Philosophical Magazine Vol. 92, No. 23, 11 August 2012, 2947–2956
Stable edge dislocations in finite crystals Arun Kumar and Anandh Subramaniam*
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Materials Science and Engineering, Indian Institute of Technology, Kanpur-208016, India (Received 13 September 2011; final version received 18 March 2012) Dislocations have been considered as mechanically unstable defects in bulk crystals, ignoring the Peierls oscillations. Eshelby [J. Appl. Phys. 24 (1953) p.176] had showed that a screw dislocation can be stable in a thin cylinder. In the current work, considering Eshelby’s example of an edge dislocation in a single crystalline plate, we show that an edge dislocation can be stable in a finite crystal. Using specific examples, we also show that the position of stability of an edge dislocation can be off-centre. This shift in the stability from the centre marks the transition from a stable dislocation to an unstable one. The above-mentioned tasks are achieved by simulating edge dislocations using the finite element method. Keywords: edge dislocation; mechanical stability; finite element method
1. Introduction The Gibbs free energy of a crystal increases in the presence of a dislocation and this scenario does not change till the melting point of the crystal and, hence, dislocations are thermodynamically unstable defects [1]. The energy of a dislocation can be approximately divided into elastic energy and core energy [2]. The core energy of the dislocation is expected to be a constant additive term to the elastic energy until the dislocation is positioned within a few Burgers vectors from a free surface [3]. In a semi-infinite body, the elastic energy associated with the long-range stress fields of a dislocation continuously decreases as the dislocation approaches a free surface; thus making dislocations mechanically unstable defects [2]. Superimposed on this energy is the energy due to the Peierls oscillations, which is small compared to the elastic energy of the dislocation [2,4]. When a single dislocation in a free-standing crystal (i.e. unloaded and unconstrained) is positioned in a Peierls valley, the system is in a metastable state and the dislocation does not leave the crystal unless the Peierls stress is exceeded by the configurational force towards the free surface [4]. The question: ‘‘Can a dislocation be stable inside a crystal?’’, is an old one and has been investigated in two distinct contexts. The first context is related to a screw dislocation in a thin cylinder, and Eshelby [5] has shown that the screw dislocation can be stable for a range of positions centred about the axis of the cylinder. The second context relates to the stability of edge dislocations near free surfaces [6,7]. This question in the second context originates from the fact that, as a dislocation *Corresponding author. Email:
[email protected] ISSN 1478–6435 print/ISSN 1478–6443 online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/14786435.2012.682176 http://www.tandfonline.com
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approaches a free surface, it causes distortion of the surface, thus increasing the surface energy [7]. Hence, the specific question which was addressed was: ‘‘Can there be a position of the dislocation near the free surface where energy of the system is a minimum?’’ [7]. In the second context, current understanding is that the free surface is the lowest energy position (i.e. there is no minimum in energy close to the free surface) [7]. The stability of edge dislocations in a crystal finite in one dimension (and infinite along the other two dimensions) has been investigated by Wu and Chiu [8]. They have shown that, for positions of the dislocation along the finite dimension (climb motion), the dislocation can be stable at the centre of the domain. It should be noted that the dimension along the dislocation line is irrelevant to the problem, except in crystals with high anisotropy [8]. Important aspects not considered in the work include: (i) the deformation of the domain (which is more pronounced in thin ones) in the presence of the edge dislocation and its effect on the stress state of the system and (ii) the stability of an edge dislocation in finite domains. In this investigation, we try to answer the following important questions: (i) can a dislocation be stable in a finite crystal? (and what is the region of stability?), and (ii) can the position of stability be off-centre (for climb in the domain)? Keeping in mind the statement made previously, it should suffice to consider crystals finite in two dimensions (as the dimension along the dislocation line is irrelevant in most cases). If one is concerned about the global energy minimum, then the term ‘metastable’ should be used (and not stable), as the edge dislocation can leave the crystal to form a surface step, thus lowering the energy of