Active slip in aluminum multicrystals

Acta metall, mater. Vol. 41, No. 2, pp. 451-468, 1993 Printed in Great Britain. All rights reserved

0956-7151/93 $6.00+ 0.00 Copyright © 1993Pergamon Press Ltd

ACTIVE SLIP IN A L U M I N U M MULTICRYSTALS Z. Y A O t and R. H. W A G O N E R Department of Materials Science and Engineering, 116 West 19th Avenue, Columbus, OH 43210-1179, U.S.A. (Received 8 November 1991; in revised form I July 1992)

Abstract--Plate-type tensile specimens composed of 1-6 large pure-aluminum grains were characterized carefully for grain shape and orientation. They were then deformed to strains between 1-8%, the stress-strain curves were recorded, and the slip traces or etch pits were analyzed using optical microscopy. In parallel, the internal stress fields in the specimens were calculated using anisotropic elastic finite element modeling. These techniques allowed a resolved shear stress (RSS) to be associated with each observed slip system. Observed were 84 {111} and 4 {110} slip planes. Dislocation nucleation was inferred to have taken place at grain boundaries, specimen edges, existing slip bands, and at grain interiors. Three general kinds of nucleation events were inferred: (1) a wholly RSS-driven event at specimen edges or grain boundaries and extending for long distances in the crystals, (2) an RSS-assisted/local dislocation mechanism at grain boundaries that also extends for large distances but which nucleates at moderate RSS's, and (3) local mechanisms which can occur at very low RSS and which usually remain local, near grain boundaries, existing slip bands, and other microstructural features. Other generalizations are drawn from the large body of data generated from these experiments and calculations.

of slip is revealed. (This is the dominant source of macroscopic stress inhomogeneity in the specimen.) Conversely, slip systems appearing in regions of low local resolved shear stress are inferred to have been otherwise activated. The quantity and complexity of the heterogeneous data obtained in both the experiments and analysis is formidable, making simple, conclusive interpretation of all of the behavior impossible. Instead, we present many original data, examples of certain common behavior, and generalizations based on observations and stress analysis of slip in the controlled specimens. For a complete accounting of all data obtained in the original study, we refer the reader to Ref. [14].

INTRODUCTION Low-temperature deformation of metal polycrystals proceeds principally by the motion of crystal dislocations. The macroscopic flow stress, however, depends on a variety of dislocation obstacles and sources (static and kinetic) ranging in scale from the electronic level to the nearly macroscopic. The selection of activated slip systems [1-8] and the flow stress of polycrystals [9-13], for example, have long been known to depend on the crystal orientations and sizes although the form and strength of the dependence differs from system to system. Detailed analysis of large assemblies of crystals (i.e. typical polycrystals) remains beyond today's techniques, except when non-physical assumptions are made to make possible the averaging of properties across the polycrystal. In order to analyze graingrain interactions, smaller assemblies of crystals with known boundary conditions must be observed. In the present work, the crystalline deformation of large, well-characterized, aluminum multicrystals (1-6 grains) deformed in uniaxial tension is investigated. Resolution of the spatial dependence of slip activation is facilitated by the large grain size. Slip trace analysis, stress-strain measurement, and anisotropic elastic finite-element modeling (FEM) are employed. By comparing the results of the numerical stress analysis with observed slip traces on the surfaces, the role of elastic mismatch across the grain boundaries on the nucleation and propagation

BACKGROUND

tPresent address: National Steel Corporation, Technical Research Center, 1745 Fritz Drive, Trenton, MI 48183, U.S.A.

Classical investigations of polycrystalline plasticity rely on two basic approaches. The first and oldest approach is based on statistical averaging over many grain orientations while ignoring local interactions at grain boundaries. This allows a statistical picture of the evolution of macroscopic textures and yield surfaces without addressing local grain-grain interactions. Alternatively, the microstructural approach is based on single-crystal behavior with adjustments for the presence of local grain boundaries. It has a close connection with microstructural observations and allows a qualitative understanding of work hardening. Unfortunately, it has usually not been possible to determine the internal stress fields in polycrystals needed for a quantitative model of slip activation and multiplication. Statistical averaging relies on a simple macroscopic constraint, such as uniform imposed strain [15-17]

451

452

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

or stress [18], applied to each grain in the assembly. More recent versions [19-21] allow various kinds of mixing of the constraints. Then, by suitable averaging at a given value of the controlled variable, the average conjugate response may be computed after assuming an idealized (or fictitious) single-crystal law. An alternative formulation which yields very similar predictions, the self-consistent method [22], considers a single crystal embedded in a uniform, infinite matrix. Such methods have shown success [23-31] in predicting texture and yield surface evolution, even though the multiplicity of local interactions and the effect of grain size are ignored. There is little possibility to predict local microstructures, work hardening (except relative rates among various strain states), or grain size effects. For this kind of information, the role of grain boundaries must be addressed. The microstructural approach depends implicitly on the micro-plastic behavior of metallic single crystals. In uniaxial tension, initial yield generally follows Schmid's law [32-35], which states that slip is activated when the tensile stress, as resolved on a given slip system, reaches a critical value, the critical resolved shear stress (CRSS). While verified in many cases (in particular for f.c.c, metals [36-43]), the "law" is violated [44-45] under conditions conducive to initial multiple slip, tensile deformation with the tensile axis near the corners of the stereographic triangle [42, 43, 46]. The work hardening and crystalline slip of metal single crystals often follows a three-stage, sequential pattern [35,47-52]. Stage I, with nearly no work hardening and little net increase in dislocation density, is associated with single slip, or easy glide. Stage II shows a rapid increase in work hardening, often nearly linear, associated with multiple slip. Stage III shows a decreasing work hardening rate associated with cross slip, usually near the fracture strain. Factors that promote multiple slip, such as tensile axis orientation [43], complex stress states induced by specimen gripping [53, 54] or higher stacking fault energies [46, 55-57], can reduce or eliminate Stage I. When multi- and poly-crystalline deformation is interpreted as a modified single-crystal response, the grain boundaries are viewed as agents promoting multiple slip from nearly the start of deformation. By a variety of mechanisms [58-64] based on elastic and plastic interactions and crystalline effects, multiple slip is activated near the grain boundaries [2-3, 58-61, 64], Stage I is suppressed [46, 65-67] and a zone of higher dislocation density [8, 68, 69] (or occasionally lower [70--71]) is established relative to the grain interiors. Thus, the initial work-hardening rates of polycrystals are similar to Stage II hardening of single,crystals with no Stage I [2, 11, 67, 72-73]. Subsequent work hardening is usually interpreted by recognizing that the response of grain-boundary zones is different from that of grain interiors [73-76]. In any case, a grain-size dependence is predicted and

