of dislocations. Thus one may consider a multiple-slip-band model as shown in Fig. 6 for evaluating the s.c.f. The plane of the drawing corresponds to Z = O. The grain boundary (GB) acts as the barrier against which dislocations on parallel slip planes pile up. The center of the locked dislocation on any slip plane is located at the point of intersection of that slip plane with the grain boundary (0], 02,..., Q ). The origin of a right-handed T rectangular Cartesian coordinate system is chosen to coincide with the point Q(T + ])/2, where T is the total number of slip lines in the group. In grain A the dislocation source on each slip band is located at a distance L fron the locked dislocation on that plane. The Y axis is nonnal to the slip planes. The dislocations in the group are edge dislocations of the same sign and directed towards the positive 2 axis. The Burgers vector is in the positive X direction. 2.3.1. Equations for the stress concentration factors To obtain the equations for the s.c.f. due to a group of slip bands it is assumed that the principle of superposition is applicable. 19 The s.c.f. equation for the 0 component obtained from equation (4) is xx then 1 g (1.62)DYj(k) [1.92Xj2(k) + 1.62Yj2(k)] 0xx = I .4 - (26) k=1 3:1 [Xj2(k) - 1.62Yj2(k)]2 + (2.92)(l.62) Xj2(k) Y32(k) In the above expression, n is the number of dislocations in each slip band, T is the total number of slip bands, and the coordinates Xj(k) and Yj(k) are measured in cm from the center of the j-th dislocation on the k-th slip band to the point under consideration. Similarly, one can obtain the stress concentration factor equations for the xy and yy components. The coordinates Xj(k) and Yj(k) in equation (26) are functions of the deformation variables of a particular sample. In the following sections several variables affecting the s.c.f. are considered separately. 2.3.2. Transformation of the coordinate system Let S be the spacing of the slip bands in on and uniform, A. the angle between the slip plane and the tensile axis, and 6, the angle between the slip direction and the tensile axis. The orientation of the grain A is so chosen that A = 9 = 45°. It is assumed that the grain boundary is parallel to the tensile axis. The s.c.f. will be analyzed as a function of N, n, and S. Fig. 7 (a) shows the transformed coordinate system for this problem. The variables S, Xj(k), and Yj(k) are transformed, respectively, according to the following equations: 20 s = s 29.’ (27) XJ-(k) = xj(k) 5D; (28) Y-(k) = y-(k) 9— (29> J J 21 where s, xj(k), and yj(k) are dimensionless parameters. In Fig. 7 (a) each slip band is identified by a number which is positive for the slip bands situated on the positive side of the Y axis, and is negative for those on the negative side. The variable k assumes the integer values from -N to +N including 0, and the total number of slip bands T is T=2N+l . (30) Let the equilibrium position Ej(k) of the j-th dislocation on the k—th slip band be defined as the distance in cm from that dislocation to the locked dislocation on its slip plane. The transformation of Ej(k) to a dimensionless parameter ej(k) 15 according to the equation E-(k) = e-(k) 70— . (31) Substituting equations (28), (29), and (30) in equation (26) and applying the procedures similar to those used in sec. 2.2.1 one obtains the s.c.f. equations for the various cases as following: i, 21 AnisotrOpic case +N . .2 .2 (S'C'f')xx = _3.24 l " y3(k) [1.92).J (k) + 1.62y3 (k)] (32) k=-N i=1 [x.2(k) - 1.62y.2(k)]2 + 4-73 x.2(k)y.2(k) J J J J N . .2 _ .2 (s.c.f.) = 2 + ” yJ(k) [*3 (k) 1'62yl (k)] (33) XX k=-N i=1 [xj2(k) - 1.62yj2(k)]2 + 4.73 Xj2(k)Yj2(k) +N ~(k) [ -2(k) — 1.62 -2(k)] (s.c.f.)Xy = 2 2 E XJ x3 yJ (34) k=-N j=l [xj2(k) - 1.62yj2(k)]2 + 4.73 xj2(k)yj2(k) Isotropic case N ‘k 3'2k .Zk (s.c.f.) = -2 E E yJ( ) [ XJ ( ) + yJ ( )J (35) xx k=-N j=1 [xj2(k) + yj2(k)]2 , ,2 _ .2 (s.c.f.) = 2 +N n yJ(k) [xJ (k) yJ (k)] (36) XX k=-N j=l [xj2(k) + yj2(k)]2 , ,2 _ .2 k (s.c.f.) = 2 EN E XJ(k) [XJ (k) yJ ( )1 (37) xy k=-N j=l [xj2(k) + yj2(k)]2 The s.c.f. at the tip 0 of the central slip band (k = O) can be obtained in the following manner. In Fig. 7 (a) Xj(k) and yj(k) are related to ej(k), s, and k by the equations 22 xj(k) = ej(k) - k 5 II l 7? yJ-(k) s The stress concentration factor as a function of any one of the three variables 5, N, or n can be obtained by assigning proper values to the other two variables. Fig. 9 shows the s.c.f. as a function of N for N = 5 to 100 obtained by choosing s = .01 and n = 99. The contribution of each slip band to the total stress concentration is shown in Fig. 10 (a) and Fig. 10 (b) for the shear and the normal stress components respectively. The s.c.f. as a function of 109105 (for s = 5 to 10'3) obtained by assuming N = 50 and n = 99 is shown in Fig. 11. By choosing N = 50 and s = .01, the s.c.f. as a function of n is shown in Fig- 12. The stress concentration factors for the various stress components as a function of N, n, and 5 obtained by applying isotropic and anisotropic elasticity theories are listed in Tables 2 (a), 2 (b), and 2 (c) respectively. In the above calculations the equilibrium positions corresponding to those of the single-slip-band model (Appendix B) have been used for ej(k). The effect of the mutual interactions between the dislocations on different slip bands will be considered later in sec. 2.4.1. 2.3.3. Effect of the orientation of slip bands In a bicrystal, the stresses in one grain due to the slip bands in the other depend upon the relative orientation of the two grains and the modes of deformation in these grains. It is in general a three- 23 dimensional problem and complicated. However, in special cases such as symmetric bicrystals the stress distribution becomes a two-dimensional one and manageable. Let +k the orientations of the two grains be so chosen that A = ¢ (as defined in sec. 2 3.2) in each grain, Vb. //+\ k the two grains are mirror images of each other across the boundary plane, and the dislocation lines in these grains are parallel to the Z axis Figure 7 (b) Symmetric bicrystal as shown in Fig. 7 (b). The coordinates xj(k) and yj(k) with respect to the tip Q of the central slip band as the origin are given by X A 7? V H j ej(k) - k s (cotA) y-(k) = -k s The s.c.f.'s for the various stress components as a function of A and s are calculated for n = 99 and N = 50. The s.c.f.'s for the shear component as a function of A obtained by assuming 5 = .01 are shown in Fig. 13, and the corresponding normal stress components are shown in Fig. 14. The s.c.f.'s as a function of 109105 for several values of A are shown in Figs. 15, 16, and 17 for the xx, xy, and yy components respectively. 2.3.4. Distribution of shear and normal stresses along the central slip band For the calculation of the stress distribution for the various stress components along the central slip band, the orientation of the 24 grain A is so chosen that A = ¢ = 45°. Let AX(i) by the x-coordinate of the i-th point measured with respect to the XYZ coordinate system, and AH, the increamental distance along the X axis. The transformation of AX(i) and AH to the dimensionless parameters AX(i) and Ah is according to the equations m HID AX(i) = AX(i) AH = Ah 2? The parameters xj(k) and yj(k) for the i—th point along the X axis are obtained from the equations x(i) = iAh (38) where the variable i assumes integer values which are negative on the same side as the pile-up, and positive ahead of the pile—up. By the substitution of the equations (38) in equation (37), the s.c.f. for the shear component as a function of AX(i), obtained for n = 99, s = .01, N = 50, and Ah = .001, is shown in Fig. 19 (a), and the corresponding s.c.f.'s for the normal stress components are shown in Figs. 19 (b) and 19 (c). 2.3.5. Transferred shear stress of grain A onto grain B Finally, the transferred shear stresses due to the slip bands in 25 grain A onto the basal plane in grain B as a function of the position along that plane will be investigated. In Fig. 7 (c), a right-handed rectangular Cartesian coordinate system X'Y’Z“ is so defined that Y' = 0 is the basal plane and the axis 2' is parallel to Z. Let A2 be the angle between the basal plane and the tensile axis in grain B, and e, the angle between the X' and X axes measured in the counter—clockwise direction. Let R(i) be the radius of a circle with Q as the center, and AY(i), the ordinate of the i—th point measured with respect to the XYZ coordinate system. The variables R(i) and AY(i) are transformed according to the equations R(i) = r(i) . . D AY(l) = Ay(i) i; where Ay(i) and r(i) are dimensionless parameters. The coordinates xj(k) and yj(k) are obtained from the equations xj(k) = eJ-(k) —k s + AX(i)— AX(i) = r(i)COSe (39) yj(k) = -k s + AY(T) Ay(i) = r(llsme —( Substituting equations (39) in equations (35). (35): and (37): the 5°C=Tcls for the various components at the i-th polht Wlth reSpeCt t0 the XYZ coordinate system are obtained. The transformation of the stress 26 components obeys the law for the transformation of a second rank tensor “13' : aik ajt “kt (1, J" k’ (i = l. 2. 3) where the direction cosines 911's for the coordinate transformation are given by cose sine 0 —sine cose O O 0 1 Therefore, 1 = 2 _ ‘ 2. u ' ... o 12 (COS 6 SH] 6) oxy s1ne COSB (oXX ny) where 012 is the transferred shear stress of grain A onto the basal plane in grain B. The s.c.f.'s for the shear component as a function of position along the X' axis for the various orientations of the basal plane in grain B obtained for N = 50, n = 99, s = .01, and r(i) = (.01)i are shown in Fig. 19 (d). 2.4. Dislocation pile-ups in the multiple-slip-band-model When a bicrystal of zinc oriented for basal glide is deformed, groups of dislocations on parallel slip planes may pile up against the grain boundary. Each dislocation will interact with other dislocations on the same slip plane and those on different planes° AS a result the equilibrium positions of the dislocations will be changed. In the 27 above calculations of the s.c.f.