the system. However, in this case, the dislocation is eliminated and the current work is concerned with a crystal containing a dislocation. Additionally, the local Peierls oscillations (which are superimposed on the elastic energy due to long-range stress fields) have been ignored in addressing the above questions, as their magnitude is small [4]. Towards this objective (to investigate the stability of edge dislocations in crystals), a thin plate of a crystal (aluminium is taken as an example) is considered, which bends in the presence of an edge dislocation (as originally conceived by Eshelby [9]). This case of a thin plate with a dislocation is very old and has been investigated from both experimental [10] and theoretical points of view [9,11,12]. To solve a broader class of dislocation-related problems, a numerical technique, such as a finite element method (FEM), is better suited compared to purely analytical techniques [13,14], which are often very cumbersome (e.g. see comment by Eshelby [9]). Additionally, standard analytical formulae can prove inadequate in the case of nanocrystals [14]. Keeping this in mind, FEM has been used in the current work to simulate an edge dislocation in finite domains. In the context of the use of FEM for studying dislocations, it is important to note investigations in the field of crystal plasticity, dislocation mechanics and dislocation dynamics. These studies have thrown considerable light on the evolution of dislocation distribution under external loading. The work of Acharya [15], Dluzewski et al. [16], Sasaki et al. [17], Dluzewski et al. [18], Roy and Acharya [19], Bulatov and Cai [20], and Belytschko and Gracie [13] can be consulted for alternate methods of simulating dislocations (especially the distribution of dislocations and their time evolution).
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2. Finite element methodology A brief outline of the simulation methodology is presented here, the details of which can be found elsewhere [14,21]. An edge dislocation in aluminium (a0 ¼ 4.04 A˚; slip system: h110i{111}, b ¼ 2a0/2 ¼ 2.86 A˚, G ¼ 26.18 GPa, ¼ 0.348 [22]) is simulated by imposing eigenstrains corresponding to the introduction of a ‘half-plane’ of atoms, as illustrated schematically in Figure 1. These eigenstrains ("xx ) are imposed uniformly in region A (Figure 1) as thermal strains in the model ("xx ¼ DT, where is the coefficient of thermal expansion and DT is the change in temperature). The difference in the current approach with respect to other prescriptions (e.g. Mura [23] uses shear eigenstrains) should be noted. This methodology to simulate an edge has been validated previously [14]. Isotropic plane strain conditions are assumed and it is also assumed that bulk material properties can be used at the lengthscales of the simulation. The domain was meshed with four-noded bilinear quadrilateral elements. Plane stress simulations were also performed to check the effect of the third dimension (if any). Standard ‘ABAQUS’ software was used to compute the stress fields and the energy of the system (displacement fields being the fundamental field variable), using linear theory of elasticity (Cook et al. [24] may be consulted for details related to finite element implementation). Total energy of the system, along with the deformed configuration, was determined for various positions of the edge dislocation in the domain (by varying x and/or y in Figure 1). In Figure 1, the point E is constrained both in the X and Y directions (displacement constraints) and E 0 in X direction. G is a general point on the mid-plane of the domain. The boundary conditions applied are sufficient to eliminate rigid body (zero energy) modes. The boundary conditions imposed imply that the domain is free to bend in the presence of the edge dislocation. The extension of the model to anisotropic conditions is straightforward and has not been considered in the present work. Additionally, Al has a low anisotropy factor (A ¼ 2C44/(C11–C12) ¼ 1.23) and, hence, the assumption of isotropic conditions is expected to give reasonable results. Core structure and its energy have not been considered in the model. The core has been replaced by an elastic region, the dimension of which is of the order of the Burgers vector in the x and y directions (this is similar to the approach used by some investigators [17,25]). This does not alter the conclusions made, as it will be a constant additive term to the total energy (i.e. this will not change the shape of the plot of energy versus position of the dislocation).
Figure 1. Finite element model (schematic; not to scale) to simulate an edge dislocation in a domain of length (L) 100 b. G is a general point on the mid-plane. Point E is constrained in both X and Y directions and E 0 in the X direction. The thickness (t) is a variable. The slip plane is kept fixed perpendicular to the Y direction.
