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High-strain dislocation patterning, texture formation and shear banding of wavy glide materials in the LEDS theory
ScriptaMaterialia,
Vol. 36, No. 2, pp. 173-181, 1997 Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in the USA. All rights reserved 13594462197 $17.00 + .OO
Pergamon
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HIGH-STRAIN DISLOCATION PATTERNING, TEXTURE FORMATION AND SHEAR BANDING OF WAVY GLIDE MATERIALS IN THE LEDS THEORY Doris Kuhlmann-Wilsdorf Department of Materials Science and Engineering University of Virginia, Charlottesville, VA 22901, USA (Received April 23, 1996) (Accepted July 8, 1996) Abstract
In the course of plastic deformation, dislocations are trapped into positions of mutual stress-screening, i.e. Low-Energy Dislocation structures (LEDS’s), in which their energy per unit line length is UD z [Gb*( l-v/2)/4x (I-v)]hr(RL&b) with RLE~Sthe near-neighbor dislocation distance. As a class, the most common LEDS’s are dislocation rotation boundaries separating misoriented volume elements. They form the so widely observed mosaic block alias dislocation cell structures of “wavy glide” materials. However, because the glide paths of dislocations terminate where they are trapped, dislocation rotation walls are the locations of abrupt strain gradients whose associated stresses increase with their misorientation angle als well as angle of inclination against the averaged shearing direction, i.e. the slip lines in the sense of continuum plasticity theory. With increasing misorientation angle, therefore, the dislocation boundaries rotate towards the macroscopic slip planes, e.g. in sheet rolling initially to be inclined about f40’ to the rolling plane (10). They delineate irregular slabs which eventually may become homologous with the macroscopically imposed strain (5), i.e. in rolling nearly parallel to the rolling plane, in torsion normal to the torsion axis. As the 2nd law of thermodynamics favors the lowest possible Ua,, and thus the lowest possible R Lack and largest rotation angle, there is no limit to the dislocation density in the walls, but new low-angle walls, seen as cell walls roughly normal to the high angle walls, are continuously formed through bending strains in the slabs (8). On account of their relatively long link lengths, the fresh cell walls constantly provide an unlimited supply of glide dislocations at a relatively small flow stress, assuring a low workhardening rate. However, as soon as they are formed, also the new cell walls trap dislocations, thereby increasing their rotation angle, and begin to rotate towards parallellity with the slip lines. In the process the average thickness, D, of the slabs progressively diminishes in line with the empirical formula D = KGb/(T-2,). Meanwhile the higherangle walls channel the glide in the slabs between them which is the cause of metallographic slip bands and can lead to shear-banding. Thus latently shear-banding is always present without any mechanical instability, The lmutually misoriented lattice orientations in the slabs comprise the texture components. Since the rotation boundaries are overwhelmingly composed of trapped glide dislocations, their morphology, and thus the resulting textures, depend on availability of slip systems, three-dimensional 173
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dislocation mobility and initial crystal orientation. It is presumed, but still requires proof, that on account of the 2ndlaw the actually realized textures embody the largest possible misorientation angles compatible with the boundary conditions. Introduction Following the considerable advances of the LEDS theory during the past seven years, the present paper shall fill in the last obvious remaining gap, the question as to the morphology of the high-strain dislocation patterns and resulting texture formation. Specifically, 0 the LEDS theory was comprehensively surveyed up to the status of 1989 including the role of Taylor lattices (1); 0 subsequently worksoftening and Hall-Petch hardening were explained (2-4); 0 the dislocation patterning underlying workhardening stages I to IV (and tentatively V) in both “planar” and “wavy” (or “distributed”) glide were elucidated (5); 0 the difference between dislocation behavior in “planar” and “distributed” glide was traced to differences in dislocation/vacancy interactions (5); 0 the shape of typical workhardening curves as consisting of piece-wise joined Vote curves was documented and shown to arise from the theoretically expected linear decrease of the dislocation retention parameter, l3, with rising effective resolved shear stress, 2 -7, (4); 0 the nature of dislocation walls as LEDS dislocation rotation boundaries was established without any reasonable remaining doubt (6,7); 0 and the important role of straingradients in the continual formation of new cell walls, of low angular misorientation and therefore controlling the flow stress, was discovered (8). The remaining task, then, is to consider dislocation morphology and texture formation in light of the LEDS theory. As this will be seen to depend on the inclusion of longer-range internal strain energy, while internal stresses have been previously largely disregarded except in (8), some basic considerations shall be presented first. Basic Considerations The LEDS theory is firmly founded on Newton’s third law, i.e. action equals reaction, and the second law of thermodynamics, i.e. maximum conversion of mechanical work into heat or, equivalently, minimization of stored energy. Fig. 1 exemplifies the approach. It envisages a simple tensile test in an isolated system via the step-wise, automated addition of weights to the load on a sample. While the applied load remains in the elastic range, force equilibrium is maintained by the elastic increase of the atomic spacings in the sample parallel to the axis. In addition, as soon as the applied stress exceeds T,, the resolved friction shear stress, the dislocations in the sample will bow out quasielastically or otherwise move lightly in local energy troughs. Thereby the corresponding anelastic complement is added to the strain, which manifests as a seemingly lowered elastic modulus (sect. 18 of (1)). Up to the FrankRead stress or bypassing among neighboring dislocations, the equilibrium is reversible both for the elastic lattice strain and the dislocations (except for a minor part due to 23, and the 2nd law of thermodynamics needs not to be invoked. The situation changes dramatically as soon as the yield stress is exceeded. Now, in addition to the elastic strain in proportion with the applied stress, irreversible plastic deformation takes place by the super-critical bowing of dislocations and/or bypassing of dislocations in dipolar configurations. Thus irreversible work is done by the load while the sample lengthens plastically. That work is overwhelmingly transformed into heat but, while workhardening progresses, a minor percentage of the expended work is stored, predominantly in the form of additional dislocations which have been trapped in the material. Their density rises in accordance with the empirical relationship 7 = F, +
crGb+
(I)
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Figure 1. Thought experiment of tensile testing in a bell jar via step-wise load application. While the stored energy in the sample rises, the free energy of the system as a whole, composed of sample and applied traction, continuously decreases as mechanical work done by the traction (i.e. the load P) is mostly converted into heat. Meanwhile the dislocation density rises so as to maintain force equilibrium in accordance with eq. 1. Following the LEDS hypothesis the free energy of the system as a whole is minimized through minimizing the stored energy in the sample.
