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On the stress fields of crystal dislocations with fractal geometry
-"
NAIlEIIAL~ SalENCE &
ENGINEERING ELSE VIE R
Materials Science and Engineering A203 (1995) 203 207
A
On the stress fields of crystal dislocations with fractal geometry D. Raabe Institut Jgir Metallkunde und Metallphysik, Kopernikusstrasse 14, Rheinisch-Wes(/gilische Technische Hochschule Aachen, 52056 Aachen, Germany Received 16 January 1995; in revised form 31 March 1995
Abstract
The elastic stress fields of single irregularly tangled dislocations were simulated numerically. The dislocations had fractal dimensions within the range 1.0 < D < 1.5. The results were compared with the stress fields associated with straight dislocations (D = 1). To calculate the stress fields with a high spatial resolution the dislocation lines were subdivided into small segments. The results confirm that the stress fields of tangled dislocations depend considerably on their fractal dimension. The maximum shear stress generated per unit length of a single dislocation decreases with an increase in fractal dimension of its dislocation line.
Keywords: Elastic stress fields; Crystal dislocations; Fractal geometry
I. Introduction
For computation of the contribution of dislocations to the yield stress crystals, the internal stresses generated by a given dislocation distribution must be determined (dislocation statics). Together with the external stresses and stresses imposed by other lattice defects they enter into the calculation of the evolution of the dislocation structure during plastic deformation (dislocation dynamics). The first subject belongs to the field of flow stress theory and is thus an ingredient of the kinetic law of crystal plasticity. The second subject falls into the domain of the structural evolution law [e.g. 1-3]. To model crystal plasticity physically, a profound theoretical backbone for computation of the linear elastic stress fields of dislocations is required [e.g. 4]. Although crystal dislocations represent one-dimensional (1D) lattice defects in terms of the Burgers circuit, they arrange themselves in intricate 3D networks. The geometrical complexity and the large number of dislocations assembled in such arrays has led to the use of analytical methods for investigation of dislocation statics and dynamics in the past [5-8]. Except for some recent studies [9,10] these analytical approaches did not include the discretization of single dislocations in space. Promoted by the rapid improvement of computer facilities, numerical approaches are now increasingly 0921-5093/95/$09.50 ~? 1995 - Elsevier Science S.A. All rights reserved
S S D I 0921-5093(95)09856-9
being developed which allow for the discrete description of single dislocations in both time and space. Most of these studies were concerned with numerical analysis of the statics and dynamics of 2D dislocation arrays [e.g. 11-13]. The mathematical treatment of single dislocations also in 3D was pioneered by Hokanson [14] and Yoffe [15]. They showed that to calculate the stress fields associated with simple dislocation arrays, the dislocation lines must be decomposed into isolated segments. However such techniques neglected the facts that first dislocation cannot terminate within an otherwise perfect region of crystal, and second, equilibrium was not maintained in such arrangements. As was shown by Li [16] and described in greater detail by deWit [17,18], a consistent mathematical 3D treatment of the stress fields of piecewise straight dislocation segments is attained when the Burgers vector is conserved at each node. This means that dislocation segments which are not entirely integrated within the network must be semi-infinite or terminate at a free surface, a grain boundary, or some other defect, where stress equilibrium can be maintained. On the basis of these studies Kubin et al. [19,20] and Devincre and Condat [21] developed 3D models in which dislocations are discretized in both space and time. Whereas considerable success was achieved by both analytical and numerical approaches in predicting flow
204
D. Raabe / Materials' Science and Engineering A203 (1995) 203- 207
curves and non-linear aspects such as dislocation patterning, the problem of dimensionality has not yet been treated adequately in the literature. Nearly all investigations tackling dislocation statics and dynamics appear to follow a similar understanding that the complex geometry of dislocation lines may be treated generally in the framework of Euclidean geometry. Indeed, most honeycomb-type networks and even bimodal distributions consisting of dislocation walls and regions which are nearly void of dislocations may be described in terms of Eucildean dimensions. However, it was shown by Hornbogen [22] that individual dislocation lines often do not have a Euclidean dimension ( D - - 1 ) but have a fractal dimension (1 < D < 2) [23]. This applies especially for dislocations which intersect with other dislocations or interact with point defects [22]. The description of the geometry of non-straight dislocation lines within the framework of fractal geometry provides a useful means of deriving the degree of deviation from the initial straight dislocation in a quantitative manner. However, it should be underscored that the employment of fractal dimensions is physically pertinent in a rigorous sense only in cases where strict self-similarity of the object described is observed [23]. Although dislocations and dislocation arrays at first bear a certain resemblance to self-similar contours [22], they do not really fulfil this criterion in a mathematical sense. Consequently, in the present study nonEuclidean dimensions are simply used to provide a quantitative method for describing non-straight dislocation lines without implying strict self-similarity. In a recent study it was shown for the first time that the stress field associated with a small angle tilt boundary indeed depends on the fractal dimension of the dislocations involved [24]. In a more systematic approach, the current study briefly addresses the influence of fractal geometry with dimensions 1.0 < D < 1.5 on the stress fields of single dislocations.
