- Email: [email protected]
Sequences of dislocation patterns
Materials Science and Engineering A317 (2001) 12 – 16 www.elsevier.com/locate/msea
Sequences of dislocation patterns F.R.N. Nabarro a,b,* a
Condensed Matter Physics Research Unit, Uni6ersity of the Witwatersrand, Pri6ate Bag 3, Johannesburg WITS 2050, South Africa b Di6ision of Materials Science and Technology, CSIR, PO Box 395, Pretoria 0001, South Africa
Abstract Understanding the formation of sequences of dislocation patterns involves first understanding why certain patterns are in an appropriate sense stable, and then understanding why one stable pattern is favoured over another. Similitude implies that if pattern A is favoured over pattern B at one dislocation density, it will also be favoured at a higher dislocation density. A change of pattern must be induced either by a breakdown of similitude or by kinematic considerations. Kinematic considerations alone may cause one dislocation pattern to develop into another. The change occurs under a prescribed flow stress. Since the work done by the external stress in moving a dislocation across the pattern is proportional to the scale of the pattern, the scale of the pattern is inversely proportional to the applied stress. While the breakdown of similitude allows the energetic relations of different patterns to vary as the dislocation density increases, increase in dislocation density may also break down the similitude of kinematic behaviour by altering the likelihood of cross slip or of climb. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Dislocations; Pattern formation; Work hardening
1. Introduction At a workshop last year [1], Ali Argon gave a paper entitled Dislocation Patterning and Cell Formation in FCC Metals: Often Overlooked Facts. The abstract ended with a paragraph in italics: ‘Thus, the central question in patterning as well as in strain hardening itself is less one of understanding the interactions of clustered dislocations with the mobile flux, but more are of understanding why the clusters are where they are.’ Typically, as deformation proceeds, a sequence of dislocation patterns appears. To quote again from Argon’s abstract: ‘It has been studied most widely in Cu single crystals. It manifests itself, starting in Stage I by the formation of kink walls, transforming into quasi-periodically placed open braids during Stage II, and develops into well formed closed cells upon the onset of dynamic recovery in Stage III, whereupon the cells undergo a remarkable self-similar reduction in size inversely proportional to the increasing plastic resistance. The dislocation content in the braids and cell walls is overwhelmingly redundant’. Argon did not * Tel.: +27-117164420; fax: +27-113398262. E-mail address: [email protected] Nabarro).
(F.R.N.
mention that in the analysis of some observers [2–4] Stage III develops into Stage IV, in which the redundant dislocations disappear, leaving predominantly ‘‘geometrically necessary’’ tilt boundaries, while Stage IV, composed of edge dislocations, in turn gives way to Stage V [5], in which screw dislocations again become prominent. While Stages I and II are characteristic of single crystals stressed to induce glide essentially on one system, the later stages seem to occur also in single crystals stressed to produce multiple glide, and in polycrystals. These multiple-glide configurations are more important technically, and, because they are more symmetrical, they may perhaps prove easier to interpret. We first need to understand why each individual dislocation pattern is stable, in the sense that it persists with little change except perhaps one of scale as the applied stress and the dislocation density increase. The more difficult problem is to see why, when the stress and dislocation density increase beyond some critical values, the dislocation pattern changes from one form to another. There are three basic possibilities. First, it may be that, as the applied stress increases, the dislocation pattern which has in some sense the lowest energy or the highest stability is overtaken by another pattern
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F.R.N. Nabarro / Materials Science and Engineering A317 (2001) 12–16
which now has a lower energy or a higher stability. We call this the energetic model. This is analogous to a thermodynamic first-order transition. Second, it may be that if a certain dislocation pattern is present, and dislocations multiply and react under an increasing stress, the pattern necessarily changes its form. As the form approaches another stable pattern, the change of form accelerates, analogously to a second-order transition. We call this the kinematic model. Thirdly, it may be that, as the stress and the dislocation density increase, modes of dislocation motion such as cross slip and climb, which were previously unimportant, become more effective. We call this the mechanistic model.
Fig. 1. A section dissected from a Taylor lattice of edge dislocations with Burgers vectors 9 (bx,0,0) which relaxes into a stable cluster [8].