‘s for the various cases it was assumed that the positions of the dislocations in the multiple-slip-band were the same as in the single—slip-band model. The error introduced by such an assumption can be estimated in the following manner. 2.4.1. Equilibrium positions of dislocations in the multiple-slip—band model The equilibrium positions of the dislocations on the different slip bands are affected by the applied stress, the boundary, and the mutual interactions between the dislocations. The dislocations are in equilibrium when the total force acting on each dislocation is zero. The solution of the equilibrium positions requires the solving of (n-T) - 1 simultaneous algebraic equations. The solution becomes impracticable when the values of n and T are large. Therefore, in the calculation of the s.c.f.’s the equilibrium positions corresponding to those of a single—slip-band (Appendix B) have been used for ej(k) as an approximation. Consider the effect on the equilibrium position of the leading dislocation in the central slip band (k = 0) due to its interactions with all the other dislocations. The shear stress components around a positive edge dislocation for an isotropic medium is shown in Fig. 8 (a). When two edge dislocations lying on two separate slip planes interact, their interaction force is noncentral, and depending upon their relative positions it changes sign. By transposing the center of the dislocation In Fig. 8 (a) to that of the j—th dislocation on the k—th slip band, the force exerted by this dislocation on the first dislocation in the central sh‘p band can be calculated. The force due to the dislocations along E — E 28 is zero since ( 0. The dislocations in the regions R], R2,and R4 Ixy)x=y= in Fig. 8 (b) exert a force on the first dislocation in the positive X direction, and those in the region R3 exert in the negative X direction. Hence the net force tends to push the leading dislocation in the central slip band closer to the grain boundary. Similarly, one can show that a few dislocations in the vicinity of the point Q should also be pushed closer to the grain boundary. For a particular value of k in the equations (32) through (37) the s.c.f. is inversely proportional to xj(k), and hence the actual s.c.f. can be higher than those obtained in the present calculations but not smaller. 2.4.2. Effect of a group of slip bands The above discussion shows that the dislocations on one plane exert forces on those in neighboring planes. In other words, the stress fields of the dislocations on the different slip bands overlap. The amount of overlap is a function of the deformation variables. The s.c.f.'s obtained in the various cases are in fact the result of this overlap. This can be shown by examining the effect of a group of slip bands on the s.c.f. For example, when the resolved shear stress is small, the number of slip bands per unit length of the crystal will also be small. The spacing of the slip bands will then be large and may be several times the mean spacing of Frank's net (lO‘icm) in an undeformed crystal. Using the values of G, b, and 0 listed in Appendix A, and assuming r = 25 g/mmZ, the quantity D/2r =5u is obtained. Therefore, the value of the parameter s in equation (27) is approximately 0.2 to 0.4. Fig. 11 shows that the magnitude of the s.c.f. for the shear component 29 corresponding to the above value of s, and n = 99 is about 400. Even this value of the s.c.f. is appreciably larger than what is obtainable when the stress fields of dislocation on nearby slip planes is neglected, n. This overlap must, therefore, be responsible for the increase in the s.c.f. Thus,when the multiple—slip-band model is considered there will be highly-stressed regions in the crystal near the piled up groups of dislocations, much higher than those obtainable with the single-slip-band model. The intense stress concentration as a result of the pile-up of dislocations on neighboring slip bands may play a major role in fracture. 2.4.3. Effect of N, n, s, and k As shown in Chapter I, the s.c.f. should be of the order of a thousand for the nucleation of a crack in zinc. Therefore, the above value of 400 of the s.c.f. is insufficient to initiate a crack. However, an increase in N and a decrease in s will mean that the number of slip bands per unit length of the crystal is increased. The effect will be an increase in the stress concentration factor as suggested in Figs. 9 and 12. These figures show that the s.c.f. increases with the increase in the number of slip bands and the number of dislocations in each slip band. Fig. 11 shows that a decrease in s will also increase the stress concentration. It may be noted that the s.c.f.'s for the shear component as a function of each of the three variables N, n, and s are larger and build up more rapidly than those for the normal stress components. To examine if the theoretical stress to initiate a crack could be reached with the multiple-slip~band model, a value of N = 50 may be 30 used for zinc at room temperature, in accordance with the observations that many slip bands are formed prior to fracture (39) (5). The appropriate value of S may be estimated from the results of Mader (40), or from the dislocation density p of a heavily deformed crystal. It is generally observed that p = lOlz/cm2 is typical of cold worked crystals, so that the value of S may be taken as 10'5cm, or 100 A°. Using the shear stress at fracture r = 80 g/mm2, one finds that D/ZT = l p, f and s = .010. For the above values of N and s, the (s.c.f.)xy can be found from Fig. 9 to be approximately 2300. From equation (24), one finds that at = 180 kg/mmz. That is, the theoretical shear strength " G/30 is exceeded. This suggests that with about 100 slip bands, and approximately 100 dislocations in each band under such an applied stress level, the stress concentration can initiate a crack. The contribution of each slip band to the total stress concentration as shown in Fig. 10 (a) suggests partial cancellation of the stress fields of the dislocations on different slip bands. The s.c.f. contribution is the maximum for k = 0 (central slip band) and it decreases with the increase in Ikl to about 7 when |k| = 100. Also, the s.c.f. for k = )5] drops to approximately one-half of the value corresponding to that of the central slip band. These results indicate that the slip bands near the point 0 aid each other in increasing the s.c.f. 0n the other hand, a partial cancellation of the stress fields of the dislocations takes place for those slip bands farther away from that point. As a result, the stress concentration decreases. Thus, a comparison of Figs. 10 (a) and 10 (b) shows that the s.c f. for the normal stress components is small with respect to the corresponding shear 31 component. Fig. 10 (b) suggests that the tensile and compressive stresses due to the various groups of dislocations result in a decrease in the normal stress components. 2.4.4. Effect of bicrystal orientation and slip modes Fig. 16 shows the effect of the spacing of the slip bands on the s.c.f. for each of the several orientations of symmetric bicrystals. For the same value of s, the (s.c.f.)xy is the maximum for A = 45°. The decrease of the s.c.f. when A deviates from 45° can be visualized with the help of Fig. 18. For example, when the orientation of the slip bands is as shown in Fig. 18 (a), the number of dislocations in the region R5 is increased. Hence, as discussed in sec. 2.4.1, the negative contribution to the shear stress is increased and the s.c.f. decreases. The same conclusion follows a similar argument for the case when A = 60° as shown in Fig. 18 (c). Fig. 13 shows that the (s.c.f.)xy is small for the symmetric bicrystals with A < 5°. For a definite tensile stress the r.s.s. as well as the internal shear stress will also be small. Therefore, the initiation of a crack in such a bicrystal would require a large r.s.s. if it is assumed that basal slip is the only operative deformation mode. However, it is known that zinc crystals can deform by the second order pyramidal slip when the r.s.s. on the pyramidal slip system is sufficiently large and the basal slip is constrained either by the geometry or the orientation of the specimen. In the symmetric bicrystals with A < 5° these conditions are satisfied and pyramidal slip is likely to occur. The dislocations on the pyramidal slip planes can pile up against the grain boundary. The stress concentration due to these piled 32 up groups of dislocations can be obtained in a manner similar to that described for the basal dislocations. However, the interaction of the dislocations on the pyramidal plane with those on the basal plane may be important. As will be shown in sec. 2.5, this interaction can produce a stress concentration high enough to initiate a crack. The results show that the s.c.f.'s in the various cases obtained by applying the anisotropic elasticity theory are smaller than the corresponding values obtained from isotropic elasticity, about 10% for the shear component and 20 - 30% for the normal stress components. While these differences are not large, the choice of the slip plane and the fracture may very well depend upon the crystal anisotropy. 