Crack nucleation and stage I propagation in high strain fatigue—II. mechanism

CRACK NUCLEATION IN HIGH STRAIN \Y.H.

AND STAGE I PROPAGATION FATIGUE-II. MECHANISM KIM* and C. LMRD

Dcpartmenr of MeralIurgy and Materials Science University of Pennsylvania. Philadelphia. PA 19104 U.S.;\.

Abstract-In order to establish the properties of the grain boundaries at which cracks nucleate in high strain fatigue. groups of grains adjacent to the free surface of polycrystalline copper have been analysed by S-ray and trace techniques prior to testing. and subsequently observed during testing to check the nucleation sites. It has thus been found that vulnerable boundaries are associated ++i*in highly misoriented grains, the dominant slip systems of which are directed over large slip distances at the intersection of the boundary with the surface. The role of deformation incompatibiiity in
ISTRODUCTION Detailed microscopic and interferometric observations designed to understand the fundamental aspects of crack nucleation in high strain fatigue were reported in the prekious paper [I] (henceforth called Part I). where it was demonstrated that fatigue cracks form by the development of steps at the intersections of grain boundaries with the specimen surface and that the grain boundaries which are preferred for crack nuclei are also favored for Stage I propagation. The purpose of the present investigation is to determine the nature of the grain boundaries which favor the development of steps and the subsequent cracks. The nature of the boundaries has been studied by crystallographic analysis of adjoining grains in polycrystalline specimens. ESPERIMESTAL The preparation of specimens. the material and the methods of testing were identical to those described * Presently; at Department of Materials Engineering. Drexel Cniversity, Philadelphia. PA 19104. U.S.A.

in Part 1. The crystallographic orientations of individual grains in the polycrystalline copper specimens were determined by the Laue back reflection method. To use this method, an average grain size of 0.4 mm was obtained by the strain-annealing technique described in Part I. Since the diameter of the Laue X-rav beam was equal to a few grain diameters, making it extremely difficult to analyse the orientation of a specific grain of interest. a lead tape mask with a hole of approximately 0.9 mm dia. was placed over the specimen so that a particular grain could be exposed to the X-ray beam through the hole. An optical microscope with two-dimensional translating stage was found useful for locating grains of interest at the hole. Figure 1 shons the lead mask (the black area around the exposed specimen surface) placed on a gain of a specimen and the back reflection diffraction pattern from this grain. When the -wins of interest were too small to have their orientations determined by the Laue method. the method of slip trace analysis by Drazin and Ortz [2] was used. However. this method is not appiicab!e to many grains due to rhe restrictive condition that four slip traces [or tra~xs of (111)

Kl%f A\;D LAIRD:

?+fECHAMSMS OF HIGH STRAZN FATIGUE

Fig. 1. Determination of the orientation of a single grain in a polycrystalline specimen: (a) A -rain on the specimen surface selected through a pin-hole made in a lead tape (the dark area). (b) Laue back reflection X-ray photograph taken from the area shown in (a)

planes] induced by cycling should appear on the specimen surface in order to determine the crystal orientation without ambiguity. Therefore, only a few grams could be oriented by this method RESULTS The micrographs of typical gage surface areas and the crystallographic orientations of each grain in those areas are shown in Figs. 2 and 3. Slip traces which appeared on the surface of each grain are labelled P and have a subscript which denotes the relative magnitude of the S&mid factor. I the highest, 2 the next highest and so on. Each grain whose orientation was determined is numbered. Where grains

contained twins. the twins were denoted by the letters ‘A and B in association with the grain number. For instance. grain 3A is in twin orientation with grain 3B in Fig. 2. The orientation of each gram is defined by the relationship between the reference axes x (surface normal). J (stress a_xis) and c (perpendicular to the X. J plane in a right-handed coordinate system) and the cubic crystal axes. i.e. three (OOl> directions so that the misorientation between two adjacent grains can be determined in terms of the orientation differences between the x axes (Ax), y axes (AJJ)and z axes (AI) of the two grains. In order to describe the misorientation of adjacent grains by a single parameter, the definition of the

KIM .%>n L.4IRD:

MECHX’XSSfS

OF HIGH STR.-\IN FATIGCE

79:

axis l:y axis x:X

03

axis

Fig. 2. (a) The surface area of specimen CM (plastic strain amplitud: O..i7?,) showing the individual grains whose orientation relationships with neighboring grains were determined. The grains as WU as the slip traces which appeared on the surface are labelled (b) The orientations of each grain shown in (a) is represented by the orientations of the reference asrs x, y. 3 in the stereographic triangle. R= misorientation parameter is taken as t Ax’ + A_$ + A? (angular differences measured in degrees) following the definition used by Clark [3] on the basis of Aaronson’s previous work [-I]; i.e. R is chosen as the minimum of the possible values it can take. The observations of cracking at specific grain boundaries are reported in Table 1. As can be seen from this table, a large degree of misorientation (R 2 25.4) is favored for crack nucleation unless the boundary lies approximately parallel to the stress axis. That between grain 4 and grain 5 (-I ! 5) in specimen C34 is an example of a high angle boundar>which did not crack for this reason. These boundaries which did develop cracks were found to be angled in the range of j&65: between their traces on the specimen surface and the stress axis. There is oi course no reason why suitable boundaries in the range 65-90’ should not crack; it just happened that the areas subjected to detailed analysis contained no such examples. However. boundaries forming an angle with the stress axis in this vulnerable range did not develop cracks if the boundary misorientation parameter was less than 25.

Micrographs shown in Figs. 4 and 5 provide more evidence for the crack nucleation process and the boundary requirements associated with it In the two micrographs which were taken from di&rent specimens. if one end of a grain (marked Al developed into a crack the opposite end of the grain (marked B) \vas never observed to do so. Furthermore, the dominant slip traces (the traces of the most active slip system in that grain) on the surface of a grain are weak and scarce near the boundary opposite to the vulnerable boundary. but the traces are numerous and intense at the vulnerable boundary. These observations clearly suggest that a vulnerable grain boundary is selected for crack nucleation because of the slip direcred at the boundary over a long tip distance, whereas the boundary opposite has no slip directed at it. and the adjacent slip operates over short distances. These observations are consistent with the observation that the crack nucleation process is associated with the formation of a grain-boundary step [I]. because a grain boundary step will be developed most effectively where the slip motions. operating over a long slip distance. are directed at the

792

b

x:x axis l:y axis o:r axis

Fig. 3. (a) The surface area of specimen CZ3 (piastic strain amplitude 0.76:<) showing the indi+
Table 1. Observed microcracks and misorientations

of adjacent grains .Angle between the stress axis and the

Ax

Ay

A2

IA 2B 2A / 3B

9 10

11 7

30 19

33.2 22.5

65 $3

3814 415

6 14.5

10 21

7 19.5

13.6 32.1

65 i

c23

1A 2 1BjZ

18 18.5

6 2

17 6

2% 19.0

33 60

Cl0

3B 38

17

24

13

31.1

51

Specimen

Grain boundary*

R = \-‘A2

+-JJ + r2

bOU7hryt

c34

l Denotes the boundary where microcrack developed. t The angtes were measured between the grain boundary trances on the specimen surface and the stress axis.

Fig. 4._.. (a) Fatigue crack occurred at the boundary, marked A, while the boundary at the opposite end of this gram. marked B. shows relatively little evidence of deformation. (b) Higher magnification of the area marked B. Specimen C23. cycled at ep = &0.76?& After 81 cycles, stopped at the tensile plastic strain limit-replication technique.

intersection of the boundary with the specimen surfete. In order to check the slip direction of the active slip system in a typical grain involved in crack forma-

tion. the actual configuration of a very active slip system of grain 1A of the specimen C23 (P2 in Fig. 3) MS determined. The determination was made by a two surface analysis (the free surface and the longitudinal section normal to the free surface) using slip traces and twin traces, and the measured angle between the slip direction, (110) pole, and the slip trace on the surface. obtained from the stereographic projection of the grain 1A (Fig. 6a). The actual configuration of the slip system P2 in grain I.4 of the specimen C23 (Fig. 6bj indeed reveals that the

Burgers vector of this system meets the cracked boundary at an angle of 42‘ to the surface. It also should be noted from Fig. 6(b) that the slip system Pz has a long slip distance toward the cracked boundary 1.4 13. whereas the slip distance is close to zero near the opposite side of the grain, which agrees exactly with the observation (Fig. 5) that slip traces are intense at one side of the grain, but they are very weak at the opposite side. DISCI_iSSIO?i (A) Crnck rrucIeutioti In Part I. thr detailed microscopic deformation processes near various grain boundaries on specimen