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2. The Taylor lattice [6] It is natural to begin with a uniform distribution of dislocations, and to enquire whether this will tend to develop a structure. Since we have difficulty in visualizing a three-dimensional array, it is tempting to begin with an array of rectilinear dislocations lying along the z-axis, and free to move in the (x, y) plane. Most simply, as in the pioneering work of Holt [7], these are screw dislocations with equal numbers of Burgers vectors + bz and − bz. Logically, these will ultimately annihilate, and the obvious answer is to proceed to the Taylor lattice with Burgers vectors equally +bx and − bx (Fig. 1). The lattice is rectangular, and the infinite lattice is clearly in equilibrium if the axial ratio is chosen appropriately. If the dislocations can move only by glide, with displacements ux, the lattice has two homogeneous deformations represented by #ux /#x and #ux /#y, and the corresponding elastic constants are positive. What happens if the section of the lattice in Fig. 1 is extracted from its matrix? Neumann [8] showed that it will relax into a distorted, but mechanically stable, configuration. However, he also considered the behaviour of the section of Fig. 2(a) extracted from the matrix. He showed that this section is not stable, it dissociates into the dipole walls of Fig. 2(b), which repel each other to infinity. The question ‘‘are finite pieces of the Taylor lattice stable?’’ has no answer. Neumann showed that the dipole walls of Fig. 2(b) are themselves very stable. If we consider a real finite dislocation array which approximates to a portion of a Taylor lattice [9], the question of its stability becomes delicate. It is curious that the configuration of Fig. 2(b) does not seem to occur in practice, although it might form a model for the redundant dislocations in cell walls.
3. Mughrabi’s model for double glide
Fig. 2. (a) A similar dissected segment, which relaxes into the dipole walls of (b). These dipole walls repel one another to infinity [8].
Taylor’s model is essentially 112-dimensional. Dislocations are represented by points in the (x,y)plane, z =0. These points are distributed in two dimensions, but they can move only in one dimension. In Mughrabi’s model [10], dislocations are again represented by points in the (x,y) plane, z =0, but they are of four Burgers vectors 9 (b,b,0) and 9 (b, −b,0)(Fig. 3). The total density of dislocations at (x,y) is z(x,y). In Mughrabi’s application, which we shall follow, z depends only on x. While the points representing the dislocations lie in the plane z= 0, their motion is controlled by interactions with forest dislocations which occur outside this plane, and which are not discussed in detail. The model is 212 dimensional. As a result of these interactions, dislocations are subjected to a frictional
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3.1. A purely kinematic model of patterning
Fig. 3. Dislocations forming Mughrabi’s model of double glide, having Burgers vectors 9(b,b,0) and 9 (b,− b,0).
stress hbvz 1/2. If the applied stress in monotonic deformation is |a, the effective stress | is | =|a −hbvz 1/2.
(1)
We may extend the model by defining two dislocation fluxes. The first flux, f(x), represents the flux of dislocations of types 1 and 2 in Fig. 3 to the right plus the flux of dislocations of types 3 and 4 to the left. This flux f(x) contributes to the plastic strain, and is directly driven by the effective stress |. We assume that the flux f(x) is given by f(x)=mz(x)|(x).
(2)
As a result of forest interactions outside the plane z =0, there is a local rate of increase of dislocation density z; + (x)= pz(x)f(x) =mpz 2(x)|(x).
(3)
There is also a rate of dislocation annihilation given by z; − (x)= qf 2(x)= qm 2z 2(x)| 2(x).
(4)
The total rate of change of z(x)is, thus, z; (x)= z; + (x)−z; − (x) = mz 2|(p −qm|).
(5)
The second flux, g(x), is the total flux of all dislocations to the right. It does not contribute to the plastic strain, and is not directly driven by the effective stress. It is essentially a flux of dislocation dipoles of the kind considered by Kratochvı´l [11], and at present we neglect it. Using the concepts z(x), |(x) and f(x), we can develop two models of dislocation patterning [12], one purely kinematic and one essentially energetic.
This model follows immediately from Eqs. (1) and (5). In monotonic deformation, the factor mz| 2 in Eq. (5) is always positive, so the rate of change of dislocation density has the sign of p−qm|. But from Eq. (1), | is a decreasing function of z. As a result, for a given value of |a, z will be positive in regions where z is large and negative in regions where z is small, so that spatial irregularities in z will amplify. While this example has shown that kinematic considerations will in general intensify fluctuations in dislocation density, there are also demonstrations that one specific dislocation pattern will lead kinematically to the formation of another specific pattern. Thus, Gassenmeier and Wilkens [13] considered a slip band in single glide. The glide process is one of simple shear, with a rotational component, which causes the slip band to make a small angle with the slip plane. As a consequence, compressive fibre stresses are set up in the slip band. These activate secondary slip systems, and the dislocations of these systems interact with those of the primary system to produce dislocation tangles along the borders of the slip bands, while the interior of the slip bands has a low density of dislocations. Similar hollow-cored slip bands can occur in multiple-glide situations [14].