total length of 3-26 000 nm. Whereas the straight dislocations examined consist exclusively of glissile edge type segments parallel to [001], the tangled dislocations contain additionally randomly distributed segments parallel to [010] (edge) and [100] (screw). From the length of the vector v combining the starting and the end coordinate of each dislocation line and the total length of the dislocation line t, the fractal dimension of the dislocation is calculated using D = l o g ( t / b ) / l o g ( v / b ) [23,24] where b is the Burgers vector. The dislocation lines are assumed to terminate at free surfaces. Denoting the number [001] segments by n o that of [010] with n~+, that of [0i0] with nB_, that of [100] with no+, and that of [TOO]with nc , the fractal dimension may be written D t o t a I = log(n0 + nB+ + nn + nc+ + nc )/log(v/b). Although the introduction of no+ and no with a constant value of (no+ - no ) would have also been pertinent in this context, this variation was avoided in order to keep the number of geometrical parameters involved as low as possible. Furthermore, it is conceivable that the variation of no+ and no_ does not lead to substantial changes compared with simulations with variation of n~ and no. The dislocation lines investigated are parallel to [001], starting from the origin of a Cartestian coordinate system. The Euclidean dislocations consisted of 3-4500 segments, each having the length of a scaling vector of length 1 nm. To enable comparison of both types of dislocation, the fractal dislocations contain the same number of [001] segments as their straight counterparts. However, in addition they contain randomly distributed portions parallel to [010] and [100]. Enriched with these additional segments, the fractal dislocations are usually much longer than their straight counterparts. For both types of dislocation the components of the 3D stress tensors are calculated according to the notation of Devincre and Condat [21] in the (001) plane in the middle of the dislocation lines by superimposing the stresses generated by each dislocation segment at the coordinate examined.
2. Simulation technique 3. Results and evaluation The dislocations are described as line defects embedded in an otherwise isotropic elastic medium. The dislocation lines are subdivided into tiny segments each having a length of 1 nm. For a given Burgers vector parallel to [100], three types of segment can be defined, i.e. screw type, glissile edge type and jog type edge dislocations [19-21]. The vectors involved generate a simple cubic lattice. For computation of the stresses the tensorial notation of Devincre and Condat [21] is employed. As shear modulus a value of 48 GPa is stipulated. Two types of dislocation are investigated, one with Euclidean and one with fractal dimension. Dislocations of the latter type have dimensions within the range 1.0 < D < 1.5. The dislocations examined have a
Figs. 1 and 2 show the maximum and minimum values of the a,,.(x,y, z) stress component of both straight (open triangles) and irregularly tangled (open squares) dislocations as calculated within the (001) plane for a distance Ay = 100 nm from the middle of the dislocation line. The Ax-positions of the extreme values varied between 240 nm and 480 nm. The maximum and minimum O-,y(X,y, z) stresses are presented as a function of the length of the dislocation line parallel to [001]. However owing to their content of [010] amd [100] segments the total length of the fractal dislocation lines is on average about 6 times longer than that of their straight counterparts.