increased hardness corresponds to increased dislocation densities, in accord with experiments [9, 10, 13] and simple dislocation models [77-79]. Bicrystal experiments have verified the general trends outlined above. Bicrystal deformation reveals the reduction of Stage I expected [2, 67], the increased work hardening [2, 66], and an increased flow stress [2, 11, 66, 72] relative to single crystals. More important for mechanistic models, an increased misorientation [58-61, 63-66, 70, 74] of the crystals enhances each effect while single crystals and similar bicrystals oriented for multiple slip show similar patterns [11, 72]. In nearly all cases, the grain boundary region is identified as highly-stressed [80] and many dislocations are introduced there, serving as local obstacles for further slip [81, 82]. Finally, the boundary configuration itself is important, with long, planar boundaries parallel to the tensile axis requiring only 4 slip systems for compatibility while embedded grains require 6 slip systems [3] and more nearly approximate polycrystals [3, 74-76]. Tricrystals approximate the hardening of polycrystals better than bicrystals [65]. Many of these effects may be related to the largely-unknown internal stress fields in these specimens. However, completely unloaded slip systems have been observed [83] to operate in tungsten bicrystals, which are elastically isotropic and therefore have nearly homogeneous stress fields. In order to make predictions of polycrystalline deformation from known single-crystal responses, the dominant mechanisms inducing the extra slip at each grain boundary must also be identified, analyzed and understood quantitatively. Although many microstructural observations of activated slip at boundaries have been made, as noted above, it usually is not possible to determine the local stress state. Thus, the exact mechanism for additionallyactivated slip is subject to conjecture. The stress analysis needed to remove this ambiguity has not been possible in closed form for multierystals (bicrystal, tricrystals . . . . ). Recently, the rise of nonlinear numerical stress analysis has improved the possibility of such analysis. Only recently has finite element modeling (FEM) been used to interpret local stresses in multicrystals. Early work reproduced single-crystal behavior [84-87], then bicrystals [88-91], and, more recently, polycrystals [92-95]. Comparisons with microstructures have been usually rather qualitative, but highlystressed regions, usually at grain boundaries, have shown high dislocation densities [92]. Furthermore, local orientations have been shown to be important, as opposed to the average orientation, or texture [95-97]. Because many kinds of activation occur at the grain boundaries, many slip systems are observed which have low apparent RSS's [2, 98]. Few highlyloaded slip systems are absent, however [93, 98]. This observation suggests that some systems are stress-activated at the continuum level while others are activated by local compatibility requirements.

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS Recently, detailed elastic FEM analysis of Fe-Si bicrystals [90] was able to predict not only slip systems identified as elastically activated, but nearly all of the ones identified as secondary slip systems originating at boundaries. This result suggests that continuum elastic effects may predominate (for strains greater than about 1%) the activation of slip in grain boundary regions, even among mechanisms previously thought to be purely of geometrical or crystalline origin. Conversely, at strains the order of elastic strains, slip systems have been shown to depend on very local geometries of the slip systems and boundary plane involved [26, 62]. In summary, the main obstacle to constructing a unified, quantitative model of plastic polycrystal deformation lies in the ability to predict the number and density of non-primary slip systems activated near the grain boundaries. The current work is a first attempt at correlating computed anisotropic elastic fields in carefully-characterized multicrystals with observed slip activity. The relative importance of continuum stress fields versus ones of crystalline origin can thus be approximately assessed. PROCEDURES

Experiments Pure aluminum (99.99%), hot-rolled, polycrystalline plates provided by the Aluminum Company of America were strain-annealed to obtain a very large grain size suitable for this study. Aluminum was chosen because it is easily strain annealed and because it (along with other f.c.c. metals) has been the subject of many prior investigations, as outlined in the Background section of this paper. Aluminum has an unusually high stacking fault energy (about 200erg/cm 2) [99, 100], and the stacking fault energy for b.c.c, materials is usually below 100 erg/cm 2 [101], such that cross-slip is relatively easy. This is presumably the reason that flow stress and work hardening rates are nearly independent of grain size in the range of 0.252.5mm [65, 67, 73]. Single-crystal flow curves for

453

AI have been presented at several stereographic orientations [72]. The deformation of aluminum, as for other f.c.c. metals, is dominated by slip on {111} planes along (110) directions, but microscopic slip bands (which generally lie in the dislocation glide planes) have occasionally been found along other directions [11]. In one study [81], {100} slip was identified in aluminum at 473 K. {100} slip was also found in aluminum tricrystals at high temperature [102] and room temperature [103]. {110} slip was observed in another study [104]. Five polycrystalline plates (up to ~250 x 40 × 3.5 mm), labeled with capital letters "B" through to "F", were strained to approximately 2% in tension and then annealed at 600°C for 7-10 days, until very large grains were obtained. The long axis of each plate is oriented along the original rolling direction. Figure 1 shows a typical strain-annealed plate (plate "B") after etching in (1 part HC1, 1 part H20, 1 part HNO3, several drops of HF) and Fig. 2 shows schematically the grain configurations for all five plates. Tensile specimens were machined from the plates in orientations and locations shown in Fig. 2 and were stress-relieved at 250°C for 3 h. They were then numbered sequentially within each plate for identification and each grain was assigned a unique lowercase letter within each plate. The tensile specimen surfaces were mechanically polished with l/4/~m diamond paste and then electro-polished in a solution of 20% perchloric acid in methanol at 0°C, 15 V, and 1-2 A. The orientation of each grain and the misorientation across each grain boundary was determined using Laue X-ray back reflection (except for the B plate, which was analyzed using electron channeling) employing standard analytical techniques presented in detail elsewhere [14]. The orientation was measured before electropolishing to preserve a more distinct specimen edge to facilitate accurate alignment. Table 1 shows the deformation and analysis schedules for the various tensile specimens. Specimens B and C1-C4 were examined for etch-pits after an etch of 50% HC1, 47% HNO3, and 3% HF for 2-3s.

Fig. 1. Strain-annealed aluminum plate, prior to cutting for tensile specimens (Plate B).

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

454 Plate B

-

Plate C

(,

Plate D

s

0

Grain a

Grain b

Grain~ Plate E

Plate F

Grain a

2

G

" a

3

4

/ Wls

Grain c

10 I

mm

Grain

b

\

Grain a

I

Fig. 2. Aluminum plates B--Fwith tensile specimenlocations, drawn schematicallyto show the orientation relationships and configurations. Specimen dimensions are to scale, but plate edges are not. However, little useful information was obtained by this technique, so it was abandoned in favor of direct examination (without etching) in subsequent experiments.

Finite element modeling All finite element calculations were carried out using ABAQUS [105]. Anisotropic elastic material properties were assumed: C l l = 108.2 GPa, C12 = 61.3GPa, C 4 4 = 2 8 . 5 G P a [106], with coordinate axes oriented along the conventional cubic cell edges. The elastic constant matrix fixed on the crystal axes was transformed to the problem coordinate system (tensile axis, transverse direction, and through-thickness direction) for use in the numerical model. Eight-node, 3D, isoparametric-interpolation elements were used for all calculations. Numerical integration is carried out at 8 Gauss points in these elements. Rather coarse meshes, samples of which are shown in the Results section, were employed because of computational storage and time limitations. In a few cases, refined meshes were used around internal features. The boundary conditions reflect those of the tensile test: the upper and lower gripped regions were

displaced in the tensile direction (to a total amount of 1% of the original tensile length) and had zero displacements in the orthogonal directions. The sides and large surfaces were assumed traction-free. Compatibility and equilibrium are automatically satisfied, in the finite element sense, throughout the specimen. The computed stress tensor, expressed in problem coordinates and in arbitrary units, is available at each Gauss point• The units are arbitrary because the 1% grip displacement is far greater than the elastic limit, which, in any ease, was not measured accurately. No attempt was made to correlate the measured stressstrain curves to "calibrate" the FEM-stresses, since the FEM was based on a purely elastic model. Therefore, only relative values of stress (among various locations and orientations) are significant. Under these boundary conditions, the simulated RSS in each crystal can differ markedly not only because of slip system orientation, but also because of the orientation of the elastic anisotropy axes. The resolved shear stress (RSS) acting on each of the twelve {111} (110) slip systems at each Gauss point was calculated using the known orientation relationships between the problem coordinates and the crystal coordinates, and between the crystal