the shear stress, G the shear modulus, b the Burgers vector, p the dislocation density anda - 0.4 within a factor of 32. Evidently, then, workhardening is due to an increasing dislocation density. At this point the 2nd law is invoked in two ways. 0 Firstly via the “LEDS Princinle”, saying that, at least in stages II, III and IV, dislocations are trapped into positions of local eneruy minima in which dislocations mutually screen their stresses. The resulting configurations are, by definition, LEDS’s &ow-Energy Qislocation &uctures) characterized by an average dislocation line energy of
with z
Un = [Gb*(1-v/2)/4~( 1-v)Jln(&&b)
(2)
with v Poisson’s ratio and RLEDsthe average near-neighbor dislocation distance. 0 Secondly, via& I, S I-IvDothq&” e.g. (4) and (7). According to it, among all possible LEDS’s which are in force equilibrium with the tractions and are accessible under the constraints of dislocation mobility and available slip systems, the one which is actually realized most nearly minimizes the energ stored in the sample. In fact, while the import of the LEDS hypothesis is clear enough and it dates back to 1968 (rlef. 9, see footnote on p. 227 of ref. 7), its precise phrasing is not evident. The form given in (7), namely requiring minimization of the energy per unit length of dislocation line, i.e. minimization of RLEnsof eq. 2 wherein all forms of internal energy are included in Un , is compatible also with interfacial ‘dislocation formation in epitaxy and phase transformations, as it ought to be. But this form may be flawed when strain-mdient-relief dislocations are included (8). If this is remedied by restricting the LEDS hypothesis to && dislocations, the LEDS hypothesis is no longer applicable to interfacial dislocations. It seems therefore that the above formulation given in bold letters is the best possible. At any rate, the writer is not aware of any violations of either the LEDS principle or the LEDS hypothesis.
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Non-LEDS Theories of Workhardening, Dynamic Effects and “Forest” Dislocations Dislocation patterning, mostly in the form of dislocation segregation, or “bunching”, into regions of high density leaving most of the volume almost dislocation-free, e.g. as in stage I “braids”, stage II “carpets”, the mosaic block structures of stages III and IV, and in fatigue in the form of “mazes” and “ladder structures”, is very wide-spread indeed. Given adequate three-dimensional dislocation mobility as in “wavy glide” materials (5) and based on the 2nd law, this is a direct consequence of eq. 2. Namely, beginning with a hypothetical random distribution, any statistical fluctuation of dislocation density so as to locally reduce dislocation spacing through mutual dislocation stress-screening at the expense of dislocation rarefaction in adjacent areas, will lower the stored energy. LEDS’s will also be formed if three-dimensional dislocation mobility is low, but not necessarily characterized by bunching. In that case Taylor lattices may arise instead (5). They are commonly mistaken for “random” dislocation “tangles” but in fact are LEDS’s with well-defined albeit still poorly understood structures (1, 10). It is their hallmark that any representative local group of dislocations has zero net Burgers vector, so that a Burgers circuit about such would be closed. However, in wavy glide and uni-directional deformation, wherein dislocations of opposite sign are persistently driven in opposite directions, net Burgers vector surpluses accumulate in representative local groups. In that case, and once more than one Burgers vector is locally available, the LEDS’s will be dislocation rotation boundaries, forming in a wide variety of mosaic block structures of misoriented volumes (11). The discussed dislocation bunching can also be attributed to various forms of mutual repulsive dislocation interactions, as is done in all non-LEDS theories of workhardening. However, mutual obstruction of dislocation movement can only be caused by local mutually repulsive stresses. These would necessarily lead to increased stored energy, in conflict with the second law and, hence, the LEDS hypothesis. They may be ruled out for that reason, apart from the mundane fact that all observed dislocation structures appear to be LEDS’s, as already indicated. For the same reason one may also rule out computer models which assume conditions far from equilibrium, following Prigogine’s thermodynamics for “energy flow-through” systems. Finally, “rer>ulsive” which have also been often invoked to account for at least part of workhardening can nlav no simriticant role vg of “wavy glide” materials. This is for the reason that the interactive stresses among approaching dislocations ready to intersect, thereby momentarily forming a four-fold node if an intersection jog was to be created, rotate them to be locally parallel, always such as to yield an “attractive” reaction. The result of that reaction is a dislocation link between two three-fold nodes, as if forming a new link in a network, always accompanied by a lowering of stored energy (12, compare Fig. 17 of 11). Short of explosive forming and as long as climb and/or recovery at ordinary rates are negligible, strain rate changes have almost no effect on workhardening and only little on the momentary flow with or with r t . stress (1, 13). ThustheI,S ED annlied stress The LEDS theory, then, is in the lineage of G. I. Taylor’s early theory of workhardening (14) and in stark contrast to Becker’s theory of dynamic, strain-rate controlled deformation (15, 16). Although long discarded as such and now-a-days virtually unknown to researchers, Becker’s theory has been perpetuated principally via the work of Orowan and Haasen. It rests on the basic assumption that, not only in creep but also in ordinary low-temperature plastic deformation, the flow stress is controlled through a competition between hardening and recovery events (compare ref. 17). Correspondingly, the concept of “dynamic recovery” , i.e. the presumed greatly enhanced speed of recovery during straining which is derived from Becker’s theory, has no room in the LEDS theory. Nor is there room for the concept of “forest” dislocations of ill-defined origin which somehow obstruct glide dislocation motion. In fact the presumed “forest” dislocations are due to dislocation reactions as already introduced and
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simply participate in the formation of LEDS’s, subject to the LEDS principle and the LEDS hypothesis. The Approximate Magnitude and Elastic Energy of Internal Non-Dislocation Stresses While it had been recognized from the outset that long-range stresses could not be the cause of workhardening but that instead dislocations would be mutually stress-screened (18), it has never been doubted that longer-range (i.e. of a range comparable to the closest wall-spacings) and/or long-range stresses would always be present. Namely, structures which are composed of dislocations trapped in energy wells are bound to be stable, in glide, up to the flow stress magnitude that generated them. Unloading, load reversal and/or reloading below the flow stress correspondingly gives rise only to some minor dislocation rearrangements which, as already mentioned, manifest as anelasticity and simulate an apparently lowered elastic modulus, typically by the factor of about one half (1,17). Hence, since longer-range and/or long-range stresses between f (z - ~3 cannot be removed in glide, and since internal stresses must vanish on average, the internal stress magnitude is expected to be zi I (T - 2,)/2, which is in agreement with the measurements of Mughrabi and coworkers (19). Even so U, , the elastic energy density per unit dislocation line length of those stresses, is not negligible. Namely with eqs. 1 and 2, RLEDs= I,‘$, v = 0.3 and b = 3 x lP* cm U, = Gzi2/2= Gb* a’/8 3 UD(a’l[O.8 In&&b)]}