205
D. Raabe / Materials" Science and Engineering A203 (1995) 203 207
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10,000
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Fig. 1. Maximum shear stress or. of straight (open triangles) and irregularly tangled (open squares) dislocations, calculated in the (001) plane Ay = 100 nm in the middle of the dislocation line. The stresses are given as a function of the length of the dislocation line parallel to [001]. Owing to their content of [010] and [100] segments, the total length of the fractal dislocation lines is on average about 6 times longer than that of their straight counterparts.
As is evident in Figs. 1 and 2, the maximum and minimum values computed for the straight dislocations are in very good accord with analytical predictions. In contrast, the corresponding fractal dislocations reveal considerable deviations with respect to these values. For fractal dislocation lines with more than 50-100 segments parallel to [001], we find both positive and negative deviations from the stresses predicted for their straight counterparts (Figs. i, 2). Short fractal dislocations containing less than 30 segments parallel to [001] generally reveal larger absolute values than their straight counterparts. On the basis of this first analysis it appears that the stress fields imposed by the short fractal dislocations are systematic in nature. At first sight it seems that for short fractal dislocations the stresses increase as a function of straight dislocations. However, the stresses associated with long fractal dislocations seem to be less systematic. The transition from
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Fig. 2. Minimum shear stress ~ , of straight (open triangles) and irregularly tangled (open squares) dislocations. Same conditions as in Fig. 1.
x [nm]
0
~5 0
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Fig. 3. Seven dislocations with identical length (1000 nm) parallel to [001] but different fractal dimensions (1.0 < D < 1.25).
the systematic to the less-systematic regime seems to relate to the distance from the dislocation lines within which the stresses were computed (Ay = 100 nm). Since in this simulation both the number of segments parallel to [001] and the fractal dimension, i.e. the total length of the dislocation line, was varied, the deviations observed cannot easily be attributed to either of these parameters. Hence in the following we investigated whether the deviations observed can be attributed simply to the total length of the dislocation line or to the fractal geometry of the dislocations. The deviations of the fractal dislocation from Euclidean geometry stem from both additional [101] and [100] type sections. To investigate the influence of fractality induced by either type in greater detail, seven dislocations with identical length parallel to [001] but different fractal dimensions were examined (Fig. 3). For this purpose three types of fractal dimension were defined: Dtota ! = log(n 0 +nB+ + nn + nc+ + nc )~log(rib), which counts the deviation imposed by both [010] and [100] segments; DB = log(n0+nB+ + nB_)/log(v/b), which only considers the influence of the [010] segments; Dc = log(n0+ nc+ + n c_ )/log(rib), which only includes the [100] segments. It has to be emphasized that D B hence accounts for the contribution of edge and Dc for the contribution of screw type dislocation segments. Fig. 4 shows the resulting maximum stresses ( A y = 100nm), expressed in units of (p'b)/(2.n.(Al)), where /~ is the shear modulus and A/ an average dislocation spacing (here 30 nm). From this analysis two results emerge. First, on average the maximum stress values increase with fractal dimension of the dislocations if only the number of segments parallel to [001] is considered on the abscissa (Fig. 4). At first sight this result does not correspond to analytical predictions. In a dislocation with fractal ge-
206
D. Raabe Materials Science and Engineering A203 (1995) 203-207 0,055
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Fractal Dimension of the Dislocation Fig. 4. Maximum stresses within a distance Ay = 100 nm from the middle of the dislocation line, expressed in units of (t~b)/ (2. ~r. (AI)), where/2 is the shear modulus and Al an average dislocation spacing (here 30 nm): total (open circles), D,o~l = log(no + nB+ + n B_ + no+ + n c )/log(v/b), total dimension as computed from the contribution of all additional [010] and [100] dislocation segments; type B (open squares), DB=log(no+nB+ +nB_)/log(v/b), partial dimension as computed from the contribution of all additional [010] edge dislocation segments; type C (open triangles), D c = log(no + nc + + nc_ )/log(v/b), partial dimension as computed from the contribution of all additional [100] screw dislocation segments.