2.4.5. Process of fracture Several modes of fracture can be observed in crystalline solids. Depending upon the mode of fracture, the role of the various stress components may vary in the different stages of the fracture process. In the case of iron, Cottrell (41) and Petch (42) consider that shear stress is responsible for the nucleation of cracks, and tensile stress, for their growth. Since zinc, like iron, exhibits brittle fracture, it appears possible that a similar fracture process may also happen in zinc. As discussed earlier, an appreciable amount of plastic deformation usually precedes the initiation of—a crack in zinc. Also, the nucleation of a crack takes place along one of the basal slip bands even when the basal plane is not subjected to the maximum tensile stress. Thus, zinc crystals seem to be weaker in shear than in tension. Furthermore, the previous calculations show that a large shear stress, 33 high enough to initiate a crack, can be produced by groups of dislocations on nearby slip planes. It appears possible that the process of crack nucleation in zinc is a shear process. The fracture of crystalline solids is a two stage process requiring the nucleation of a crack and its subsequent growth. The condition that a crack, after it is formed in a material, would grow, may depend upon the nature of the normal stress components in the vicinity of the crack. From the measurements of the surface energy of crystals under different conditions, Gilman (43) suggests that the extension of a crack is aided by tension and hindered by compression. Figs. 15 and 17 show that the s.c.f.'s for both xx and yy components are also considerably large, and hence the internal tensile stress will be high. As Cottrell (41) and Petch (42) have suggested, if the tensile stress is large enough to cause a crack to propagate as a Griffith crack, the fracture will be brittle in nature. Tensile stresses of such an order of magnitude are attainable according to the present multiple- slip—band model, thus fracture in zinc could be of such a nature. In contrast, the internal stresses for the xx and yy components are zero according to the single—slip-band model. The applied tensile stress which is usually small at fracture, alone would not be large enough to cause a catastrophic failure, and brittle fracture in zinc would be hardly explainable with the single-slip-band model. The formation of a crack and its location are important features to be explained by any successful theory of fracture. Fig. 19 (a) shows that the (s.c.f.)X is large at the tip of the central slip band. y It decreases rapidly and oscillates away from the tip on the side of the 34 pile—up. It means that the internal shear stress due to a group of slip bands is short-ranged on the side of the pile-up. 0n the other hand, the shear stress decreases relatively slowly and does not change sign ahead of the pile-up. Or, in other words, the shear stress field ahead of the pile-up is long-ranged. These results will affect the formation and the location of the crack in the following way. Since two new surfaces are created when a crack develOps, the increase in surface energy would have to be supplied by the strain energy of the system if the work done by the external forces during the fracture process is negligibly small. The energy available on the side of the pile-up is small because of the short-ranged nature of the stress field. In contrast, a large amount of strain energy is available ahead of the pile-up. Therefore, a crack is likely to form ahead of the pile-up. The same is true for other orientations as can be seen in Fig. 19 (d). 2.4.6. Application In order that a crack can be formed, the shear stress should be of the order of the theoretical shear strength of the material. Therefore, the larger the (s.c.f.)xy, the smaller the r.s.s. at fracture will be. As discussed previously, the (s.c.f.) is the largest for a xy 45°-symmetric bicrystal. Furthermore, the shear components in grain A and in grain B are compatible; i.e., the total (s.c.f.)x‘y should be doubled when slip bands in both grains meet at the grain boundary. For other orientations, the shear stress in one grain is not exactly compatible with that in the other grain. They can act Opposing each other depending upon the relative orientations of the two grains. 35 5 Therefore, the r.s.s. required to initiate a crack will be the smallest for the 45°-symmetric bicrystal. This is in agreement with the fracture stresses reported by Gilman (5) and by Chyung (44) for a wide variety of orientations of zinc bicrystals. Current theories of fracture based on the single-slip-band model suggest several possible locations of crack formation with respect to the barrier. It may be possible to test the validity of these theories and the present one based on the multiple—slip-band model in crack nucleation by the following experiment. Consider a bicrystal with the basal plane in grain A at 45° to the tensile axis, and in grain B at 90°. If the tensile stress is o, the r.s.s. r on the basal plane in grain A will be 0/2, and that in grain B will be zero. According to the single-slip—band model, (s.c.f.)xy is equal to n, (s.c.f.)XX and (s.c.f.)yy are both equal to zero at the tip of the pile-up. Therefore, the transferred shear stress would have stress components acting on the basal plane in grain B: Oxy = 0, Oyy = nr. This would mean that a shear crack could not form on the basal plane in grain B. The three possible locations of a crack could be (a) the basal plane in grain A (5) (16), (b) the prism plane {1010} in grain A (9) (10), and (c) a plane at 65° to the tensile axis in grain B (8) (11). In contrast, according to the multiple-slip-band model, Figs. 13 and 14 show that (s.c.f.)X = 2333, (s.c.f.)XX = 1250, (s.c.f.) = y yy 872. The transferred stress would have stress components on the basal plane in grain B: Oxy = 1901., oyy = 3390T, and Fig. 19 (d) shows that the shear stress increases along this plane to a value of 740T. It can L—— A 36 also be seen in Figs. 19 (a) and 19 (d) that the shear stress field is short-ranged on the side of the pile-up and long-ranged ahead of it, indicating that a crack is likely to initiate along the basal plane in grain B. Thus, it may be possible that the validity of the foregoing analysis can be verified experimentally by fracture studies on bicrystals with large differences in the orientations of the two grains. 2.5. Crack nucleation in single crystals Bell and Cahn (27) suggested that a dislocation on the basal plane can interact with that on the pyramidal plane under appropriate conditions. They considered the nucleation of twins in zinc single crystals to be a consequence of the intersection of basal and pyramidal slip bands, and obtained experimental evidence substantiating their theory. However, the stress concentration resulting from the intersection of the slip bands was not quantitatively calculated in their work. By considering each slip band as being associated with a pile-up of dislocations, the stress concentration produced by such a configuration can be analyzed for the role it may play in the fracture process of zinc. Extending an idea originated by Cottrell (20) for the fracture in b.c.c. metals, the dislocation configuration shown in Fig. 20 (a) may be considered for evaluating the s.c.f. that it may produce. This configuration will be referred to in the future as a double pile-up of dislocations. 2.5.1. Basal-pyramidal interaction When two rows of dislocations, one gliding on (0001) with a Burgers vector.§ [ 1120 ], and the other on (1122) with a Burgers vector 3 37 g [1123], meet under the action of the applied stress, the two leading dislocations can coalesce to form a new dislocation with a Burgers vector a[0001]. This dislocation reaction can be written as %[ iizo ] + %[1123] . a[0001] (40) In so combining, 37% of the elastic strain energy is released (see Appendix C). The new dislocation with Burgers vector a[0001] lies parallel to the line of intersection of the basal and pyramidal planes and is sessile (32). Therefore, it can act as a stable barrier against which the dislocations on the two intersecting slip planes may pile up. As shown in Fig. 20 (a), the origins of three right-handed rectangular Cartesian coordinate systems XYZ, UVZ, and R52 are chosen to coincide with the centers of a basal, a C, and a second order pyramidal dislocation respectively. The plane of the drawing corresponds to Z = O. The point 0 is the tip of the double pile-up of dislocations and coincides with the center of the C dislocation. The basal and pyramidal planes intersect at an angle 6 = 61°. The Burgers vectors b, c, and p of the above dislocations direct towards the positive X, U, and R axes respectively, and the Y, V, and S are normal to their respective slip planes. The signs of the dislocations are chosen to be consistent with Frank's convention (36). The dislocation lines are parallel to the Z axis. 2.5.2. Equilibrium positions of dislocations in a double pile-up of dislocations Let the position of the j—th dislocation on the basal plane be 38 E(j), and that of the k-th dislocation on the pyramidal plane be P(k), both measured in cm from the point 0 to the centers of the corresponding dislocations. The dislocations are in equilibrium when the resultant force acting on each dislocation is zero. Using the Peach-Koehler equation (45), the force F per unit length of a dislocation line is 'I'll ll rrl >< CDI where t is a unit tangent of the dislocation line at the point of consideration, and G is a vector whose i-th component is given by G1= 201.3. J In the above express10n, Oij = OXX’ Oxy’ etc. are the stress components and bi = bx, by and bz are the components of the Burgers vector of the dislocation. The force equilibrium equations for the basal and pyramidal dislocations are derived in the following section by applying isotropic elasticity theory and the principle of superposition. The force acting on a pyramidal dislocation is the resultant of three components: the one due to the applied shear stress, the repulsion between those on the same slip plane, and the attraction as well as repulsion due to the dislocations on the basal plane. The R component 0f the shear force acting on the k—th dislocation in the R—Z plane is derived in the following manner. 