3.2. An essentially energetic model of patterning A basic difficulty in any energetic model of dislocation patterning is to explain the breakdown of the principle of similitude. If dislocation pattern A is favoured over pattern B under a stress |and with a mean dislocation density z¯ , how can B be favoured over A if the stress is increased to k| and the mean dislocation density to k 2z¯ ? Another basic difficulty is that the configuration of lowest energy is that in which all the dislocations have escaped through the free surface of the sample. It is essential [15] to introduce some constraint on the permissible distributions. A further difficulty, which can probably be resolved only by introducing kinematic considerations, is that of determining the intensity of the concentrations of dislocations in a given type of pattern. Patterns form because the dislocations screen each other’s stress fields. If the local dislocation density is z(x), the energy per unit length of dislocation is of order −(vb 2/4y) ln(bz 1/2). Here, bz 1/2 is a small fraction, and its logarithm is numerically large unless z is very large. Suppose we consider a random array of edge dislocations. The energy per unit length can be reduced by allowing the dislocations to multiply and form a Taylor lattice of very high density. Even with the additional constraint of constant length of dislocation line, these dislocations
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can reduce their energy by forming alternating tilt boundaries, in which the elastic field of each dislocation is effectively confined to a region of dimensions equal to the nearest-neighbour separation. The only limit to this process is when the angles of tilt become of the order of a radian; the energy continues to decrease as the dislocations arrange themselves into boundaries of ever-increasing tilt angles separated by ever-increasing distances. The constraint introduced in Ref. [15], that configuration ‘‘is approached which minimizes the free energy, per unit length of dislocation line’’, is clearly inadequate. The experimental constraint is that any change in the dislocation configuration occurs under a prescribed flow stress. While the clustering of dislocations under a constant mean dislocation density can reduce the total energy, it follows [10] from Eq. (1) that this clustering also reduces the flow stress. If the mean density is increased to keep the flow stress constraint, it can be shown [12] that the elastic energy actually increases. However, for a system under stress, the thermodynamic potential is not the energy but the enthalpy, the energy less the work done by the applied stresses. We now consider a given degree of clustering in cells of different sizes. The increase in energy for each dislocation initially present turns out to be independent of the cell size, but the work done by the external stresses is proportional to the distance the dislocation moves, that is, to the cell size, and also to the applied stress. For the enthalpy to decrease, the size of the cell must exceed a critical size, which is inversely proportional to the applied stress. Again we face the problem that the thermodynamic driving force for cell formation is larger, the larger the cell. The solution is that the growth is an amplification of an existing fluctuation in the dislocation density. The larger the initial fluctuation both in amplitude and in wavelength, the more rapid the growth. The statistical probability of a fluctuation is a decreasing function both of the amplitude of the fluctuation and of the size of the region in which the fluctuation occurs. Those cells will predominate for which the product of the statistical probability of a fluctuation and its rate of growth is largest. It turns out that these cells have twice the linear dimensions of the smallest cell which can grow under the given applied stress, and grow from a statistically likely fluctuation. The side of the cell is about 12 times the mean separation of dislocations in a random distribution.
4. The validity of an energetic model A plastically deforming metal is a system driven far from equilibrium. Under isothermal conditions, usually
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less than a tenth of the work done on the sample remain stored in it [16]. How can we apply quasi-equilibrium energetic considerations to such a system? The answer may appear by analogy with the phenomenon of the acceleration of creep by a phase change or by thermal expansion in a polycrystal of anisotropic grains. The phase change or thermal expansion causes large microstresses, which produce local plastic deformation without any macroscopic change of shape. Even a small imposed homogeneous stress biases these plastic deformations, producing macroscopic deformation. The present problem is the converse of this. External stress renders the whole sample plastic, and produces a homogeneous dislocation flux which does not produce any change in the local dislocation density. Once the external stress has rendered the dislocations mobile, local fluctuations in dislocation density can locally bias this flux in the way described in Section 3.2.