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Figure 2. Stress state of the system ( x plot) when the edge dislocation is positioned at x ¼ 30b and y ¼ 5b (x and y are as in Figure 1). The approximate angle of bending in the presence of the edge dislocation is also marked in the figure. Dimensions marked are for the undeformed shape.
3. Results and discussion We first consider the results of the simulation for a domain of thickness (t) of 20b. The stress state of the system ( x plot) when the dislocation is positioned at x ¼ 30b and y ¼ 5b (x and y as in Figure 1) is shown in Figure 2. The energy of the system for various values of x and y is plotted in Figure 3. It is seen that, for considerable positions along the X-axis (varying location of point G), the dislocation is stable with respect to arbitrary displacements. For a path along the X-axis up to a value of x ¼ 35b (point R in Figure 3), the system displays neutral equilibrium (within a numerical accuracy of 0.1%). Hence, with respect to arbitrary displacements from point G, the system can be termed ‘quasi-stable’. The locus of unstable positions is marked as PQ (and P‘Q’) in Figure 3 and the curve PQRQ‘P’ encloses the stable region. This implies that for G at 35b, the displacements are restricted to point into the region enclosed by the curve. It is worthwhile noting that, although the term ‘about’/‘approximate’ should qualify all results presented in this section (as they arise from finite element computations), these have often been left out. For x 30b the unstable position is at a position of about y ¼ 5.5b (i.e. at 0.55 t/2 ¼ 55% of the half the thickness of the domain). This agrees well with the results of Wu and Chiu [8], who calculated a value of the unstable position to be 0.53 t/2 (x being an irrelevant parameter in their work). For the case of the screw dislocation in a cylinder, Eshelby had calculated the unstable position to be at 0.54 r (where, r is the radius of the cylinder) [5]. For 30b5 x 535b, the energy of the unstable position reduces until the minimum at y ¼ 0 ceases to exist (Figure 3). With increasing thickness (t) of the domain, there are no qualitative changes to the energy landscape (i.e. regarding maxima, minima and stability), until t approaches about 80b. In this regime (20b 5 t 5 80b), the range of neutral equilibrium decreases with increasing thickness (when t ¼ 80b, this value reduces to 5b). However, in thicker domains (t 2 [80b,100b], L ¼ 100b), important features begin to emerge with respect to stability of the edge dislocation. For t ¼ 90b, the system displays no neutral equilibrium along the X-direction. Qualitatively, t ¼ 80b and t ¼ 100b represent two ends of the spectrum of thicknesses. For t ¼ 100b, there is no stable region in the domain and the centre of the domain is an unstable position (Figure 4a). For t ¼ 80b, the situation is similar to the domain with t ¼ 20b (for positions with x 2 [0b, 13b]), with the central position being a stable point and the point of instability is at about 0.54 t/2 (Figure 4b). For t 2 [90b, 94b], the stable
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Figure 3. Energy of the system for different positions of the edge dislocation (varying x and y) in the domain (t ¼ 20 b). The region of stability of an edge dislocation is the region enclosed by the curve PQRQ‘P’. Point R is at (35b, 0). Within the region of stability for a path along the X-direction, the system displays neutral equilibrium.