ometry one would expect the maximum stresses to decrease rather than to increase owing to the selfscreening of dislocation segments having opposite sign. However, if the stresses predicted are normalized by the total length of the fractal dislocation line, which is much longer than the length of its straight counterpart (Fig. 3), it is found that the internal stresses indeed decrease with an increase in the fractal dimension of the dislocation (Fig. 5). This finding is attributed to selfscreening effects. It must be emphasized that this result only applies for stresses per unit length. It can thus be concluded that for a given dislocation density, i.e. for a fixed total length of all dislocation lines per volume, the generation of fractal dislocations can reduce internal stresses. However it must be considered that in addition to the influence of the length of the dislocation line and 0.04
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Fig. 5. Maximum stresses in units of (/~. b)/(2.n.(Al)) per unit length of the fractal dislocation line, symbols as in Fig. 4.
the self-screening effects, a third effect is superimposed, i.e. the character of the dislocation segments involved (edge, screw). The latter two parameters superimpose in the present calculation. For this reason the partial dimensions D B and D c, which account for either type of dislocation segment, were both considered. Figs. 4 and 5 show the normalized stresses as a function of the total dimension, of the dimension imposed exclusively by the addition of [010] edge segments (type B, open squares), and of the dimension imposed exclusively by the addition of [100] screw segments (type C, open triangles). As is evident from Figs. 4 and 5, both types of dislocation segment affects the stresses predicted in a similar manner. However, straight dislocations usually arrange themselves in elastically more or less favourable networks. The conclusion that internal stresses are degraded with increasing fractal dimension of the dislocations involved is thus to a certain extent academic. In teal materials it will often occur that the gain in i~ternal stresses via self-screening is compensated by the increase in the total length of the fractal dislocation line (Figs. 1-3). However, the situation of successful selfscreening may indeed apply if the dislocation density is very low, i.e. when the dislocations do not affect each other or when the mobility of the dislocations is impeded. Furthermore, this result could be revelant in interpreting metastable arrangements such as dislocation tangles. The second observation is that the degra¢ dation of the stresses with the increase in fractal dimension discussed above is only weakly affected by the type of fractal dimension considered (Dtotal, D B or Dc). This means that in the present simulation the deviation of the maximum stresses does not relate systematically to the types of dislocation segment which contribute to the increase in the fractal dimension.
References [1] A.S. Argon (ed.), Constitutive Equations in Plasticity, MIT Press Cambridge MA, 1975. [2] H. Mughrabi (ed.), in R.W. Cohn, P. Hassen and E. J. Kramer (eds.), Materials Science and Technology, Vol. 6, VCH 1993. [3] U.F. Kocks, A.S. Argon and M.F: Ashby, in B. Chalmers, J.W. Christian and T.B. Massalski (eds.), Thermodynamics and Kinetics of Slip, Progress in Materials Science, Vol. 19, Pergamon, 1975. [4] J.P. Hirth and J. Lothe, Theory of Dislocations, McGraw-Hill, 1968. [5] D.B. Holt, J. Appl. Phys., 41 (1970) 3197. [6] F.B. Prinz and A.S. Argon, Acta Metall., 32 (1984) 1021. [7] H. Mughrabi, Acta Metall., 31 (1983) 1367. [8] A.S. Argon and P. Hassen, Acta Metall., 41 (1993) 3289. [9] P. HS.hner, PhD Thesis, Universitiit Stuttgart, 1992, pp. 74-91. [10] G. Saada and E. Bouchaud, Acta Metall., 41 (1993) 2173. [1 I] N.M. Ghoniem and R.J. Amodeo, Phys. Rev. B, 41 (1989) 6958, 6968.