z 0.3 a* UD
(3)
The Role of Elastic Strain Gradient Energies in Determining the Dislocation Wall Spacing An obvious remaining problem is why dislocation walls are not more widely spaced than according to the empirically well established (20) relationship between flow stress and average cell diameter (and presumably wall spacing in stage IV), D, i.e. z-z,=KGb/D
(4)
with K = 10 but on occasion as low as 2.5 (8,20-22). Certainly, distributing the same dislocation density over fewer walls will decrease R LED~ and thereby decrease the dislocation line energy. Attempts to explain eq. 4 by means of long-range stress energy associated with terminating dislocation walls failed after an apparent initial success (11,23), as it has proven impossible to theoretically account for numerically satisfactory parameters. Only very recently has the answer to the puzzle of eq. 4 been found as follows: Unlike dislocation walls extending through whole crystals as in kinking (compare 24), inevitably longe:r-range stresses arise along more or less parallel dislocation rotation walls of finite length formed from trapped glide dislocations in accordance with the preceding considerations. They will thus create the corresponding elastic strain gradients within the slabs between the walls, most commonly causing elastic bending. If the maximum tensile/compressive stress in the outermost fibers of the bent slab defined by the parallel wall sections is c~,,,,and the slab has, say, a thickness of D and length of L, then the associated maximum elastic tensile/compressive strain will be F,,,= o,/E = DX/2L where X is the resulting elastic bending angle. Hence, with D/L = h, CT,,, = EX/2X
(5)
The associated. elastic strain energy is proportional to Q,,,‘, i.e. to X2. However, the bending could be alternatively generated by a tilt wall, of spacing
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h = b/(2 sinX/2) = b/X
(6)
and thus by an energy proportional to X In( l/X). As a consequence, the LEDS hypothesis requires that, beyond some critical value, elastic bendings (and similarly also elastic twists) will be partly relieved by the corresponding dislocation walls. The lower limit of beginning conversion of elastic strain gradients into new dislocation walls depends somewhat on geometry but it would seem that at maximum h I D/5. Moreover, for the bending to be completely converted into dislocation rotation walls would require an infinite elastic strain gradient. The details of the derivation are given in (8). According to it, minimum stored energy is obtained for Xmin= [3 In( l/X& - 1]/(3D/b) = const bD
(7)
Therefore, on the reasonable assumption that cr,,,compares with the flow stress so that (z - 2, ) = Gb/h = GX, eq. 4 follows. Ref. (8) documents that this simple theory is in full accord with the measurements of Langford and Cohen (21) as interpreted by means of the mesh-length theory (22), and thus is in full accord with the LEDS theory which has developed out of it (l), wd that the momentary flow stress is controlled by the new walls. This proviso does not only follow from the data but is a necessary consequence of the theory. This is so because the theory identifies the flow stress with the Frank-Read super-critical bowing stress of always the longest most frequent links in the structure (whence incidentally also follows the mild low-temperature strain rate dependence of the flow stress (13)). In summary, then, the spacing between parallel walls is limited by the continual creation of new cell walls which relieve unavoidable strain gradients in accord with eq. 4. At any moment the newest walls control the flow stress via their low rotation angles and thus long link lengths. Meanwhile, high-angle rotation walls can be eliminated through merging when the specimen diameter shrinks faster with stress than D, as is the case in wire drawing (2 1) and probably also in sheet rolling. Kinematics of the Dislocation Structure Evolution in Uni-Directional Straining of “Wavy-Glide” Materials The evolution of the dislocation structure during urn-directional, low-temperature straining of “wavy” (or “distributed”) glide materials is rather well understood. It progresses from “braids” in stage I, to “carpets” in stage II, to roughly equiaxed dislocation cell structures with small angles of misorientation in stage III, and to roughly parallel dislocation boundaries which are increasingly homologous with the imposed strain in stage IV. The LEDS transformations separating these stages are in line with the LEDS hypothesis in that Un , the dislocation line energy disregarding non-dislocation stresses, decreases more or less discontinuously from stage to stage, while within any one stage Un decreases continuously. Further, the shape of the stress-strain curves conforms to the theoretically expected linear decline with stress of fi, the fraction of glide dislocations which are trapped in the dislocation structure at the end of their paths. In regard to these facts the reader is referred to the pertinent papers (1,4,5,7,11,13,25). Taking, then, up at the end of the preceding section, consider that, clearly, “new” dislocation rotation boundaries cannot stay “new” for long. They capture dislocations and thereby increase their angles and as a result rotate towards the orientation of the slab plane so as to propagate and refine the preexisting structure. The reason for this wall alignment with increasing misorientation angles are the strain gradients which are associated with cell walls formed from trapped glide dislocations (7,26). These impart the corresponding elastic strain energy over and above that of the self-stresses of the