794

Fig. 5. (a) Fatigue crack occurred at the boundary. marked A, while the boundary at the opposite end of this grain. marked B, shows little evidence of deformation. (b) Higher magnification of the area marked A. Specimen C34, cycled at cp = iO.577;. After 100 ct-cles, stopped at the tensile p&tic strain limit-replication technique.

surfaces were reported and thus it has been shown that the fatigue crack nucleation process at high strain amplitudes is simply a process of forming steps with a height of I - 2 ym The steps have a sharp root radius. The grain boundaries at which cracks nucleate have now been shown to have the following properties: (1) their traces lie at an angle in the range of j&90” with respect to the stress asis: (2) the boundary joins highly misoriented grains and (3) a dominant slip system in one or both of the adjoining grains. operating over a long slip distance, is directed at the intersection of the boundary with the surface. Such a nucleation process. showing the boundary re-

quirements. is illustrated schematically in Fig. 7. Thus, a step can form most effectively at the grain boundary 112, but no noticeable surface step can occur at the boundary 2 I3 (Fig. 7) and accordingly crack nucleation takes place at the boundary 112, but not at 2!3. If the slip in the most active slip system in grain 1 is directed at the boundary 1 f t (as indicated with dotted lines in Fig. 7) in a similar way as that in grain ZI step formation should not be seriously disrupted at this boundary because the degree of damaging activity in grain 1 is not the same as that in grain 2. However. the rate of step growth at the boundary

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OF HIGH STR,%IN F.ATIGKE

PLANE OF SYSTEM P2 GRAIN IA ,lWIN BOUNDARY ON / / SURFRCE

ibl

u- -

TWIN

BOUNDARY

ON

SECTION

Fig. 6. (a) Stereographic projection and (111) traces on the surface of grain IA in specimen C23 (see Fig. 3 for the surface area). (111) and (100) poles of grain 19 which is in twin orientation with grain IA are also shown in solid symbols. (b) Actual configuration of the slip system Pz of grain 1A in specimen C23. cannot be considered simply to depend on the differences in the degree of slip activity between the two grains. Probably the degrees of cross slip, as has been discussed by Margolin et al. [j-7], is one of the important factors affecting the rate of step growth at grain boundaries. Margohn and his coworkers have shown that in polycrystalline materials [5] and also in &axial bicrystals [6] the stress in the gram boundary region is higher than that in the volumes away from the grain boundary. This is because (1) different displacements of the two crystals at the boundary under the apptied stress produces an elastic distortion and (2) secondary slip systems near the grain boundary are activated to accommodate the strain incompatibility between two differently oriented crystals. This stress gradient promotes cross slip in dislocations moving from a repion of low stress (the grain interior) to the region of higher stress

adjacent to the grain boundary. Finally they [7-J concluded that the cross slip resulting from the presence of a stress gradient enhances the irreversibility of slip at relatively high strain amplitude. Certainly. slip irreversibility can be expected to increase the rate of step growth. Consequently, the formation of a grain boundary step and its growth rate ought .to be affected in a complicated way by both the degree of misorientation of the two neighboring grains and by the details of the slip processes within them In addition, grain boundary dislocation sources might well contribute to grain boundary step formation even after the dislocation cell structure, typical of the saturated state at the high. strains of interest here. is established in the grains. The stress gradient and the associated operation of secondary slips near grain boundaries are, however, expected to change as the saturation condition is

TRACE OF THE MOST ACTIVE SLIP SYSTEM

GRAIN I

GRAIN

2

TRACE OF THE MOST ACTIVE SLIP SYSTEM

GRAIN

3

(a)

CRACK

NUCLEUS

(b) Fig. 7. Schematic representation of the crack nucleation process in high strain fatigue. (a) The active slip s)-stems in either one of the grains (I) and (2) or in both of them are directed at the boundary between grain (I) and (3). (b) After cycling. a step is formed at the boundary (I)! (2).