1) The force due to a glide dislocation in the same pile-up on the pyramidal plane is i l 39 where Ors is the shear component of the stress field of the dislocation. The quantities b, X and Y in equations (1), (2), and (3) can be replaced by p, R, and S respectively to obtain the expressions for the various stress components. Substituting D = Gb/2W(l-V), p = 2.108b, and S = 0 in the expression for Ors obtained from equation (3), the force F(p) acting on the k-th dislocation due to the pyramidal dislocations is 1 m m r-H— lib/IN 7r»I where 2.15 the number of dislocations on the pyramidal plane. 2) The force exerted by the dislocation with Burgers vector c is He) = (0.1.” (42) where o' is the shear component of the stress field of the C dislocation. rs The transformation of the stress components obeys the law for the transformation of a second rank tensor Oij = aik ajt Okli where the direction cosines aij‘s in the present case are g1ven by sine —cose 0 [aij] 2 cose sine 0 o 0 1 Equation (42) becomes F(C) = -{ (C0526 - sinQe) ouv + cose sine (avV - OUU) } p , (43) 40 Again the quantities b, X and Y in equations (1), (2), and (3) can be replaced by c, U, and V reSpectively. One obtains 4G C V U2 (44) Ovv ‘ °uu = 2n (1 — .) (U2 + v2) Substituting c = 1.856b in the various expressions for the stress components, and equation (44) in equation (43), the force due to the C dislocation becomes 1.856 D 2 2 2 _ 2 4 . v u2 45) = _-————~————— - — + . F(C) (U2 + V2)2 {U (U V ) (cos e s1n e) s1ne cose l p ( 3) Let T be the resolved shear stress acting on the pyramidal plane in the R direction. The force due to the shear stress T is F(a) = -r p . (46) 4) The force exerted by a dislocation on the basal plane is F(b) = 0:5 p is the shear component of the stress field of a basal location. where “$3 The aijls for the coordinate transformation in this case are -cose —sine 0 . = sine —cose 0 [ajjl 0 0 1 Therefore, F(p) = { (cosze — sinze) Oxy + cose sine (Oyy - Oxx) } p 41 Let n be the number of dislocations on the basal plane. The force due to the basal-dislocations is given as n . F(b) = D p jzlfi XZU )+Y2U )[:Xm (cosze - sin26)(X2(j) - Y2(j) ) (47) + (4sine cose Y(j)x2(j) E] where X(j) and Y(j) are the coordinates of the center of the k-th dislocation on the pyramidal plane. The total force acting on the k-th pyramidal dislocation is F(k) = F(p) + F(c) + F(a) + F(b) . and the force equilibrium equation for the dislocations on the pyramidal plane may be written as F(k) = O (k = 1, 2,..., Z) Dividing equations (41), (45), (46) and (47) by 21p, the force equilibrium equation for the pyramidal dislocations becomes = _Q_ n ________l__.___—— x ')(x2(') - Y2(j) )(cosze - sinze) . F(k) 2. {(3.21 ( x2(J)+Y2(i) )2 [(3 J ' ' ' —-—lL§§§- [: 2 - V2 cosze - sinze) + 4s1ne cose Y(j) X2(J):D (U2+V2)2 U(U )( Z 1 1 + 4sine cose V U2:] + 2-108 m2] P k)—P m _”§ = 0 . (48) . ‘ . ._ _,. . .... - 94-O°"""" ' "' - .Q "“ ‘ 42 2.5.3. Transformation of the coordinate system The coordinates of the k-th pyramidal dislocation with respect to the three coordinate systems shown in Fig. 20 (b) are transformed according to the equations xu) = x1.) ,9,- Y(i) = y(j) .9; u = 1.5—, (49) D V = Vi P(m) = p(m) -§; —‘ where X(j), y(j), u, v, and p(m) are dimensionless parameters. Substituting u = p(k) sine v = -p(k) cose y(j) = -u 92(k) = u2 + v2 . and equations (49) in equation (48), the force equilibrium equation for the pyramidal dislocations reduces to = 1 - 2 - _ 2 - 2 _ - 2 F(k) jg] ( x2(j)+y2(j) )2 [:}(J)(x (J) y (J) )(cos a S1n 6) '3 2 '2 + 4sine cose y(i) x2(j):])+%iE7§'[::(Cos1ig§;" e) (1-(l.856)2) 43 K . 1 1 + 4s1ne cose:] + 2.108 X _ - —-= 0 . "1:1 plkl p(m) 2 (50) m+k (k = 1, 2, ..,£) Let R(k) and S(k) be the coordinates of the j-th basal dislocation with respect to the k-th dislocation. Transfonhing E(j), R(k), and S(k) according to the equations E(J) = e(j) 2% R(k) = r(k) % S(k) = S(k) 2?; where r(k), S(k), and e(j) are dimensionless parameters, one can derive the force equilibrium equation for the basal dislocations as Z ' = _._]—_ 2 _ 2 2 _ ' 2 F(j) 2.108 ( kgl ( r2(k)+sz(k) )2 [:r(k)(r (k) s (k) )(cos 9 Sln e) (51) - 4sine cose S(k) r2(k):])+ where 1' is the r.s.s. on the basal plane in the x direction. 2.5.4. Stress components at the tip of the double pile-up of dislocations Following the definition of the s.c.f. in sec. 2 2.1, the s.c.f. components due to the pyramidal dislocations at the tip 0 are 44 f1 (5 c.f.) = 2 r5 k=l P(kl (52) (5“C°f°)rr = (s.c.f.)SS = 0 , and those due to the basal dislocations are n (s.c.f.) = 2 z 1 xy j=l 9U (53) ..f. = Hf. = 0 (sc )XX (sc )yy The total shear stress (due to applied stress and internal stress) acting on the basal plane at the tip of the double pile-up is given by k .— _ l 2 _ . 2 —J _ . Oxy — (T [:l + 2 £21 E(Ei:] [:cos 6 Sln e ) (Sine cose)o (54) E '2‘ —y‘ l +'r' 1+2 - i=1 e(J _ where o is the applied stress on the pyramidal plane. Similarly, K 1 .1 = _ . v - - 2 Oxx 2. [E + 2 kg] ETE7__J (Sine cose) + (Sin 6)o (55) Z l = I 2 ' Oyy 0 + 0 cos 9 + 41 Sine c056 kZl ET?) (56) where o“ is the applied tensile stress on the basal plane. In the derivation of equations (54), (55), and (56), the contribution of the C dislocation to the total stress is excluded. 45 2.5.5. Calculation of p(k) and the s.c.f. The dislocations are in equilibrium when the shear force acting on each dislocation is zero, i.e., F(k) = 0 and F(j) = 0 for each value of k and j. To obtain the solution of such a problem,(n + z) simultaneous algebraic equations are to be solved. The solution also depends on the ratio q of the r.s.s. on the pyramidal plane to that on the basal plane, where q varies with the experimental conditions. A general analysis becomes complicated. However, a numerical solution by using an appropriate approximation method is possible. In calculating the positions of the pyramidal dislocations one may use the data reported by Bell and Cahn (27) obtained from their experiments in zinc single crystals with the angle between the basal plane and the tensile axis of l.5° (i.e., q = 16.5) according to the procedure outlined below. Assume that the dislocations on the basal plane are in equilibrium and fixed, and their positions in the double pile—up model are the same as in the single-slip-band model. It is also assumed that the initial positions of the pyramidal dislocations are the same as in the single—slip-band model, but taking into account the ratio of q. The final position of each of the pyramidal dislocations is obtained by including the force due to its interaction with all the dislocations in the configuration and applying iterative procedures. The values of p(k) for n = 49, 69, and 99 and z = 3 to 99 obtained after several iterations are listed in Table 3. The positions of the dislocations in the various cases were obtained by applying the principle of virtual work (3c). The s.c.f.'s for the shear component at the tip 0 on the pyramidal plane as a function of n and z are listed in Table 4. 46 2.5.6. Stress concentration due to a double pile-up of dislocations In a double pile—up of dislocations, each dislocation will interact with the other dislocations on the same slip plane and those on the other plane. The shear force acting on a dislocation due to these interactions is a function of n, the population of dislocations on the basal plane, and Z , that on the pyramidal plane. For most orientations of zinc crystals, the number of dislocations on the basal plane may be larger compared with that on the pyramidal plane since the critical resolved shear stress for basal slip is about 18 g/mm2 (46) whereas the c.r.s.s. for pyramidal slip is l—l.5 kg/mm2 (47). The effect of dislocation interaction on the positions of the dislocations along each of the two planes w.r.t. the barrier may be examined by considering, for example, the case when n = 69 and 2.: 24. As shown in Table 3, the positions of the first three dislocations on the pyramidal plane are p(l) = 0.0498, p(2) = 0.1804, and p(3) = 0.4614. For a single-pile-up of 24 pyramidal dislocations, the values will be .1469, .4932, and 1.039. This suggests that a pyramidal dislocation is pushed closer to the obstacle by the stress fields of the basal dislocations. Similarly, the positions of the basal dislocations will also be affected by the stress fields of the pyramidal dislocations° Thus, the assumption that the basal dislocations are fixed would cause some error in the calculated positions of the pyramidal dislocations. By resolving the various force components acting on the basal dislocations due to their interactions with the pyramidal dislocations, one can show that the basal dislocations in the vicinity of the tip Q will also be pushed towards the barrier. Since the positions of the 47 dislocations w.r.t. the barrier along each of the two planes determine the resulting stress concentration, and the stress components in equations (54), (55), and (56) are inversely proportional to e(j) and p(k), the actual s.c.f. can be higher than that obtained in the present approximation but not smaller. Two features of a double pile—up of dislocations may be noted in view of the data shown in Table 3. With the increase in the number of dislocations on the basal and pyramidal planes, the spacing between the leading dislocations becomes much smaller than that between the dislocations further back in the sequence. Secondly, all the dislocations are pushed closer together, and hence the length of the slip band on the pyramidal plane is decreased. Consequently, the s.c.f. will increase with the increase in the number of dislocations on the two planes. For a single pile-up of dislocations against an obstacle, the s.c.f. for the shear component at the tip of the pile-up is equal to the number of dislocations in the pile-up. Thus, for twenty-four dislocations on the pyramidal plane a shear stress of 