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A theory of dipolar dislocation wall structures
Materials Science and Engineering, A188 (1994) 69-79
69
A theory of dipolar dislocation wall structures M. Saxlovfi Faculty of Mathernatics and Physics, Charles University, 121 16 Prague (Czech Republic) J. Kratochvll Faculty of Civil Engineering, Czech Technical University, 166 29 Prague (Czech Republic) (Received September 20, 1993; in revised form January 31, 19941
Abstract The goal of the paper is to answer three questions: (1) to investigate the way in which the dipolar dislocation walls are formed, (2) to explain their tendency to regular arrangement and (3) to determine their geometry. The answers are based on the linear stability analysis of the proposed model of sweeping and dragging mechanisms of dislocation dipole clustering. From the set of wall orientations provided by the analysis the most stable dipolar dislocation structure actually formed is selected by an energetically motivated geometrical criterion. The model is employed for prediction of geometry of the dipolar wall structures produced by collinear slip belonging to one or two slip planes, coplanar or non-coplanar symmetric double slip, and symmetric multislip. In all cases the predicted wall geometry is consistent with the available observations.
1. Introduction
The most expressive feature of plastic behaviour on a microscale is a formation of a dislocation structure and its changes during plastic deformation. The underlying phenomenon is that many more dislocations are produced than is necessary to carry plastic deformation. The excess production of dislocations can be attributed to the uncorrelated bursts of dislocations by activated dislocation sources [1]. As a result, too many dislocations are generated. Some of them disappear at the surface but most of them become stored in the solids. In that process the deformed solid tends to reduce the internal energy by a mutual screening of elastic fields of stored dislocations. There are two basic types of ductile solid: dislocation-cell-forming materials and solids where cells are not formed. The principal difference between them is the way in which dislocations can screen their fields. In cell-forming materials the leading mechanism is the individual screening where dislocations are stored in the form of dipoles. The screening is very effective as the stress field of one dislocation is compensated by the stress field of the neighbouring dislocation with the opposite Burgers vector. Typically long bundles of primary nearly edge dislocations interact pairwise over part of their length (see Fig. 2 in ref. 2; for a review see ref. 3). From the bundles, progressively shorter dipolar loops are formed by the chopping mechanism [3, 4]. (1921-5093/94/$7.00 SSD10921-5093(94109542-5
Another mechanism of the loop production is the mutual annihilation of screw parts of mixed dislocation dipoles (see Fig. 4 in ref. 5). A large quantity of dipolar loops created behind jogged screw dislocations was reported as a dominant feature in Fe-Si crystals [6]. Observations on a wide range of materials and deformation modes prove that dipolar loops are accumulated, forming a cell structure of denser regions, commonly called cell walls, separated by regions of a much lower density [7]. On the contrary, in the solids where cells are not formed the stored dislocations screen their elastic field, collectively forming an arrangement which can be roughly modelled as a Taylor lattice [8, 9]. In this article the discussion is restricted to cellforming solids. In such materials the most typical stored dislocation arrangements are cell walls of the prevailing dipolar character with the sum of Burgers vectors close to zero. Another type of arrangement consists of thin layers of dislocations of one sign which are coupled with misorientation of the crystal lattice. A typical example is subgrain boundaries mainly observed at higher homologous temperatures or very high strain [10]. Increasing misorientations of the crystal lattice in the cell-forming materials are observed in advanced stages of deformation [11]. The walls between misoriented dislocation cells can be understood as dipolar walls with a subgrain boundary component. Another type of dislocation arrangement that © 1994 - Elsevier Sequoia. All rights reserved
70
M. Saxlovd, J. Kratochvil
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Theory of dipolar dislocation wall structures
is clearly connected with misorientations is dislocation grids (sheets) [12-14]. More extended features are bands of localized shear and persistent slip bands. The subgrain boundaries, grids and bands of localized shear which are of similar physical origin [15] but different from the origin of the dipolar walls are not discussed in the present paper. The geometry of dipolar wall arrangements was theoretically studied by Dickson et al. [16, 17]. These workers have assumed the existence of a double pseudo-polygonization arrangement of dipolar loops to predict the energetically most stable walls observed in fatigued f.c.c, metals. Comparison of the predictions of Dickson et al. with experiments is fairly good, but some discrepancies exist (they will be mentioned in Section 3). A different approach was proposed by Kratochvfl and Saxlov~i [18]. It was based on the drift and sweeping mechanisms of the dislocation wall formation. The energetically motivated criterion of Dickson et al. has been built in the proposed theory as a subsidiary selection rule. The geometry predicted by our method agrees well with observations on f.c.c. crystals deformed by double slip with the tensile axes oriented near the [100]-[110] stereographic line [18]. Similarly good results have been reached for f.c.c. crystals deformed along the tensile axis oriented in the [001] direction [19]. In the present paper the proposed theory is reviewed and applied in a systematic study of dipolar wall structures produced by single, double or multiple slip. The mathematical description of the model where one or several slip systems operate simultaneously is summarized in Section 2. The theory developed in Section 2 is applied for various cases of dipolar wall structure in Sections 3 and 4; their geometry is predicted and compared with the available observations. We used mainly experimental results on f.c.c, metal single crystals, where a variety of slip system combinations may operate according to loading conditions. In these cases, observed dislocation structures, i.e. at advanced stages of plastic deformation, are often well documented and analysed. Only the most typical papers with a detail description of the wall geometry were selected in our list of the references. However, several other relevant publications were checked and to our knowledge no experimental result seems to be in contradiction with our predictions of the dipolar wall geometry proposed in the present paper.
2. Model of dipolar wall structure formation
In the model of the cell formation proposed in the recent publications [18-28] the stored dislocations in the form of dipolar loops are swept by gliding dis-
locations or drifted by stress gradients into dense regions called cell walls [7]. In the dense regions at a sufficiently high density the dislocations annihilate, causing dynamic recovery. The glide dislocations enter the theory through the plastic strain. The proposed theory consists of three basic steps. As the first step we describe the mechanism of the dipolar wall formation and thus give the answer to the first question in the abstract, namely the way in which dislocation walls are formed. We start with the simplest one-dimensional model of single slip and then extend it to three dimensions and several active slip systems. Only cases of approximately same slip activity in all participating slip systems are considered. The reason is that this restriction simplifies substantially the subsequent analysis of the model. The stability analysis of the model given in Section 2.2. represents the second step. It is shown that a homogeneous dipolar loop distribution is unstable and tends to form a periodic arrangement of dipolar walls. Hence this section answers the second question in the abstract, i.e. an explanation of their tendency to regular arrangement. Only the linear stability of the model is analysed which is not fully adequate for the later stages of plastic deformation where the cell walls are sufficiently developed. Nevertheless, the analysis seems to provide a firm base for the prediction of the dipolar wall geometry. However, the present form of the analysis does not predict the geometry of walls uniquely; a spectrum of wall orientations which depends on the number and combinations of slip systems activated is predicted. Therefore, as the third step, two subsidiary selection rules are formulated (Section 2.3) which select out of the spectrum the most stable wall orientations with the highest probability to be actually formed. The linear stability analysis of eqn. (3) supplemented with the selection rules determines the dipolar wall geometry. 2.1. The model Consider first a one-dimensional model. The balance law governing the evolution of the local density p(x, e) of stored dipolar loops can be expressed in the form
0p Oe
0 Ox J + g
(1)
where x is the coordinate and the plastic strain e serves as the evolution parameter. Equation (1) means that the rate of accumulation of stored dislocations per unit strain at place x must be equal to their net flux - OJ/Ox plus the generation rate g of stored dislocations per unit strain. The rate g is positive (generation) for a low density p and becomes negative (annihilation) for a
M. Saxlov6,J. Kratochvil / Theoryof dipolar dislocation wallstructures sufficiently high p. The expression for the flux J was deduced from the fact that dipolar loops are drifted on the average towards stress concentrators [20, 21]. The stress concentrators can be created by glide dislocations stuck at already-formed clusters. Because of this effect, a fluctuation in the dipolar loop density can serve as an embryonic cluster from which a dislocation pattern may develop. In ref. 21 it was shown that the drift mechanism is an elementary consequence of the dislocation theory. The alternative and probably a more effective mechanism based on the sweeping of dipolar loops by glide dislocations was proposed in refs. 18 and 27. Perhaps both mechanisms promote the dislocation pattern formation and support each other. They are of a similar nature and in the first approximation lead to the same expression for the flux J: +co
J=D ~ f M(a)p(x+a)da Ox
(2)
In both mechanisms the flux is of a dislocation line tension origin. In the dipole drift theory [21] the integral term represents the bowing stress needed to get glide dislocations through the space between clusters. The flux of dipole loops is caused by the gradient of this stress. In the sweeping mechanism [27] the line tension is responsible for the shape of glide dislocations through the integral term and their slope is proportional to the driving force of the loop flux. The differences in the expression for the flux J are of the second order; the factor D is positive ("uphill diffusion") in both theories and generally depends on p. The generalization of the model described by eqns. (1) and (2) for three dimensions and several slip systems was proposed by Kratochvfl and Saxlovfi [18]. In that case the balance law for the total dislocation loop density p (the sum of the densities of loops produced by the N active slip systems) is
Op(r, e) + D Z div J grad j × ~
,1
I
o3
J M(a)p(r+ ab ~, e) da = Ng
(3)
cO
where b J is the unit vector in the Jth slip direction, r is the position vector, and div J and grad j mean the standard differential operations with respect to Jth slip direction. Taking the common evolution parameter e, we suppose that the slip activity of all slip systems considered is the same and uniform. Moreover, for simplicity it is assumed in balance law (3) a certain degree of isotropy; the function M(a) and the factors D and g are supposed to be the same for all active slip systems. The linear stability analysis employed in the present paper admits only D and g constant.