because the dislocation cell structure developed constitutes a near-minimum energy configuration of the stored dislocations [S,9]. Indeed the stress gradient due to plasticity considerations may be eliminated entirely, but the elastic component of the stress gradient can be expected to persist because the grain misorientation will not change under cyclic plastic deformation. Thus this stress-induced contribution to step irreversibility should persist. Moreover, the small step developed during rapid hardening may play a role as a stress raiser in keeping active the dominant slip system directed at the boundary. and the waviness of slip associated with the dislocation cell structure Can be expected to promote step irreversibility. Furthermore it was observed that the deformation marks on a specimen surface produced during saturation are more intense than those produced in the same number of cycles during rapid hardening [lo]. i.e. the development of general notch-peak topography on a free surface per cycle is enhanced by achieving saturation. Therefore, the growth rate of the surface step at a grain boundary can also be expected to increase during saturation. A test of this prediction is reported in a companion paper with affirmative result [IO]. It is interesting that a parallel can be drawn with observations made on the effect of slip orientation on crack nucleation but in long life fatigue and at twin boundaries [Il. 121. For example. Boettner rf al. [ll] found that among a series of parallel twin boundaries in copper specimens. only alternate twin boundaries developed cracks in association with estrusions. They accordingly suggested that extrusion achieved

of material is favored if the slip vectors of each crystal adjoining the tv,in boundary converge and meet the boundary at the free surface. Although the results of the present investigation indicate that dominant, directed slip in either one of the grains across a grain boundary is sufficient for step formation and crack nucleation. the importance of the boundary-slip-surface system is the same in both studies. It is important to note. however. that in the present study, where the plastic strain amplitudes ranged from 0.3 to 0.75?;, no crack was ever observed at a twin boundary. In fact the surface topography associated with twin boundaries was different from that at grain boundaries. The interferograms shotn in Fig. S taken seriatim in tension and compression iliustrate typical twin boundary geometry. Note that the surface profile varies smoothly (one never sees a step with 90’ flank angle). It is also clear from the interferograms that twin bout&r& change their surface contours upon load revers.4 from a smooth step-like feature in tension (Figs. Sa and c) to a shallow valley in compression (Figs. Sb and d) indicating that the slip and surface topography is reversible to a major extent. The reader must be reminded that the great vertical resolution and magnification of the interference fringes in comparison with the horizontal magniftcation gives a highly exaggerated impression of the normal displacements of the surface. For example. the notch-like feature in Fig. S(b) is really quite an innocuous depression. The interferogram (Fig. SC)taken after 60 cycles (by this number of c\-cles the fringes crossing a nearby gmin boundary showed a clear discontinuity at the boundary [I] 1 indicates essentially the same features

Fig. S. Interferoprams tensile plastic (3OC). (cl 60T. at vulnerable e, = *0.76’,.

taken ol a twin boun&r)-. marked T.B. la) Aiter 30 cycles. stopped at the strain limit ilOT). (b) After 50 q&s, sopped at the compression plastic strain limit id) 6OC. Note that the boundary topography is quite diikxnt from that which develops see Figs. 5 and 6 in Ref. ii. Specimen CX cycled at boundaries ~for example. depresions-replication Fringes shifted in the direction of the arrow head indists technique.

as those shown in Fig. Y(a) except for a slight increase in step height. Further cycling produced no further changes in the twin boundary configuration but merely intensification of the slip traces in the adjoining grains. However. at low strain amplitudes and in particular when a twin boundary contains the active slip system. the twin boundary is preferred for crack nucleation to slip bands within the bulk [11] because of the plastic anisotropy associated with the misorientation of the twin boundary. This causes extra high slip activity localized to a few slip bands near the boundary and consequently severe notchpeak topography on the surface. As the strain range increases, however, slip activity extends to some distance or to the whole volume of the grain depending on the applied stress level. Thus the surface topographical changes become less localized and the bulk geometrical aspects of slip are dominant in crack nucleation. Of course, the possibility that slip on slip planes parallel to the twin boundary helps in maintaining continuity between the two grains explains why ttvin boundaries are not sclectsd for crack nucleation in high strain fatigue. This again suggests that compatibility difficulties are important in contributing to the formation of a sharp discontinuity (such as a 90’ step) at high angle boundaries.