point is located off-centre along the Y direction (with x ¼ 0) (Figure 4c). For t ¼ 90b, the minimum position is at 6b and, for t ¼ 94b, it is located at 17b. The global energy maximum shifts between these two thicknesses: for t ¼ 94b, the global energy maximum lies at the centre of the domain (Figure 4c), while, for t ¼ 90b, it lies offcentre (Figure 4c). In Eshelby’s analysis of screw dislocations in cylinders, the stable position was found to be at the centre (accompanied by a twist of the cylinder) [5]. In experiments conducted on Pb crystallites on Ru substrates, an off-centre position for a screw dislocation has been observed [26], but the reason for this has not been explained. The transition from a domain with a stable position of an edge dislocation to one with no such position takes place at t ¼ 96b (Figure 4a). A domain of thickness of t ¼ 96b shows a point of inflection in the energy versus position plot (point I in Figure 4a). For t ¼ 80b, an interesting feature appears for x 2 [14b, 16b]: the minimum located at y ¼ 0 splits into two minima, as shown in Figure 5 for x ¼ 14b and x ¼ 15b. This ‘x’ position for off-centre minima tends to zero with increasing thickness and, for t ¼ 90b, this is realized at x ¼ 0. Figure 6 shows the energy landscape for the domain with thickness (t) ¼ 90b. Loss in stability with increasing ‘x’ is clearly seen from this plot. These results are summarized in Table 1, which highlights the following points: (i) in domains thicker than 90b, the stability region shrinks to a line (along Y), and (ii) the dislocation is unstable in domains thicker than 96b.
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Figure 4. Energy of the system with varying y (keeping x ¼ 0) for domains of different thicknesses (t): (a) t ¼ 80 b, t ¼ 88 b, (b) t ¼ 90 b, t ¼ 94 b, (c) t ¼ 96 b, 100 b. Axes for the insets are the same as that for the main figures.
The features observed can be qualitatively understood as follows. When L ¼ t (or L t), the bending of the domain is small and the behaviour of the dislocation is similar to that of a dislocation in a semi-infinite body (i.e. always attractive to the free surface and there is no stable position in the domain). It is to be noted that the bending increases with a decreasing thickness of the plate. When t 5 L (t 5 0.8 L), the domain bends in the presence of the dislocation, thus partially relaxing the stress field (and energy) of the dislocation. This bending decreases as the dislocation moves towards the free surface along Y (e.g. for x ¼ 0, the angle of bending for y ¼ 0 is 4.8 , which decreases to 3.6 at y ¼ 5b). This partial relaxation due to bending is superimposed on the usual trend of a decreasing energy of the dislocation towards the free surface. The competition between the two effects leads to an off-centre maximum (as reflected in position of instability for y 0.55 t/2). The X-direction is a direction of neutral equilibrium for considerable positions of the dislocation (0.6 L/2 for thin beams), beyond which the usual effect of the free-surface takes over. The transition regime from thick beams (t 0.8 L) to square domains (t ¼ L) takes place via a sequence of energy landscapes, exhibiting interesting features, as outlined above. From the above-mentioned figures, it is seen that the y ¼ 0 plane is a mirror plane with respect to energy; i.e. positions of þy are the same as positions of y
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Figure 5. Energy of the system for t ¼ 80 b, for x ¼ 14b and x ¼ 15b. Change in the position of the global maximum is to be noted.
Figure 6. Energy landscape for different positions of the edge dislocation in a domain of thickness (t) ¼ 90b. This is to be compared with the landscape obtained in Figure 3.
(for any ‘x’). This plane is only an approximate mirror plane for stresses (black to white mirror plane – taking compressive stresses to tensile stresses) due to the bending of the domain. Computations for the case of a domain in plane stress condition (results not presented here) show no qualitative differences with the
L/t
5
1.25
1.11
1.06
1.04
1
Thickness (t)
20
80
90
94
96
100
–
–
–
Decreasing energy with x
Neutral equilibrium till x ¼ 5b
Neutral equilibrium till x ¼ 35b
Varying x (y ¼ 0)
Varying y (x ¼ 0)
Decreasing energy with y
Maximum at y ¼ 0 Point of inflection at y ¼ 21 b
Minimum at y ¼ 17 b Maxima at y ¼ 0, y ¼ 25.4 b
Minimum at y ¼ 0 Maximum at y ¼ 5.5 b Minimum at y ¼ 0 Maximum at y ¼ 21.6 b Minimum at y ¼ 6 b Maxima at y ¼ 0, y ¼ 24.3 b
Table 1. Summary of the results for domains of varying thickness.