71
The active slip systems enter balance law (3) only through the slip vectors b J, J= 1, ..., N. This is the consequence of the assumption that dipolar loops are treated as point objects, which can be moved only in their own slip direction. The main reason for introduction of such simplification is that then the threedimensional theory of dislocation clustering is easily formulated. On the one hand the simplification is justified as the size of the loops is usually an order of magnitude smaller than the wavelength of the formed dislocation pattern. On the other hand there is a disadvantage; as shown at the end of this section, the linear analysis of model (3) does not determine the wall orientations uniquely. From the predicted spectrum of wall orientations the walls actually formed are selected by two subsidiary rules. The rules incorporate implicitly the finite size of dipolar loops, their interaction and slip plane geometry; in this way the rules subside the simplification introduced into model (3). The main tool in our study is the linear stability analysis of model (3). As more apparent dipolar wall structures exist typically at later stages of the dislocation pattern development, a non-linear analysis of the model of the wall formation would be more adequate. A non-linear approach would require incorporation of the effect of the annihilation of dipolar loops within clusters. The annihilation starts when the density of loops within a cluster becomes sufficiently high. A simple model of the role of the annihilation in the process of the dislocation pattern formation was proposed by Fran~k et al. [26] and studied in ref. 28. The results indicate that the most distinguished consequence of the annihilation is thinning of the dislocation clusters and the formation of dipolar walls. Two accompanying effects, namely a change in the wavelength of the dislocation pattern [26, 28] and a cell misorientation analysed by Kratochvfl [23, 24], do not seem to modify the basic geometry of the pattern as derived from the linear stability analysis. As the model used by Fran6k et al. [26] and Zatloukal [28] is much simplified, we are aware of the limited applicability of this conclusion. Nevertheless, we try to explore what information about the dipolar wall geometry we are able to obtain using the available simple linear means.
2.Z Linear stability analysis of the model When investigating the linear stability of the uniform distribution of loops described by eqn. (3), one is looking for the fastest-growing perturbation in the form of an infinitesimal wave. The perturbed solution of eqn. (3) is assumed in the form
p(r, e)= fi(e) + Ap exp(ik-r+ we)
(4)
where fi(e) represents the homogeneous solution of eqn. (3), Ap the infinitesimal amplitude of the per-
M. Saxlovti,J. Kratochvil / Theoryof dipolar dislocation wallstructures
72
turbation, k the wavevector and ~o the amplification factor, which determines the rate of growth (~o >0) or decay (~o<0) of the perturbation. Using eqn. (4), eqn. (3) yields
where/i is a non-zero solution of
2R(p)+/~ dR(/~) dp
=0
(13)
N
w(k) = D ~, (pk~)ZR(pkJ)
(5)
Note that in this case condition (10) is satisfied as
J=l
where pk J is the scalar product pk J= k. b J and R(p) represents the Fourier transform of M(a): oo
R(p) = f M(a) e x p ( i p a ) d a
(6)
As in physically realistic cases the function M(a) gives a positive value of R(p) and D is positive, the perturbation grows exponentially and the uniform distribution of dipolar loops is unstable. In the first approximation the wave with maximum oJ given by eqn. (5) determines the geometry of the dislocation pattern. The conditions for the maximum 09 are
0 o(k)
Ok-----/-=O
i = 1, 2, 3
3
Z b / K i = s pJ ^
h
cJLs L
i = 1, 2, 3
(8)
where bi J are the components of the unit slip vector b J, X ~= X(p J) and
X(p) =p(2R(p) + pdR(p)l dp ]
(9)
Conditions (7)2 of the maximal co are
(16)
where h is the number of linearly independent vectors b J and c JL are the coefficients of linear dependence: h
b~= ~, cJLb L
(17)
The reason for the restriction is that the condition Is J[ = 1 can be satisfied for certain combinations of sL only.
The symmetry of the slip directions b J determines the symmetry of K. Let ~m be the symmetry axis of the set of vectors {sJbJ}m for a given combination {s J}m. Then the component Kem of the vectors K m in the ~m direction is t~
K~ m= +- "m Z ((b/) 2dX(p) /<0 dp p=pJ/
J=l
3
J = 1,..., N
i=1,2,3
(10)
(11)
i=1
where pJ are determined by eqn. (9) for X(p J), which is a solution of system (8). System (11) has a solution if and only if X(p J) = 0 for all J and J = l ..... N
(18)
be
The unknown components K~ of the wavevector K can be found from the relations defining p J, i.e.
IPJl=P#0
J=h+l,...,n
L=I
L=I
J=l
Z biJKi =PJ
(15)
for all possible combinations of signs s J = + 1. However, for linearly dependent b J the choice is restricted by the condition
N
Z b/X ~= 0
J=l,...,n
i=l
(7)
Three equations (7)1 for three unknown components of the wavevector K corresponding to the maximal ~o can be written using eqn. (5):
(14)
From eqns. (11) and (12) it follows that for n noncollinear slip directions bJ(n<~N) we are looking for the set {K} of vectors K, which satisfy the relation
sJ= Z
0%(k) 0k~<0
dX(p) = -2R(/~) < 0 dp t,=~
(12)
where b~m= sJbJ with s j from {s J}" and the component b J of the unit vector b J in ~m direction. The vector Ko m parallel to ~m is the smallest of {Kin}, K0 m= K~". For the single-slip direction the axis ~ is identified with that direction, K~ is determined by eqn. (18) and the other two components perpendicular to ~ are arbitrary. In the case of double slip with h = 2, eqn. (18) is valid for the component K~ in the plane of the vectors b 1 and b:, and the component of K perpendicular to that plane is arbitrary. For multiple slip with h = 3 the discrete wavevectors K m coincide with the vectors K 0m.
M. Saxlov6, J. Kratochvil
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Theory of dipolar dislocation wall structures
In summary, the linear stability analysis of model (3) yields the set of the wavevectors {K} which determines the geometry of the dislocation pattern; the potentially possible orientations of the dipolar walls identified with the fastest-growing perturbation waves are perpendicular to the wavevectors K from {K}. 2.3. Selection rules Generally, the stability analysis in Section 2.2 provides a whole spectrum of wall orientations given by the set {K}. The set {K} consists of the solutions {K m} for the permitted different combinations {sS}m in eqn. (11) with 3 - h arbitrary components, i.e. arbitrary magnitudes K m~>K0" of the wavevectors. The distances between the walls are equal to the wavelength 2 given by the magnitude K of the wavevector K, i.e. 2 = 2Jr/K. Hence, very small or even zero wavelengths of the dislocation pattern are permitted by the model for the case of one or two active slip directions. This is not physically realistic. According to eqns. (6), (9) and (13) the wavelength is controlled by the shape function M. As noted by Kratochvil and Saxlov~i [18] we can expect that a more sophisticated form of M should exclude this possibility of too short wavelengths. Therefore (i) we admit in {K m} only the wavevectors of magnitudes not far from the minimum K0m. The second effect, which influences the determination of the walls actually formed, is a preferred formation of dipolar walls with a lower energy. Since the walls considered are dipolar in nature, they all correspond to low energy configurations and the difference between the energy of different walls are of the second order. Therefore, following Dickson et al. [16], we take as an energy criterion the tightness of the stacking of the dipolar loops for the walls. The tightest stacking of the loops can be obtained by the connection of loops along their short segments. The walls which contain the axis ~J1 bisecting the acute angle between the directions e j and e z of edge (longer) parts of dipolar loops of the slip systems J and 1, i.e. for which K m. ~ l J = 0, are stabilized by the energetically favoured stacking of the building loops b J, e J and b 1, e l (Fig. 1). The angle r/gl
A
Fig. 1. The tightest stackingof dipolar loops egand e ~along their short segmentsin walls containingthe axis ~ Jt bisectingthe acute angle between them.
73
between the normal K to the wall and the axis ~ sl can be taken as a measure of the tightness. Therefore (ii) we suppose that angles ~/not far from ½x favour that wall orientation. The method of determination of wall orientations based on the linear stability analysis of eqn. (3) supplemented by selection rules (i) and (ii) will be analysed in detail for one or two active slip systems in Section 3. Multiple-slip wall orientations are discussed in Section 4.
3. Single- and double-slip walls
For the single-slip direction b belonging to one or two slip planes (primary system or primary and crossglide systems) the axis ~ is identified with the vector b. The most favoured walls with the smallest wavevector Ko(K o =/3) and also the tightest loop stacking (t/= l~r) are perpendicular to b. This is exactly the optimal orientation observed in the ladder structure of persistent slip bands in f.c.c, single crystals [29-32] and walls in cyclically deformed Fe-Si crystals which results after local dissolution of veins [33]. For the two different slip directions b ~¢ b 2 (the most common form of glide deformation) the set {K} consists of two continuous spectra {K 1} and {K 2} (Fig. 2). In the coordinate system x, y, z with the z axis parallel to the axis ~ bisecting the acute angle 2~b between the slip directions b ~and b 2 we obtain
+ K2 = + ~ P ,Kv, 0 - sin ¢
(19)
where the component Ky of K is arbitrary and /3 is given by the influence function through eqn. (13). The distances of the walls measured along the axes ~1 and ~2 (i.e. in the plane of Burgers vectors b I and b 2) are (2er cos #)//3 and (2rising)//3 respectively, in agreement with the two-dimensional model for double slip studied by Kratochvfl [23]. This is indicated by the traces of the walls shown in the picture on the righthand side in Fig. 2. The walls corresponding to the set {K} (19) are represented by the planes (~2) for the vector {Kl}, and the planes (~1) for the vectors {K 2} (the symbol (abc) means planes containing the direction [abc]). The restrictions of the set {K} imposed by supplementary rules (i) and (ii) will be specified separately according to the mutual orientation of active slip systems. The resulting spectrum of the predicted wall orientations for a given double slip is governed by the orientation of
74
M. Saxlov6, J. Kratochvil /
Theoryof dipolar dislocation wall structures
x II{2 /~/sin ¢
I
I y -
./," .
I I
,
Ko
IK,\
fj" , \
zlI~S symmetry of glide
I {R1}//
,
K;
/1 ,-
{rd}t
, oo~-
\~/COS ¢ {~ }={~1}+{~2}
traces of watts {K} in section ('bl.6z)
Fig. 2. Geometry of the walls K for the double slip b 1and b 2.
!
{1
[0111
[112] 1 [101]
//•11 21
--T .... .
fK'0
-tl 01 t
|'¢'31t/~
e''~
trace (111) traces of walls
1
{a) (b) Fig. 3. Geometry of the coplanar slip b ~and b 2 in the common slip plane. (a) The traces of walls K02 are parallel to ~ 2, i.e. (b) to the traces of the critical slip plane in f.c.c, crystals.
100 -"
_ 210
110
Fig. 4. Stereographic triangle with the main slip system (11i~101].
the axis ¢ (bisecting the acute angle between the edge dislocation directions) with respect to axes ~1 and ~ 2 of the slip directions. Four different cases, which cover all possible combinations of double slip in f.c.c, metals, are considered and shown together with collinear slip later in Fig. 7, where the coordinate system ~1, y, ~2 of Fig. 2 is used. In all cases the equal slip activities are assumed in both considered slip systems.
It is interesting to note that the activation of the third coplanar system (in f.c.c, or h.c.p, metals) provides no new wall geometry different from that produced by the combinations of the double slips (b 1, b2), (b l, b 3) and (b 2, b3). The reason is that, in the case of three coplanar slip directions, condition (16) is not fulfilled for any set of signs s J= + 1 considering all three slip directions, J = 1, 2, 3. The traces of walls in the slip plane should lie along the slip directions (compare for example Fig. 10 in ref. 13, Fig. 8 in ref. 40, and Fig. 4 in ref. 41).