Table 1. Deformationand examination schedules I II IlI IV V 250°C/3h 1% straine t c h re-polish +7% strain 250°C/3h 1-1.5% strain re-polish +6% strain etch Dl, E2, E3 150°C/4h 4% strain 200°C/96h 1% strain +1% strain re-polish D2, El, E5, F 150°C/4h 4% strain FA 150°C/4h 8% strain Note: Except for specimensets B and C, whichwereetched after about 1% strain,all specimenswereexamined directly after each deformation. Specimen B C1-C4

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS coordinates and the 12 allowed slip systems. For cases in which non-{ 111 } < 110> slip was observed, the RSS's for observed slip system possibilities were also computed. Since the experimental procedure could not distinguish between the various slip directions in a given plane, only the most highly-loaded slip directions on each plane were considered in ranking the slip planes by RSS. RESULTS The detailed results of both the experiments and the numerical analyses are voluminous. They can be found in their entirety in Ref. [14] In order to keep this presentation to a manageable length, we will concentrate on summary data and will present results for two typical specimens in some detail. This information outlines the general responses of the multicrystals in terms of activated slip systems and comparison with numerical results. Activated slip systems and RSS's Thirteen (13) specimens having 32 grains were tested and analyzed. Of the 32 grains, 7 were internal grains and 25 had external surfaces. Slip activity on 88 slip planes was observed away from the gripped regions of the specimens, 84 of which were { 111 } type and 4 of which were on { 110} planes. All of the { 110} slip planes appeared to be registered with nearby {111} slip bands. (These either show a one-to-one correspondence across grain boundaries or at edges, or the {110} slip bands terminate at {111} bands.) Of the 84 {111 } slip planes, 11 of the probable nucleation sites appear to have been at grain centers, 20 at specimen edges, 33 at grain boundaries, and 17 at both edges and boundaries (indistinguishably). Note: The probable nucleation site of each slip system was inferred by its appearance on the optical micrographs. "Edge" systems had traces emanating from specimen edges with very few lines intersecting a grain boundary. "Boundary" systems were observed at boundaries but did not extend to the specimen edge. "Edge/boundary" systems had most traces in contact with both boundaries and specimen edges, such that the origin could not be determined unambiguously. Furthermore, "center" systems had traces in contact

Slip plane (84){111} t y p e

(4){110} t y p e

455

with neither boundaries or specimen edges. Only three slip systems were clearly "registered" one-forone with slip systems across grain boundaries although many others had partial registry and were counted simply as boundary systems. The "center" systems are certainly undercounted by this method because, after sufficient strain, they intersect with other features. Thus, finding a center system depends on unloading at the correct strain. "Center" nucleation does not necessarily mean true bulk nucleation, because the specimens are generally only 3 mm thick and nucleation presumably occurred at the large specimen faces, even in the "center" systems. A summary of activated slip system data appears in Table 2, along with the average resolved shear stress calculated by FEM for the active slip systems. The distributions of resolved shear stress (calculated by anisotropic elastic FEM) for each type of slip system are presented in Fig. 3. The relationship between the probable nucleation site and the FEMcomputed resolved shear stress (at the slip site) is very suggestive. The distribution for all {111 } slip planes (excluding three clearly-registered systems across boundaries), Fig. 3(a), shows preferred resolved shear stresses at ~250 and 325a.u. (a.u. = "arbitrary units," see Procedures section), with a smaller peak around 75 a.u. In general, there is a wide range of RSS, from 60-390 a.u. When segregated by probable nucleation site, the distributions take on additional character. Systems apparently originating at specimen edges [Fig. 3(b)] have a rather clear distribution of RSS's centered at ~325 a.u. with a distribution width of +50a.u. and minimum/maximum values of 230 and 390 a.u., respectively. Boundary system.s have a rather broad, uniform range of RSS's between 60-375 a.u. except for a peak at ~225 a.u. Edge/boundary systems, i.e. those which intersected with both boundaries and edges at their first observation, have a similar average RSS to the edge systems but have two distinct peaks, at ~225a.u. and 325a.u. The upper peak may be conjecturally correlated to the edge-nucleated peak and the lower peak to the boundary-nucleated peak for systems which were sufficiently loaded to extend to the specimen edges. (Thus, in this picture, the lightly-loaded boundary slip systems

Table 2. Classificationof slip in 32 grains Probable nucleation site No. Boundary-general 33 Boundary-registered 3 Edge 20 Center 11 Edge/boundary 17

~rRssa(a.u. a) 225 _+85 70 + 60 300 + 50 150 + 70 290 _+55

Boundary-registered 2 85 ± 5 Edge I 35 Center (at slip band) 1 270 Note: aResolvedshear stressesare presented in arbitrary units correspondingto an elasticextension of each specimen by 1%. The scatter shown represents a singlestandard deviation(standard error of fit).

YAO and WAGONER:

456

ACTIVE SLIP IN A L U M I N U M MULTICRYSTALS

(a) 0.4

0.4

All Activated {111} Planes

0.3

¢

0.3

0.2

0.2

0.1

0.1

0

25

75 125 175 225 27532537S 425 Resolved Shear Stress (a.u.)

0

0.4

Edge Nucleation

0.3

0.2

0.1

0.1

25

75 125 175 225 275 325375 425 Resolved Shear S t r e ~ (a.u.)

75 125 175 225 275 325375 425

(e) Canter

0,3

0.2

0

25

Resolved Shear Stress (a.u.)

(b) 0.4

(d)

0

(c)

25

75 125 175 225 275 325375 425 Resolved Shear S t r e ~ (a.u.)

(f)

0.4 Boundary

(short r ~ e )

0.3 0.2 0.1 0

25

75 125 175 225 275 325375 425 Resolved Shear 8 t r e ~ (era.)

0.70.5 0.6 ~ S l i 0.4 0.3 0.2 0.1 0

25

P

[]

Registered {110}

[]

Other{110}

75 125 175 225 275 325375 425 R e ~ l v o d S h e a r S t r e ~ (a.u.)

Fig. 3. Distributions of observed slip systems as functions of FEM-calculated resolved shear stresses acting on each slip system: (a) all activated {111} slip systems; (b) edge-nucleated { 111} slip systems; (c) boundary-nucleated { 111} slip systems; (d) edge and boundary (indistinguishable) {111} slip systems; (e) center-nucleated { 111} slip systems; (f) { 110} slip systems.

which contribute to the broad boundary distribution don't extend across the specimen and are not counted as edge/boundary systems.) "Center" systems show a distribution of RSS's with two widely-separated peaks, one at ~75 a.u. and one at ~225 a.u. None higher than 240 a.u. was observed. With one exception, these slip systems were observed at the same locations as intense primary slip (i.e. # 1 ranked RSS) and sometimes terminated at primary slip bands. For example, in specimen E2/grain a, slip systems B and C appeared simultaneously at the grain center. System B is the primary one (237 a.u.) and system A is very lightly loaded (72 a.u.). This kind of split suggests the origin of the two peaks in Fig. 3(e), presumably related to a geometrically-required dislocation nucleation mechanism. The one exception occurred in specimen E3, grain a. Slip system C appeared at the grain center without significant nearby slip activity observed even though the RSS was calculated to be only 89 a.u.