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constituting dislocations which the cell would have in the thoroughly annealed state, e.g. as computed in (27) and in :many much more recent computer simulations of dislocation arrays. Those previously neglected strain gradients result from the fact that the shear strain imparted by the glide dislocations along their path ends where they are trapped, i.e. in the boundaries (26). For example, a tilt wall of dislocation spacing h and formed of trapped glide dislocations is associated with the abrupt shear strain change Ay, = bs/h= 2sin(@/2), with Q,the wall misorientation angle, and this independent of the height of the wall or how long was the free path of the glide dislocations before meeting the boundary. In a slab between high-angle walls which is subdivided by tilt walls on account of bending strain relief as explained above, this Aywgives rise to the corresponding elastic stress component parallel to A’ywof magnitude A%+= G 2sin(0/2). By contrast, seeing that glide parallel to dislocation rotation walls is always possible through minor dislocation rearrangements, a set of dislocation walls parallel to the macroscopic “slip lines” in the sense of continuum theory, is free of such strain gradients. As a result the LEDS hypothesis predicts increasing such alignment of dislocation boundaries as their rotation angles increase (compare (5)). The above goes far to answer the question of why there w to be two families of dislocation rotation walls, previously named (28) “incidental” (ordinary dislocation cell boundaries, ID’s), and “cell block boundaries” (CBB’s), with variable misorientations up to being indistinguishable from grain boundaries (7,lO). In fact, as seen there is a continual creation of new cell walls almost normal to approximately parallel higher-angle walls. The new walls rotate towards the common plane of the earlier, stronger walls and in the process replenish the bending (and/or torsional) strain gradients which cause the formation of new low-angle ID walls. It is only too natural for observers to interpret such a continuum, ranging from long, more or less parallel high-angle walls to low-angle cell.boundaries normal thereto, in terms of two populations (29), compare (7). However, careful measurements as in (30) and (3 l), do not show a bi-modal distribution of angles. Rather if the corresponding histograms of presumed ID’s ;and CBB’s are combined, a smooth distribution is found. Similarly, TEM micrographs, e.g. Fig. 2., show a continuum of wall rotation angles. Even so, the initial idea that there should be two distinct families of walls (28,32) is almost certainly correct, wherein CBB’s delineate “cell blocks” with different selections of active slip systems, falling short of the five required by the Taylor criterion. In that interpretation, the CBB’s (but ti the ID’s) are “geometrically necessary” in accommodating the angular difference in lattice orientation due to slip on different slip system selections. However, it is now tentatively concluded that CBB’s as defined, are microscopically observed Q& as “dense dislocation wal1.s”(DDW’s, 32) of long extent, whereas the great majority of higher- and high-angle boundaries which in past publications (including refs. 1,7,8,26,28,30-32 and Fig. 2, for example) were believed to be C:BB’s, are in fact matured ID’s.
........
(3
5
4”
- - - - - IO” < e I 18”
-***-- 4”
ND 4
0.5km -
Figure 2. TEM micrograph of 98% cold-rolled Ni (99.990/o),courtesy of Darcy A. Hughes (Fig. 2 of ref. 30). Section normal to the rolling plane whose normal is indicated as ND, while tke rolling direction is vertical, parallel to the average wall direction. Note the large spread of misorientations and the “bamboo structure” due to cell walls between the stronger boundaries.
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Shear Bands and Texture Formation After severe sheet rolling, the high-angle boundaries are roughly parallel to the rolling plane as shown in Fig.2, but, after mild to moderate rolling, i.e. in stage III, they are oriented - & 40” to the rolling plane (32). This is in line with the expected alignment of strong boundaries with the “slip lines” of continuum’s theory, derived in the previous section. And in this orientation the strong walls are bound to channel glide, which not only accounts for slip bands but for shear banding whenever the workhardening rate is critically low. Latently, shear banding is thus always present. However, the conditions under which it becomes a technological problem still require elucidation. At any rate, considering that polyslip is active throughout, neither the ID’s nor the “true” CBB’s need to follow crystallographic orientations. Thus true CBB’s can readily cross grain boundaries. Fig. 10 of (33) is believed to show this effect in the form of a roughly 20pm wide striping inclined about 40” to the rolling plane. The stripe pattern in Fig. 11 of (34) is probably also due to CBB’s, whereas H. Raabe’s own interpretation in terms of super-dislocations is untenable. That the walls represent boundaries not only of relative misorientations but also of discontinuous changes of plastic strain has been recently addressed also by Leffers (35) for rolling and by Sedl&ek (36) for fatigue. It is as yet not clear to what extent the preceding considerations may support or be in conflict with that work. At any rate, the stage IV reorientation of the boundaries more closely to the plane of rolling, e.g. as in Fig. 2 and typically in the form of a herringbone-type of pattern (e.g. Fig. 2 of (7) due to D. A. Hughes), presumably occurs because the free dislocation paths in steeply inclined cells become geometrically severely shortened. This, then, increases the flow stress and therewith zi, so that in accord with the LEDS hypothesis the boundaries rotate towards the rolling plane, thereby diminishing the flow stress and the internal stress energy. Collectively the ranges of lattice orientations in the slabs between the walls constitute textures, albeit in contrast to the Taylor model different components occur in close proximity within any one grain. Experimentally this phenomenon has most recently been brilliantly documented by Panchanadeeswaran, Doherty and Becker (33). As a result, stabilization of the lattice misorientations is inevitably slow and imprecise, which is perhaps the decisive reason for the well-known shortcomings of the Taylor model (33). Finally, the above LEDS interpretation of texturing is strongly supported also by the work of Smallman and Lee (37) who have studied texture evolution in rolled a-brass. By contrast to the wavy-glide materials discussed so far, a-brass deforms by “planar-glide”, exhibiting Taylor lattices, and contains dislocation rotation boundaries only at the domain boundaries (1). Even so, the well-known { 1lo}<1 12> a-brass rolling texture was also found as fine lamellae, namely in twinorientations separated by high-angle dislocation rotation boundaries (37). From this one may conclude with some modest degree of confidence that the LEDS hypothesis operates also in planar-glide materials, namely on the dislocations in the domain boundaries. In very much the same manner as already outlined for the wavy-glide case, this causes a sub-division of the domains and consequent proliferation of domain walls composed of closely spaced dislocations with small energy per unit length of dislocation line, eq. 2. The resulting texture is different from that in wavy glide as, constrained by the planarity of the glide within the Taylor lattices, the domain walls are much less free to reorient, and the lamellae in mutual twin orientation form a distinct texture. It is similarly evident that the texture here and in wavy glide must depend on strain geometry. Acknowledgment Many discussions with my husband, H. G. F. Wilsdorf, and his unwavering support are most gratefully acknowledged.