5. Mechanisms which become more effective at high dislocations densities In an energetic approach, one preferred dislocation pattern may give way to another as the dislocation density increases because similitude has broken down. The line energy of a dislocation is not constant, but depends logarithmically on the dislocation density. In a kinematic approach, not only may one pattern lead geometrically to another as in the work of Gassenmeier and Wilkens [13], but increasing dislocation density may influence the relative ease of different types of dislocation motion. Specific cases are climb and cross slip. Mecking and Estrin [17] calculated that with a uniform dislocation density and a strain rate of 10 − 2 s − 1, the concentration of vacancies produced by plastic deformation would equal that present in thermal equilibrium at a temperature of half the melting point Tm. They added ‘‘Attention should be drawn to the possible role of mechanically produced vacancies below 0.5Tm in case of extremely localized deformation.’’ Considering that 0.5Tm for copper is 380°C, and that the dislocation density in the cell walls is some 10 times the mean density, it seems likely that the mechanical production of point defects in cell walls could permit the redundant edge dislocations to annihilate, and so induce the transition from Stage III to Stage IV. In the case of cross slip Kuhlmann-Wilsdorf [18] suggested that ‘‘similitude must break down eventually … for example, cross slip, conservative climb or climb becomes easy’’. There is a simple mechanism for this in the case of cross slip in f.c.c. metals. Dislocations are dissociated on the glide plane. They cross slip when a combination of the glide stress and of thermal activation unites the two partials over a short segment, allowing the dislocation to dissociate on the cross-slip
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plane. Once a segment has cross slipped, the nodes between the portions dissociated on different planes can easily separate. In addition to the glide component of the stress, both the applied stress and the internal stresses, which are of comparable magnitude, have components in the glide plane and perpendicular to the glide direction. These components, proportional to the square root of the dislocation density, may either increase or decrease the separation of the partials. If the separation of two partials which are unlikely to cross slip is increased, they will still be unlikely to cross slip; if the separation is decreased, cross slip may be induced.
6. Discussion There are many attempts to apply to the problem of dislocation patterning techniques which have been successful in solving other problems. One of these techniques is that of chemical reaction– diffusion theory, which successfully predicts spatial and temporal oscillations. This approach meets with two major problems. The first is to find a component in dislocation dynamics which can be formally represented as a diffusion of dislocations. If we have, for example, only edge dislocations of one sign, their material repulsions lead to a diffusion-like term. The reality is more like a weakly charged Taylor lattice, with local small preponderances of dislocations of one or the other sign. These preponderances will relax by an effectively diffusional process, but there is no tendency for local variations in the total dislocation density to relax. (Consider, for example, a crystal containing isolated clusters of the form of Fig. 2(a).) A further serious error is the neglect of the tensor nature of the forces between edge dislocations. How can one build a theory of dislocation patterning using a model which does not recognize that a tilt boundary is a configuration of low energy? The incorporation of the real tensor forces between dislocations would probably make even two-dimensional simulations intractable analytically. The other technique is the finite-element computer modelling of dislocation motion using reasonably realistic laws of motion. There is little doubt
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that this modelling will lead to interesting results. However, suppose that such a simulation successfully models the formation of a dislocation pattern which is observed in practice, or even the transition from one pattern to another. How much insight have we gained? Suppose our model incorporates cross slip and climb. Are either or both of these processes essential to the observed behaviour? This insight can be gained only by making a series of computations with varying tendencies to cross slip and to climb, finding the range of these tendencies in which the observed pattern formation occurs, and showing that this range corresponds to the range in which this particular pattern formation occurs in practice. Such explanations probably lie only slightly beyond the present power of computers. Finally, the role of experiment must not be neglected. Experiment [19] seems more likely than theory to give an acceptable answer to the question ‘‘how do interstitial edge dipoles annihilate?’’