S t r e s s - s t r a i n curves

The engineering stress-strain curves for specimens C 1 - F are presented in Fig. 4(a-f). (The curve for specimen B was not recorded.) As will be shown in the detailed examples which follow, the strain is generally quite inhomogeneous over the tensile region such that the average strain measured has little Table 3. Stress (at 3% strain) and work hardening rate (at 1.5% and 3% strain) Specimen E2 E3 E5 DI FA C2 E1 F C1 D2 C3 C4

~q./.(MPa) 19.6 19.3 18.7 18.6 18.4 17.7 14.4 13.1 12.7 I 1.5 I 1.3 9.7

hi.SOlo

n3°:.

0.53 0,10 0.16 0.21 0,26 0,40 0.47 0.22 0.38 0.70 0.42 0.76

0.40 0.09 0.22 0.16 0.18 0.60 0.18 0.38 0.21 0.49 0.23 0.49

YAO and WAGONER:

ACTIVE SLIP IN ALUMINUM

(a) 25

MULTICRYSTALS

457

(b) .

,

.

.

.

.

.

.

.

.

25

-

20

D1

"~ 2O C2

15 lO

~c3C

1 10

...---- C4

5

%

= 0.16

D2 n8% = 0.49 0

.

.

1

.

.

.

.

.

,

•

,

2 3 4 5 T r u e S t r a i n (%)

0

.

,

-

,

i

.

2

.

.

.

.

3

,

4

-

5

T r u e S t r a i n (%)

(c) 25

a

1

0

(d) ,

-

,

.

.

.

.

-

,

25

-

•

,

- ,

.

,

-

,

.

,

.

E2

~'~ 2O

2O

== 15

E

~

1

5

n3%

E2 n3% = 0.40 E3 n3% = 0.09

= 0.18

E4 n 3 % = 0.18 E 5 n 3 % = 0.22

0

.

0

,

.

1

,

.

2

,

.

3

,

.

4

t

.

.

.

.

1

.

.E4

2

3

n3. %

4

,

0..lS

5

T r u e Strain (%)

True Strain (%) (f)

(e) 25

.

.

5

•

,

-

,

.

,

.

.

25

-

.

,

.

,

.

,

.

,

.

,

20 "~

==

.

C2~

2O

15

= 1o

j

15

F

C1 ,."

c/2

n3%

=

0.3~

'1 ' 2 ' 3 ' 4 " 5 " T r u e S t r a i n (%)

10

C3

C4

'~

1 C2 ns% = O.60 C3 n3% = 0.23

5

~

0

....... 1

ca. , ~ 7 °'.49~

2 3 4 5 T r u e S t r a i n (%)

Fig. 4. Stress-strain plots obtained for various multicrystal tensile specimens: (a) specimen C1-C4 (second straining); (b) specimen D1 and D2; (c) specimen El, E4, E5; (d) specimen E2-E4; (e) specimen F; (f) specimen C 1 ~ 4 (first straining).

material significance. In trying to find a correlation between the stress-strain curves and microstructure, a number of boundary configuration and mechanical response measures were examined: Mechanical response measures: • flow stress at 0.2, 1.5 and 3% strain • work hardening rates, logarithmic and linear, at 1.5 and 3.0% strain • duration and slope of apparent Stage II. Boundary configuration measures: • grain shape and boundary orientation (internal, edge, across grains, etc.) • grain boundary area/volume ratio • RSS ratio in most highly-loaded slip systems.

All attempts to correlate the measures from the two groups failed. However, the tensile axes of the large grains in each specimen were plotted onto a standard stereographic triangle in order to examine qualitatively the role of the predominant single-crystal orientations. The single-crystal stress-strain curves for these orientations (from Ref. [72]) were then compared with the multicrystal curves. In nearly all comparisons of greatly different orientations, the ranking of single-crystal results matched the multicrystal ones in terms of flow stress and strain hardening. Among very similar principal grain orientations, however, the effects of grain boundary configuration could be discerned. (Thus, it appears that the crystalline texture will dominate the macroscopic mechanical response in pure aluminum, without regard to boundary configurations. This agrees with previous,

458

YAO and WAGONER:

ACTIVE SLIP IN ALUMINUM MULTICRYSTALS of work hardening at strains greater than about 1%. A similar effect can be seen in comparing curves for E2, E3, E4. While E3 and E4 are very similar, in spite of the single-crystal vs bicrystal configuration, E2, with an internal grain between the two external grains, has a much higher rate of work hardening at higher strains ( > ~ 1.5%). At a strain of 1.5%, E2 exhibits an n value (bin a / b i n E) of 0.53 vs a value of 0.10 for E3, and 0.26 for E4. Finally, comparison of specimen E1 with E5 or D2 with D I shows that long grain boundaries roughly parallel to the tensile axis

more limited results which show very little dependence flow stress on grain size, over a wide range of grain sizes.) Comparison of some pairs of the tensile curves is more revealing than that of the whole ensemble. For example, specimens C3 and C4 are nearly identical specimens and show very similar hardening curves, extent of Stage II, etc. Conversely, C1 and C2 are very similar (the large grains are identically oriented) but the presence of the small internal grain near the flat grain boundary seems to cause a much higher rate

Specimen C1 Cla

-'Zr"3201 l Cle

X

-.643

I-.-//

.-.7oo-

X

r-.713]

-.2,,

Y F-.767l

L ". ,"]

Y

L.lO7/ .635 -J

r e291 1"1741

Clb X

I-.13d .383,

-.301

: , , i Y[--.2761

[-.9061

~ .767~

L .330J

(a) ill

Clb

/

(b)

(c) 3567

8

9

8

7

Contour range: 1 - 240 a. u., 10 - 390 a.u.. A(352) *(212)

C(244)

(d)

B(362)

* (256) C~47)

~la

~331) *(311) C~30)

* (26)

*(73)

/ /

/

~390) *(278)

*(213)

~ I

~1B5)

C(18B)

B(356)

]3(306) Clb [

A(342) * (321) C (32) * (76)

Fig. 5. Experimental and FEM simulation results for specimen Ch (a) schematic plot of observed slip activity (with crystal orientations shown); (b) FEM mesh (250 elements, 914 nodes, 2742 d.f.); (c) contour plot of maximum resolved shear stress throughout the specimen (note: the maximum RSS refers to different slip systems at different sites); (d) ranking of slip systems by local RSS and typical values in arbitrary units (*refers to unobserved slip systems).

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS tend to promote low flow stresses relative to single crystals or internal/external grain assemblies. Presumably, the long grain boundary provides an easy source for dislocation nucleation and multiplication, but does not constrain dislocation motion greatly because it lies parallel to the principal stress direction. An interrupted tensile test sequence was measured for the specimens from the "C" plate. Figure 4(f) shows the original curves up to the first examined strain, 1.5%, for comparison with the second sequences [Fig. 4(a)], up to a new strain of 6%. During the second loading, the initial flow stresses have returned to nearly the values (in three of the four cases, even lower stress) at the start of the first loading, far below the values at the finish of the first test. This indicates nearly perfect annealing at room temperature for this aluminum, at least at small strains. However, the order of hardening and flow stress between the four specimens are unchanged in the two tests, so consistent, reproducible results are obtained in this sense.