D. Raabe / Materials Science and Engineering A203 (1995) 203 207
[12] A.N. Gulluoglu, D.J. Srolovitz, R. LeSar and P.S. Lomdahl, Scr. Metall., 23 (1989) 1347. [13] V.A. Lubarda, J.A. Blume and A. Needleman, Acta Metall., 41 (1993) 625. [14] J.L. Hokanson, J. Appl. Phys., 34 (1963) 2337. [15] E.H. Yoffe, Philos. Mag., 5 (1960) 161. [16] J.C.M. Li, Acta Metall., 13 (1965) 931. [17] R. DeWit, Phys~ Statis. Solidi., 20 (1967) 567, 575. [18] R. DeWit. Aeta Metall., 13 (1965) 1210. [19] L.P. Kubin, G.R. Canova, M. Condat. B., Devincre, V. Pontikis
2fl7
and Y. Br6chet, SolM State Phenom. 23 24 (1992) 445 472.1. [20] L.P. Kubin, in H. Mughrabi (ed.), in R.W. Cahn, P. Haasen and E.J. Kramer (eds.), Materials Science and Teehnology, Vol. 6, VCH, 1993, p. 137. [21] B. Devincre and M. Condat, Acta Metall., 40 (1992) 2629. [22] E. Hornbogen, Int. Mat. Rev., 34 (1989) 277. [23] B.B. Mandelbrot, Tlle Fraetal Geometry qf Nature. Freeman, New York, 1982. [24] D. Raabe, F. Roters and G. Gottstein. Comput Mater. Sci., in press.
NAIlEIIAL~ SalENCE &
ENGINEERING ELSE VIE R
Materials Science and Engineering A203 (1995) 203 207
A
On the stress fields of crystal dislocations with fractal geometry D. Raabe Institut Jgir Metallkunde und Metallphysik, Kopernikusstrasse 14, Rheinisch-Wes(/gilische Technische Hochschule Aachen, 52056 Aachen, Germany Received 16 January 1995; in revised form 31 March 1995
Abstract
The elastic stress fields of single irregularly tangled dislocations were simulated numerically. The dislocations had fractal dimensions within the range 1.0 < D < 1.5. The results were compared with the stress fields associated with straight dislocations (D = 1). To calculate the stress fields with a high spatial resolution the dislocation lines were subdivided into small segments. The results confirm that the stress fields of tangled dislocations depend considerably on their fractal dimension. The maximum shear stress generated per unit length of a single dislocation decreases with an increase in fractal dimension of its dislocation line.
Keywords: Elastic stress fields; Crystal dislocations; Fractal geometry
I. Introduction
For computation of the contribution of dislocations to the yield stress crystals, the internal stresses generated by a given dislocation distribution must be determined (dislocation statics). Together with the external stresses and stresses imposed by other lattice defects they enter into the calculation of the evolution of the dislocation structure during plastic deformation (dislocation dynamics). The first subject belongs to the field of flow stress theory and is thus an ingredient of the kinetic law of crystal plasticity. The second subject falls into the domain of the structural evolution law [e.g. 1-3]. To model crystal plasticity physically, a profound theoretical backbone for computation of the linear elastic stress fields of dislocations is required [e.g. 4]. Although crystal dislocations represent one-dimensional (1D) lattice defects in terms of the Burgers circuit, they arrange themselves in intricate 3D networks. The geometrical complexity and the large number of dislocations assembled in such arrays has led to the use of analytical methods for investigation of dislocation statics and dynamics in the past [5-8]. Except for some recent studies [9,10] these analytical approaches did not include the discretization of single dislocations in space. Promoted by the rapid improvement of computer facilities, numerical approaches are now increasingly 0921-5093/95/$09.50 ~? 1995 - Elsevier Science S.A. All rights reserved
S S D I 0921-5093(95)09856-9
being developed which allow for the discrete description of single dislocations in both time and space. Most of these studies were concerned with numerical analysis of the statics and dynamics of 2D dislocation arrays [e.g. 11-13]. The mathematical treatment of single dislocations also in 3D was pioneered by Hokanson [14] and Yoffe [15]. They showed that to calculate the stress fields associated with simple dislocation arrays, the dislocation lines must be decomposed into isolated segments. However such techniques neglected the facts that first dislocation cannot terminate within an otherwise perfect region of crystal, and second, equilibrium was not maintained in such arrangements. As was shown by Li [16] and described in greater detail by deWit [17,18], a consistent mathematical 3D treatment of the stress fields of piecewise straight dislocation segments is attained when the Burgers vector is conserved at each node. This means that dislocation segments which are not entirely integrated within the network must be semi-infinite or terminate at a free surface, a grain boundary, or some other defect, where stress equilibrium can be maintained. On the basis of these studies Kubin et al. [19,20] and Devincre and Condat [21] developed 3D models in which dislocations are discretized in both space and time. Whereas considerable success was achieved by both analytical and numerical approaches in predicting flow