References [1] 1998 NIST/LLNL Workshop on Work Hardening and Dislocation Patterning in Metals. [2] T. Unga´ r, M. Zehetbauer, Scripta Mater. 35 (1996) 1467 –1473. [3] A.S. Argon, P. Haasen, Acta Metall. Mater. 41 (1993) 3289 – 3306. [4] F.R.N. Nabarro, Scripta Metall. Mater. 30 (1994) 1085 –1087. [5] P.N.B. Anongba, J. Bonneville, J.L. Martin, Acta Metall. Mater. 41 (1993) 2897 – 2906. [6] G.I. Taylor, Proc. Roy. Soc. Lond. A 145 (1934) 362 –388. [7] D.L. Holt, J. Appl. Phys. 41 (1970) 3197 – 3201. [8] P. Neumann, Mater. Sci. Eng. A81 (1986) 465 – 475. [9] D. Kuhlmann-Wilsdorf, Mater. Sci. Eng. A39 (1979) 231. [10] H. Mughrabi, S. Afr. J. Phys. 9 (1986) 62 – 68. [11] J. Kratochvı´l, Mater. Sci. Eng. 164 (1993) 15 – 22. [12] F.R.N. Nabarro, Phil. Mag. A80 (2000) 759 – 764. [13] P. Gassenmeier, M. Wilkens, Phys. Stat. Sol. 30 (1968) 833 – 843. [14] D.A. Hughes, W.D. Nix, Mater. Sci. Eng. A122 (1989) 153 –172. [15] D. Kuhlmann-Wilsdorf with, H.G.F. Wilsdorf, J.A. Wert, Scripta Metall. Mater. 31 (1994) 729 – 734. [16] M.B. Bever, J.L. Holt, A.L. Titchener, Progr. Mater. Sci. 17 (1973) 1 – 190. [17] H. Mecking, Y. Estrin, Scripta Metall. 14 (1980) 815 –819. [18] D. Kuhlmann-Wilsdorf, Mat. Res. Innovat. 1 (1998) 265 –297. [19] M. Niewczas, Z.S. Basinski, J.D. Embury, Mater. Sci. Eng. A234 –236 (1997) 1030 – 1032.
Sequences of dislocation patterns F.R.N. Nabarro a,b,* a
Condensed Matter Physics Research Unit, Uni6ersity of the Witwatersrand, Pri6ate Bag 3, Johannesburg WITS 2050, South Africa b Di6ision of Materials Science and Technology, CSIR, PO Box 395, Pretoria 0001, South Africa
Abstract Understanding the formation of sequences of dislocation patterns involves first understanding why certain patterns are in an appropriate sense stable, and then understanding why one stable pattern is favoured over another. Similitude implies that if pattern A is favoured over pattern B at one dislocation density, it will also be favoured at a higher dislocation density. A change of pattern must be induced either by a breakdown of similitude or by kinematic considerations. Kinematic considerations alone may cause one dislocation pattern to develop into another. The change occurs under a prescribed flow stress. Since the work done by the external stress in moving a dislocation across the pattern is proportional to the scale of the pattern, the scale of the pattern is inversely proportional to the applied stress. While the breakdown of similitude allows the energetic relations of different patterns to vary as the dislocation density increases, increase in dislocation density may also break down the similitude of kinematic behaviour by altering the likelihood of cross slip or of climb. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Dislocations; Pattern formation; Work hardening
1. Introduction At a workshop last year [1], Ali Argon gave a paper entitled Dislocation Patterning and Cell Formation in FCC Metals: Often Overlooked Facts. The abstract ended with a paragraph in italics: ‘Thus, the central question in patterning as well as in strain hardening itself is less one of understanding the interactions of clustered dislocations with the mobile flux, but more are of understanding why the clusters are where they are.’ Typically, as deformation proceeds, a sequence of dislocation patterns appears. To quote again from Argon’s abstract: ‘It has been studied most widely in Cu single crystals. It manifests itself, starting in Stage I by the formation of kink walls, transforming into quasi-periodically placed open braids during Stage II, and develops into well formed closed cells upon the onset of dynamic recovery in Stage III, whereupon the cells undergo a remarkable self-similar reduction in size inversely proportional to the increasing plastic resistance. The dislocation content in the braids and cell walls is overwhelmingly redundant’. Argon did not * Tel.: +27-117164420; fax: +27-113398262. E-mail address: [email protected] Nabarro).
(F.R.N.
mention that in the analysis of some observers [2–4] Stage III develops into Stage IV, in which the redundant dislocations disappear, leaving predominantly ‘‘geometrically necessary’’ tilt boundaries, while Stage IV, composed of edge dislocations, in turn gives way to Stage V [5], in which screw dislocations again become prominent. While Stages I and II are characteristic of single crystals stressed to induce glide essentially on one system, the later stages seem to occur also in single crystals stressed to produce multiple glide, and in polycrystals. These multiple-glide configurations are more important technically, and, because they are more symmetrical, they may perhaps prove easier to interpret. We first need to understand why each individual dislocation pattern is stable, in the sense that it persists with little change except perhaps one of scale as the applied stress and the dislocation density increase. The more difficult problem is to see why, when the stress and dislocation density increase beyond some critical values, the dislocation pattern changes from one form to another. There are three basic possibilities. First, it may be that, as the applied stress increases, the dislocation pattern which has in some sense the lowest energy or the highest stability is overtaken by another pattern
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 1 9 2 - 3
F.R.N. Nabarro / Materials Science and Engineering A317 (2001) 12–16
which now has a lower energy or a higher stability. We call this the energetic model. This is analogous to a thermodynamic first-order transition. Second, it may be that if a certain dislocation pattern is present, and dislocations multiply and react under an increasing stress, the pattern necessarily changes its form. As the form approaches another stable pattern, the change of form accelerates, analogously to a second-order transition. We call this the kinematic model. Thirdly, it may be that, as the stress and the dislocation density increase, modes of dislocation motion such as cross slip and climb, which were previously unimportant, become more effective. We call this the mechanistic model.