3.1. Case 1: Coplanar slip Coplanar slip on a dominant glide plane has been reported by numbers of workers. It often prevails over glide activity of the other secondary dislocations (see for example refs. 2, 11-13 and 34-36). Coplanar slip predominates locally even in some crystals oriented for multislip ( ~ ref. 37 for [110J-oriented Cu crystals) or in polycrystals [38, 39]. For coplanar slip the bisecting axis ~ is parallel to the axis ~2 and the preferred wall orientations <~2>(i.e. for which r/=½~z) selected by rule (i) are close to the orientation of the wall perpendicular to the axis ~1 (Fig. 3(a)). The predominant wall orientations are of the type {112} in f.c.c, metals. The traces of the predicted walls in the common slip plane are parallel to ~2, i.e. to the traces of the critical slip plane of f.c.c. metals (Fig. 3(b)), in agreement with the observations mentioned above.
3.2. Case 2: Symmetric non-coplanar slip: perpendicular Burgers vectors For symmetric non-coplanar double slip, no active slip plane contain both slip directions. Choosing perpendicular slip directions we consider two active slip systems ( 11 ] )½[101 ] and ( 111 )½[10 i] as shown below in Fig. 5. They represent the predominant primary and critical slip systems for a f.c.c, metal crystal oriented near the center [210] of the [100]-[110] stereographic line (Fig. 4). The two axes of symmetry are ~1 -[100] for s z = s 2 and ~2--[001] for s 1= - s 2. For the considered slip systems the directions of the dipolar loops are e 1[1[i21], e 21[[121] (Fig: 5) and the axis ~ of both loop directions lies along [120], i.e. perpendicular to ~2, as shown in Fig. 6. From the (100) spectrum the wall of orientation (001) is the most favoured by the stability criterion (ii)
M. Saxlov(~, J. Kratochvil
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Theory of dipolar dislocation wall structures
~ •
i~21~[ool] ~
~o)
[
/I
Ie'j"-tL..../1 J Fig. 5. Geometry of the double slip (Ill)½[101] and (1 1 1)~[10i].
75
./i
~
tl00}[
~'v4
....
J
/
/~'11[100] Fig. 6. Ideal stacking of dipolar loops in the walls.
(r/= 90°). The (001) wall is also the wall corresponding to the shortest wavevector Ko 2 from that spectrum. In the (001) spectrum the tightest stacking of the dipolar loops occurs for the (210) wall (~/= 90°), and the magnitude of the wavevector corresponding to the (210) wall is 1.11K01. From the (001) spectrum, the (100) wall belongs to the shortest wavevector with the magnitude K0 ~ and the measure of tightness ~/= 63.4 °. The dipolar loop arrangement in the (001 ), (210) and (100) walls are shown in Fig. 6. It is interesting to note that, according to L'Esp6rance et al. [42] (for a summary see ref. 17), the (001) walls and the walls intermediate between the orientations (210) and (100) are actually observed in dipolar wall structures formed by considered double slip. 3.3. Case 3." Symmetric non-coplanar slip: nonperpendicular Burgers vectors The third type of double slip, which causes a marked structuralization in f.c.c, metals, is the simultaneous action of primary and conjugate slip systems with the slip direction b 2 out of the primary slip plane and the angle 2~b = ]Jr between both slip directions [40, 42-45]. Considering (11])11101] plus (111)½[110] slip systems with the axes ~l, ~2 and ~ along [211], [01]] and [233] respectively, we predict similarly as in case 2 three prominent wall orientations. The tightest stacking of the dipolar loops occurs for the (011 ) walls from the (211) spectrum and for the (311) walls from the {01 ]) spectrum. The application of rule (i), which selects the wall orientation corresponding to the shortest wavevector, to the (211) and (01 i) spectra indicates another possibility. All walls between (311) and (211) walls from the (01 ]) spectrum have a relatively high stability. Their measures of tightness are greater than ,/(211)=80 ° and the magnitudes of the wavevectors are less than 1.01K0 ~. All these predicted walls are actually observed (see the papers mentioned above). 3.4. Comment I: On the rule of Dickson et al. The preferred wall orientations in three considered typical cases of double slip in f.c.c, metals deformed in tension along the directions indicated in Fig. 4 are
TABLE 1. Dipolar walls in f.c.c,crystals deformed by symmetricdouble slip Dipolar wallsfor the following tensile axes
Predicted from {Kt} Predicted from {K2} Dickson et al., wallsW2 Dicksonetal.,wallsWl
[221] Case 1
[2101 Case2
[211] Case 3
(112)
(210)-(100) (001 ) (210) (001)
(311)-(211) (01 ] ) (311 ) (01])
(112)
summarized in Table 1. The orientations of walls of labyrinth structures for the considered double slips as predicted by Dickson et al. [16, 17] are also listed. As was shown for all three cases of symmetric double slip, the wall orientations predicted by the proposed method are in full agreement with the available observations. On the contrary, as Table 1 indicates, the wall geometry prediction by Dickson et al. [16, 17] for non-coplanar slip is too restricted in comparison with the available experimental data. It should be noted that the principal difference between our approach and the considerations of Dickson et al. [16] is that in our case the basic wall orientations are determined by the pattern-forming process, and we are not a priori restricted to polygonized walls only. 3.5. Case 4." Non-coplanar slip: both Burgers vectors lying in one active slip plane The formation of the most preferred wall orientations with the shortest wavevectors K01 or K02 and simultaneously with the tightest stacking of loops (r/=½:r) is excluded for all less symmetrical oriented slip systems where the axis ~ is not perpendicular to any axis ~. An example is double non-coplanar slip, for which the slip plane of one system contains both slip directions b 1 and b 2. Such pairs of slip systems can be equally activated in f.c.c, crystals deformed along the corner orientations (100) or (111).
76
M. Saxlov6, J. Kratochvil
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Theory of dipolar dislocation wall structures
Let us consider for example two active slip systems (lli)½[101] and (ill)½[ll0] with the axes ~1, ~2 and along the directions [211], [01]] and [231], respectively. Similarly as for the symmetric non-coplanar slip analysed above we predict, from the (~2) spectrum, wall orientations intermediate between the (211) walls having the smallest wavevector K01 and a relatively high stability such as r/= 77.4 ° and the (111) walls with the tightest stacking of dipolar loops (r/=~ ) and K = 1.06K01. The formation of (~ 1) is, however, rather suppressed because, for the (01 i) walls preferred by condition (i) the stability measure r/ is only 40.9 ° and the most stabilized (102) walls from that spectrum has relatively high K = 1.58K02. The consequence of this situation is a compromise: walls in the vicinity of the (113) walls with K = 1.17/(02 and r/=71.2 ° are assumed to be observed (Fig. 7). The predictions of case 4 cannot be checked directly as it appears only as a component of multiple slip. Case 7 of the following section, Section 4, shows that case 4 represents just these pairs of slip systems which cause rather a low stability of multiple slip walls in (111)oriented crystals.
tions. Consequently, in these regions the occurrence of multiple-slip walls can be expected. In the case of multiple slip the system of n equations (15) (n~>3) yields four solutions K m or less than four under restriction (16). The actual formation and stability of the multiple-slip wall with the orientation K m depends on the additional conditions stated in selection rules (i) and (ii). To check these conditions we use the fact that the solutions K m are also solutions for all pairs J, I of slip systems J, I = 1.... , N. This is evident from the structure of eqn. (18). For each pair J, I of slip systems we can determine how close is the magnitude of a particular K m to the corresponding smallest one--selection rule (i)--and how K m- ~zs is close to zero, i.e. if the corresponding r/ is close to lzr--selection rule (ii). The formation of the K m wall and its inherent stability is determined by the number of double-slip systems for which K m are included in the spectra of the predicted and effectively formed wall orientations in the sense of Section 3 (Fig. 7). The dipolar loops produced by such slip systems are building blocks of the multiple-slip wall K m. The described method will be illustrated by cases 5-7 of multiple-slip in f.c.c, crystals.
4. Multiple-slip walls
4.1. Case 5: Multiple slip o f (lOO)-oriented fc.c. crystals
In the case of several equally stressed slip systems, e.g.f.c.c, crystals with corner orientations, some regions of the crystal are deformed in more than two slip direc-
Consider a [100]-oriented f.c.c, crystal with eight equally stressed slip systems deformed by_multislija with four slip directions b s along [110], [110], [101] and [101] ( N = 8 ; n = 4 , h = 3 ) and four slip planes
i~=
!~LG
[ ~ ,~2
1
I
I
.7 •
I
/ " {lib
I,~z
/ i / q :90 ° 61=~ z
I¢ 2
Ko q=90"
-'l
j ..:__--
//{1
"~90"
CASE 1
~=60"
I~ 2
n=/*l"
n=90"
K,
Klo •" ~ " "
__
/
/ " q=90"
/'{1
•
/
=
.
CASE 2
I
c'/'77"-
i
...
CASE 3
90" ! I
CASE 4
Fig. 7. The sets of the predicted walls orientations for different cases of two slip systemsin f.c.c,metals.