204
D. Raabe / Materials' Science and Engineering A203 (1995) 203- 207
curves and non-linear aspects such as dislocation patterning, the problem of dimensionality has not yet been treated adequately in the literature. Nearly all investigations tackling dislocation statics and dynamics appear to follow a similar understanding that the complex geometry of dislocation lines may be treated generally in the framework of Euclidean geometry. Indeed, most honeycomb-type networks and even bimodal distributions consisting of dislocation walls and regions which are nearly void of dislocations may be described in terms of Eucildean dimensions. However, it was shown by Hornbogen [22] that individual dislocation lines often do not have a Euclidean dimension ( D - - 1 ) but have a fractal dimension (1 < D < 2) [23]. This applies especially for dislocations which intersect with other dislocations or interact with point defects [22]. The description of the geometry of non-straight dislocation lines within the framework of fractal geometry provides a useful means of deriving the degree of deviation from the initial straight dislocation in a quantitative manner. However, it should be underscored that the employment of fractal dimensions is physically pertinent in a rigorous sense only in cases where strict self-similarity of the object described is observed [23]. Although dislocations and dislocation arrays at first bear a certain resemblance to self-similar contours [22], they do not really fulfil this criterion in a mathematical sense. Consequently, in the present study nonEuclidean dimensions are simply used to provide a quantitative method for describing non-straight dislocation lines without implying strict self-similarity. In a recent study it was shown for the first time that the stress field associated with a small angle tilt boundary indeed depends on the fractal dimension of the dislocations involved [24]. In a more systematic approach, the current study briefly addresses the influence of fractal geometry with dimensions 1.0 < D < 1.5 on the stress fields of single dislocations.
total length of 3-26 000 nm. Whereas the straight dislocations examined consist exclusively of glissile edge type segments parallel to [001], the tangled dislocations contain additionally randomly distributed segments parallel to [010] (edge) and [100] (screw). From the length of the vector v combining the starting and the end coordinate of each dislocation line and the total length of the dislocation line t, the fractal dimension of the dislocation is calculated using D = l o g ( t / b ) / l o g ( v / b ) [23,24] where b is the Burgers vector. The dislocation lines are assumed to terminate at free surfaces. Denoting the number [001] segments by n o that of [010] with n~+, that of [0i0] with nB_, that of [100] with no+, and that of [TOO]with nc , the fractal dimension may be written D t o t a I = log(n0 + nB+ + nn + nc+ + nc )/log(v/b). Although the introduction of no+ and no with a constant value of (no+ - no ) would have also been pertinent in this context, this variation was avoided in order to keep the number of geometrical parameters involved as low as possible. Furthermore, it is conceivable that the variation of no+ and no_ does not lead to substantial changes compared with simulations with variation of n~ and no. The dislocation lines investigated are parallel to [001], starting from the origin of a Cartestian coordinate system. The Euclidean dislocations consisted of 3-4500 segments, each having the length of a scaling vector of length 1 nm. To enable comparison of both types of dislocation, the fractal dislocations contain the same number of [001] segments as their straight counterparts. However, in addition they contain randomly distributed portions parallel to [010] and [100]. Enriched with these additional segments, the fractal dislocations are usually much longer than their straight counterparts. For both types of dislocation the components of the 3D stress tensors are calculated according to the notation of Devincre and Condat [21] in the (001) plane in the middle of the dislocation lines by superimposing the stresses generated by each dislocation segment at the coordinate examined.