All the observations and discussions made so far in the present paper arc based on experimental results obtained on ‘vat-J’ slip material. copper. It is clear. however, that the major factors so far identified as governing the crack nucleation process in wavy slip materials should also be applicable to planar slip materials because the factors are not slip modz dependent. However? the rate of step growth during the early stages of fatigue life is expected to be much slower. even at higher strain amplitudes. in planar slip materials due to difficulty in the cross slip necessary to promote irreversibility of slip. Observation of intergranular cracks in such planar slip materials as 70-30 brass [13. l-t] and Cu-i”, Al [13] tssted in high strain fatigue supports the conclusion that crack nucleation processes in planar slip materials are practically the same as those in wavy slip materials. Impurity particles segregated along grain boundaries. particularly in brass specimens as has been reported by Boetmer, I&d and SlcEvily [13]. could have a significant elect in accelerating intergranular cracking. IB) Sruge I crclck fropaprion

It was shown in Part I that grain boundaries which are preferred for crack nucleation are also preferred

KIM AND LAIRD:

%fECHANISMS

/GRAIN BOUNDARY t TRACE (a)

SMALL

TENSION

,J”GRAIN BOUNDARY TRACE lb) MAXIMUM TENSION

:GRAIN BOUNDARY TRACE (C) COMPRESSION

Fig. 9. Schematic representation of Stage I crack growth along a grain boundary in high strain fatigue by the plastic

blunting process. Note that the crack tip deformation can be asymmetric with respect to the boundary along which a crack propagates. depending on the orientations of the adjacent grains.

for Stage I propagation. Therefore we suggest that any mechanism explaining the process of Stage I propagation, at least in high strain fatigue,- should take account of the requirements for the crack nucleation process. namely, high angle boundary and slip motions directed at the intersection of a boundary with the surface. In the light of these requirements the process of Stage I crack propagation along a grain boundary in high strain fatigue is depicted in Fig. 9. Since a crack is already initiated at the boundary 1 12, the requirements for crack nucleation must be satisfied and the associated slip motions are indicated with arrows in grain 2. Grain 2 is assumed to be oriented favorably for directed slip at the boundary; whether or not grain 1 is so oriented will influence the crack propagation in detail but is not of fundamental importance. Because the slip vactor marked in grain 2 is associated with the most active slip system in that grain it is naturally oriented to provide the greatest plastic deformation at the crack tip without requiring the stress concentration of the crack to initiate slip on other systems. When the tensile limit of the cycle is reached. extensive plastic shear has occurred along the active slip system emanating from the crack tip, and the crack has advanced by

OF HIGH STRAfS

F.4TICUE

&I (Fig. 9bi At this stage the crack growth process is essentially identieaf with the plastic blunting process proposed for Stage II propagation by Laird [Is] and recently re&rmed by Laird and de la Veaux [ I6]. except the degree of strain concentration is so low that crack tip blunting occurs simply by bulk slip on the most active slip system in grain 2. When the load is reversed, the slip at the crack tip will also be reversed. but the new surface gained in tension is conserved. and an increment of crack growth per cycle is achieved. This type of plastic blunting process for Stage I crack growth has been proposed previously by Kaplan and Laird [17] for low strain fatigue. but those workers did not take into account the elect of grain orientation and the possibility of intergranular Stage I propagation. In Fig. 9, the active slip system of grain 1, even if unfavorably oriented for plastic blunting at the crack tip, will eventually contribute to crack propagation either because work-hardening occurs in grain 2. or because the increasing length of the crack raises the stress intensity in both grains, If the active slip system in grain 1 is boundary directed in a similar way to that in grain 2, the blunt crack tip formed at the tension limit of each cycle would be roughly symmetrical with respect to the boundary. because the degree of plastic deformation at the crack tip can be expected to reach the same level on both sides of the boundary However. if the active slip system in one grain (sax grain 1 in Fig. 9) is not directed favorably for crack tip blunting. it would be possible for the- crack tip to form an asymmetric geometry with respect to the grain boundary because the favorably oriented active slip system in grain 2 could cause more extensive deformation on its side of the boundary. Accordingly. a longjtudinal section normal to the crack should reveal a path which is asymmetrical with respect to the boundary, i.e. a major portion of the crack path would belong to grain 2. A typical example of this effect is shown in Fig. 9 of Part 1. Where a grain boundary is inclined at approximately 45’ to the stress axis the degree of crack tip blunting will be minimized because it is likely that the active slip system in one of the adjoining grains will have its slip vector lying approximately parallel to the boundary. In this case, according to the experimental evidence reported in Part I. grain boundary sliding appears to be the most likely mechanism for Stage I grouth. However, this grain boundary sliding mechanism can be considered as a special case of the plastic blunting process described in Fig. 9. Since the active slip system usually has its slip vector close to 45’ with respect to the stress axis. such a grain boundary sliding mechanism is a likely possibility for crack propagation. The above discussion thus leads to a definition of Stage I propagation in high strain fatigue, namely, as that stage of crack propagation where cracks grow along preferred grain boundaries by the plastic blunt-