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
A--(111), B=(11I), C ~ - ( l i l ) , D-=(111). * From eqn. (15) choosing {s J}'n in agreement with eqn. ( 1 6 ) w e obtain S 1 = S 2 = S 3 =S 4 s'
=
-s
2 =
-s
3
~
=s 4
s l = - S 2 = $ 3 = --S4 ~
K'
I1[100]
K 1 = 21/2~
[011]
K 2 = 2~
K311101]-]
K 3 = 2[~
K 2 II
Three mutually perpendicular walls (100), (011) and (01l) form cells with the aspect ratio 21/2 along the stress axis [100] (Fig. 8). The combinations of two slip systems which fulfil the selection rules (i) and (ii) in the sense of Section 2 for the transverse walls K 1 and the longitudinal walls K 2 and K 3 are listed in Table 2. Other pairs of the active slip systems do not fulfil either selection rule (i) as K l >2.4K 0 or (ii) as q < 50 ° for walls K 2 and K 3. By inspection of Table 2, we see that all three wall orientations listed are effectively formed and stabilized; in the transverse walls formed by all eight slip systems, each loop type is stabilized by the tightest stacking with two other types. The longitudinal walls, however, are formed non-concurrently. Each consists of loops of four slip systems and is stabilized by the stacking of loops in one coupling pair. The cell structure in deformed {100}-oriented f.c.c. crystals is documented by transmission electron microscopy observations (see for example 47-51) and agrees well with our prediction. The dipolar wall geometry of the considered case is conditioned by a nearly equal local slip activity in all eight slip systems. This condition was satisfied for the deformed [001] copper crystals by Takeuchi [52] for which Kawasaki and Takeuchi [49] actually found the cell structure of
TABLE 2. The sets of acceptable pairs of equally stressed slip systems in [100] f.c.c, crystals which effectively form and stabilize the walls Sets of pairs for the following walls Transverse (100)
B2+CI A2 + D1 C3 + B4 A3 + D4 Ko 2 q=½~ Case 2
Longitudinal
A2+C1 B2 + D1 A3 + B4 C3 + D4 Kll I t/= 63.4 ° Case 2
(011)
(Oli)
A3 + D1 A2 + D4 Ko 2 ~/=½~ Case 3
B4 + C1 B2 + C3 KII2 ~/=~n Case 3
the predicted geometry. Their examination of the foils perpendicular to the tensile axis shown that the apparent circular traces consist of short sections of predicted (110) and ( l i 0 ) walls. Moreover, in the longitudinal foils the shape of the cells was elongated in the direction of the tensile axis, as indicated in Fig. 8. The appearance of a fine distribution of the longitudinal walls corresponds to a non-concurrent formation of both walls. A more effective formation of the transverse walls is documented by the observation of condensate (100) walls in [100] fatigued copper [50]. The apparent spherical shape of individual cells with all six slip directions (110) uniformly distributed in an almost isotropic cell structure reported by Van Drunen and Saimoto [47] and Goettler [48] can be easily explained. The activation of all six slip directions provides no new geometry different from that corresponding to any combination of four non-coplanar slip directions. Basically the nonconcurrent formation of walls of types [100/and {0111 implies only short walls in regions with all six (110) slip directions and thus makes the cell shapes apparently circular.
4.2. Case 6: Multiple slip of (Oll)oriented fc.c. crystals The wall geometry of [011] and [100] deformed crystals is similar. In both cases the slip directions 1, 2, 3 and 4 are activated, giving rise to the same wall geometry of the picture on right-hand side of Fig. 8. However, unlike the [100] case, the [011] orientation of applied stress activates only four equally stressed slip systems A2, A3, D1 and D4. Thus the (01i) wall (longitudinal for the [011] stress axis) cannot be formed. In fact, in sample sections (011) of [011] deformed crystals a "banded" structure with traces along [01i] (i.e. only those of the longitudinal (100) walls) was observed by Monteiro and Kestenbach [53].
4.3. Case 7."Multiple slip of (l l l)-oriented fc.c. crystals We consider a [1111-oriented f.c.c, crystal deformed in three non-coplanar directions b J along [110], [101] and [0111 (n = h = 3) by the activity of six slip systems B4, B5, C1, C5, D1, D4. The solutions K m for {s J}m in
[100] If oppl.iedstress f.c.c.c. !~1
[o1~]
*The notation used for the slip planes (A, B, C and D) and the slip directions ( 1, 2 ..... 6) follows that of Schmid and Boas [46].
77
i
.~(
~100) ,~ I
~_ _ x[~ ~/~
Fig. 8. Multiple-slip dipolar walls in the [ 100] f.c.c, crystal.
~/'2II/p
78
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
eqn. (18) are S 1 =$2=S
3
KIll[Ill]
~
sl=-s 2=-s 3 ~
K211[113]
s l = - s 2=s 3
~K3II[I~I]
S 1 :S
~
2 ~. - - S 3
K4I[[~ll]
Here K a =/~6~/2/2; g 2 = K 3 = g 4 = / 9 ( 2 2 ) 1 / 2 / 2 . The double-slip combinations for which K m belongs to the predicted effectively formed stable wall orientations are listed in Table 3. Similarly as in case 5, the transverse ( 111 ) walls, which correspond to the vector K ~, are formed by all six slip systems and each type of building loop is stabilized by coupling to other two types. For the longitudinal (113), (131) and (311) walls the situation is less favourable. For each wall, only four slip systems participate and the loops are coupled to only one other type of loop. The rest of the loop types can form double-slip walls or contribute to other multiple-slip walls. This situation probably explains the inherent geometrical variability of dipolar loop accumulations documented in [111J-oriented grains of the polycrystal [53] or in [111] crystals [49].
5. Discussion
The proposed theoretical model is expressed through single eqn. (3). It describes in mathematical terms how dipolar dislocation walls are formed by the sweeping and drift mechanisms, as was suggested and analysed in the series of papers in refs. 18-28 and is summarized in Section 2.1. The condensed form of the theory was achieved by introducing three main simplifying assumptions: (a) the slip rates of all active slip systems are the same and uniform throughout the deformed crystal; (b) dipolar loops are treated as point TABLE 3. The sets of acceptable pairs of slip systemsforming walls in [111] f.c.c,crystals Sets of pairs for the followingwalls Transverse (111)
Longitudinal (113)
B4+D1 B5+C1 C5 + B4 C1 +D4 D4+B5 D 1 + C5 1.06K01 q=½ar Case 4
(131)
(311)
B4+D1 B5+C1 C5 + B4 C1 +D4 D4+B5 D 1 + C5 1.17K02 q=71.2 ° Case 4
1.17 K02 r/= 71.2° Case 4
Case 4
objects, (c) no mutual interaction among dipolar loops is considered. If we could relax simplifying assumption (a), the stress and the corresponding continuum mechanics equations have to be incorporated into the theory as the stress field controls the local slip [21, 25]. The assumption of the uniform slip rates is meaningful in later stages of plastic deformation, where a sufficiently strong internal stress is already built and compensates the influence of inhomogeneities caused by dipolar walls. The considered wall structures represent a nearly stabilized situation; hence the smaller role of the stress field is not too surprising and the simplification (a) introduced can be tolerated. The stability analysis of eqn. (3) shows that the uniform distribution of stored dislocations is unstable and there is a tendency to form regularly arranged dipolar walls. The mathematical method of the analysis expressed through eqns. (4)-(17)looks technically complicated, but it is just a logical extension of the very simple procedure of determination of dipolar wall geometry suggested in ref. 23. In ref. 23 the somewhat artificial two-dimensional model of symmetric double slip was analysed. The proposed extension of the procedure to the more realistic three-dimensional model valid for any combination of active slip systems is geometrically more complicated. However, there is another problem. Because of the simplifying assumptions (b) and (c) the determination of dipolar wall geometry by the linear stability analysis of three-dimensional model (3) is not sufficiently specific. A whole spectrum of dipolar wall orientations is permitted. This inadequacy of the model is compensated by introduction of two selection rules in Section 2.3, which determine from the spectrum of permitted wall orientations the most stable dipolar wall configuration with the highest probability of actually being formed. As was indicated in Section 2.3, the first rule which admits only short wavevectors compensates simplification (b). The second rule which prefers the tightest stacking of dipolar loops for the walls compensates simplification (c). In that way the linear stability analysis of eqn. (3) supplemented with the two selection rules determines the wall geometry uniquely. A natural question arises why the simplifying assumptions are not relaxed and the effects of the nonuniform slip rates, the finite length of dipolar loops and the interaction among them are not built directly in a unified model. Such an attempt can be made, but we would lose the principal advantage of the present simplifying approach: the possibility of analytical solution of the problem and the corresponding deeper physical insight in dipolar wall geometry which is demonstrated in the seven cases analysed in Sections 3 and 4.
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
A n o t h e r limitation of the present m e t h o d is the restriction to the linear stability analysis. A non-linear analysis, which would include annihilation of dipolar loops, should predict realistic d e v e l o p m e n t of the dipolar wall profile and the d e p e n d e n c e of the distance between walls on the applied stress. T h e a p p r o x i m a t e uniformity of slip rates assumed in the present p a p e r should be a consequence of the theory. It is interesting that despite all the simplifications the p r o p o s e d m o d e l is successful in the prediction of the correct wall geometry as d o c u m e n t e d by the results in Sections 3 and 4.
20 21 22 23 24 25 26 27 28 29 30 31
References
32
1 J. Kratochvfl, Czech. J. Phys. B, 38 (1988)421. 2 C. S. Pande and E M. Hazzledine, Philos. Mag., 24 (1971) 1039. 3 S. J. Basinski and Z. S. Basinski, in E R. N. Nabarro (ed.), Dislocations in Solids, North-Holland, Amsterdam, 1979, p. 261. 4 U. Essmann, Phys. Status Solidi, 12 (1965) 707. 5 U. Essmann and M. Rapp, Acta Metall., 21 (1973) 1305. 6 M. Mori, M. Tokuwanu and T. Miyazaki, Philos. Mag. A, 40 (1979)409. 7 J. Kratochvfl and M. Saxlov~i, Phys. Scr., T49 (1993) 399. 8 G.I. Taylor, Proc. R. Soc. London, Set. A, 145 (1934) 362. 9 D. Kuhlmann-Wilsdorf, Mater. Sci. Eng., 86 (1987) 53. 10 D.A. Rigney, R. Divakar and S. M. Kuo, Scr. Metall. Mater., 27(1992) 975. 11 I. Baker, B. Ralph and N. Hansen, in P. O. Kettunen and T. K. Lepist6 (eds.), Proc. 8th Int. Conf. on the Strength of Metals and Alloys, Pergamon, Oxford, 1988, p. 277. 12 J.W. Steeds, Proc. R. Soc. London, Ser. A, 292 (1966) 343. 13 U. Essmann, Phys. Status Solidi, 12 (1965) 723. 14 H. Mughrabi, in A. S. Argon (ed.), Constitutive Equations in Plasticity, MIT Press, Cambridge, MA, 1975, p. 199. 15 J. Kratochvfl, Mater. Sci. Eng., A164 (1993) 15. 16 J. I. Dickson, J. Boutin and G. UEsp6rance, Acta Metall., 34 (1986) 1505. 17 J. I. Dickson, L. Handfield and G. UEsp6rance, Mater. Sci. Eng., 81 (1986) 477. 18 J. Kratochvfl and M. Saxlov~i, Solid State Phenom., 23-24 (1992)369. 19 M. Saxlov~, Phys. Status Solidi ( b), 181 (1994) 321.