247 should be attained. However, for a double pile-up of dislocations with n = 69 and Z = 24 Table 4 shows that a shear stress of 631 is reached. Hence, a double pile-up of dislocations produces a greater stress magnification which increases the stress by a factor of about 2.6. To examine if the theoretical stress to initiate a crack can be reached with the double pile—up model, consider the observations of Bell and Cahn (27), namely the almost longitudinal cracks occurring in zinc crystal wires with the basal planes at an angle of l.5° w.r.t. the 48 tensile axis. The highly stressed pyramidal planes in those crystals were (1122) and ( 1122 ) on which an appreciable amount of slip was observed. Therefore, the interaction of the dislocations on the basal and pyramidal planes according to equation (40) was favorable. Bell and Cahn reported a tensile stress of about 6 kg/mm2 at fracture, implying a r.s.s. of about 2.6 kg/mm2 on the pyramidal plane and 0.15 kg/mm2 on the basal plane. Under such an applied stress, a reasonable value of n may be approximately 50 to 100, and that of 2, 20 to 50. Using n = 69 and 2 = 24, one obtains an internal shear stress at the tip Q on the pyramidal plane of 63T or 164 kg/mmz, and that on the basal plane of 10 kg/mmz. Therefore, the shear stress at the tip Q on the basal plane obtained from equation (54) is approximately 102 kg/mmz. Hence, the theoretical shear strength of ~G/30is reached, and a crack can nucleate. As discussed in sec. 2.4.5, a large tensile stress normal to the cleavage plane would be necessary to propagate a Griffith crack. In the case of double pile—up in iron, Cottrell (20) suggested that the dislocation with [001] Burgers vector will produce a large tensile stress normal to the (001) cleavage plane, since the configuration around this dislocation is similar to an extra half—plane of atoms inserted in the manner of a lcleavage knife'. At the double pile-up in zinc, which is geometrically similar to that envisaged by Cottrell for b.c.c. metals, a tensile stress on the (0001) cleavage plane of such an order of magnitude as to cause a crack to propagate may also be produced by the dislocation with [0001] Burgers vector. The foregoing analysis may also be applied to the fracture process 49 in bicrystals. The assumption that the basal-pyramidal interaction, equation (40), precedes the formation of double pile-up will not be essential if the dislocations on the two planes pile up against the grain boundary. Thus, a crack in a bicrystal might nucleate within a grain or at the grain boundary depending upon the site of the double pile-up. // // // / / // , V)~—-Boundary 1 Figure 3. Single—slip-band model. “q“? r —— 2.: 4W2; 21.7 51 xx .mwxm > w>wpmmoc wgp acoFm :owpznvgpmwv A.m.u.mv .v mesmvm 'lj A- ow. mm. om. mm. om. m_. CH. mo. "4." r. “" .'- Gus-“aw _ . x xx .: we cowpoczm m mm “.m.u.mV ucm xxA.w.u.mV .m wgzmwm c oo_ om om on om cm ow om ow OF 52 Ax xx IQF Grain A Source—* . // I /ll Q (T+1 )/2 _/’ // / 2 Figure 6. Multiple-slip-band model. Grain B 54 Grain A G Grain B QN g8 Qk ‘0 / +3 Q1 F1'gure 7 (a). Transformation of the coordinate system for the - 50 multiple- slip- -band model, A — 55 Gm“ A G Grain B Figure 7 (c). Coordinate systems for the superposition of stresses of grain A onto grain B. in different slip bands. slocations di between Figure 8. Interactions 3500 3000 2500 2000 ' 1500 ' 1000 ‘ Figure 9. s.c.f. as a function of N (n Xy x 00— om ow on om om ow om _ _ _ _ 2 _ _ _ ON 0_ 0 OF- 58 ON- om- oe- om- om- ON- ow- om- 00—- _ fl _ _ _ _ _ _ _ ax AooIWaUcmV 120 100 80 60 4o 20 -4o 59 Figure 10 (b). (s.c.f.)xx and (s.c.f.) as a function of k (A = 45, N = 50, S = .01, n = 99). 6000 _ xy 5000 4000 1 3000 XX 2000 yy 1000 1 -3 —2 -l 0 1 10910 s Figure 11. s.c.f. as a function of 10910 s (n = 99, N = 50, A = 45 ). 2400 - xy 2000 - 1600 ‘- XX 1200 - 800 - 400 —‘ 0 10 20 30 40 50 60 70 80 90 100 n Figure 12. s.c.f. as a function of n (N = 50, S = .01, 4 = 450)° 62 2500 1 (s.c.f.)X 2000 - 1500 ’ 1000 500 Figure 13. (s c.f.) as a function of A (n = 99, N = 50, s = .01). xy _. ____.-... -. 2500 s.c.f. 2000 1500 1000 500 Figure 14. (s.c.f.) 63 and (s.c.f.)yy as a function of A (n = 99, N = 50, s - .01). XX 64 4000 I (s.c.f.)xx 3500 - 3000 2500 2000 t 1500 — 1000 - 500 - /5 80° —3 -2 -1 0 log10 s Figure 15. (s.c.f.)xx as a function of log10 s (n = 99, N = 50, s = .01). 65 6000 «— (s.c.f.)xy 5000 w 4000 m 3000 v- 2000 <- 1000 ~ 45. 75° 10° 50 1 1 I 1 -3 -2 -1 0 Figure 16. (s.c.f.)xy as a function of 10910 s (n = 99, N 50, S 10910 S 66 1400 ‘ s.c.f. ( )yy 1200 - 1000 ’ 8001’ 600 T 400 ' 200 — A = 45° A = 30° x = 20° I I I I -3 -2 -1 0 log1O s Figure 17. (s.c.f.) as a function of log 5 (n = 99. N = 50, S = o01l- yy 10 .mcovmwg msorxw> Cw mcorpwoo_m_v do mUFmvw mmmcpm mo covpm__mocmu .w_ wesmwm I I + + I + >.>. I + + + + I xx I I + I + + %X mm mm am mm mm _m _ pcmcanou mcovcmg msowem> c? mpcwcomEoo mmmepm wsp yo :cwm _ mmwgpm E E E 67 .A_oo. u c< .om u 2 .mm 1 : .ome n xv wean acfiw Fmebcmu web meo_m co.bsncepm.u xxfl.c.o.mv .Aav m_ mtsm_a oomI- e .4. ._.. e. a . :::/7Z//777P 3.. 4:24? 68 comm 69 mo. #0. z .omv u xv vcmn awrm amepcmo msp mcorm cowpznwgpm?v >2 A.c.o.mv .Anv m_ wazm.a O I) 7:777:77 s 2 :44: / oom coo— x» A.%.U.mv 70 .AFQQ. u ;< .mm .om¢ u xv econ QWFm Fmgpcmu asp mCQFm covpspvgpmwv xxA.m.u.m .on mF mssmwu l com QOOF XX A.%.U.mv oom— 2000 1600 1200 800 400 71 15° = 45° Figure 19 (d). Transferred shear stress of grain A onto grain B (A = 45°, n = 99, N = 50, S = .01). 72 ( S'OUY‘CQ @Z (0001 ) Figure 20 (a). Double pile-up of dislocations in zinc. 73 .\ x514 s \. ‘0 /Q *- S Q Hi) u 11 9 J v“ c 0 -¢— 1 -<-— V —>4-<— X(j)—>— b (000]) oZ ~<+—————— e(j) ——-————>— Figure 20 (b). Transformation of the coordinate system for the double pile-up of dislocations in zinc. mo.o- mF. m.~ mwm.~- Np.o- N.o F._ mm._- P m.o- m.m v.m NNN.o- o..- m.m m.m wm~.o- m N.F- o.m o.m_ mm_.o- o.m- m.m N.¢_ m¢_.o- m_ m._- um.o_ m._N mo_.o- m.N- m.__ «.m_ mp_.o- em m.~- m._~ N.N¢ mmo.o- o.m- m.mm m.om mo.o- me M ~.m- o.m~ “.mm ao.o- 0.5. _.mm u._m m¢o.o- mo mm.m- N.~v m.mm mmo.o- m.m- c.0e m.mm Fmo.o- mm , n. a. >.>-A..+.U.WV >XA.u—..Uamv XXA.LV.U.mV .>. NEAL..U.MV NXA.W.U.WV KXA..+.U.W .>. C uwuogpomwc< uvgoepomH a:-mpwa mcu mo m:m_q ecu o» Faeroe Ac » xv mcmpa men :0 mucmcoapau .m.u.m ._ open» 75 Table 2 (a). s.c.f. as a function of N (A = 45°, 5 = .01, n = 99) Isotropic Anisotropic N (s.c.f.)xx (s.c.f.)xy (s.c.f.)yy (s.c.f.)XX (s.c.f )xy (s.c.f.)yL 5 262 645 153 269 615 118 15 593 1217 385 583 1141 316 25 823 1613 555 802 1506 458 35 1012 1933 695 981 1801 575 50 1250 2333 873 1207 2170 722 60 1390 2565 977 1340 2383 808 75 1580 2877 1118 1520 2669 926 100 1857 3329 1325 1783 3084 1098 Table 2 (b). s.c.f. as a function of n (A = 45°, 5 = .01, N = 50) Isotropic Anisotropic n (s.c.f.)XX (s.c.f.)§¥ (s.c.f.)yy (s.c.f.)xx (s.c.f.)xy (s.c.f.)yy 9 241 518 138 248 492 106 19 437 865 266 434 811 220 24 512 1004 326 507 940 267 1 29 582 1132 378 574 1057 310 39 708 1357 469 693 1265 387 i 49 818 1555 551 798 1449 454 1 59 917 1734 625 892 1615 516 69 1009 1899 693 979 1767 572 79 1094 2053 756 1061 1909 625 89 1174 2197 816 1136 2043 675 99 1250 2333 872 1207 2170 722 mm _m_ mm ow RMF N3 m cm mk— mm mm mm_ mm N Fm mmm w__ No omm om_ F mpp me mx— mm_ 05m mw_ Huor x m omm mmm mmm NFm emu o_¢ HIOF mom mmm mmm Fee amop “mm N-o_ x m mm NNN oN—N NoN_ wa mmmm 0mm” NIoF mom mmmm mmmF moPF momm NMQF m-o_ x m 00oF comm womm mwm— memo FMVN m-o_ zxA.m.u.mv xxA.m.u.mV xxA.$.u.mv zxfl.m.u.mv xxfi.».u.mv xxAxfu.WV mEu m mucus uwaocuomwc< 62qoep0mH QT_m mo mcwomam .Aome n x .om u z .33 u CV ma_tow;p xpwovpmmpo quoeuom_cm new uraoguomw ms“ mcwaQam an cchmpno mp_:mmg mzu do xgmcezm .Auv N mpnmp ' - V'"‘*‘ ("Ann-1: v," » .. . _., , 77 Table 3. Positions of pyramidal dislocations in a double pile-up of dislocations p(k) (unit of length [Gp/Zn (1 - v)t]) n 4 PH) 0(2) 11(3) 13(4) 13(5) p(f.) 49 3 0.2756 1.403 4.215 49 9 0.1206 0.4596 1.225 2.431 4.171 20.35 49 19 0.0677 0.2478 0.6255 1.201 1.983 52.94 49 24 0.0569 0.2081 0.5130 0.98655 1.614 70.37 49 49 0.0335 0.1240 0.2918 0.5467 0.8903 162.7 49 69 0.0257 0.0957 0.2226 0.4103 0.6656 179.5 49 99 0.0191 0.0718 0.1658 0.3021 0.4851 356.8 69 3 0.2260 0.1987 3.774 69 9 0.1013 0.3895 1.064 2.168 3.777 19.115 69 19 0.0584 0.2116 0.5559 1.0760 1.802 50.78 69 24 0.0498 0.1804 0.4614 0.8900 1.474 67.85 69 49 0.0304 0.1116 0.2674 0.5098 0.8297 158.9 69 69 0.0237 0.8768 0.2063 0.3865 0.6294 235.2 69 99 0.0179 0.0668 0.1555 0.2863 0.4642 352.0 99 3 0.1821 1.001 3.303 99 9 0.0808 0.3100 0.8907 1.873 3.333 17.66 99 19 0.0480 0.1709 0.4722 0.9357 1.593 48.19 99 24 0.04185 0.1489 0.3993 0.7802 1.313 64.82 99 49 0.0269 0.0975 0.2405 0.4638 0.7581 154.3 __9977 69 0.0214 0.0782 0.187 0.358 0.583 229.9 99 99 0.0169 0.0628 0.147 0.275 0.448 347.1 78 Table 4. (5°C“f')rs at the tip Q on the pyramidal plane as a function of 2 for various values of n 2 n = 49 n = 69 n = 99 3 9.15 11.05 13.58 9 24.62 29.01 36.0 19 46.21 53.18 64.15 24 55.60 63.07 74.60 49 96.53 105.80 118.95 69 126.83 136.85 150.97 99 171.05 182.0 196.0 III. EXPERIMENTAL PROCEDURES AND RESULTS The purpose of the experimental part of this work was to investigate the positions of the cracks with respect to the grain boundary in zinc bicrystals of controlled orientations to see whether they agree with those described in Chapter II based on the multiple-slip-band model, in the hope that a criterion for the fracture process in zinc might be deduced from such experiments. Single crystals were also tested in order to compare with the fracture behavior of the bicrystals. 