Fig. 6. Refined FEM mesh around grain e in specimen C1. (217 elements, 518 nodes, 1544d.f.) Boundary conditions at edge of mesh were taken from FEM results for original mesh [Fig. 5(b)]. contours at arbitrary RSS levels marked 1-10 and Fig. 5(d) shows the order of the slip systems ranked by RSS throughout the specimen along with typical RSS Values (in arbitrary units) in the region. Taken together, Fig. 5(c) and (d) indicate the distribution of RSS on the various observed slip planes. (Note that the RSS's presented refer to the most highlyloaded slip system operating on the slip planes among the allowed (110) slip directions.) Figure 5 shows that the densest and least-localized slip activity is observed in the regions of highest RSS and that the primary slip systems (or those within a few percent of the primary RSS) were activated: A in grain a (classified "edge/bounary"), A in grain b ("edge"), B and A in grain c ("boundary"). In particular, the regions near the center of grain b and around the internal grain e show the strongest slip traces, at the highest RSS contours. All of these slip

Detailed observations and comparison with FEM--two specimens Specimen C1 consists of three grains, a and b at the ends of the tensile specimen (with a planar grain boundary oriented about 60 ° from the tensile axis) and e an elliptical one near the center of a, Fig. 2. The grain orientations, observed slip activity after 6% strain, finite element mesh, and resolved shear stress contours calculated by F E M are shown in Fig. 5(a~l). In particular, Fig. 5(c) shows RSS Cla

Clc

Clb x

l •

(a)

459

GB

""

(b) Fig. 7. Micrographs taken from specimen CI: (a) at grain boundary a--¢ after 1% strain (~400X); (b) at same location after 6% strain (~ 100X).

AM 41/2--J

460

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

systems have RSS's in the high range, from 330390 a.u., suggesting simple stress activation at edges and boundaries. Two slip systems are moderately loaded, B in b (332a.u. at the grain boundary, about 10% less than the primary system A) and C in c (252 a.u., about 30% less than primary systems B and A). B in b cannot be distinguished between edge and boundary nucleation, but it becomes more highly loaded near the boundary relative to A. It was classified of the "edge/boundary" type and fits the pattern of large-extent slip systems having high calculated RSS's corresponding to the distribution peak at ~325 a.u. C in c remains localized near the grain boundary, is clearly classified a "boundary" system, and has a calculated RSS 3 0 0 below the primary ones (252 a.u.), corresponding with the second distribution peak at ~ 225 a.u. Finally, slip system C in grain b clearly nucleated at the grain interior (in plan view), at the primary slip bands labeled A. It occurs on the least-loaded plane (165a.u., ~ 6 0 % below the primary RSS) and appears fairly localized at the region of intense slip activity corresponding to the most

highly-stressed region. Slip system C is a bit atypical for center-nucleated systems in that it has a RSS between the two peaks in the distribution. While the origin/termination at the A slip bands clearly suggests a local dislocation mechanism of nucleation, the orientation of grain b means that even the leastfavored plane has a significant RSS. It appears that system C was nucleated by a dislocation/geometric mechanism at a stress well below that for primary slip and the fortuitously high RSS allowed it to extend to large dimensions, on the order of the specimen width. The mesh used for calculating the stress fields was fairly coarse because of computer time and space limitations for the 3D anisotropic calculation. In order to check whether the results were mesh-dependent, particularly around the internal grain c, a finer sub-mesh was generated, Fig. 6. The nodal displacements at the small mesh boundaries were taken to be the values from the large-mesh results, Thin elements were utilized at the grain boundary because this was shown in an earlier F E M study often to be important [90]. However, in the case of the plate geometries studied here, the finer boundary meshes made little

Specimen E2

X

E2b ~ - - - ~ Z r .793

E2c X .--.---~. Zr

-

O.O -1

E2a X ]~-4~

1.,853 L..456

Z['--609

,:,,

]

.~7o /

(a)

D(111~ ~ 1 ~ . ~ ~"/L:'/'b-~~

E2b

E2a

(b)

E2a

(c)

C(111) E2b

"

'

-

-

~

l

l

~

J

/

E2a

Fig. 8. Schematic plots of observed slip activity for specimen E2 (with crystal orientations shown): (a) after a strain of 1%; (b) after a total strain of 2%; (c) after a total strain of 4%.

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

in b started at the edges of the specimen. After an additional 1% strain, Fig. 8(b), two of the "center" slip systems appear to be "boundary" or "boundary/edge" systems and at a total strain of 4%, some of the "center" systems remains identifiable. This sequence illustrates the difficulty in identifying the origin of slip systems unambiguously from micrographs taken after strains of a few percent. This effect certainly produces a marked undercounting of "center" slip systems in this study. Specimen E2 was also analyzed using FEM. The mesh is shown in Fig. 9(a) and the RSS contours and rank order by region are shown in Fig. 9(b, c). As in specimen C1, the slip which is first seen [Fig. 8(a)] occurs near the regions of highest stress (near the centers of the two end grains) and then begins to multiply at the grain boundary regions. Five of the observed 7 initial slip systems have the highest RSS's calculated: B in a (226), B in b (302), A in b (276), A in e (251) and B in c (220). The two remaining original slip systems are very lightly loaded, C in a (85a.u.) and D in b (ll0a.u.), and are found in regions of primary slip at grain centers. In fact, both slip systems have slip traces nearly parallel to the tensile axis and thus have very low RSS's.

difference, with no change in slip plane ranking inside or out of the crystal, and RSS's within 8% of the ones calculated originally. All of the experimental results discussed above were obtained with specimen C1 deformed to an average engineering strain of 6% in one continuous test. Figure 7(a) is a typical optical micrograph from which the schematic results were taken. Figure 7(b) is an optical micrograph of the same specimen region after a prestrain of 1% and etching. The etch pits at the smaller strain have a rather random, unbanded appearance, usually up to a critical strain of about 1-1.5%. The bands, resolved without etching, appear at higher strains. Specimen E2 consists of two end grains, a and b, with an internal grain, e, at the a-b grain boundary. The specimen was deformed to 4% in one step, examined, polished, and annealed. Then, it was subjected to a strain of 1% and examined, and another 1% and examined. The schematic figures of the slip activity at the end of each of these deformations appear in Fig. 8(a-c). Because of the interrupted testing at small strains, it is possible to see clearly that three slip systems originated in the "center", two systems in internal grain e at the grain boundaries, and two slip systems (a)

x A

!

FT-(b) 2

3

4

5

5

6

7

8

E2a -

Contour range: 1 - 179 a.u., 10 - 308 a.u..

(c) B(288) .4.(183)

B(192) *(138) D(132)

.," s~

E2b

A(188) B(302)

fE2¢

A(276)

I

*(224)

D(ll0)

\..

461

B(225)

A(127) ~ *(128) \ C(97)

~B( -\'

X

B(191)

~ll - *(181)

C(74)

A(105) *(106)

C(851

220) A(251) *(252) C(219) D(80)

Fig. 9. Experimental and FEM simulation results for specimen E2: (a) FEM mesh (138 elements, 322 nodes, 966 d.f.); (b) contour plot of maximum resolved shear stress throughout the specimen (note: the maximum RSS refers to different slip systems at different sites); (c) ranking of observed slip systems by local RSS and typical values in arbitrary units (in parentheses).