KIM

ASO

LAIRD:

MECHANISMS

ing process which is controlled mainly by shear along the active. bulk slip system in the grains adjacent to the boundary. Stage I growth gives way to Stage II growth either when the crack reaches the triple point of the boundary in which it started thus encountering a crystal with a different orientation, or else when the increasing stress intensity of the growing crack is sufficient to initiate slip on new systems symmetrically on both sides of the crack. CONCL#USIONS In continuation of the phenomenological study reported in Part I, further investigations have been carried out from the crystallographic point of view on crack nucleation and Stage I propagation processes and we draw the following conclusions. (1) The requirements for crack nucleation at a grain boundary are as follows. (a) The boundary has to be a high angle boundary with misorientation parameter R z 25. (b) The slip on the active slip system in either one or both of the adjoining grains should be directed at the intersection of the boundary with the specimen surface and (c) the trace of the boundary in the free surface should lie at an angle in the range 30-90” with respect to the stress axis. Requirements (a) and (b) indicate that incompatibility between two adjoining grains operates to prevent reversibility of step formation at the boundary and thus causes crack nucleation. (2) The surface topography developed near twin boundaries is different from that developed near high angle grain boundaries in being smoother and more shallow. Slip in planes parallel to the twin boundary relieves incompatibility effects and prevents crack nucleation at such boundaries. (3) The requirements for crack nucleation were found favorable for Stage I crack propagation in which the crack advances by the plastic blunting process in a manner similar to that in Stage II, but with the slip less localized to the crack tip. (4) The requirements identified for crack nucleation are general to a very wide range of metals and alloys.

OF HIGH STRAIN FATIGUE

79’)

However, the kinetics of crack propagation can be expected to be slower for planar slip materials. since slip tends to be more reversible in these materials and thus the ratcheting process which builds up the nucleating step at the grain boundary is less efficient than in wavy slip materials. Acknowledgements-We are very grateful for critical comments on the manuscript from J. M. Finney, H. Margolin, P. Neumann and S. M. Wolf. The work was supported by the National Science Foundation under Grant No. DMR 76-00678 through the Laboratory for Research on the Structure of Matter, University of &nnsylvania.

REFERENCES 1. W. H. Kim and C. Laird, rlcra Met. 26, 777 (1978). 2. M. P. Drazin and H. M. Otte Tables for Determining Cubic Crystal Orientation from Surface Traces of Octahedral Planes. Harrod, Baltimore (1965). 3. J. B. Clark, Metallurgy Society Conference (edited by H. I. Aaronson and G. S. Ansell), Vol. 38. Gordon & Breach, New York (1967). 4. H. I. Aaronson, Decomposition of Austenite by Diflusional Processes (edited by Z. F. Zackay and H. I. Aaronson), p. 387 (1962). 5. H. Margolin and M. S. Stanescu, Acra Met. 23, 1411 (1975). 6. Y. Chuang and H. Margolin, Mets/f. Trans. 4, 1905 (1973). 7. H. Margolin, Y. Mahajan and Y. Saleh. Scripta Met. To be published. 8. C. E. Feltner and C. Laird, Acta Met. 15, 1633 (1967). 9. D. Kuhlman-Wibdorf and C. Laird, Mater. .Sci. Engng 27, 137 (1977). 10. W. H. Kim and C. Laird, Mater. Sci. Engng. To be published. 11. R. C. Boettner, A. J. McEvily and Y. C. Liu. Phi/. Mag. 10. 95 (196-t). 12. A. W. Thompson, Acta Met. 20, 1085 (1972). 13. R. C. Boettner, C. Laird and A. J. McEvily, Trans. AIME 233, 379 (1965). 14. W. A. Wood, Phil. &fag. 3. 693 (1958). 15. C. Laird, ASTM STP 415 131 (1967). 16. C. Laird and R. de la Veaux, Metall. Trans. 8A, 657 (1977). 17. H. I. Kaplan and C. Laird, Trans. AIME, 239. 1017 (1967).