3.1. Preparation of the specimens Single crystals and bicrystals of zinc (99 99+ pure) were grown from the melt using a modified Bridgeman technique (48). A detailed account of the furnace setup and the instrumentation is given by Chyung (44). Polycrystalline zinc blanks as shown in Figs. 21 (a) and (b) were machined from rolled slabs. Seed crystals of desired orientations were then welded into place. During the crystal growth, a dry nitrogen atmOSphere was maintained in the furnace. The crystals were etched in dilute hydrochloric acid, and their orientations determined by using the Laue back-reflection technique. An acid saw was used to cut the crystals to appropriate dimensions. Tensile specimen holders were cemented to both ends of the crystal by using Armstrong cement and curing at 200°F. The crystals were chemically polished to eliminate possible surface damage with the solution suggested by Brandt et a1. (49). The specimens were 79 80 connected to the loading fixture of an Instron testing machine by means of steel pins. A cross-head speed of .1 cm/min correSponding to a strain rate of 4%/min for a 25 mm gage length of the specimen was selected for the tensile testing. Three types of bicrystals were grown. The orientation relationships of the component grains are shown in Figs. 21 (c), (d) and (e). The first type consists of symmetric bicrystals in which the component grains are mirror images of each other across the boundary plane. These bicrystals are designated as (s — A) where A is the angle between the tensile axis and the basal plane. The two types of asymmetric bicrystals are designated as (A1 - A2) and (41 - Ha). The orientations of both the grains for (A1 - A2) are ”soft”, while for (41 - Ha) the orientation of grain A is soft and that of grain B is ”hard”. As shown in Table 5, three of the bicrystals of the (A1 - A2) category were so grown that A was greater than 45° in each grain, while the fourth was designated to have A1>45° and A2<45°. The (41 - Ha) bicrystal has A<45° in both grains. Altogether thirteen specimens, seven bicrystals and six single crystals, were tested. The observations were carried out with a Bausch and Lomb Research Metallograph. 3.2. Results 3.2.1. Asymmetric bicrystals The bicrystals with orientations (46 — 89), (55 — 83), and (59 - 72) deformed, under the applied stress, by basal slip in their component grains prior to fracture. In each of these bicrystals, 81 a crack was found nucleated at the grain boundary along one of the basal slip bands in grain B where A = 89°, 83°, and 72° but not in grain A. Fig. 22 shows the crack in the bicrystal with orientation (46 - 89) formed at a tensile stress 0 of 277 g/mm2 after 1.4% elongation. The bicrystal with orientation (55 - 83) fractured at o = 160 g/mm2 after 60% elongation (see Fig. 23). The crack shown in Fig. 24 in the bicrystal with orientation (59 - 72) was nucleated at o = 315 g/mmzafter 6.2% elongation. In the bicrystal with orientation (49 - 31) twins were nucleated at a tensile stress of 311 g/mmz, and fracture was initiated at o = 422 g/mm2 after 15.8% elongation. Fig. 25 shows a crack formed in this bicrystal along the basal plane in grain B. The slip lines in grain B are seen to be fine, whereas fine slip lines are grouped into coarse slip bands in grain A. Table 5 lists the data obtained from the tensile tests together with the observed deformation modes prior to fracture. The r.s.s. at fracture in the two grains of each bicrystal is calculated according to the actual orientations of the slip bands determined from the photomicrographs. A series of parallel twins were formed in grain B of the bicrystal with orientation (41 - Ha) as can be seen in Fig. 26 (a). In this bicrystal, basal slip is constrained in grain 8, while it is possible in two < 1120 > directions in grain A. The twins were nucleated at a tensile stress of 50 g/mmz, and a crack was initiated in grain B at o = 437 g/mmz. The crack propagated rapidly in grain B, and it also extended across the grain boundary nearly all the way 82 through grain A. The specimen separated into two halves while being removed from the Instron Unit. Figs. 26 (a) and (b) show the upper- half of the specimen and the cleavage face of grain B respectively. The proximity of both (1010) and (0001)twin in grain B matrix normal to the tensile axis was recognized, and a Laue back-reflection pattern taken with the beam normal to the cleavage plane in grain B showed that this plane was (0001)twin. 3.2.2. Symmetric bicrystals with orientation (5 - A) In the bicrystal with A = 63° a low angle boundary inclined at an angle of 55° to the grain boundary was present in grain B prior.to testing. Fig. 27 shows two cracks formed in grain B, one at the grain boundary and the other at the low angle boundary. The bicrystal with A = 40° twinned at a tensile stress of 190 g/mmz, and a crack was found nucleated at o = 400 g/mm2 along the basal plane of the twin in grain B. The crack seen in Fig. 28 propagated rapidly across the grain B and then partially through grain A. 3.2.3. Single crystals In the crystals with A = 53° and 40°, the deformation continued by basal slip, and turned inhomogeneous prior to fracture. Cracks were found to have nucleated in the shoulder regions of these specimens. Fig. 29 shows a crack along a basal slip band and the associated kink band in the crystal with A = 53°. A similar crack was also observed in the crystal with A = 26°. The crystals with A = 26° and 20° showed a marked difference in their deformation behavior as compared with that of the other 83 specimens. As shown in Figs. 30 and 31 (b), a set of non-basal slip traces making an angle of approximately 61° with the basal slip bands was fonned. A rapid increase in the work hardening rate at a later stage of deformation in these crystals, Fig. 31(c) for example, indicated that pyramidal slip had probably occurred. Fig. 31 (a) shows the crack observed in the crystal with A = 20°. The twins seen in Fig. 30 and Fig. 31 (a) were formed prior to fracture. The orientation of the crystal (Hb) in Table 5 is equivalent to a rotation of 90° of the crystal (Ha) about the tensile axis. In these crystals, cracks were found to be associated with intersecting twins of {1012} < 1011 > type fonned prior to fracture. The crack fonned along the (0001)twin in the crystal with orientation (Hb) is shown in Fig. 32. A set of basal slip traces parallel to the crack can also be seen. Figs. 33 (a) and (b) are photomicrographs of the cleavage face of the crystal with orientation (Ha) taken with bright field and dark field illumination respectively. The orientation of the cleavage face of this crystal was found by using the Laue back-reflection technique to be (000])tw1n° 84 an— 25fi3r16+43r+ 25 —>—’3|—<— 25 9-1 9. 5 .5 1 j 1L: 100 —————*-| (+23 +45|-+16+|3|-<— 28 ~»-|3|-+ 22 ’4 '35 JL L I 2A I I / «I? ”(3‘ (d) 1 ”7'0 1 A l A \ }165 I | 1 1 (AZ 2121 I A— J—— ——— — — T [1100]/_ 5 1 (1120) 1 (s-A) A-l-JfAZ (A1-A2) (a) Figure 21. Dimensions and orientations of the specimens. (a) Zinc blank for bicrystals (mm) (b) Zinc blank for single crystals (mm) (c) (d) and (e) Bicrystal orientations and their designations. 85 A (1010) B Figure 22. Crack nucleation in specimen No. 1 (46 - 89). 200x. Figure 23. Crack nulceation in specimen No. 2 (55 - 83). 100x. Figure 24. Crack nucleation in specimen No. 3 (59 - 72). 100x. Figure 25. Crack nucleation in specimen No. 4 (49 - 31). 100x. (1120) A (0001) 0 Figure 26 (a). Upper half of the fractured specimen (41 - Ha). 100x. (b) Cleavage face of grain B. 100x. 90 nzngcson w_mcm zo_ mgp p< An hxgmvczon :wmgm mgp p< Am DXQOF ,Amo I mv o uoz cmeuwam cw cowumw_o:: xuwgu “mm mesmwm m 8%: Jfl/{ilu xgmuczon mecm 304 Figure 28. Crack nucleation in specimen No. 7 (s - 40). 100x. (min) Figure 29. Crack nucleation in specimen Ho. 8 (A = 53). 100X‘ [1510] / Twin 93 61° (1010) Basal and pyramidal slip traces in specimen 10 (A = 26°). 100x. No. Figure 30. Pyramidal slip «4' /' f Figure 3l° (a) Crack nucleation in Specimen Noe ll (A = 20). lOOx. (b) Basal and pyramidal slip traces. lOOx. .._ _ ‘o s...- ~~_ 95 .Aoom u «v FF .0: qumxgo mzp mo m>c=o :wmgpm . mmmgym .Auv Pm mesmam compmmCOFM & om om ow om _ fl _ om OF mgspumgm ooF com com oov oom ANEE\mV mmmgpm mFPmcmh in specimen No. l2 (Hb) lGOx. Figure 32. Crack nucleation along (0001)twin (000l) twin (a) (b) Figure 33. Crack nucleation along (0001)tw1-n in specimen No. 13 (Ha). a) Bright field. lOOx. b) Dark field micrograph of the upper left corner in (a). 250x. mm mam—a AOFQPV op Fm—Fmgma Fmpmxgu as» we mumw vmogm .on _N .mwa cw + “Aav _N .mwa :_ mm mcqu Aowp_v op FmFFmLma c_mcm msp we mumm umocm ++ lc_3pfl_ooov co am.m m * mm» . mm» - soc moo Non moo o m: m— am» - mm» - *mN mom N,_ moo o DI N_ mm» mm» mm» - mmF wmq _.mm _.o om+ _P mm» mm» max - mm_ qqm mm _‘o om+ o— - - mm» - we 00— N.Om Flo oq++ m I - mm» - om_ mwm m._m mo.o mm+1 i w -iul rlilll .IIIIIiil II ..... i- i , m_mumxcu mchwm -::, i , iiizil mm:A . mm» «mm_ mm_ oov wV¢F o._ oa - m N % u - mm; «m «w m9 o.N To me - m m mm» . mm» amm mo_ va «.0 05” myI . _q++ m mm» . mm» _o_ oPN mmv w.m_ Fuo Pm - ma v - . mmx m__ qu mflm N.o mo.o NR - mm m u - mm» mm mm 00— oo _.o mm - mm N - . mm» N.