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
79
J. Kratochvfl and S. Libovick2~,Scr. Metall., 20 (1986) 1625. J. Kratochvfl, Rev. Phys. Appl., 23 (1988) 419. J. Kratochvfl, J. Mech. Behav. Met., 2 (1990) 353. J. Kratochvfl, Scr. Metall. Mater., 24 (1990) 891. J. Kratochvfl, Scr. Metall. Mater., 24 (1990) 1225. A. Fran6k, R. Kalus and J. Kratochvfl, Philos. Mag. A, 64 (1991)497. A. Fran6k, J. Kratochvfl, M. Saxlov~iand R. Sedl~i6ek, Mater. Sci. Eng., A137(1991) 119. J. Kratochvfl and M. Saxlov~i, Scr. Metall. Mater., 26 (1992) 113. J. Zatloukal, M.S. Thesis, Faculty of Mathematics and Physics, Charles University, Prague, 1993. N.Y. Jin, Acta Metall., 37(1989) 2055. Z. S. Basinski and S. J. Basinski, Prog. Mater. Sci., 36 (1992) 89. F. Ackermann, L. P. Kubin, J. Lepinoux and H. Mughrabi, Acta Metall., 32 (1984) 715. K. Mecke and C. Blochwitz, Cryst. Res. TechnoL, 17 (1982) 743. B. gest~ik, V. Nov~ik and S. Libovick)~, Philos. Mag. A, 57 (1988) 353. Z. S. Basinski, Discuss. FaradaySoc., 38 (1964) 93. H. Mughrabi, Philos. Mag., 23 (1971 ) 897. J. D. Ashton, O. T. Woo and B. Ramaswami, Metall. Trans. A, 6 (1975) 1957. W. Vorbrugg, H. Ch. Goetting and Ch. Schwink, Phys. Status Solidi B, 46 (1971)257. U. Essmann, M. Wilkens and M. Rapp, Acta Metall., 16 (1968) 1275. A. T. Winter, O. B. Pedersen and K. V. Rasmussen, Acta Metall., 29 ( 1981 ) 735. N. Y. Jin and A. T. Winter, Acta Metall., 32 (1984) 989. Y.A. Pokhil, Krist. Tech., 9 (1974) 1179. G. L'Esp6rance, J. B. Vogt and J. I. Dickson, Mater. Sci. Eng., 79(1986) 141. Y. Kawasaki, J. Phys. Soc. Jpn., 36 (1974) 142. N.Y. Jin, Philos. Mag. A, 48 (1983) L33. R. Wang and H. Mughrabi, Mater. Sci. Eng., 63 (1984) 147. E. Schmid and W. Boas, in F. A. Hughes (ed.), Plasticity of C~stals, London, 1950, p. 88. G. Van Drunen and S. Saimoto, Acta Metall., 19 ( 1971 ) 213. E. Goettler, Philos. Mag., 28 (1973) 1057. Y. Kawasaki and T. Takeuchi, Scr. Metall., 14 (1980) 183. N. Y. Jin and A. T. Winter, Acta Metall., 32 (1984) 1173. H. Mughrabi, T. Ungfir, W. Kienle and M. Wilkens, Philos. Mag. A, 53 (1986) 793. T. Takeuchi, J. Phys. Soc. Jpn., 40 (1976) 741. S. N. Monteiro and H. J. Kestenbach, Metall. Trans. A, 6 (1975) 938.
69
A theory of dipolar dislocation wall structures M. Saxlovfi Faculty of Mathernatics and Physics, Charles University, 121 16 Prague (Czech Republic) J. Kratochvll Faculty of Civil Engineering, Czech Technical University, 166 29 Prague (Czech Republic) (Received September 20, 1993; in revised form January 31, 19941
Abstract The goal of the paper is to answer three questions: (1) to investigate the way in which the dipolar dislocation walls are formed, (2) to explain their tendency to regular arrangement and (3) to determine their geometry. The answers are based on the linear stability analysis of the proposed model of sweeping and dragging mechanisms of dislocation dipole clustering. From the set of wall orientations provided by the analysis the most stable dipolar dislocation structure actually formed is selected by an energetically motivated geometrical criterion. The model is employed for prediction of geometry of the dipolar wall structures produced by collinear slip belonging to one or two slip planes, coplanar or non-coplanar symmetric double slip, and symmetric multislip. In all cases the predicted wall geometry is consistent with the available observations.
1. Introduction
The most expressive feature of plastic behaviour on a microscale is a formation of a dislocation structure and its changes during plastic deformation. The underlying phenomenon is that many more dislocations are produced than is necessary to carry plastic deformation. The excess production of dislocations can be attributed to the uncorrelated bursts of dislocations by activated dislocation sources [1]. As a result, too many dislocations are generated. Some of them disappear at the surface but most of them become stored in the solids. In that process the deformed solid tends to reduce the internal energy by a mutual screening of elastic fields of stored dislocations. There are two basic types of ductile solid: dislocation-cell-forming materials and solids where cells are not formed. The principal difference between them is the way in which dislocations can screen their fields. In cell-forming materials the leading mechanism is the individual screening where dislocations are stored in the form of dipoles. The screening is very effective as the stress field of one dislocation is compensated by the stress field of the neighbouring dislocation with the opposite Burgers vector. Typically long bundles of primary nearly edge dislocations interact pairwise over part of their length (see Fig. 2 in ref. 2; for a review see ref. 3). From the bundles, progressively shorter dipolar loops are formed by the chopping mechanism [3, 4]. (1921-5093/94/$7.00 SSD10921-5093(94109542-5
Another mechanism of the loop production is the mutual annihilation of screw parts of mixed dislocation dipoles (see Fig. 4 in ref. 5). A large quantity of dipolar loops created behind jogged screw dislocations was reported as a dominant feature in Fe-Si crystals [6]. Observations on a wide range of materials and deformation modes prove that dipolar loops are accumulated, forming a cell structure of denser regions, commonly called cell walls, separated by regions of a much lower density [7]. On the contrary, in the solids where cells are not formed the stored dislocations screen their elastic field, collectively forming an arrangement which can be roughly modelled as a Taylor lattice [8, 9]. In this article the discussion is restricted to cellforming solids. In such materials the most typical stored dislocation arrangements are cell walls of the prevailing dipolar character with the sum of Burgers vectors close to zero. Another type of arrangement consists of thin layers of dislocations of one sign which are coupled with misorientation of the crystal lattice. A typical example is subgrain boundaries mainly observed at higher homologous temperatures or very high strain [10]. Increasing misorientations of the crystal lattice in the cell-forming materials are observed in advanced stages of deformation [11]. The walls between misoriented dislocation cells can be understood as dipolar walls with a subgrain boundary component. Another type of dislocation arrangement that © 1994 - Elsevier Sequoia. All rights reserved
70
M. Saxlovd, J. Kratochvil
/
Theory of dipolar dislocation wall structures
is clearly connected with misorientations is dislocation grids (sheets) [12-14]. More extended features are bands of localized shear and persistent slip bands. The subgrain boundaries, grids and bands of localized shear which are of similar physical origin [15] but different from the origin of the dipolar walls are not discussed in the present paper. The geometry of dipolar wall arrangements was theoretically studied by Dickson et al. [16, 17]. These workers have assumed the existence of a double pseudo-polygonization arrangement of dipolar loops to predict the energetically most stable walls observed in fatigued f.c.c, metals. Comparison of the predictions of Dickson et al. with experiments is fairly good, but some discrepancies exist (they will be mentioned in Section 3). A different approach was proposed by Kratochvfl and Saxlov~i [18]. It was based on the drift and sweeping mechanisms of the dislocation wall formation. The energetically motivated criterion of Dickson et al. has been built in the proposed theory as a subsidiary selection rule. The geometry predicted by our method agrees well with observations on f.c.c. crystals deformed by double slip with the tensile axes oriented near the [100]-[110] stereographic line [18]. Similarly good results have been reached for f.c.c. crystals deformed along the tensile axis oriented in the [001] direction [19]. In the present paper the proposed theory is reviewed and applied in a systematic study of dipolar wall structures produced by single, double or multiple slip. The mathematical description of the model where one or several slip systems operate simultaneously is summarized in Section 2. The theory developed in Section 2 is applied for various cases of dipolar wall structure in Sections 3 and 4; their geometry is predicted and compared with the available observations. We used mainly experimental results on f.c.c, metal single crystals, where a variety of slip system combinations may operate according to loading conditions. In these cases, observed dislocation structures, i.e. at advanced stages of plastic deformation, are often well documented and analysed. Only the most typical papers with a detail description of the wall geometry were selected in our list of the references. However, several other relevant publications were checked and to our knowledge no experimental result seems to be in contradiction with our predictions of the dipolar wall geometry proposed in the present paper.
2. Model of dipolar wall structure formation
In the model of the cell formation proposed in the recent publications [18-28] the stored dislocations in the form of dipolar loops are swept by gliding dis-
locations or drifted by stress gradients into dense regions called cell walls [7]. In the dense regions at a sufficiently high density the dislocations annihilate, causing dynamic recovery. The glide dislocations enter the theory through the plastic strain. The proposed theory consists of three basic steps. As the first step we describe the mechanism of the dipolar wall formation and thus give the answer to the first question in the abstract, namely the way in which dislocation walls are formed. We start with the simplest one-dimensional model of single slip and then extend it to three dimensions and several active slip systems. Only cases of approximately same slip activity in all participating slip systems are considered. The reason is that this restriction simplifies substantially the subsequent analysis of the model. The stability analysis of the model given in Section 2.2. represents the second step. It is shown that a homogeneous dipolar loop distribution is unstable and tends to form a periodic arrangement of dipolar walls. Hence this section answers the second question in the abstract, i.e. an explanation of their tendency to regular arrangement. Only the linear stability of the model is analysed which is not fully adequate for the later stages of plastic deformation where the cell walls are sufficiently developed. Nevertheless, the analysis seems to provide a firm base for the prediction of the dipolar wall geometry. However, the present form of the analysis does not predict the geometry of walls uniquely; a spectrum of wall orientations which depends on the number and combinations of slip systems activated is predicted. Therefore, as the third step, two subsidiary selection rules are formulated (Section 2.3) which select out of the spectrum the most stable wall orientations with the highest probability to be actually formed. The linear stability analysis of eqn. (3) supplemented with the selection rules determines the dipolar wall geometry. 2.1. The model Consider first a one-dimensional model. The balance law governing the evolution of the local density p(x, e) of stored dipolar loops can be expressed in the form
0p Oe
0 Ox J + g
(1)
where x is the coordinate and the plastic strain e serves as the evolution parameter. Equation (1) means that the rate of accumulation of stored dislocations per unit strain at place x must be equal to their net flux - OJ/Ox plus the generation rate g of stored dislocations per unit strain. The rate g is positive (generation) for a low density p and becomes negative (annihilation) for a
M. Saxlov6,J. Kratochvil / Theoryof dipolar dislocation wallstructures sufficiently high p. The expression for the flux J was deduced from the fact that dipolar loops are drifted on the average towards stress concentrators [20, 21]. The stress concentrators can be created by glide dislocations stuck at already-formed clusters. Because of this effect, a fluctuation in the dipolar loop density can serve as an embryonic cluster from which a dislocation pattern may develop. In ref. 21 it was shown that the drift mechanism is an elementary consequence of the dislocation theory. The alternative and probably a more effective mechanism based on the sweeping of dipolar loops by glide dislocations was proposed in refs. 18 and 27. Perhaps both mechanisms promote the dislocation pattern formation and support each other. They are of a similar nature and in the first approximation lead to the same expression for the flux J: +co
J=D ~ f M(a)p(x+a)da Ox
(2)
In both mechanisms the flux is of a dislocation line tension origin. In the dipole drift theory [21] the integral term represents the bowing stress needed to get glide dislocations through the space between clusters. The flux of dipole loops is caused by the gradient of this stress. In the sweeping mechanism [27] the line tension is responsible for the shape of glide dislocations through the integral term and their slope is proportional to the driving force of the loop flux. The differences in the expression for the flux J are of the second order; the factor D is positive ("uphill diffusion") in both theories and generally depends on p. The generalization of the model described by eqns. (1) and (2) for three dimensions and several slip systems was proposed by Kratochvfl and Saxlovfi [18]. In that case the balance law for the total dislocation loop density p (the sum of the densities of loops produced by the N active slip systems) is
Op(r, e) + D Z div J grad j × ~
,1
I
o3
J M(a)p(r+ ab ~, e) da = Ng
(3)
cO
where b J is the unit vector in the Jth slip direction, r is the position vector, and div J and grad j mean the standard differential operations with respect to Jth slip direction. Taking the common evolution parameter e, we suppose that the slip activity of all slip systems considered is the same and uniform. Moreover, for simplicity it is assumed in balance law (3) a certain degree of isotropy; the function M(a) and the factors D and g are supposed to be the same for all active slip systems. The linear stability analysis employed in the present paper admits only D and g constant.