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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
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PII S1359-6462(96)00347-8
HIGH-STRAIN DISLOCATION PATTERNING, TEXTURE FORMATION AND SHEAR BANDING OF WAVY GLIDE MATERIALS IN THE LEDS THEORY Doris Kuhlmann-Wilsdorf Department of Materials Science and Engineering University of Virginia, Charlottesville, VA 22901, USA (Received April 23, 1996) (Accepted July 8, 1996) Abstract
In the course of plastic deformation, dislocations are trapped into positions of mutual stress-screening, i.e. Low-Energy Dislocation structures (LEDS’s), in which their energy per unit line length is UD z [Gb*( l-v/2)/4x (I-v)]hr(RL&b) with RLE~Sthe near-neighbor dislocation distance. As a class, the most common LEDS’s are dislocation rotation boundaries separating misoriented volume elements. They form the so widely observed mosaic block alias dislocation cell structures of “wavy glide” materials. However, because the glide paths of dislocations terminate where they are trapped, dislocation rotation walls are the locations of abrupt strain gradients whose associated stresses increase with their misorientation angle als well as angle of inclination against the averaged shearing direction, i.e. the slip lines in the sense of continuum plasticity theory. With increasing misorientation angle, therefore, the dislocation boundaries rotate towards the macroscopic slip planes, e.g. in sheet rolling initially to be inclined about f40’ to the rolling plane (10). They delineate irregular slabs which eventually may become homologous with the macroscopically imposed strain (5), i.e. in rolling nearly parallel to the rolling plane, in torsion normal to the torsion axis. As the 2nd law of thermodynamics favors the lowest possible Ua,, and thus the lowest possible R Lack and largest rotation angle, there is no limit to the dislocation density in the walls, but new low-angle walls, seen as cell walls roughly normal to the high angle walls, are continuously formed through bending strains in the slabs (8). On account of their relatively long link lengths, the fresh cell walls constantly provide an unlimited supply of glide dislocations at a relatively small flow stress, assuring a low workhardening rate. However, as soon as they are formed, also the new cell walls trap dislocations, thereby increasing their rotation angle, and begin to rotate towards parallellity with the slip lines. In the process the average thickness, D, of the slabs progressively diminishes in line with the empirical formula D = KGb/(T-2,). Meanwhile the higherangle walls channel the glide in the slabs between them which is the cause of metallographic slip bands and can lead to shear-banding. Thus latently shear-banding is always present without any mechanical instability, The lmutually misoriented lattice orientations in the slabs comprise the texture components. Since the rotation boundaries are overwhelmingly composed of trapped glide dislocations, their morphology, and thus the resulting textures, depend on availability of slip systems, three-dimensional 173
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dislocation mobility and initial crystal orientation. It is presumed, but still requires proof, that on account of the 2ndlaw the actually realized textures embody the largest possible misorientation angles compatible with the boundary conditions. Introduction Following the considerable advances of the LEDS theory during the past seven years, the present paper shall fill in the last obvious remaining gap, the question as to the morphology of the high-strain dislocation patterns and resulting texture formation. Specifically, 0 the LEDS theory was comprehensively surveyed up to the status of 1989 including the role of Taylor lattices (1); 0 subsequently worksoftening and Hall-Petch hardening were explained (2-4); 0 the dislocation patterning underlying workhardening stages I to IV (and tentatively V) in both “planar” and “wavy” (or “distributed”) glide were elucidated (5); 0 the difference between dislocation behavior in “planar” and “distributed” glide was traced to differences in dislocation/vacancy interactions (5); 0 the shape of typical workhardening curves as consisting of piece-wise joined Vote curves was documented and shown to arise from the theoretically expected linear decrease of the dislocation retention parameter, l3, with rising effective resolved shear stress, 2 -7, (4); 0 the nature of dislocation walls as LEDS dislocation rotation boundaries was established without any reasonable remaining doubt (6,7); 0 and the important role of straingradients in the continual formation of new cell walls, of low angular misorientation and therefore controlling the flow stress, was discovered (8). The remaining task, then, is to consider dislocation morphology and texture formation in light of the LEDS theory. As this will be seen to depend on the inclusion of longer-range internal strain energy, while internal stresses have been previously largely disregarded except in (8), some basic considerations shall be presented first. Basic Considerations The LEDS theory is firmly founded on Newton’s third law, i.e. action equals reaction, and the second law of thermodynamics, i.e. maximum conversion of mechanical work into heat or, equivalently, minimization of stored energy. Fig. 1 exemplifies the approach. It envisages a simple tensile test in an isolated system via the step-wise, automated addition of weights to the load on a sample. While the applied load remains in the elastic range, force equilibrium is maintained by the elastic increase of the atomic spacings in the sample parallel to the axis. In addition, as soon as the applied stress exceeds T,, the resolved friction shear stress, the dislocations in the sample will bow out quasielastically or otherwise move lightly in local energy troughs. Thereby the corresponding anelastic complement is added to the strain, which manifests as a seemingly lowered elastic modulus (sect. 18 of (1)). Up to the FrankRead stress or bypassing among neighboring dislocations, the equilibrium is reversible both for the elastic lattice strain and the dislocations (except for a minor part due to 23, and the 2nd law of thermodynamics needs not to be invoked. The situation changes dramatically as soon as the yield stress is exceeded. Now, in addition to the elastic strain in proportion with the applied stress, irreversible plastic deformation takes place by the super-critical bowing of dislocations and/or bypassing of dislocations in dipolar configurations. Thus irreversible work is done by the load while the sample lengthens plastically. That work is overwhelmingly transformed into heat but, while workhardening progresses, a minor percentage of the expended work is stored, predominantly in the form of additional dislocations which have been trapped in the material. Their density rises in accordance with the empirical relationship 7 = F, +
crGb+
(I)
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Figure 1. Thought experiment of tensile testing in a bell jar via step-wise load application. While the stored energy in the sample rises, the free energy of the system as a whole, composed of sample and applied traction, continuously decreases as mechanical work done by the traction (i.e. the load P) is mostly converted into heat. Meanwhile the dislocation density rises so as to maintain force equilibrium in accordance with eq. 1. Following the LEDS hypothesis the free energy of the system as a whole is minimized through minimizing the stored energy in the sample.