Fig. 1. A section dissected from a Taylor lattice of edge dislocations with Burgers vectors 9 (bx,0,0) which relaxes into a stable cluster [8].
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2. The Taylor lattice [6] It is natural to begin with a uniform distribution of dislocations, and to enquire whether this will tend to develop a structure. Since we have difficulty in visualizing a three-dimensional array, it is tempting to begin with an array of rectilinear dislocations lying along the z-axis, and free to move in the (x, y) plane. Most simply, as in the pioneering work of Holt [7], these are screw dislocations with equal numbers of Burgers vectors + bz and − bz. Logically, these will ultimately annihilate, and the obvious answer is to proceed to the Taylor lattice with Burgers vectors equally +bx and − bx (Fig. 1). The lattice is rectangular, and the infinite lattice is clearly in equilibrium if the axial ratio is chosen appropriately. If the dislocations can move only by glide, with displacements ux, the lattice has two homogeneous deformations represented by #ux /#x and #ux /#y, and the corresponding elastic constants are positive. What happens if the section of the lattice in Fig. 1 is extracted from its matrix? Neumann [8] showed that it will relax into a distorted, but mechanically stable, configuration. However, he also considered the behaviour of the section of Fig. 2(a) extracted from the matrix. He showed that this section is not stable, it dissociates into the dipole walls of Fig. 2(b), which repel each other to infinity. The question ‘‘are finite pieces of the Taylor lattice stable?’’ has no answer. Neumann showed that the dipole walls of Fig. 2(b) are themselves very stable. If we consider a real finite dislocation array which approximates to a portion of a Taylor lattice [9], the question of its stability becomes delicate. It is curious that the configuration of Fig. 2(b) does not seem to occur in practice, although it might form a model for the redundant dislocations in cell walls.
3. Mughrabi’s model for double glide
Fig. 2. (a) A similar dissected segment, which relaxes into the dipole walls of (b). These dipole walls repel one another to infinity [8].
Taylor’s model is essentially 112-dimensional. Dislocations are represented by points in the (x,y)plane, z =0. These points are distributed in two dimensions, but they can move only in one dimension. In Mughrabi’s model [10], dislocations are again represented by points in the (x,y) plane, z =0, but they are of four Burgers vectors 9 (b,b,0) and 9 (b, −b,0)(Fig. 3). The total density of dislocations at (x,y) is z(x,y). In Mughrabi’s application, which we shall follow, z depends only on x. While the points representing the dislocations lie in the plane z= 0, their motion is controlled by interactions with forest dislocations which occur outside this plane, and which are not discussed in detail. The model is 212 dimensional. As a result of these interactions, dislocations are subjected to a frictional
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3.1. A purely kinematic model of patterning
Fig. 3. Dislocations forming Mughrabi’s model of double glide, having Burgers vectors 9(b,b,0) and 9 (b,− b,0).
stress hbvz 1/2. If the applied stress in monotonic deformation is |a, the effective stress | is | =|a −hbvz 1/2.
(1)
We may extend the model by defining two dislocation fluxes. The first flux, f(x), represents the flux of dislocations of types 1 and 2 in Fig. 3 to the right plus the flux of dislocations of types 3 and 4 to the left. This flux f(x) contributes to the plastic strain, and is directly driven by the effective stress |. We assume that the flux f(x) is given by f(x)=mz(x)|(x).
(2)
As a result of forest interactions outside the plane z =0, there is a local rate of increase of dislocation density z; + (x)= pz(x)f(x) =mpz 2(x)|(x).
(3)
There is also a rate of dislocation annihilation given by z; − (x)= qf 2(x)= qm 2z 2(x)| 2(x).
(4)
The total rate of change of z(x)is, thus, z; (x)= z; + (x)−z; − (x) = mz 2|(p −qm|).