71
The active slip systems enter balance law (3) only through the slip vectors b J, J= 1, ..., N. This is the consequence of the assumption that dipolar loops are treated as point objects, which can be moved only in their own slip direction. The main reason for introduction of such simplification is that then the threedimensional theory of dislocation clustering is easily formulated. On the one hand the simplification is justified as the size of the loops is usually an order of magnitude smaller than the wavelength of the formed dislocation pattern. On the other hand there is a disadvantage; as shown at the end of this section, the linear analysis of model (3) does not determine the wall orientations uniquely. From the predicted spectrum of wall orientations the walls actually formed are selected by two subsidiary rules. The rules incorporate implicitly the finite size of dipolar loops, their interaction and slip plane geometry; in this way the rules subside the simplification introduced into model (3). The main tool in our study is the linear stability analysis of model (3). As more apparent dipolar wall structures exist typically at later stages of the dislocation pattern development, a non-linear analysis of the model of the wall formation would be more adequate. A non-linear approach would require incorporation of the effect of the annihilation of dipolar loops within clusters. The annihilation starts when the density of loops within a cluster becomes sufficiently high. A simple model of the role of the annihilation in the process of the dislocation pattern formation was proposed by Fran~k et al. [26] and studied in ref. 28. The results indicate that the most distinguished consequence of the annihilation is thinning of the dislocation clusters and the formation of dipolar walls. Two accompanying effects, namely a change in the wavelength of the dislocation pattern [26, 28] and a cell misorientation analysed by Kratochvfl [23, 24], do not seem to modify the basic geometry of the pattern as derived from the linear stability analysis. As the model used by Fran6k et al. [26] and Zatloukal [28] is much simplified, we are aware of the limited applicability of this conclusion. Nevertheless, we try to explore what information about the dipolar wall geometry we are able to obtain using the available simple linear means.
2.Z Linear stability analysis of the model When investigating the linear stability of the uniform distribution of loops described by eqn. (3), one is looking for the fastest-growing perturbation in the form of an infinitesimal wave. The perturbed solution of eqn. (3) is assumed in the form
p(r, e)= fi(e) + Ap exp(ik-r+ we)
(4)
where fi(e) represents the homogeneous solution of eqn. (3), Ap the infinitesimal amplitude of the per-
M. Saxlovti,J. Kratochvil / Theoryof dipolar dislocation wallstructures
72
turbation, k the wavevector and ~o the amplification factor, which determines the rate of growth (~o >0) or decay (~o<0) of the perturbation. Using eqn. (4), eqn. (3) yields
where/i is a non-zero solution of
2R(p)+/~ dR(/~) dp
=0
(13)
N
w(k) = D ~, (pk~)ZR(pkJ)
(5)
Note that in this case condition (10) is satisfied as
J=l
where pk J is the scalar product pk J= k. b J and R(p) represents the Fourier transform of M(a): oo
R(p) = f M(a) e x p ( i p a ) d a
(6)
As in physically realistic cases the function M(a) gives a positive value of R(p) and D is positive, the perturbation grows exponentially and the uniform distribution of dipolar loops is unstable. In the first approximation the wave with maximum oJ given by eqn. (5) determines the geometry of the dislocation pattern. The conditions for the maximum 09 are
0 o(k)
Ok-----/-=O
i = 1, 2, 3
3
Z b / K i = s pJ ^
h
cJLs L
i = 1, 2, 3
(8)
where bi J are the components of the unit slip vector b J, X ~= X(p J) and
X(p) =p(2R(p) + pdR(p)l dp ]
(9)
Conditions (7)2 of the maximal co are
(16)
where h is the number of linearly independent vectors b J and c JL are the coefficients of linear dependence: h
b~= ~, cJLb L
(17)
The reason for the restriction is that the condition Is J[ = 1 can be satisfied for certain combinations of sL only.
The symmetry of the slip directions b J determines the symmetry of K. Let ~m be the symmetry axis of the set of vectors {sJbJ}m for a given combination {s J}m. Then the component Kem of the vectors K m in the ~m direction is t~
K~ m= +- "m Z ((b/) 2dX(p) /<0 dp p=pJ/
J=l
3
J = 1,..., N
i=1,2,3
(10)
(11)
i=1
where pJ are determined by eqn. (9) for X(p J), which is a solution of system (8). System (11) has a solution if and only if X(p J) = 0 for all J and J = l ..... N
(18)
be
The unknown components K~ of the wavevector K can be found from the relations defining p J, i.e.
IPJl=P#0
J=h+l,...,n
L=I
L=I
J=l
Z biJKi =PJ
(15)
for all possible combinations of signs s J = + 1. However, for linearly dependent b J the choice is restricted by the condition
N
Z b/X ~= 0
J=l,...,n
i=l
(7)
Three equations (7)1 for three unknown components of the wavevector K corresponding to the maximal ~o can be written using eqn. (5):
(14)
From eqns. (11) and (12) it follows that for n noncollinear slip directions bJ(n<~N) we are looking for the set {K} of vectors K, which satisfy the relation
sJ= Z
0%(k) 0k~<0
dX(p) = -2R(/~) < 0 dp t,=~
(12)
where b~m= sJbJ with s j from {s J}" and the component b J of the unit vector b J in ~m direction. The vector Ko m parallel to ~m is the smallest of {Kin}, K0 m= K~". For the single-slip direction the axis ~ is identified with that direction, K~ is determined by eqn. (18) and the other two components perpendicular to ~ are arbitrary. In the case of double slip with h = 2, eqn. (18) is valid for the component K~ in the plane of the vectors b 1 and b:, and the component of K perpendicular to that plane is arbitrary. For multiple slip with h = 3 the discrete wavevectors K m coincide with the vectors K 0m.
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
In summary, the linear stability analysis of model (3) yields the set of the wavevectors {K} which determines the geometry of the dislocation pattern; the potentially possible orientations of the dipolar walls identified with the fastest-growing perturbation waves are perpendicular to the wavevectors K from {K}. 2.3. Selection rules Generally, the stability analysis in Section 2.2 provides a whole spectrum of wall orientations given by the set {K}. The set {K} consists of the solutions {K m} for the permitted different combinations {sS}m in eqn. (11) with 3 - h arbitrary components, i.e. arbitrary magnitudes K m~>K0" of the wavevectors. The distances between the walls are equal to the wavelength 2 given by the magnitude K of the wavevector K, i.e. 2 = 2Jr/K. Hence, very small or even zero wavelengths of the dislocation pattern are permitted by the model for the case of one or two active slip directions. This is not physically realistic. According to eqns. (6), (9) and (13) the wavelength is controlled by the shape function M. As noted by Kratochvil and Saxlov~i [18] we can expect that a more sophisticated form of M should exclude this possibility of too short wavelengths. Therefore (i) we admit in {K m} only the wavevectors of magnitudes not far from the minimum K0m. The second effect, which influences the determination of the walls actually formed, is a preferred formation of dipolar walls with a lower energy. Since the walls considered are dipolar in nature, they all correspond to low energy configurations and the difference between the energy of different walls are of the second order. Therefore, following Dickson et al. [16], we take as an energy criterion the tightness of the stacking of the dipolar loops for the walls. The tightest stacking of the loops can be obtained by the connection of loops along their short segments. The walls which contain the axis ~J1 bisecting the acute angle between the directions e j and e z of edge (longer) parts of dipolar loops of the slip systems J and 1, i.e. for which K m. ~ l J = 0, are stabilized by the energetically favoured stacking of the building loops b J, e J and b 1, e l (Fig. 1). The angle r/gl
A
Fig. 1. The tightest stackingof dipolar loops egand e ~along their short segmentsin walls containingthe axis ~ Jt bisectingthe acute angle between them.
73
between the normal K to the wall and the axis ~ sl can be taken as a measure of the tightness. Therefore (ii) we suppose that angles ~/not far from ½x favour that wall orientation. The method of determination of wall orientations based on the linear stability analysis of eqn. (3) supplemented by selection rules (i) and (ii) will be analysed in detail for one or two active slip systems in Section 3. Multiple-slip wall orientations are discussed in Section 4.