the shear stress, G the shear modulus, b the Burgers vector, p the dislocation density anda - 0.4 within a factor of 32. Evidently, then, workhardening is due to an increasing dislocation density. At this point the 2nd law is invoked in two ways. 0 Firstly via the “LEDS Princinle”, saying that, at least in stages II, III and IV, dislocations are trapped into positions of local eneruy minima in which dislocations mutually screen their stresses. The resulting configurations are, by definition, LEDS’s &ow-Energy Qislocation &uctures) characterized by an average dislocation line energy of
with z
Un = [Gb*(1-v/2)/4~( 1-v)Jln(&&b)
(2)
with v Poisson’s ratio and RLEDsthe average near-neighbor dislocation distance. 0 Secondly, via& I, S I-IvDothq&” e.g. (4) and (7). According to it, among all possible LEDS’s which are in force equilibrium with the tractions and are accessible under the constraints of dislocation mobility and available slip systems, the one which is actually realized most nearly minimizes the energ stored in the sample. In fact, while the import of the LEDS hypothesis is clear enough and it dates back to 1968 (rlef. 9, see footnote on p. 227 of ref. 7), its precise phrasing is not evident. The form given in (7), namely requiring minimization of the energy per unit length of dislocation line, i.e. minimization of RLEnsof eq. 2 wherein all forms of internal energy are included in Un , is compatible also with interfacial ‘dislocation formation in epitaxy and phase transformations, as it ought to be. But this form may be flawed when strain-mdient-relief dislocations are included (8). If this is remedied by restricting the LEDS hypothesis to && dislocations, the LEDS hypothesis is no longer applicable to interfacial dislocations. It seems therefore that the above formulation given in bold letters is the best possible. At any rate, the writer is not aware of any violations of either the LEDS principle or the LEDS hypothesis.
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Non-LEDS Theories of Workhardening, Dynamic Effects and “Forest” Dislocations Dislocation patterning, mostly in the form of dislocation segregation, or “bunching”, into regions of high density leaving most of the volume almost dislocation-free, e.g. as in stage I “braids”, stage II “carpets”, the mosaic block structures of stages III and IV, and in fatigue in the form of “mazes” and “ladder structures”, is very wide-spread indeed. Given adequate three-dimensional dislocation mobility as in “wavy glide” materials (5) and based on the 2nd law, this is a direct consequence of eq. 2. Namely, beginning with a hypothetical random distribution, any statistical fluctuation of dislocation density so as to locally reduce dislocation spacing through mutual dislocation stress-screening at the expense of dislocation rarefaction in adjacent areas, will lower the stored energy. LEDS’s will also be formed if three-dimensional dislocation mobility is low, but not necessarily characterized by bunching. In that case Taylor lattices may arise instead (5). They are commonly mistaken for “random” dislocation “tangles” but in fact are LEDS’s with well-defined albeit still poorly understood structures (1, 10). It is their hallmark that any representative local group of dislocations has zero net Burgers vector, so that a Burgers circuit about such would be closed. However, in wavy glide and uni-directional deformation, wherein dislocations of opposite sign are persistently driven in opposite directions, net Burgers vector surpluses accumulate in representative local groups. In that case, and once more than one Burgers vector is locally available, the LEDS’s will be dislocation rotation boundaries, forming in a wide variety of mosaic block structures of misoriented volumes (11). The discussed dislocation bunching can also be attributed to various forms of mutual repulsive dislocation interactions, as is done in all non-LEDS theories of workhardening. However, mutual obstruction of dislocation movement can only be caused by local mutually repulsive stresses. These would necessarily lead to increased stored energy, in conflict with the second law and, hence, the LEDS hypothesis. They may be ruled out for that reason, apart from the mundane fact that all observed dislocation structures appear to be LEDS’s, as already indicated. For the same reason one may also rule out computer models which assume conditions far from equilibrium, following Prigogine’s thermodynamics for “energy flow-through” systems. Finally, “rer>ulsive” which have also been often invoked to account for at least part of workhardening can nlav no simriticant role vg of “wavy glide” materials. This is for the reason that the interactive stresses among approaching dislocations ready to intersect, thereby momentarily forming a four-fold node if an intersection jog was to be created, rotate them to be locally parallel, always such as to yield an “attractive” reaction. The result of that reaction is a dislocation link between two three-fold nodes, as if forming a new link in a network, always accompanied by a lowering of stored energy (12, compare Fig. 17 of 11). Short of explosive forming and as long as climb and/or recovery at ordinary rates are negligible, strain rate changes have almost no effect on workhardening and only little on the momentary flow with or with r t . stress (1, 13). ThustheI,S ED annlied stress The LEDS theory, then, is in the lineage of G. I. Taylor’s early theory of workhardening (14) and in stark contrast to Becker’s theory of dynamic, strain-rate controlled deformation (15, 16). Although long discarded as such and now-a-days virtually unknown to researchers, Becker’s theory has been perpetuated principally via the work of Orowan and Haasen. It rests on the basic assumption that, not only in creep but also in ordinary low-temperature plastic deformation, the flow stress is controlled through a competition between hardening and recovery events (compare ref. 17). Correspondingly, the concept of “dynamic recovery” , i.e. the presumed greatly enhanced speed of recovery during straining which is derived from Becker’s theory, has no room in the LEDS theory. Nor is there room for the concept of “forest” dislocations of ill-defined origin which somehow obstruct glide dislocation motion. In fact the presumed “forest” dislocations are due to dislocation reactions as already introduced and
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simply participate in the formation of LEDS’s, subject to the LEDS principle and the LEDS hypothesis. The Approximate Magnitude and Elastic Energy of Internal Non-Dislocation Stresses While it had been recognized from the outset that long-range stresses could not be the cause of workhardening but that instead dislocations would be mutually stress-screened (18), it has never been doubted that longer-range (i.e. of a range comparable to the closest wall-spacings) and/or long-range stresses would always be present. Namely, structures which are composed of dislocations trapped in energy wells are bound to be stable, in glide, up to the flow stress magnitude that generated them. Unloading, load reversal and/or reloading below the flow stress correspondingly gives rise only to some minor dislocation rearrangements which, as already mentioned, manifest as anelasticity and simulate an apparently lowered elastic modulus, typically by the factor of about one half (1,17). Hence, since longer-range and/or long-range stresses between f (z - ~3 cannot be removed in glide, and since internal stresses must vanish on average, the internal stress magnitude is expected to be zi I (T - 2,)/2, which is in agreement with the measurements of Mughrabi and coworkers (19). Even so U, , the elastic energy density per unit dislocation line length of those stresses, is not negligible. Namely with eqs. 1 and 2, RLEDs= I,‘$, v = 0.3 and b = 3 x lP* cm U, = Gzi2/2= Gb* a’/8 3 UD(a’l[O.8 In&&b)]}