¢ wm_ Rum «‘F _,o mm - we _ chCszh a__m QWFm m cemcw < :wwcw ANEE\av AcwE\EoV quwEmLxm _mmmm mczpumcm mgzpumcm ummqm .o: Aum>gwmnov ANEE\ um meowmn vow: cowpmpcmwco Fmpmxcu mczpumgw m:_uwomcg m:m_a memn mmmcpm :owumoCOFm & -mmocu cowumELowmu we mmaxh mg» :0 .m‘m.m wFTmcmP mumu mpwmcmp we zgmsszm .m anmH IV. DISCUSSION 4.1. Location of cracks 4.l.l. Bicrystals The orientations of the basal planes in the two grains in Fig. 22 are nearly the same as in the bicrystal discussed in sec. 2.4.6, where the basal plane in grain A is at 45° to the tensile axis, and in grain B, at 90°. The nucleation of the crack along the basal plane in grain B ahead of the pile-up in grain A agrees with the location suggested by the multiple-slip-band model developed in Chapter II. The same conclusion can be drawn for the bicrystal with orientation (4l — Ha) in which the crack is developed along the basal plane in the twin of grain B that is at 86° to the tensile axis as shown in Fig. 26 (a). The cracks initiated at the grain boundary have propagated across grain B as seen in Fig. 26 (b). The cracks shown in Figs. 23 and 25 are also nucleated along the basal planes in grain B, in agreement with the fracture process discussed in sec. 2.4.5 (see Figs. l9 (a) and (d) L In each of these bicrystals the r.s.s. acting on the basal dislocations in grain A is nearly the maximum for the given tensile stress since A] is close to 45°. Furthermore, the stress concentration factor due to the slip bands in grain A is also nearly the highest (see Fig. l3). A transferred stress high enough to initiate a crack in grain B will be reached before there can be any significant stress concentration developed at the grain boundary due to the slip bands in grain B. 99 lOO The above analysis explains the location of cracks in the bicrystals with other orientations as well. As can be seen in Fig. 24, the orientations of the basal planes in both of the grains are larger than 45°; however, the crack is formed only in grain B in which the basal plane deviates more from 45° than that in grain A. At a later stage of deformation, in the bicrystal with orientation (S - 40), twins were nucleated in grain B. The crack seen in Fig. 28 has been shown to be along the (OOOl)tw1-n in grain B. No slip traces parallel to the cleavage plane have been observed. It is most likely that the stress concentration due to the slip bands in grain A has initiated the crack in grain B. The cracks shown in Figs. 27 (a) and (b) are nucleated at the grain boundary and at the low angle boundary where there is a known stress concentration. Since the bicrystal is symmetric, the stresses in the two grains are more or less equal when slip bands in both grains meet at the grain boundary. Hence,it is difficult to establish whether a crack in a grain is nucleated by the stress concentration due to the l slip bands in the neighboring grain or in itself. It seems that fracture study with asymmetric bicrystals as described previously could be more informative than with symmetric ones. The mechanism developed by Stroh (l4) as shown in Fig. l (d) attributes crack nucleation at a low angle boundary to the tensile stress at the end of the boundary. However, Stofel and Wood (l7) found no evidence of fracture in their experiments with zinc single crystals containing low angle boundaries supporting the Stroh mechanism. According to these workers the boundary is equivalent to being cut up lOl into short segments by the pile-up of dislocations against it, whereby the stress concentration due to the boundary is reduced. Instead, Stofel and Wood suggest that the cracks are nucleated by the coalescence of pyramidal dislocations as outlined in Chapter I. In the present case, only basal dislocations are involved and the slip planes on the two sides of the boundary are nearly continuous. One may use an analysis similar to that described in secs. 2.3.5 and 2.4.5, i.e., the stress concentration due to a group of slip bands on one side of the boundary may be high enough to initiate a crack in the neighboring grain ahead of the slip bands. The nucleation of the crack in Fig. 27 (b) can, therefore, be the result of the stress concentration due to the dislocation pile-ups at the low angle boundary. The crack can propagate on both sides of the boundary, since the basal planes on the two sides of the boundary are nearly continuous. 4.l.2. Single crystals In the case of bicrystals, the nucleation of cracks takes place at points of high stress concentration such as the grain boundary. For the crack to nucleate in a single crystal in which the grain boundary is absent, some other obstacles for the formation of piled—up groups of dislocations strong enough to withstand the large stress resulting from the pile-ups are required. If no such obstacles are present, cracks may be nucleated near the specimen grip (Fig. 29) where there is a known stress concentration. The cracks shown in Figs. 32 and 33 (a) are associated with the junctions of the tips of two conjugate twins on opposite sides of the 102 crack in accordance with the mechanism proposed by Bell and Cahn (27). The slip traces parallel to the crack as seen in Fig. 32 are due to the basal slip within the twin indicating that the crack was formed along (000])twin° In the crystals discussed so far, no second order pyramidal slip traces were detected. Figs. 30 and 31 (b) show the pyramidal slip traces that intersect the basal slip bands. The stress concentration due to the interaction of basal and pyramidal dislocations, as discussed in sec. 2.5, is evidenced by the rapid increase in the work hardening rate prior to fracture ( see Fig. 31 (c) ). However, the profuse twinning near the crack, shown in Fig. 31 (a), might have obscured the pyramidal slip traces. Upon careful examination a faint set of pyramidal slip traces can be seen in the lower left corner of the photanicrograph. The slip lines nearly perpendicular to the crack are due to basal slip within the twins. Fig. 3l (b) shows one of the regions in the vicinity of the crack where twins are absent; intersecting basal and pyramidal slip traces are clearly revealed. As discussed in sec. 2.5, the stress concentration necessary to initiate a crack can be produced by such an interaction of the basal and pyramidal dislocations. 4.2. Crack nucleation A comparison of Figs. 19 (a), (b) and (c) shows that while the nonnal stress components of the stress field of the pile—up are short- ranged on the side of the pile-up as well as ahead of it, the shear stress canponent ahead of the pile-up is long-ranged although that on the side of the pile-up is also short-ranged. Therefore, as discussed in sec. 2.4.5, l03 the strain energy available for the crack nucleation from the stress field of the pile-up will be small on the side of the pile-up as compared with that ahead of the pile-up. Hence the crack nucleation in zinc should occur mainly ahead of the pile-up and should be of shear type in nature as observed in the present experiments. If crack nucleation were caused by the tensile stress component, a change in the indices of the fracture plane is expected (50). In the present work, however, cracks were initiated along the basal plane in all the crystals with soft as well as hard orientations. The conclusion that the crack nucleation in zinc is primarily a shear process is thus substantiated. 4.3. River pattern The wavy lines, called “river pattern", seen in Figs. 26(b), 33 (a), and 33 (b) are steps on the cleavage plane formed during the propagation of the crack. As pr0posed by Gilman (5l) for the formation of the river pattern, when a crack cuts a screw dislocation, the cleavage surface will be stepped, with the height of the step being approximately one Burgers vector. These unit steps tend to merge and build up to visible steps in the form of the river pattern. According to this mechanism, the river pattern on the basal plane should be caused by the intersection of the crack with pyramidal or C dislocations. 4.4. A criterion for the fracture process in zinc Shchukin and Likhtman (52) studied the fracture behavior of amalgamated and pure zinc crystals. Based on the single-slip-band model they pr0pose that CT = K2 can be used as a criterion for fracture. A "brittle” or embrittled crystal fractures whenever the product of the lO4 applied normal stress 0 and shear stress T acting on the cleavage plane reaches a critical value K2. In the above expression, K2 = k (Gy/L), where k is a dimensionless coefficient of the order of unity, G the shear modulus, y a tenn related to the surface energy, and L the length of the pile-up equivalent to the grain size or the diameter of the crystal. Shchukin and Likhtman found the value of K to be 95 g/mm2 for amalgamated zinc crystals at 77°K, and 205 g/mm2 for pure zinc crystals at room temperature. Similar criteria for fracture were derived by Stroh (l4) that K2 = 4Gy/nL, based on the mechanism illustrated in Fig. l (d), and by Gilman (5) that K2 = 6y [E6 (1 - u)]%/NL, for the dislocation pile-up mechanism shown in Fig. l (c). However, Stofel and Wood (l7) found from their experiments with zinc crystals that none of the above fracture criteria were obeyed. 0n the other hand, the present multiple—slip-band model suggests that the controlling factors in the fracture process could be the internal stresses as discussed in sec. 2.4.5. Thus, one may consider a criterion for fracture based on the product of the internal nonnal and shear stresses, such as O. T. = K2, (57) where K is of the order of the theoretical shear strength of the material, and °i and 11 are respectively the transferred nonnal and shear stresses due to the slip bands in one grain onto the cleavage plane in the other grain obtained by applying the analysis described in secs. 2.3.5 and 2.4.5. Following the definition of the s.c.f. in sec. 2.2.], equation (57) may be written as ll 7< ix) 2 s.c.f. s.c.f. T ( )Xy( )yy 105 0r Tq=K , where q = v/7(s.c.f.)xy (s.c.f.)yy Using the tensile data obtained from the present work together with those reported by Gilman (5), the values of K have been calculated and are listed in Table 6. The crystals which twinned prior to fracture have not been included since the problem becomes a three-dimensional one. It is seen that the values of K are of the order of the theoretical strength of the material ~ G/30 and vary within a factor of 2. Whether such a criterion based on the internal stresses and the multiple—slip—band model has its applicability to brittle fracture in general is a question of significance that remains to be answered. 