3. Single- and double-slip walls
For the single-slip direction b belonging to one or two slip planes (primary system or primary and crossglide systems) the axis ~ is identified with the vector b. The most favoured walls with the smallest wavevector Ko(K o =/3) and also the tightest loop stacking (t/= l~r) are perpendicular to b. This is exactly the optimal orientation observed in the ladder structure of persistent slip bands in f.c.c, single crystals [29-32] and walls in cyclically deformed Fe-Si crystals which results after local dissolution of veins [33]. For the two different slip directions b ~¢ b 2 (the most common form of glide deformation) the set {K} consists of two continuous spectra {K 1} and {K 2} (Fig. 2). In the coordinate system x, y, z with the z axis parallel to the axis ~ bisecting the acute angle 2~b between the slip directions b ~and b 2 we obtain
+ K2 = + ~ P ,Kv, 0 - sin ¢
(19)
where the component Ky of K is arbitrary and /3 is given by the influence function through eqn. (13). The distances of the walls measured along the axes ~1 and ~2 (i.e. in the plane of Burgers vectors b I and b 2) are (2er cos #)//3 and (2rising)//3 respectively, in agreement with the two-dimensional model for double slip studied by Kratochvfl [23]. This is indicated by the traces of the walls shown in the picture on the righthand side in Fig. 2. The walls corresponding to the set {K} (19) are represented by the planes (~2) for the vector {Kl}, and the planes (~1) for the vectors {K 2} (the symbol (abc) means planes containing the direction [abc]). The restrictions of the set {K} imposed by supplementary rules (i) and (ii) will be specified separately according to the mutual orientation of active slip systems. The resulting spectrum of the predicted wall orientations for a given double slip is governed by the orientation of
74
M. Saxlov6, J. Kratochvil /
Theoryof dipolar dislocation wall structures
x II{2 /~/sin ¢
I
I y -
./," .
I I
,
Ko
IK,\
fj" , \
zlI~S symmetry of glide
I {R1}//
,
K;
/1 ,-
{rd}t
, oo~-
\~/COS ¢ {~ }={~1}+{~2}
traces of watts {K} in section ('bl.6z)
Fig. 2. Geometry of the walls K for the double slip b 1and b 2.
!
{1
[0111
[112] 1 [101]
//•11 21
--T .... .
fK'0
-tl 01 t
|'¢'31t/~
e''~
trace (111) traces of walls
1
{a) (b) Fig. 3. Geometry of the coplanar slip b ~and b 2 in the common slip plane. (a) The traces of walls K02 are parallel to ~ 2, i.e. (b) to the traces of the critical slip plane in f.c.c, crystals.
100 -"
_ 210
110
Fig. 4. Stereographic triangle with the main slip system (11i~101].
the axis ¢ (bisecting the acute angle between the edge dislocation directions) with respect to axes ~1 and ~ 2 of the slip directions. Four different cases, which cover all possible combinations of double slip in f.c.c, metals, are considered and shown together with collinear slip later in Fig. 7, where the coordinate system ~1, y, ~2 of Fig. 2 is used. In all cases the equal slip activities are assumed in both considered slip systems.
It is interesting to note that the activation of the third coplanar system (in f.c.c, or h.c.p, metals) provides no new wall geometry different from that produced by the combinations of the double slips (b 1, b2), (b l, b 3) and (b 2, b3). The reason is that, in the case of three coplanar slip directions, condition (16) is not fulfilled for any set of signs s J= + 1 considering all three slip directions, J = 1, 2, 3. The traces of walls in the slip plane should lie along the slip directions (compare for example Fig. 10 in ref. 13, Fig. 8 in ref. 40, and Fig. 4 in ref. 41).
3.1. Case 1: Coplanar slip Coplanar slip on a dominant glide plane has been reported by numbers of workers. It often prevails over glide activity of the other secondary dislocations (see for example refs. 2, 11-13 and 34-36). Coplanar slip predominates locally even in some crystals oriented for multislip ( ~ ref. 37 for [110J-oriented Cu crystals) or in polycrystals [38, 39]. For coplanar slip the bisecting axis ~ is parallel to the axis ~2 and the preferred wall orientations <~2>(i.e. for which r/=½~z) selected by rule (i) are close to the orientation of the wall perpendicular to the axis ~1 (Fig. 3(a)). The predominant wall orientations are of the type {112} in f.c.c, metals. The traces of the predicted walls in the common slip plane are parallel to ~2, i.e. to the traces of the critical slip plane of f.c.c. metals (Fig. 3(b)), in agreement with the observations mentioned above.
3.2. Case 2: Symmetric non-coplanar slip: perpendicular Burgers vectors For symmetric non-coplanar double slip, no active slip plane contain both slip directions. Choosing perpendicular slip directions we consider two active slip systems ( 11 ] )½[101 ] and ( 111 )½[10 i] as shown below in Fig. 5. They represent the predominant primary and critical slip systems for a f.c.c, metal crystal oriented near the center [210] of the [100]-[110] stereographic line (Fig. 4). The two axes of symmetry are ~1 -[100] for s z = s 2 and ~2--[001] for s 1= - s 2. For the considered slip systems the directions of the dipolar loops are e 1[1[i21], e 21[[121] (Fig: 5) and the axis ~ of both loop directions lies along [120], i.e. perpendicular to ~2, as shown in Fig. 6. From the (100) spectrum the wall of orientation (001) is the most favoured by the stability criterion (ii)
M. Saxlov(~, J. Kratochvil
/
Theory of dipolar dislocation wall structures
~ •
i~21~[ool] ~
~o)
[
/I
Ie'j"-tL..../1 J Fig. 5. Geometry of the double slip (Ill)½[101] and (1 1 1)~[10i].
75
./i
~
tl00}[
~'v4
....
J
/
/~'11[100] Fig. 6. Ideal stacking of dipolar loops in the walls.
(r/= 90°). The (001) wall is also the wall corresponding to the shortest wavevector Ko 2 from that spectrum. In the (001) spectrum the tightest stacking of the dipolar loops occurs for the (210) wall (~/= 90°), and the magnitude of the wavevector corresponding to the (210) wall is 1.11K01. From the (001) spectrum, the (100) wall belongs to the shortest wavevector with the magnitude K0 ~ and the measure of tightness ~/= 63.4 °. The dipolar loop arrangement in the (001 ), (210) and (100) walls are shown in Fig. 6. It is interesting to note that, according to L'Esp6rance et al. [42] (for a summary see ref. 17), the (001) walls and the walls intermediate between the orientations (210) and (100) are actually observed in dipolar wall structures formed by considered double slip. 3.3. Case 3." Symmetric non-coplanar slip: nonperpendicular Burgers vectors The third type of double slip, which causes a marked structuralization in f.c.c, metals, is the simultaneous action of primary and conjugate slip systems with the slip direction b 2 out of the primary slip plane and the angle 2~b = ]Jr between both slip directions [40, 42-45]. Considering (11])11101] plus (111)½[110] slip systems with the axes ~l, ~2 and ~ along [211], [01]] and [233] respectively, we predict similarly as in case 2 three prominent wall orientations. The tightest stacking of the dipolar loops occurs for the (011 ) walls from the (211) spectrum and for the (311) walls from the {01 ]) spectrum. The application of rule (i), which selects the wall orientation corresponding to the shortest wavevector, to the (211) and (01 i) spectra indicates another possibility. All walls between (311) and (211) walls from the (01 ]) spectrum have a relatively high stability. Their measures of tightness are greater than ,/(211)=80 ° and the magnitudes of the wavevectors are less than 1.01K0 ~. All these predicted walls are actually observed (see the papers mentioned above). 3.4. Comment I: On the rule of Dickson et al. The preferred wall orientations in three considered typical cases of double slip in f.c.c, metals deformed in tension along the directions indicated in Fig. 4 are
TABLE 1. Dipolar walls in f.c.c,crystals deformed by symmetricdouble slip Dipolar wallsfor the following tensile axes
Predicted from {Kt} Predicted from {K2} Dickson et al., wallsW2 Dicksonetal.,wallsWl
[221] Case 1
[2101 Case2
[211] Case 3
(112)
(210)-(100) (001 ) (210) (001)
(311)-(211) (01 ] ) (311 ) (01])
(112)
summarized in Table 1. The orientations of walls of labyrinth structures for the considered double slips as predicted by Dickson et al. [16, 17] are also listed. As was shown for all three cases of symmetric double slip, the wall orientations predicted by the proposed method are in full agreement with the available observations. On the contrary, as Table 1 indicates, the wall geometry prediction by Dickson et al. [16, 17] for non-coplanar slip is too restricted in comparison with the available experimental data. It should be noted that the principal difference between our approach and the considerations of Dickson et al. [16] is that in our case the basic wall orientations are determined by the pattern-forming process, and we are not a priori restricted to polygonized walls only. 3.5. Case 4." Non-coplanar slip: both Burgers vectors lying in one active slip plane The formation of the most preferred wall orientations with the shortest wavevectors K01 or K02 and simultaneously with the tightest stacking of loops (r/=½:r) is excluded for all less symmetrical oriented slip systems where the axis ~ is not perpendicular to any axis ~. An example is double non-coplanar slip, for which the slip plane of one system contains both slip directions b 1 and b 2. Such pairs of slip systems can be equally activated in f.c.c, crystals deformed along the corner orientations (100) or (111).
76
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
Let us consider for example two active slip systems (lli)½[101] and (ill)½[ll0] with the axes ~1, ~2 and along the directions [211], [01]] and [231], respectively. Similarly as for the symmetric non-coplanar slip analysed above we predict, from the (~2) spectrum, wall orientations intermediate between the (211) walls having the smallest wavevector K01 and a relatively high stability such as r/= 77.4 ° and the (111) walls with the tightest stacking of dipolar loops (r/=~ ) and K = 1.06K01. The formation of (~ 1) is, however, rather suppressed because, for the (01 i) walls preferred by condition (i) the stability measure r/ is only 40.9 ° and the most stabilized (102) walls from that spectrum has relatively high K = 1.58K02. The consequence of this situation is a compromise: walls in the vicinity of the (113) walls with K = 1.17/(02 and r/=71.2 ° are assumed to be observed (Fig. 7). The predictions of case 4 cannot be checked directly as it appears only as a component of multiple slip. Case 7 of the following section, Section 4, shows that case 4 represents just these pairs of slip systems which cause rather a low stability of multiple slip walls in (111)oriented crystals.
tions. Consequently, in these regions the occurrence of multiple-slip walls can be expected. In the case of multiple slip the system of n equations (15) (n~>3) yields four solutions K m or less than four under restriction (16). The actual formation and stability of the multiple-slip wall with the orientation K m depends on the additional conditions stated in selection rules (i) and (ii). To check these conditions we use the fact that the solutions K m are also solutions for all pairs J, I of slip systems J, I = 1.... , N. This is evident from the structure of eqn. (18). For each pair J, I of slip systems we can determine how close is the magnitude of a particular K m to the corresponding smallest one--selection rule (i)--and how K m- ~zs is close to zero, i.e. if the corresponding r/ is close to lzr--selection rule (ii). The formation of the K m wall and its inherent stability is determined by the number of double-slip systems for which K m are included in the spectra of the predicted and effectively formed wall orientations in the sense of Section 3 (Fig. 7). The dipolar loops produced by such slip systems are building blocks of the multiple-slip wall K m. The described method will be illustrated by cases 5-7 of multiple-slip in f.c.c, crystals.