z 0.3 a* UD
(3)
The Role of Elastic Strain Gradient Energies in Determining the Dislocation Wall Spacing An obvious remaining problem is why dislocation walls are not more widely spaced than according to the empirically well established (20) relationship between flow stress and average cell diameter (and presumably wall spacing in stage IV), D, i.e. z-z,=KGb/D
(4)
with K = 10 but on occasion as low as 2.5 (8,20-22). Certainly, distributing the same dislocation density over fewer walls will decrease R LED~ and thereby decrease the dislocation line energy. Attempts to explain eq. 4 by means of long-range stress energy associated with terminating dislocation walls failed after an apparent initial success (11,23), as it has proven impossible to theoretically account for numerically satisfactory parameters. Only very recently has the answer to the puzzle of eq. 4 been found as follows: Unlike dislocation walls extending through whole crystals as in kinking (compare 24), inevitably longe:r-range stresses arise along more or less parallel dislocation rotation walls of finite length formed from trapped glide dislocations in accordance with the preceding considerations. They will thus create the corresponding elastic strain gradients within the slabs between the walls, most commonly causing elastic bending. If the maximum tensile/compressive stress in the outermost fibers of the bent slab defined by the parallel wall sections is c~,,,,and the slab has, say, a thickness of D and length of L, then the associated maximum elastic tensile/compressive strain will be F,,,= o,/E = DX/2L where X is the resulting elastic bending angle. Hence, with D/L = h, CT,,, = EX/2X
(5)
The associated. elastic strain energy is proportional to Q,,,‘, i.e. to X2. However, the bending could be alternatively generated by a tilt wall, of spacing
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h = b/(2 sinX/2) = b/X
(6)
and thus by an energy proportional to X In( l/X). As a consequence, the LEDS hypothesis requires that, beyond some critical value, elastic bendings (and similarly also elastic twists) will be partly relieved by the corresponding dislocation walls. The lower limit of beginning conversion of elastic strain gradients into new dislocation walls depends somewhat on geometry but it would seem that at maximum h I D/5. Moreover, for the bending to be completely converted into dislocation rotation walls would require an infinite elastic strain gradient. The details of the derivation are given in (8). According to it, minimum stored energy is obtained for Xmin= [3 In( l/X& - 1]/(3D/b) = const bD
(7)
Therefore, on the reasonable assumption that cr,,,compares with the flow stress so that (z - 2, ) = Gb/h = GX, eq. 4 follows. Ref. (8) documents that this simple theory is in full accord with the measurements of Langford and Cohen (21) as interpreted by means of the mesh-length theory (22), and thus is in full accord with the LEDS theory which has developed out of it (l), wd that the momentary flow stress is controlled by the new walls. This proviso does not only follow from the data but is a necessary consequence of the theory. This is so because the theory identifies the flow stress with the Frank-Read super-critical bowing stress of always the longest most frequent links in the structure (whence incidentally also follows the mild low-temperature strain rate dependence of the flow stress (13)). In summary, then, the spacing between parallel walls is limited by the continual creation of new cell walls which relieve unavoidable strain gradients in accord with eq. 4. At any moment the newest walls control the flow stress via their low rotation angles and thus long link lengths. Meanwhile, high-angle rotation walls can be eliminated through merging when the specimen diameter shrinks faster with stress than D, as is the case in wire drawing (2 1) and probably also in sheet rolling. Kinematics of the Dislocation Structure Evolution in Uni-Directional Straining of “Wavy-Glide” Materials The evolution of the dislocation structure during urn-directional, low-temperature straining of “wavy” (or “distributed”) glide materials is rather well understood. It progresses from “braids” in stage I, to “carpets” in stage II, to roughly equiaxed dislocation cell structures with small angles of misorientation in stage III, and to roughly parallel dislocation boundaries which are increasingly homologous with the imposed strain in stage IV. The LEDS transformations separating these stages are in line with the LEDS hypothesis in that Un , the dislocation line energy disregarding non-dislocation stresses, decreases more or less discontinuously from stage to stage, while within any one stage Un decreases continuously. Further, the shape of the stress-strain curves conforms to the theoretically expected linear decline with stress of fi, the fraction of glide dislocations which are trapped in the dislocation structure at the end of their paths. In regard to these facts the reader is referred to the pertinent papers (1,4,5,7,11,13,25). Taking, then, up at the end of the preceding section, consider that, clearly, “new” dislocation rotation boundaries cannot stay “new” for long. They capture dislocations and thereby increase their angles and as a result rotate towards the orientation of the slab plane so as to propagate and refine the preexisting structure. The reason for this wall alignment with increasing misorientation angles are the strain gradients which are associated with cell walls formed from trapped glide dislocations (7,26). These impart the corresponding elastic strain energy over and above that of the self-stresses of the