(5)
The second flux, g(x), is the total flux of all dislocations to the right. It does not contribute to the plastic strain, and is not directly driven by the effective stress. It is essentially a flux of dislocation dipoles of the kind considered by Kratochvı´l [11], and at present we neglect it. Using the concepts z(x), |(x) and f(x), we can develop two models of dislocation patterning [12], one purely kinematic and one essentially energetic.
This model follows immediately from Eqs. (1) and (5). In monotonic deformation, the factor mz| 2 in Eq. (5) is always positive, so the rate of change of dislocation density has the sign of p−qm|. But from Eq. (1), | is a decreasing function of z. As a result, for a given value of |a, z will be positive in regions where z is large and negative in regions where z is small, so that spatial irregularities in z will amplify. While this example has shown that kinematic considerations will in general intensify fluctuations in dislocation density, there are also demonstrations that one specific dislocation pattern will lead kinematically to the formation of another specific pattern. Thus, Gassenmeier and Wilkens [13] considered a slip band in single glide. The glide process is one of simple shear, with a rotational component, which causes the slip band to make a small angle with the slip plane. As a consequence, compressive fibre stresses are set up in the slip band. These activate secondary slip systems, and the dislocations of these systems interact with those of the primary system to produce dislocation tangles along the borders of the slip bands, while the interior of the slip bands has a low density of dislocations. Similar hollow-cored slip bands can occur in multiple-glide situations [14].
3.2. An essentially energetic model of patterning A basic difficulty in any energetic model of dislocation patterning is to explain the breakdown of the principle of similitude. If dislocation pattern A is favoured over pattern B under a stress |and with a mean dislocation density z¯ , how can B be favoured over A if the stress is increased to k| and the mean dislocation density to k 2z¯ ? Another basic difficulty is that the configuration of lowest energy is that in which all the dislocations have escaped through the free surface of the sample. It is essential [15] to introduce some constraint on the permissible distributions. A further difficulty, which can probably be resolved only by introducing kinematic considerations, is that of determining the intensity of the concentrations of dislocations in a given type of pattern. Patterns form because the dislocations screen each other’s stress fields. If the local dislocation density is z(x), the energy per unit length of dislocation is of order −(vb 2/4y) ln(bz 1/2). Here, bz 1/2 is a small fraction, and its logarithm is numerically large unless z is very large. Suppose we consider a random array of edge dislocations. The energy per unit length can be reduced by allowing the dislocations to multiply and form a Taylor lattice of very high density. Even with the additional constraint of constant length of dislocation line, these dislocations
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can reduce their energy by forming alternating tilt boundaries, in which the elastic field of each dislocation is effectively confined to a region of dimensions equal to the nearest-neighbour separation. The only limit to this process is when the angles of tilt become of the order of a radian; the energy continues to decrease as the dislocations arrange themselves into boundaries of ever-increasing tilt angles separated by ever-increasing distances. The constraint introduced in Ref. [15], that configuration ‘‘is approached which minimizes the free energy, per unit length of dislocation line’’, is clearly inadequate. The experimental constraint is that any change in the dislocation configuration occurs under a prescribed flow stress. While the clustering of dislocations under a constant mean dislocation density can reduce the total energy, it follows [10] from Eq. (1) that this clustering also reduces the flow stress. If the mean density is increased to keep the flow stress constraint, it can be shown [12] that the elastic energy actually increases. However, for a system under stress, the thermodynamic potential is not the energy but the enthalpy, the energy less the work done by the applied stresses. We now consider a given degree of clustering in cells of different sizes. The increase in energy for each dislocation initially present turns out to be independent of the cell size, but the work done by the external stresses is proportional to the distance the dislocation moves, that is, to the cell size, and also to the applied stress. For the enthalpy to decrease, the size of the cell must exceed a critical size, which is inversely proportional to the applied stress. Again we face the problem that the thermodynamic driving force for cell formation is larger, the larger the cell. The solution is that the growth is an amplification of an existing fluctuation in the dislocation density. The larger the initial fluctuation both in amplitude and in wavelength, the more rapid the growth. The statistical probability of a fluctuation is a decreasing function both of the amplitude of the fluctuation and of the size of the region in which the fluctuation occurs. Those cells will predominate for which the product of the statistical probability of a fluctuation and its rate of growth is largest. It turns out that these cells have twice the linear dimensions of the smallest cell which can grow under the given applied stress, and grow from a statistically likely fluctuation. The side of the cell is about 12 times the mean separation of dislocations in a random distribution.