106 Table 6. K values for bicrystals (N = 50, n = 99, S = 100 A°) R.s.s. on the basal plane in q = T q = K Orientation grain A, r /(s.c.f.) (s.c.f.) (k /mm2) (g/mmz) xy Y4 g 46 - 89 138 915 126.2 55 - 83 79 2224 175.6 59 - 72 84 1954 164.1 s - 63 143 l380 197.3 43 - 45* 25 3480 174** 53 - 54* 46 3036 139.6 * Ref. Gilman (5) ** Since both A] and A2 are nearly 45°, K value is doubled. V. CONCLUSIONS A multiple-slip-band model is pr0posed for the fracture process in zinc based on the stress concentration produced by piled-up groups of dislocations on many nearby slip planes. A calculation of the stress concentration factor as a function of the geometric and deformation variables shows that the overlap of the stress fields of the dislocations in the different slip bands leads to a stress concentration near the grain boundary high enough to initiate a crack. The shear stress field due to a group of slip bands is found to be short-ranged on the side of the pile-up and long-ranged ahead of it, suggesting that a crack is more likely to nucleate ahead of the Pile-up. The results also show that a tensile stress large enough to propagate a crack as a Griffith crack can be produced by a group of slip bands. Observations on crack formation in zinc bicrystals grown with controlled orientations and tested in tension show that fracture always takes place along the basal plane in the neighboring crystal grain ahead of the pile-up, in accordance with the preposed model. The fact that the basal plane is invariably the cleavage plane in the bicrystals as well as single crystals tested suggests that fracture in zinc is primarily a shear process. An analysis is made of the interaction between piled-up groups of l07 108 dislocations on the basal and pyramidal slip planes. A calculation of the stress concentration factor as a function of the number of dislocations on the two planes indicates that the stress concentration at the tip of the double pile-up can be large enough to initiate a crack. OiTi = K2 may be suggested as a criterion for fracture in zinc, where Oi and T1 are respectively the internal nonnal and shear stresses acting on the cleavage plane at a point, and K is of the order of the theoretical strength of the material. Fracture commences when this product exceeds K2. l0. ll. l2. l3. l4. l5. l6. l7. l8. I9. 20. 2l. BIBLIOGRAPHY Cottrell, A. H., Proc. Roy. Soc., 2_7_6, 1 (I963). Pugh, s. F., Brit. .1. Appl. Phys., 13, 129 (I967). Cottrell, A. H., Dislocations and Plastic Flow in Crystals, a) p. 9, b) p. 34, c) p. l05, Oxford Univ. Press, London (I953). Dieter, G. E., Mechanical Metallurgy, p. 96, McGraw Hill Inc., New York (1961). Gilman, J° J., Trans. A. I. M. E., 313, 783 (I958). Deruyttere, A. and Greenough, G. B., J. Inst. Metals, 34, 337 (I956). Zener, C., Phys. Rev., 33, l28 (l946). Zener, C., Fracturing of Metals, p. 3, A. S.i4., Cleveland (I948). Koehler, J. 5., Phys. Rev., 33, 480 (I952). Stroh, A. N., Proc. Roy. Soc., A223, 404 (I954). Stroh, A. N., Proc. Roy. Soc., A232, 548 (I955). Orowan, E., Dislocations in Metals, p. 69, A. I. M. E., New York (1954). Chou, Y. T., J. Appl. Phys., 33, 2747 (I962). Stroh, A. N., Phil. Mag., 3, 597 (I958). Bullough, R., Phil. Mag., 9, 9l7 (I964). Bullough, R., Quoted in Ref. 6. Stofel, E. J. and Wood, 0. 3., Fracture of Solids, p. 52l, A. I. M. E., Gordon and Breach, New York (I962). Smith, E., Acta Met., 14, 985 (I966). Stroh, A. N., Adv. Phys., 3, 4l8 (I957). Cottrell, A. H., Trans. A. I. M. E., 313, I92 (I958). Honda, R., Acta Met., 3, 969 (l96l). lO9 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 11o Stokes, R. J., Johnston, T. L., and Li, C. H., Phil. Mag., 3, 718 (1958). Nashburn, J., Gorum, A. E., and Parker, E. R., Trans. A. I. M. E., _l3, 230 (1959). Keh, A. S., Li, C. M., and Chou, Y. T., Acta Met , Z, 694 (1959). Chou, Y. T., J. Appl. Phys., 33, 1920 (1961). Parker, E. R., Fracture, p. 181, Mass. Inst. of Tech., Swampscott (1959). Bell, R. L. and Cahn, R. H., J. Inst. Metals, 33, 433 (1958). Burr, D. J. and Thompson, N., Phil. Mag., 3, 1773 (1962). Gilman, J. J., J. Appl. Phys., 32, 1262 (1956). Greenwood, G. N. and Duarrell, A. G., J. Inst. Metals, 33, 551 (1953 — 54). Gilman, J. J., Trans. A. I. M. E., 333, 998 (1956). Price, P. 8., Electron Microscopy and Strength of Crystals, p. 41, Interscience Pub. (1963). Chou, Y. T. and Eshelby, J. D., J. Mech. and Phys. Solids, 13, 27 (1962). Baker, C., Chou, Y. T., and Kelly, A., Phil. Mag., 3, 1305 (1961). Eshelby, J. 0., Phil. Mag., 33, 903 (1949). Frank, F. C., Phil. Mag., 33, 809 (1951). Eshelby, J. 0., Frank, F. C., and Nabarro, F. R. N., Phil. Mag , 33, 351 (1951). Mitchell, T. E., Hecker, S. S., and Smialek, R. L., Phys. Stat~ Sol , 11, 585 (1965). ‘ Kratochvil, P. and Bocek, M., Phys. Stat. $01., 3, K-69 (1964). Mader, 5., Electron Microsggpy and Strgggth of Crystals, p. 183, Interscience Pub. (1963). 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 111 Cottrell, A. H., Fracture, p. 20, Mass. Inst. of Tech., Swampscott (1959). Petch, N. J., ibid., p. 54. Gilman, .1. J., .1. Appl. Phys., _3_1_, 208 (1960). Chyung. C. K., Thesis, Mich. State Univ., (1965). Peach, M. and Koehler, J. S., Phys. Rev., 33, 436 (1950). Jilson, 0. C., Trans. A. I. M. E., 133, 1129 (1950). Bell, R. L. and Cahn, R. H., Proc. Roy. Soc., 3333, 494 (1957). Noggle, T. S., Rev. Sci. Instr., 35, 184 (1956). Brandt, R. C., Adams, K. H., and Vreeland Jr., T., Cal. I. T. Res. Rep., December (1961). Petch, N. J., Progress in Metal Phys., V. 5, p. l, Butterworth Scientific Pub., London (1954). Gilman, J. J., Trans. A. I. M. E., 393, 1252 (1955). Shchukin, E. D. and Likhtman, V. I., Soviet Phys. “Doklady” (English Translation), 3, 111 (1959). Taylor, A. and Kagle, B. J., ggystallogrgphic Data on Metal and Allqy Structures, p. 263, Dover, New York (1962). Huntington, H. B., Solid State Physics, p. 213, Academic Press, New York (1958). Heannon, R. F. S., An Introduction to Apined Anisotrogic Elasticity, p. 44, Oxford Univ. Press, London (1961). Frank, F. C. and Nicholas, J. F., Phil. Mag., 33, 1213 (I953). APPENDIX A. Lattice parameters and elastic constants of zinc Zinc has a hexagonal close-packed structure with lattice parameters a = 2.6649 A° and c = 4.9468 A° (c/a = 1.8563) at roan temperature (53). Elastic compliance constants of zinc crystal (10'12 anz/dyne) (Ref. 54) S S 13 33 44 S66 = 2(511‘512) 0.838 0.053 -O.731 2.838 2.610 1.570 as c cons po ycrys (1011 dynes/anz) (Ref. 55) 0 = 1/3 (isotropy) 112 APPENDIX B. Equilibrium positions of dislocations in the single-slip—band model Eshelby et a1. (37) analyzed the equilibrium positions of n dislocations in a piled-up group under an applied stress. The equilibrium positions X = X(i) (i = l, 2,..., n) of the dislocations in Fig. 3 are detennined from the equations X(O) = O n 0 . 20 X(Jj-Xfi) +T=O (3:1,2,...,n) ": 1+.- Gb . . . . where D = 21T (1 _ v) (isotropic elast1c1ty theory) 0 = ég- (anisotropic elasticity theory) By considering X(i) as the zeros of the polynomial f(X) = ( X - X(I) ) (X - X(Z) )...( X - X00) 9 Eshelby et al. (37) found that f(X) is given by the first derivative of the n-th Laguerre polynomial The equilibrium positions for n = 1 to 99 obtained by Mitchell et a1. (38) are listed in Table 7. ELM ., .— __~..._ 114 Table 7. Equilibrium positions of dislocations in the single-slip-band model. Equilibrium positions of dislocations in pile-ups are obtained by multiplying by D/ZT ” X1, x2’ ¥33 x4:__ x53 x6’ x7, ‘x8 Xn 1 2.000 2 1.268, 4.732 3 0.935, 3.305, 7.759 4 0.743, 2.572, 5.731, 10.95 5 0.617, 2.113, 4.611, 8.399, 14.26 6 0.527, 1.796, 3.877, 6.919, 11.23, 17.65 7 0.461, 1.564, 3.352, 5.916, 9.421, 14.19, 21.09 8 0.409, 1.385, 2.956, 5.182, 8.162, 12.07, 17.25, 24.59 9 0.368, 1.243, 2.646, 4.617, 7.222, 10 57, 14.84, 20.38,...., 28.12 10 0 334, 1.128, 2.396, 4.167, 6.487, 9.428, 13.10, 17.70,...., 31.68 19 0.183, 0.616, 1.301, 2.240, 3.440, 4.911, 6.661, 8.706,...., 64.65 24 0.146, 0.493, 1.039, 1.786, 2.738, 3.899, 5.273, .868,...., 83.37 29 0.122, 0 410, 0.864, 1.485, 2.275, 3.236, 4.370, .682,...., 102.3 39 0.091, 0.307, 0.647, 1.112, 1.702, 2.417, 3.260, .231,. .., 140.4 49 0.073, 0.246, 0.517, 0.888, 1.359, 1.930, 2.601, .374, ..., 178.8 59 0.061, 0.205, 0.431, 0.740, 1.132, 1.607, 2.165, .807,...., 217.4 69 0.052, 0.175, 0.369, 0.634, 0.970, 1.376, 1.854, .403,...., 256.1 79 0.045, 0.153, 0.323, 0.555, 0.848, 1.204, 1.622, .101,..... 295.0 89 0.040, 0.136, 0.287, 0.493, 0.754, 1.070, 1.441, .867,...., 334.0 99 0.036, 0.123, 0.258, 0.444, 0.678, 0.962, 1.296, .680,...., 373.0 APPENDIX C. Decrease in strain energy due to the basal-pyramidal interaction According to Frank and Nicholas (56), the elastic strain energy per unit length of a dislocation line is proportional to the square of the absolute value of its Burgers vector. Two dislocations with Burgers vector 5] and 52 will attract to form a new dislocation with Burgers vector 63 if b32 < b]2 + b22, and will repel if 632 > b]2 + b22, where 5] + 52 = 53. In the case of zinc, the elastic strain energy of a basal, a second order pyramidal, and a C dislocation will be proportional to I, 4.45, and 3.45, respectively. Therefore, the basal—pyramidal interaction, equation (40), is favorable. The elastic strain energy decrease due to the coalescence process is ' (5.45 - 3.45) AE = 5045 x 100 = 37% 115 1—1 Mw"LRPT '3‘" IES U E T A 1 CHIGAN S M l .4 .1 4.4.4... ......\.: hwnwwl .4... .4 .9435, , . . 1 . 1 . . , . 1\ 4.4 . 1 . . . . . . . ....:.4..44..._..4.4.4...w4 , . . . . . . 4\\4i V411.\\\\\ 4 a 4 7 :31: :11). 17.1: 1 . . .44.... 1, 1 . . . . .44.1.....:1......11.4. 1 . . . . . . . . . 41511444‘4 I\.I~.!1v. , . . , . . . 45,4414. 1 1 . . 441441443 . . . . . 1 44.4 . 21.14.1411,. 1.111 4.4114 . . 1 . 4 . 4 4 4 1 . 4 4 . 14‘4'1447 LII..4- . 444... ....1v..4'v4 . . 4 . 4.4.41.4...14 3...... 41.44.44. , 4114141444 44......(4 4 464444.404 . , . .., 211:2..14. ..,, 4....4441144. 4.4.4.4.... 4.444444: . 4.1... 1..... 71.....11 1 . . 4. 44:1: ..1 . . ._4_ 4.4. 4:;..:.~.441.4_44.1.44 ... 24,141. 1...:114....1.4-.. N., .944 ._..1:,.. 417..