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constituting dislocations which the cell would have in the thoroughly annealed state, e.g. as computed in (27) and in :many much more recent computer simulations of dislocation arrays. Those previously neglected strain gradients result from the fact that the shear strain imparted by the glide dislocations along their path ends where they are trapped, i.e. in the boundaries (26). For example, a tilt wall of dislocation spacing h and formed of trapped glide dislocations is associated with the abrupt shear strain change Ay, = bs/h= 2sin(@/2), with Q,the wall misorientation angle, and this independent of the height of the wall or how long was the free path of the glide dislocations before meeting the boundary. In a slab between high-angle walls which is subdivided by tilt walls on account of bending strain relief as explained above, this Aywgives rise to the corresponding elastic stress component parallel to A’ywof magnitude A%+= G 2sin(0/2). By contrast, seeing that glide parallel to dislocation rotation walls is always possible through minor dislocation rearrangements, a set of dislocation walls parallel to the macroscopic “slip lines” in the sense of continuum theory, is free of such strain gradients. As a result the LEDS hypothesis predicts increasing such alignment of dislocation boundaries as their rotation angles increase (compare (5)). The above goes far to answer the question of why there w to be two families of dislocation rotation walls, previously named (28) “incidental” (ordinary dislocation cell boundaries, ID’s), and “cell block boundaries” (CBB’s), with variable misorientations up to being indistinguishable from grain boundaries (7,lO). In fact, as seen there is a continual creation of new cell walls almost normal to approximately parallel higher-angle walls. The new walls rotate towards the common plane of the earlier, stronger walls and in the process replenish the bending (and/or torsional) strain gradients which cause the formation of new low-angle ID walls. It is only too natural for observers to interpret such a continuum, ranging from long, more or less parallel high-angle walls to low-angle cell.boundaries normal thereto, in terms of two populations (29), compare (7). However, careful measurements as in (30) and (3 l), do not show a bi-modal distribution of angles. Rather if the corresponding histograms of presumed ID’s ;and CBB’s are combined, a smooth distribution is found. Similarly, TEM micrographs, e.g. Fig. 2., show a continuum of wall rotation angles. Even so, the initial idea that there should be two distinct families of walls (28,32) is almost certainly correct, wherein CBB’s delineate “cell blocks” with different selections of active slip systems, falling short of the five required by the Taylor criterion. In that interpretation, the CBB’s (but ti the ID’s) are “geometrically necessary” in accommodating the angular difference in lattice orientation due to slip on different slip system selections. However, it is now tentatively concluded that CBB’s as defined, are microscopically observed Q& as “dense dislocation wal1.s”(DDW’s, 32) of long extent, whereas the great majority of higher- and high-angle boundaries which in past publications (including refs. 1,7,8,26,28,30-32 and Fig. 2, for example) were believed to be C:BB’s, are in fact matured ID’s.
........
(3
5
4”
- - - - - IO” < e I 18”
-***-- 4”
ND 4
0.5km -
Figure 2. TEM micrograph of 98% cold-rolled Ni (99.990/o),courtesy of Darcy A. Hughes (Fig. 2 of ref. 30). Section normal to the rolling plane whose normal is indicated as ND, while tke rolling direction is vertical, parallel to the average wall direction. Note the large spread of misorientations and the “bamboo structure” due to cell walls between the stronger boundaries.
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Shear Bands and Texture Formation After severe sheet rolling, the high-angle boundaries are roughly parallel to the rolling plane as shown in Fig.2, but, after mild to moderate rolling, i.e. in stage III, they are oriented - & 40” to the rolling plane (32). This is in line with the expected alignment of strong boundaries with the “slip lines” of continuum’s theory, derived in the previous section. And in this orientation the strong walls are bound to channel glide, which not only accounts for slip bands but for shear banding whenever the workhardening rate is critically low. Latently, shear banding is thus always present. However, the conditions under which it becomes a technological problem still require elucidation. At any rate, considering that polyslip is active throughout, neither the ID’s nor the “true” CBB’s need to follow crystallographic orientations. Thus true CBB’s can readily cross grain boundaries. Fig. 10 of (33) is believed to show this effect in the form of a roughly 20pm wide striping inclined about 40” to the rolling plane. The stripe pattern in Fig. 11 of (34) is probably also due to CBB’s, whereas H. Raabe’s own interpretation in terms of super-dislocations is untenable. That the walls represent boundaries not only of relative misorientations but also of discontinuous changes of plastic strain has been recently addressed also by Leffers (35) for rolling and by Sedl&ek (36) for fatigue. It is as yet not clear to what extent the preceding considerations may support or be in conflict with that work. At any rate, the stage IV reorientation of the boundaries more closely to the plane of rolling, e.g. as in Fig. 2 and typically in the form of a herringbone-type of pattern (e.g. Fig. 2 of (7) due to D. A. Hughes), presumably occurs because the free dislocation paths in steeply inclined cells become geometrically severely shortened. This, then, increases the flow stress and therewith zi, so that in accord with the LEDS hypothesis the boundaries rotate towards the rolling plane, thereby diminishing the flow stress and the internal stress energy. Collectively the ranges of lattice orientations in the slabs between the walls constitute textures, albeit in contrast to the Taylor model different components occur in close proximity within any one grain. Experimentally this phenomenon has most recently been brilliantly documented by Panchanadeeswaran, Doherty and Becker (33). As a result, stabilization of the lattice misorientations is inevitably slow and imprecise, which is perhaps the decisive reason for the well-known shortcomings of the Taylor model (33). Finally, the above LEDS interpretation of texturing is strongly supported also by the work of Smallman and Lee (37) who have studied texture evolution in rolled a-brass. By contrast to the wavy-glide materials discussed so far, a-brass deforms by “planar-glide”, exhibiting Taylor lattices, and contains dislocation rotation boundaries only at the domain boundaries (1). Even so, the well-known { 1lo}<1 12> a-brass rolling texture was also found as fine lamellae, namely in twinorientations separated by high-angle dislocation rotation boundaries (37). From this one may conclude with some modest degree of confidence that the LEDS hypothesis operates also in planar-glide materials, namely on the dislocations in the domain boundaries. In very much the same manner as already outlined for the wavy-glide case, this causes a sub-division of the domains and consequent proliferation of domain walls composed of closely spaced dislocations with small energy per unit length of dislocation line, eq. 2. The resulting texture is different from that in wavy glide as, constrained by the planarity of the glide within the Taylor lattices, the domain walls are much less free to reorient, and the lamellae in mutual twin orientation form a distinct texture. It is similarly evident that the texture here and in wavy glide must depend on strain geometry. Acknowledgment Many discussions with my husband, H. G. F. Wilsdorf, and his unwavering support are most gratefully acknowledged.
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