4. The validity of an energetic model A plastically deforming metal is a system driven far from equilibrium. Under isothermal conditions, usually
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less than a tenth of the work done on the sample remain stored in it [16]. How can we apply quasi-equilibrium energetic considerations to such a system? The answer may appear by analogy with the phenomenon of the acceleration of creep by a phase change or by thermal expansion in a polycrystal of anisotropic grains. The phase change or thermal expansion causes large microstresses, which produce local plastic deformation without any macroscopic change of shape. Even a small imposed homogeneous stress biases these plastic deformations, producing macroscopic deformation. The present problem is the converse of this. External stress renders the whole sample plastic, and produces a homogeneous dislocation flux which does not produce any change in the local dislocation density. Once the external stress has rendered the dislocations mobile, local fluctuations in dislocation density can locally bias this flux in the way described in Section 3.2.
5. Mechanisms which become more effective at high dislocations densities In an energetic approach, one preferred dislocation pattern may give way to another as the dislocation density increases because similitude has broken down. The line energy of a dislocation is not constant, but depends logarithmically on the dislocation density. In a kinematic approach, not only may one pattern lead geometrically to another as in the work of Gassenmeier and Wilkens [13], but increasing dislocation density may influence the relative ease of different types of dislocation motion. Specific cases are climb and cross slip. Mecking and Estrin [17] calculated that with a uniform dislocation density and a strain rate of 10 − 2 s − 1, the concentration of vacancies produced by plastic deformation would equal that present in thermal equilibrium at a temperature of half the melting point Tm. They added ‘‘Attention should be drawn to the possible role of mechanically produced vacancies below 0.5Tm in case of extremely localized deformation.’’ Considering that 0.5Tm for copper is 380°C, and that the dislocation density in the cell walls is some 10 times the mean density, it seems likely that the mechanical production of point defects in cell walls could permit the redundant edge dislocations to annihilate, and so induce the transition from Stage III to Stage IV. In the case of cross slip Kuhlmann-Wilsdorf [18] suggested that ‘‘similitude must break down eventually … for example, cross slip, conservative climb or climb becomes easy’’. There is a simple mechanism for this in the case of cross slip in f.c.c. metals. Dislocations are dissociated on the glide plane. They cross slip when a combination of the glide stress and of thermal activation unites the two partials over a short segment, allowing the dislocation to dissociate on the cross-slip
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plane. Once a segment has cross slipped, the nodes between the portions dissociated on different planes can easily separate. In addition to the glide component of the stress, both the applied stress and the internal stresses, which are of comparable magnitude, have components in the glide plane and perpendicular to the glide direction. These components, proportional to the square root of the dislocation density, may either increase or decrease the separation of the partials. If the separation of two partials which are unlikely to cross slip is increased, they will still be unlikely to cross slip; if the separation is decreased, cross slip may be induced.
6. Discussion There are many attempts to apply to the problem of dislocation patterning techniques which have been successful in solving other problems. One of these techniques is that of chemical reaction– diffusion theory, which successfully predicts spatial and temporal oscillations. This approach meets with two major problems. The first is to find a component in dislocation dynamics which can be formally represented as a diffusion of dislocations. If we have, for example, only edge dislocations of one sign, their material repulsions lead to a diffusion-like term. The reality is more like a weakly charged Taylor lattice, with local small preponderances of dislocations of one or the other sign. These preponderances will relax by an effectively diffusional process, but there is no tendency for local variations in the total dislocation density to relax. (Consider, for example, a crystal containing isolated clusters of the form of Fig. 2(a).) A further serious error is the neglect of the tensor nature of the forces between edge dislocations. How can one build a theory of dislocation patterning using a model which does not recognize that a tilt boundary is a configuration of low energy? The incorporation of the real tensor forces between dislocations would probably make even two-dimensional simulations intractable analytically. The other technique is the finite-element computer modelling of dislocation motion using reasonably realistic laws of motion. There is little doubt
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that this modelling will lead to interesting results. However, suppose that such a simulation successfully models the formation of a dislocation pattern which is observed in practice, or even the transition from one pattern to another. How much insight have we gained? Suppose our model incorporates cross slip and climb. Are either or both of these processes essential to the observed behaviour? This insight can be gained only by making a series of computations with varying tendencies to cross slip and to climb, finding the range of these tendencies in which the observed pattern formation occurs, and showing that this range corresponds to the range in which this particular pattern formation occurs in practice. Such explanations probably lie only slightly beyond the present power of computers. Finally, the role of experiment must not be neglected. Experiment [19] seems more likely than theory to give an acceptable answer to the question ‘‘how do interstitial edge dipoles annihilate?’’
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