Comments
Reduced region of stability as compared to t ¼ 20 b (Figure 4b) Stability only along Y Loss of stability along X (Figure 6) Stability only along Y No stability along X (Figure 4c) Loss of stability along Y No stability along X (Figure 4a) Dislocation unstable (Figure 4a)
Large region of stability (Figure 3)
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above-mentioned results for plane strain condition. Plane stress condition is realized for a crystal that is thin in the third dimension and, hence, this implies that, qualitatively, the third dimension is irrelevant under isotropic conditions. It is expected that the results are also valid for crystals with low anisotropy [8]. This also implies that dislocations can be stable in crystals that are finite in two or three dimensions. The computations in the current work have been performed for a domain of fixed length (of 100b). Given the continuum nature of the simulations, the results are expected to be valid for any domain as long as the comparisons are made for a given L/t (both L and t should be greater than about 5b, the usual value assumed for the upper bound of the core radius [4])
4. Summary The mechanical stability of an edge dislocation in finite crystals (finite in the two relevant dimensions) has not been investigated so far. Using Eshelby’s classic example of an edge dislocation in a thin single crystalline plate which can bend, we show that an edge dislocation can be stable for a wide range of positions within a finite crystal, with the exception of one path along which it displays neutral equilibrium. Hence, the term ‘stable’ may be replaced with ‘quasi-stable’. Additionally, the transition from a stable dislocation to an unstable one is studied by increasing the thickness of a rectangular domain. First, the loss of stability occurs along the X-direction and the region of stability reduces to a line (along the Y direction). It is seen for domains of specific dimensions that the stable position of the edge dislocation can be off-centre (in the direction of climb). In nearly square domains, edge dislocations are unstable. The aforementioned tasks were achieved by simulating an edge dislocation using eigenstrains in a finite element model.
Acknowledgments The authors thank Professor K. Srihari and Professor Madhav Ranganathan for invaluable discussions.
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[10] S. Amelinckx, The direct observation of dislocations, in Solid State Physics (Supplement 6), F. Seitz and D. Turnbull, eds., Academic Press, London, 1964, p.405. [11] F. Kroupa, Appl. Math. 4 (1959) p.239. [12] V.L. Indenbom, Sov. Phys.-Dokl. 4 (1960) p.1125. [13] T. Belytschko and R. Gracie, Int. J. Plast. 23 (2007) p.1721. [14] P. Khanikar, A. Kumar and A. Subramaniam, Philos. Mag. 91 (2011) p.730. [15] A. Acharya, J. Mech. Phys. Solids 49 (2001) p.761. [16] P. Dluzewski, G. Jurczak, G. Maciejewski, S. Kret, P. Ruterana and G. Nouet, Mater. Sci. Forum 404-407 (2002) p.141. [17] K. Sasaki, M. Kishida and Y. Ekida, Int. J. Numer. Methods Eng. 54 (2002) p.671. [18] P. Dluzewski, G. Maciejewski, G. Jurczak, S. Kret and J.-Y. Laval, Comput. Mater. Sci. 29 (2004) p.379. [19] A. Roy and A. Acharya, J. Mech. Phys. Solids 53 (2005) p.143. [20] V.V. Bulatov and W. Cai, Computer Simulations of Dislocations, Oxford University Press, Oxford, 2006. [21] P. Khanikar and A. Subramaniam, J. Nano Res. 10 (2010) p.93. [22] E.A. Brandes (ed.), Smithells Metals Reference Book, Butterworths, London, 1983. [23] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, The Hague, 1982. [24] R.D. Cook, D.S. Malkus, M.F. Plesha and R.J. Witt, Concepts and Applications of Finite Element Analysis, Wiley, Singapore, 2003. [25] V.A. Lubarda and X. Markenscoff, Arch. Appl. Mech. 77 (2007) p.147. [26] M. Ranganathan, D.B. Dougherty, W.G. Cullen, T. Zhao, J.D. Weeks and E.D. Williams, Phys. Rev. Lett. 95 (2005) p.225505.