4. Multiple-slip walls
4.1. Case 5: Multiple slip o f (lOO)-oriented fc.c. crystals
In the case of several equally stressed slip systems, e.g.f.c.c, crystals with corner orientations, some regions of the crystal are deformed in more than two slip direc-
Consider a [100]-oriented f.c.c, crystal with eight equally stressed slip systems deformed by_multislija with four slip directions b s along [110], [110], [101] and [101] ( N = 8 ; n = 4 , h = 3 ) and four slip planes
i~=
!~LG
[ ~ ,~2
1
I
I
.7 •
I
/ " {lib
I,~z
/ i / q :90 ° 61=~ z
I¢ 2
Ko q=90"
-'l
j ..:__--
//{1
"~90"
CASE 1
~=60"
I~ 2
n=/*l"
n=90"
K,
Klo •" ~ " "
__
/
/ " q=90"
/'{1
•
/
=
.
CASE 2
I
c'/'77"-
i
...
CASE 3
90" ! I
CASE 4
Fig. 7. The sets of the predicted walls orientations for different cases of two slip systemsin f.c.c,metals.
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
A--(111), B=(11I), C ~ - ( l i l ) , D-=(111). * From eqn. (15) choosing {s J}'n in agreement with eqn. ( 1 6 ) w e obtain S 1 = S 2 = S 3 =S 4 s'
=
-s
2 =
-s
3
~
=s 4
s l = - S 2 = $ 3 = --S4 ~
K'
I1[100]
K 1 = 21/2~
[011]
K 2 = 2~
K311101]-]
K 3 = 2[~
K 2 II
Three mutually perpendicular walls (100), (011) and (01l) form cells with the aspect ratio 21/2 along the stress axis [100] (Fig. 8). The combinations of two slip systems which fulfil the selection rules (i) and (ii) in the sense of Section 2 for the transverse walls K 1 and the longitudinal walls K 2 and K 3 are listed in Table 2. Other pairs of the active slip systems do not fulfil either selection rule (i) as K l >2.4K 0 or (ii) as q < 50 ° for walls K 2 and K 3. By inspection of Table 2, we see that all three wall orientations listed are effectively formed and stabilized; in the transverse walls formed by all eight slip systems, each loop type is stabilized by the tightest stacking with two other types. The longitudinal walls, however, are formed non-concurrently. Each consists of loops of four slip systems and is stabilized by the stacking of loops in one coupling pair. The cell structure in deformed {100}-oriented f.c.c. crystals is documented by transmission electron microscopy observations (see for example 47-51) and agrees well with our prediction. The dipolar wall geometry of the considered case is conditioned by a nearly equal local slip activity in all eight slip systems. This condition was satisfied for the deformed [001] copper crystals by Takeuchi [52] for which Kawasaki and Takeuchi [49] actually found the cell structure of
TABLE 2. The sets of acceptable pairs of equally stressed slip systems in [100] f.c.c, crystals which effectively form and stabilize the walls Sets of pairs for the following walls Transverse (100)
B2+CI A2 + D1 C3 + B4 A3 + D4 Ko 2 q=½~ Case 2
Longitudinal
A2+C1 B2 + D1 A3 + B4 C3 + D4 Kll I t/= 63.4 ° Case 2
(011)
(Oli)
A3 + D1 A2 + D4 Ko 2 ~/=½~ Case 3
B4 + C1 B2 + C3 KII2 ~/=~n Case 3
the predicted geometry. Their examination of the foils perpendicular to the tensile axis shown that the apparent circular traces consist of short sections of predicted (110) and ( l i 0 ) walls. Moreover, in the longitudinal foils the shape of the cells was elongated in the direction of the tensile axis, as indicated in Fig. 8. The appearance of a fine distribution of the longitudinal walls corresponds to a non-concurrent formation of both walls. A more effective formation of the transverse walls is documented by the observation of condensate (100) walls in [100] fatigued copper [50]. The apparent spherical shape of individual cells with all six slip directions (110) uniformly distributed in an almost isotropic cell structure reported by Van Drunen and Saimoto [47] and Goettler [48] can be easily explained. The activation of all six slip directions provides no new geometry different from that corresponding to any combination of four non-coplanar slip directions. Basically the nonconcurrent formation of walls of types [100/and {0111 implies only short walls in regions with all six (110) slip directions and thus makes the cell shapes apparently circular.
4.2. Case 6: Multiple slip of (Oll)oriented fc.c. crystals The wall geometry of [011] and [100] deformed crystals is similar. In both cases the slip directions 1, 2, 3 and 4 are activated, giving rise to the same wall geometry of the picture on right-hand side of Fig. 8. However, unlike the [100] case, the [011] orientation of applied stress activates only four equally stressed slip systems A2, A3, D1 and D4. Thus the (01i) wall (longitudinal for the [011] stress axis) cannot be formed. In fact, in sample sections (011) of [011] deformed crystals a "banded" structure with traces along [01i] (i.e. only those of the longitudinal (100) walls) was observed by Monteiro and Kestenbach [53].
4.3. Case 7."Multiple slip of (l l l)-oriented fc.c. crystals We consider a [1111-oriented f.c.c, crystal deformed in three non-coplanar directions b J along [110], [101] and [0111 (n = h = 3) by the activity of six slip systems B4, B5, C1, C5, D1, D4. The solutions K m for {s J}m in
[100] If oppl.iedstress f.c.c.c. !~1
[o1~]
*The notation used for the slip planes (A, B, C and D) and the slip directions ( 1, 2 ..... 6) follows that of Schmid and Boas [46].
77
i
.~(
~100) ,~ I
~_ _ x[~ ~/~
Fig. 8. Multiple-slip dipolar walls in the [ 100] f.c.c, crystal.
~/'2II/p
78
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
eqn. (18) are S 1 =$2=S
3
KIll[Ill]
~
sl=-s 2=-s 3 ~
K211[113]
s l = - s 2=s 3
~K3II[I~I]
S 1 :S
~
2 ~. - - S 3
K4I[[~ll]
Here K a =/~6~/2/2; g 2 = K 3 = g 4 = / 9 ( 2 2 ) 1 / 2 / 2 . The double-slip combinations for which K m belongs to the predicted effectively formed stable wall orientations are listed in Table 3. Similarly as in case 5, the transverse ( 111 ) walls, which correspond to the vector K ~, are formed by all six slip systems and each type of building loop is stabilized by coupling to other two types. For the longitudinal (113), (131) and (311) walls the situation is less favourable. For each wall, only four slip systems participate and the loops are coupled to only one other type of loop. The rest of the loop types can form double-slip walls or contribute to other multiple-slip walls. This situation probably explains the inherent geometrical variability of dipolar loop accumulations documented in [111J-oriented grains of the polycrystal [53] or in [111] crystals [49].
5. Discussion
The proposed theoretical model is expressed through single eqn. (3). It describes in mathematical terms how dipolar dislocation walls are formed by the sweeping and drift mechanisms, as was suggested and analysed in the series of papers in refs. 18-28 and is summarized in Section 2.1. The condensed form of the theory was achieved by introducing three main simplifying assumptions: (a) the slip rates of all active slip systems are the same and uniform throughout the deformed crystal; (b) dipolar loops are treated as point TABLE 3. The sets of acceptable pairs of slip systemsforming walls in [111] f.c.c,crystals Sets of pairs for the followingwalls Transverse (111)
Longitudinal (113)
B4+D1 B5+C1 C5 + B4 C1 +D4 D4+B5 D 1 + C5 1.06K01 q=½ar Case 4
(131)
(311)
B4+D1 B5+C1 C5 + B4 C1 +D4 D4+B5 D 1 + C5 1.17K02 q=71.2 ° Case 4
1.17 K02 r/= 71.2° Case 4
Case 4
objects, (c) no mutual interaction among dipolar loops is considered. If we could relax simplifying assumption (a), the stress and the corresponding continuum mechanics equations have to be incorporated into the theory as the stress field controls the local slip [21, 25]. The assumption of the uniform slip rates is meaningful in later stages of plastic deformation, where a sufficiently strong internal stress is already built and compensates the influence of inhomogeneities caused by dipolar walls. The considered wall structures represent a nearly stabilized situation; hence the smaller role of the stress field is not too surprising and the simplification (a) introduced can be tolerated. The stability analysis of eqn. (3) shows that the uniform distribution of stored dislocations is unstable and there is a tendency to form regularly arranged dipolar walls. The mathematical method of the analysis expressed through eqns. (4)-(17)looks technically complicated, but it is just a logical extension of the very simple procedure of determination of dipolar wall geometry suggested in ref. 23. In ref. 23 the somewhat artificial two-dimensional model of symmetric double slip was analysed. The proposed extension of the procedure to the more realistic three-dimensional model valid for any combination of active slip systems is geometrically more complicated. However, there is another problem. Because of the simplifying assumptions (b) and (c) the determination of dipolar wall geometry by the linear stability analysis of three-dimensional model (3) is not sufficiently specific. A whole spectrum of dipolar wall orientations is permitted. This inadequacy of the model is compensated by introduction of two selection rules in Section 2.3, which determine from the spectrum of permitted wall orientations the most stable dipolar wall configuration with the highest probability of actually being formed. As was indicated in Section 2.3, the first rule which admits only short wavevectors compensates simplification (b). The second rule which prefers the tightest stacking of dipolar loops for the walls compensates simplification (c). In that way the linear stability analysis of eqn. (3) supplemented with the two selection rules determines the wall geometry uniquely. A natural question arises why the simplifying assumptions are not relaxed and the effects of the nonuniform slip rates, the finite length of dipolar loops and the interaction among them are not built directly in a unified model. Such an attempt can be made, but we would lose the principal advantage of the present simplifying approach: the possibility of analytical solution of the problem and the corresponding deeper physical insight in dipolar wall geometry which is demonstrated in the seven cases analysed in Sections 3 and 4.
M. Saxlov6, J. Kratochvil
/
Theory of dipolar dislocation wall structures
A n o t h e r limitation of the present m e t h o d is the restriction to the linear stability analysis. A non-linear analysis, which would include annihilation of dipolar loops, should predict realistic d e v e l o p m e n t of the dipolar wall profile and the d e p e n d e n c e of the distance between walls on the applied stress. T h e a p p r o x i m a t e uniformity of slip rates assumed in the present p a p e r should be a consequence of the theory. It is interesting that despite all the simplifications the p r o p o s e d m o d e l is successful in the prediction of the correct wall geometry as d o c u m e n t e d by the results in Sections 3 and 4.
20 21 22 23 24 25 26 27 28 29 30 31
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32
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