2. Simulation technique 3. Results and evaluation The dislocations are described as line defects embedded in an otherwise isotropic elastic medium. The dislocation lines are subdivided into tiny segments each having a length of 1 nm. For a given Burgers vector parallel to [100], three types of segment can be defined, i.e. screw type, glissile edge type and jog type edge dislocations [19-21]. The vectors involved generate a simple cubic lattice. For computation of the stresses the tensorial notation of Devincre and Condat [21] is employed. As shear modulus a value of 48 GPa is stipulated. Two types of dislocation are investigated, one with Euclidean and one with fractal dimension. Dislocations of the latter type have dimensions within the range 1.0 < D < 1.5. The dislocations examined have a
Figs. 1 and 2 show the maximum and minimum values of the a,,.(x,y, z) stress component of both straight (open triangles) and irregularly tangled (open squares) dislocations as calculated within the (001) plane for a distance Ay = 100 nm from the middle of the dislocation line. The Ax-positions of the extreme values varied between 240 nm and 480 nm. The maximum and minimum O-,y(X,y, z) stresses are presented as a function of the length of the dislocation line parallel to [001]. However owing to their content of [010] amd [100] segments the total length of the fractal dislocation lines is on average about 6 times longer than that of their straight counterparts.
205
D. Raabe / Materials" Science and Engineering A203 (1995) 203 207
16 z [nml
14
0.5 F
>~f=
00/ O
7~
00o
°o:I Zj2---
°I : i
2~ 1
~-~
10
:
: :l '~
100
1.2<:0<1.6[
1,000
100
10,000
Number of Segments parallel to [001] (Straight Part of Dislocation)
Fig. 1. Maximum shear stress or. of straight (open triangles) and irregularly tangled (open squares) dislocations, calculated in the (001) plane Ay = 100 nm in the middle of the dislocation line. The stresses are given as a function of the length of the dislocation line parallel to [001]. Owing to their content of [010] and [100] segments, the total length of the fractal dislocation lines is on average about 6 times longer than that of their straight counterparts.
As is evident in Figs. 1 and 2, the maximum and minimum values computed for the straight dislocations are in very good accord with analytical predictions. In contrast, the corresponding fractal dislocations reveal considerable deviations with respect to these values. For fractal dislocation lines with more than 50-100 segments parallel to [001], we find both positive and negative deviations from the stresses predicted for their straight counterparts (Figs. i, 2). Short fractal dislocations containing less than 30 segments parallel to [001] generally reveal larger absolute values than their straight counterparts. On the basis of this first analysis it appears that the stress fields imposed by the short fractal dislocations are systematic in nature. At first sight it seems that for short fractal dislocations the stresses increase as a function of straight dislocations. However, the stresses associated with long fractal dislocations seem to be less systematic. The transition from
-2
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::::::
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30 100
::~
1,000
: : ::: 10,000
Number of Segments parallel to [001 ](Straight Part of Dislocation)
Fig. 2. Minimum shear stress ~ , of straight (open triangles) and irregularly tangled (open squares) dislocations. Same conditions as in Fig. 1.
x [nm]
0
~5 0
50
Fig. 3. Seven dislocations with identical length (1000 nm) parallel to [001] but different fractal dimensions (1.0 < D < 1.25).