462

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS Thus, two very lightly-loaded slip systems are activated and registered across a grain boundary. Similar observations by Hook and Hirth [58-60] (and later found through FEM calculation by Wu et al. [90] to occur at very low continuum stresses in Fe-3%Si) were labeled "second order" slip.

After a strain of 4%, ten slip systems are observed with the highest slip density beginning to appear around the grain boundaries. The slip includes one of the four {110} planes found in this study. As usual, the {110} plane (D in grain c) is clearly registered with a {111} plane, (C in a). Most interesting, neither of these planes is highly loaded, both being in the range of 80-85 a.u., about 1/3 of the primary slip systems. Because of the sequences of micrographs at these strains, we are able to infer that slip system C in grain a nucleated at very low RSS in the same region as B [Fig. 8(a)], which was clearly primary (by a factor of about 2 over the next-most favored plane), presumably by a dislocation/geometric reaction rather than by stress activation at the continuum level. Then, as straining proceeded, the C bands extended to the a-c grain boundary (Fig. 8(b)], presumably driven by the additional dislocations generated in conjunction with continued slip on the B system in a. At the largest strain [Fig. 8(c)], the C system in grain a has apparently activated the {110} slip (D) in grain e, which is clearly registered across the boundary.

Other selected observations

Figure 10(a-c), respectively, show typical micrographs of slip systems nucleated at existing slip bands in specimen C2, at regions away from grain boundaries, and at grain boundary regions. The grain boundary region shows some local slip systems along with ones that continue large distances away from the boundary. Limited cross slip or jogs caused by the intersection of other slip bands are also apparent. Similarly, E3 is a bicrystal with a grain boundary across the specimen. FEM simulation showed low RSS at the grain boundary, and the slip trace density is consistently low at the grain boundary. Figure 1 l(a,b) show the clear registry of some slip traces (a) and the unregistered appearance of others

C2a r-.os2 L -.993

C2b F..062 L-.766

..SOl .407

.105

/..139 I_ .383

.866

Yr-.766

Yr -.276

L.lO7 ] .635

L-.9°6 ] .330

C2a

C2b ~ 2/,/j~ ..-

~-/.,~

D(101) .-

ec

X

(a)

GB

(b)

(e)

Fig. I0. Micrographs taken from specimen C2 after 6% strain" (a) at grain boundary a--b(100X); (b) away from grain boundary in grain b (100X); (¢) at slip bands in grain b (100X),

463

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

Specimen C3

x

C3d

[ -.301X

1- . . 3 6 7 l ~ - " ' D " Z 1- .910 1

..819/1 /..454 ~

~

'- .367 - '

C3b Z .954 l

i

.21o 1 -.547 I -.818 ..I

.267 YI-.o19 1 1-.906 l t...432

C3b

C3d

/

..

°°" • °°"

%°

, °°°°-°°"° ,~o°°'°

J GB GB

Fig. I 1. Micrographs taken from specimen C3 after 6% strain, at two points along grain boundary b-d (IOOX).

at a different part of the same boundary, k-d in specimen C3. Presumably the difference is caused by the boundary orientation itself and its effect on a pass-through mechanism, rather than on crystal misorientation (which is the same) or the continuum stress (which is nearly the same at the two sites). Figure 12(a-c) are micrographs from three regions of the a--b grain boundary in specimen F. The appearance of rather dense slip traces corresponding to system C in a region where the traces parallel the

boundary [Fig. 12(b)] was common to many specimens. Further along the boundary, Fig. 12(a, c), the boundary plane is no longer parallel to the slip traces and slip traces for C in a are very sparse [Fig. 12(c)] or are absent altogether [Fig. 12(a)]. The schematic in Fig. 12 shows the same result for system C in e near the f--e grain boundary. The two specimens having long grain boundaries roughly parallel to the tensile axis, D2 and El, are shown in Figs 13 and 14. The schematic view of the slip activity [parts (a)] are remarkably

464

YAO and WAGONER:

ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

Specimen

(a)

Ff

X~

"'==.s], _ r .122

Z[.4231

Fd

L.o.=z].°°,

YF '899 1

x Fe

L.515]/ .824 _ _

X Fb [:=l,t Y

] -.547J t..833

1.391 ] L.009

Yf .760 317 ] .562

/

-.o=.

/

L-.627J

r.3911

/.54Sl L.731J~ Y r..6Oll

Z r-.052 "1

I .'623 Y pL4o~1 12.8831

/ .751/

t-l.836-J

/-.238/

Ff

,..,

.634 r.,=ol

x Fa Zr'921

zi..552. I

.537

Yr .0431

[-.339 J L,.281 J

r-.2251 ~

F

Fd

i

F¢

:.

\

" \

-

d/l/4

Fig. 12. (a) Caption on facing page.

similar, showing nearly homogeneous distribution of slip activity throughout the specimen. Some slip systems remain fairly local to the boundary while others cross the specimen. The parts (b) of Figs 13 and 14 show a very uniform distribution of RSS throughout the specimen (essentially contour # 10 away from the narrow end regions). These specimens exhibited rather low initial flow stresses with high work hardening. CONCLUSIONS AND SUMMARY Thirteen tensile specimens comprised of 32 pure aluminum grains were deformed in uniaxial tension

to strains between 1 and 8%. Eighty-eight slip planes were observed by optical microscopy and anisotropic elastic finite-element modeling was used to compute the resolved shear stress (in arbitrary units suitable for quantitative, relative rankings) acting on the preferred slip system on the observed planes. The following generalizations are suggested by this body of observations. 1. Slip on 84 {111} planes and on 4 {110} planes was observed, ignoring slip systems in the gripped regions. 2. The slip on {110} planes was always connected with near by {111} slip, either registered

YAO and WAGONER:

ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

465

(b)

'oqi

GB

Fig. 12. Micrograph taken from specimen F at three points alon'g grain boundary aq}, after 4% strain.

across grain boundaries or near intense primary slip bands. 3. The { 111 } slip systems were classified by their probable sites of origin as follows, although the information available systematically undercounts grain-center nucleation, perhaps by a factor of two or more, because of subsequent intersections with other features (grain boundaries, specimen edges) prior to observation: • • • •

Boundary nucleation (total) Edge nucleation Center nucleation Edge/boundary/center indistinguishable

43% 24% 13% 20%

4. The distribution of resolved shear stresses (RSS) acting on each slip system, expressed in arbitrary units (a.u.), was markedly different according to probable site of origin: • Boundary nucleation: broad distribution of RSS between 50--400 a.u., with peak at 225 a.u. Peak about three times the background level. • Edge nucleation: single peak centered at 325 a.u. with a width of + 50 a.u.

• Center nucleation: two distinct peaks at 75 and 225 a.u. with individual widths of approximately _ 30--45 a.u. • Slip systems clearly registered across a boundary [3]: average of 70 _ 60 a.u. • {110} slip systems [4]: average of 120 + 100 a.u. 5. The RSS distribution f o r each type of slip system led to a conjecture of three basic kinds of dislocation nucleation mechanisms: • Stress-driven nucleation--occurring at specimen edges or grain boundaries but extending for specimen-scale distances, the RSS for these systems is ~200-400 a.u., with a clear peak at 325 a.u. • Stress/geometry driven nucleation--fairly highly stressed but assisted locally by stress concentrations, these systems usually do not extend for specimen-scale distances, at least before a great deal of work hardening. These systems have RSS's around 225 a.u. Peaks were seen among the boundary and center systems (presumably initiated at bands or at particles), including the lower peak for the "edge/boundary" sets.