the systematic to the less-systematic regime seems to relate to the distance from the dislocation lines within which the stresses were computed (Ay = 100 nm). Since in this simulation both the number of segments parallel to [001] and the fractal dimension, i.e. the total length of the dislocation line, was varied, the deviations observed cannot easily be attributed to either of these parameters. Hence in the following we investigated whether the deviations observed can be attributed simply to the total length of the dislocation line or to the fractal geometry of the dislocations. The deviations of the fractal dislocation from Euclidean geometry stem from both additional [101] and [100] type sections. To investigate the influence of fractality induced by either type in greater detail, seven dislocations with identical length parallel to [001] but different fractal dimensions were examined (Fig. 3). For this purpose three types of fractal dimension were defined: Dtota ! = log(n 0 +nB+ + nn + nc+ + nc )~log(rib), which counts the deviation imposed by both [010] and [100] segments; DB = log(n0+nB+ + nB_)/log(v/b), which only considers the influence of the [010] segments; Dc = log(n0+ nc+ + n c_ )/log(rib), which only includes the [100] segments. It has to be emphasized that D B hence accounts for the contribution of edge and Dc for the contribution of screw type dislocation segments. Fig. 4 shows the resulting maximum stresses ( A y = 100nm), expressed in units of (p'b)/(2.n.(Al)), where /~ is the shear modulus and A/ an average dislocation spacing (here 30 nm). From this analysis two results emerge. First, on average the maximum stress values increase with fractal dimension of the dislocations if only the number of segments parallel to [001] is considered on the abscissa (Fig. 4). At first sight this result does not correspond to analytical predictions. In a dislocation with fractal ge-
206
D. Raabe Materials Science and Engineering A203 (1995) 203-207 0,055
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ometry one would expect the maximum stresses to decrease rather than to increase owing to the selfscreening of dislocation segments having opposite sign. However, if the stresses predicted are normalized by the total length of the fractal dislocation line, which is much longer than the length of its straight counterpart (Fig. 3), it is found that the internal stresses indeed decrease with an increase in the fractal dimension of the dislocation (Fig. 5). This finding is attributed to selfscreening effects. It must be emphasized that this result only applies for stresses per unit length. It can thus be concluded that for a given dislocation density, i.e. for a fixed total length of all dislocation lines per volume, the generation of fractal dislocations can reduce internal stresses. However it must be considered that in addition to the influence of the length of the dislocation line and 0.04
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the self-screening effects, a third effect is superimposed, i.e. the character of the dislocation segments involved (edge, screw). The latter two parameters superimpose in the present calculation. For this reason the partial dimensions D B and D c, which account for either type of dislocation segment, were both considered. Figs. 4 and 5 show the normalized stresses as a function of the total dimension, of the dimension imposed exclusively by the addition of [010] edge segments (type B, open squares), and of the dimension imposed exclusively by the addition of [100] screw segments (type C, open triangles). As is evident from Figs. 4 and 5, both types of dislocation segment affects the stresses predicted in a similar manner. However, straight dislocations usually arrange themselves in elastically more or less favourable networks. The conclusion that internal stresses are degraded with increasing fractal dimension of the dislocations involved is thus to a certain extent academic. In teal materials it will often occur that the gain in i~ternal stresses via self-screening is compensated by the increase in the total length of the fractal dislocation line (Figs. 1-3). However, the situation of successful selfscreening may indeed apply if the dislocation density is very low, i.e. when the dislocations do not affect each other or when the mobility of the dislocations is impeded. Furthermore, this result could be revelant in interpreting metastable arrangements such as dislocation tangles. The second observation is that the degra¢ dation of the stresses with the increase in fractal dimension discussed above is only weakly affected by the type of fractal dimension considered (Dtotal, D B or Dc). This means that in the present simulation the deviation of the maximum stresses does not relate systematically to the types of dislocation segment which contribute to the increase in the fractal dimension.
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and Y. Br6chet, SolM State Phenom. 23 24 (1992) 445 472.1. [20] L.P. Kubin, in H. Mughrabi (ed.), in R.W. Cahn, P. Haasen and E.J. Kramer (eds.), Materials Science and Teehnology, Vol. 6, VCH, 1993, p. 137. [21] B. Devincre and M. Condat, Acta Metall., 40 (1992) 2629. [22] E. Hornbogen, Int. Mat. Rev., 34 (1989) 277. [23] B.B. Mandelbrot, Tlle Fraetal Geometry qf Nature. Freeman, New York, 1982. [24] D. Raabe, F. Roters and G. Gottstein. Comput Mater. Sci., in press.