466

YAO and WAGONER: ACTIVE SLIP IN ALUMINUM MULTICRYSTALS Specimen D2

--X~----t~Zr.4071 •623e~"1 I -.699/ D2b

r.4

L.061 J

L .301-1

"Y I--.296 1

x ~-~z

1"242] t...966

.052

"Y

/"°6'/ L .095 d

.891 "1 .462 J

I-.°61-1

.459 L-.244 J

(a)

(b) 789

987

II I\ 15 C o n t o u r range:

1 - 2 1 9 a.u., 10 - 3 6 0 a.u..

Fig. 13. Experimental and FEM simulation results for specimen D2: (a) schematic plot of observed slip activity (with crystal orientations shown); (b) contour plot of maximum resolved shear stress throughout the specimen (note: the maximum RSS refers to different slip systems at different sites).

• Dislocation-mechanism driven--can occur at very low continuum stresses. These systems were seen at grain boundaries (essentially all of the registered systems and a small peak among "boundary" systems classified as not clearly registered), and at grain interiors (especially terminating at other slip bands). These systems are usually quite local, presumably because the continuum RSS is insufficient for long-range motion. All of the {110} systems were probably generated in this way. 6. The elastic compatibility stresses at grain boundaries were computed with fairly coarse FEM meshes and in some cases were checked with finer meshes. In a few cases, where primary and secondary slip systems in the bulk had close RSS's, there was a change of ranking that agreed with the appearance of slip at the boundary. In most cases, the order remained unchanged. This result is different from the analysis of F E - 3 % S i bicrystals, where extensive reordering slip was predicted near finely-meshed grain boundaries at the intersection of specimen surface and grain boundary [90].

7. In general, regions of highest RSS showed the first slip and, later, the most widespread slip. At higher strains, the slip traces tended to concentrate in grain boundary regions. 8. No general correlation could be found between the stress-strain curves and the grain structures of the various specimens. However, comparison of similar pairs of tensile specimens (with simple differences in grain structure) showed that the presence of internal grains promotes rapid work hardening while long grain boundaries roughly parallel to the tensile axis promote low flow stresses. A qualitative analysis of simple orientation-dependent flow based on singlecrystal experiments suggests that the macroscopic mechanical response of aluminum polycrystals is dominated by texture rather than grain boundary configuration. 9. High slip trace densities were observed at several locations where the slip plane traces were nearly parallel to a nearby grain boundary. As the grain boundary orientation changed, fewer slip traces were observed. This result tends to confirm the obstacle role of grain boundaries often mentioned in the literature.

YAO and WAGONER:

ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

467

Specimen E 1 Ela

Elb

x

t

.743

.648 .980

-.629

-.147

-.182

(a)

(b) 9

8

1~

Nab

~,~-'~

~,~

7

~

Contour range: 1 - 173 a.u.,10 - 340 a.u.. Fig. 14. Experimental and F E M simulation results for specimen El: (a) schematic plot of observed slip activity (with crystal orientations shown); (b) contour plot of m a x i m u m resolved shear stress throughout the specimen (note: the m a x i m u m RSS refers to different slip systems at different sites).

Acknowledgements--The authors would like to thank John

P. Hirth for many helpful discussions. This work was supported by the National Science Foundation (DMR8614338) and the Ohio Supercomputer Center (PAS 080). The Aluminum Company of America provided the highpurity aluminum used for all experiments.

REFERENCES

1. J. D. Livingston and B. Chalmers, Acta metall. 5, 322 (1957). 2. R. Clark and B. Chalmers, Acta metall. 2, 80 (1954). 3. J. J. Hauser and B. Chalmers, Acta metall. 9, 802 (1961). 4. J. C. Suits and B. Chalmers, Acta metall. 9, 854 (1961). 5. P. J. Worthington and E. Smith, Acta metall. 12, 1277 (1964). 6. P. J. Worthington and E. Smith, Acta metall. 14, 35 (1966). 7. J. C. Swearengen and R. Taggart, Acta metalL 19, 543 (1971). 8. W. E. Carrington and D. McLean, Acta metall. 13, 493 (1965). 9. E. O. Hall, Proc. Phys. Soc. Lond. 64, 747 (1951). 10. N. J. Petch, J. lron Steel Inst. 174, 25 (1953). 11. U. F. Kocks, Metall. Trans. 1, 1121 (1970). 12. J. D. Eshelby, F. C. Frank and F. R. N. Nabarro, Phil. Mag. 42, 351 (1951).

13. A. N. J. Petch and R. W. Armstrong, Metall. Trans. 22A, 2695 (1991). 14. Z. Yao, Ph.D thesis, The Ohio State Univ. (1990). 15. G. I. Taylor, J. Inst. Metals 62, 307 (1938). 16. J. F. W. Bishop and R. Hill, Phil. Mag. 42, 1298 (1951). 17. R. Hill, Proc. Phys. Soc. Lond. 9, 351 (1952). 18. G. Sachs, Z. desV. D. I. 72, 734 (1928). 19. U. F. Kocks and G. I. Canova and J. J. Jonas, Acta metall. 31, 1243 (1983). 20. P. H. Lequeu and J. J. Jonas, Metall. Trans. 19A, 105 (1988). 21. F. Montheillet, P. Gilormini and J. J. Jonas, Acta metall. 33, 705 (1985). 22. B. Budianski and T. T. Wu, Proc. 4th Congr. Appl. Mech., p. 1175 (1962). 23. D. Daniel and J. J. Jones, Metall. Trans. 21A, 331 (1990). 24. P. Van Houtte, 7th Int. Conf. on Textures of Materials, Netherlands Society for Materials Science, Zwijndrecht, Belgium, pp. 7-23 (1984). 25. G. Kapil, K. Mathur and Paul R. Dawson, Int. J. Plasticity 5, 67 (1989). 26. T. A. Bamford, W. A. T. Clark and R. H. Wagoner, unpublished research. 27. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. 31, 1951 (1983). 28. R. J. Asaro and A. Needleman, Acta metall. 33, 923 (1985).

468

YAO and WAGONER:

ACTIVE SLIP IN ALUMINUM MULTICRYSTALS

29. P. R. Dawson, Int. J. Solids Struct. 23, 947 (1987). 30. P. R. Dawson, Int. J. Plasticity 5, 67 (1989). 31. D. C. Crites, R. K. Raghvan and B. L. Adams, MetalL Trans. 18A, 255 (1987). 32. E. Schmid, Proc. Int. Congr. Appl. Mech., p. 342 (1924). 33. P. Rosbaud and E. Schmid, Z. Phys. 32, 197 (1925). 34. W. Boas and E. Schmid, Trans. Am. Inst. Min. Engrs 61, 767 (1930). 35. E. Schrnid and G. Siebel, Z. Electrochem. 37, 447 (1931). 36. R. Karnop and G. Sachs, Trans. Am. Inst. Min. Engrs 49, 480 (1928). 37. M. Masima and G. Sachs, Z. Phys. 50, 161 (1928). 38. G. Sachs and J. Weerts, Z. Phys. 62, 473 (1930). 39. E. Osswald, Z. Phys. 83, 55 (1933). 40. J. Diehl, Z. Phys. 47, 331 (1956). 41. E. N. daC. Andrade and D. A. Aboav, Proc. R. Soc. A 240, 304 (1957). 42. P. Haasen, Phil. Mag. 3, 384 (1958). 43. T. E. Mitchell and P. R. Thornton, Phil. Mag. 8, 1127 (1963). 44. S. Mader, Z. Phys. 149, 73 (1957). 45. C.J. Bolton and G. Taylor, Phil. Mag. 26, 1359 (1972). 46. F. R. Nabarro, Advances in Physics, No. 50, p. 193 (1964). 47. E. Schmid and W. Boas, Kristallplastizitat, Fig. 93 (1935). 48. R. F. Miller and W. E. Milligan, Trans. Am. Inst. Min. Engrs 124, 229 (1937). 49. E. N. DAC. Andrade and C. Henderson, Phil. Trans. A 244, 177 (1951). 50. K. M. Carlsen and R. W. K. Honeycombe, J. Inst. Metals 83, 449 (1955). 51. J. Friedel, Phil. Mag. 46, 1159 (1955). 52. A. Seeger, J. Diehl, S. Mader and H. Rebstock, Phil. Mag. 2, 323 (1957). 53. Y. Ohtani and J. Takamura, J. Jap. Inst. Metals 43, 348 (1979). 54. F. Inoko, M. Kurimoto and T. Nagatomo, J. Jap. lnst. Metals 53, 281 (1989). 55. P. R. Thornton and P. B. Hirsch, Phil. Mag. 3, 738 (1958). 56. M. G. Whelan, Proc. R. Soc. A 249, 114 (1958). 57. P. B. Hirth, P. G. Patridge and R. L. Segall, Phil. Mag. 4, 721 (1959). 58. R. E. Hook and J. P. Hirth, Acta metall. 15, 535 (1967). 59. R. E. Hook and J. P. Hirth, Acta metalL 15, 1099 (1967). 60. R. E. Hook and J. P. Hirth, Trans. Jap. Inst. Metals 9, Suppl., 778 (1968). 61. J. P. Hirth, Phil. Mag. 40, 39 (1979). 62. Z. Shen, R. H. Wagoner and W. A. T. Clark, Acta metalL 36, 3231 (1988). 63. J. Gemperlova, V. Paidar and F. Kroupa, Czech. J. Phys. !139, 427 (1989). 64. P. Sittner and V. Paidar, Acta metall. 37, 1717 (1989). 65. C. Elbaum, Trans. Am. Inst. Min. Engrs 218, 455 (1960). 66. K. T. Aust and N. K. Chen, Acta metall. 2, 632 (1954). 67. R. L. Fleischer and W. A. Backofen, Trans. Metal 218, 243 (1960). 68. D. McLean, Conf.. on Relation o f Properties of Structure. National Physical Laboratory, London (1963). 69. H. Margolin and R. Longo, Acta metalL 23, 67 (1975). 70. S. Miura and Y. Saeki, Acta metall. 26, 93 (1978). 71. D. Vesely, Scripta metalL 6, 753 (1972). 72. R. S. Davis, R. L. Fleischer, J. D. Livingston and B. Chalmers, J. Metals 136, (1957).

73. C. Elbaum, Trans. Am. Inst. Min. Engrs 218, 444 (1960). 74. Yii-der Chuang and H. Margolin, MetalL Trans. 4, 1905 (1973). 75. T. D. Lee and H. Margolin, MetalL Trans. 8A, 157 (1977). 76. K. Hashimoto and H. Margolin, Acta metall. 31, 787 (1983). 77. J. C. M. Li, Trans. Am. Inst. Min. Engrs 227, 239 (1963). 78. M. Schankula, J. D. Embury and D. J. Lloyd, Proc. 2nd. Int. Conf. on Strength o f Metals and Alloys, p. 209. Am. Soc. Metals, Metals Park, Ohio (1970). 79. H. Yada and H. Mimura, J. Phys. Soc. Japan 24, 1269 (1968). 80. C. S. Lee and H. Margolin, Metall. Trans. 17A, 451 (1986). 81. S. Miura and K. Hamashima, J. Soc. Mater. Sci. Japan 27, 1146 (1978). 82. S. Miura and K. Hamashima and K. T. Aust, Acta metalL 28, 1591 (1980). 83. J. R. Fekete, J. E. Talia and R. Gibala, Dislocations in Solids, p. 651. Univ. of Tokyo Press (1985). 84. H. Miyamoto and M. Kikuchi, J. Fac. Eng. Univ. Tokyo Set. B 33, 463 (1976). 85. H. Miyamoto, K. Funami and M. Uehara, J. Fac. Eng. Univ. Tokyo Ser. B 34, 349 (1977). 86. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. 30, 1087 (1982). 87. X. Hu and H. Margolin, MetalL Trans. 21A, 3075 (1990). 88. K. Kitagawa, H. Asada, R. Monzen and M. Kikuchi, Trans. Jap. lnst Metals 27, Suppl., 827 (1986). 89. T. Ohashi, H. Hanzawa and M. Kishida, J. Jap. Inst. Metals 44, 876 (1980). 90. R. H. Wagoner, Z. C. Yao and Q. Wu, Advances in Constitutive Laws for Engineering Materials, Vol. I (edited by Fan Jinghong and Sumio Murakami), p. 955. International Academic Publisher, P. R. China (1989). 91. C. Wei, R.-G. Qing, S. Lin and J.-M. Xiao, Acta mech. solida sin. 2, 375 (1989). 92. H. Miyamoto, M. Shiratori, T. Miyoshi and M. Oto, Bull. J S M E 14, 893 (1971). 93. K. Hashimoto and H. Margolin, Acta metall. 31, 773 (1983). 94. S. Nagashima, M. Shiratori, K. Matsukawa and R. Nakagawa, Trans. Jap. Inst. Metals 27, Suppl., 883 (1986). 95. T. Abe, S. Nagaki and M. Furuno, Bull. J S M E 29, 1111 (1986). 96. T. Ohashi, Trans. Jap. Inst. Metals 27, Suppl., 835 (1986). 97. T. Ohashi, Trans. Jap. Inst. Metals 28, Suppl. 906 (1987). 98. C. S. Lee and H. Margolin, Metall. Trans. 17A, 26 (1986). 99. A. W. Ruff Jr, MetalL Trans. 1, 2391 (1970). 100. L. E. Murr, International Phenomena in Metals and Alloys, pp. 145-147. Addison-Wesley, Reading, Mass. (1975). 101. J. F. Devlin, J. Phys. F: Metals Phys. 11, 2497 (1981). 102. S. Hashimoto, T. K. Fujii and S. Miura, Trans. lnst. Metals 27, Suppl., 921 (1986). 103. E. P. Kvam and R. W. Balluffi, Scripta metall. 21, 1573 (1987). 104. S. Miura, Trans. Jap. Inst. Metals 27, 689 (1986). 105. Hibbit, Karlsson & Sorensen, Inc., 35 South Angell Street, Providence, R.I. 02906, U.S.A. 106. J. P. Hirth and J. Lothe, Theory o f Dislocations, 2nd edn, p. 287. Wiley-Interscience, New York.