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This article was downloaded by: [University of Cincinnati] On: 21 March 2011 Access details: Access Details: [subscription number 930567118] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK Philosophical Magazine Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713695589 Image forces on edge dislocations: a revisit of the fundamental concept with special regard to nanocrystals Prasenjit Khanikara; Arun Kumara; Anandh Subramaniama a Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India First published on: 04 December 2010 To cite this Article Khanikar, Prasenjit , Kumar, Arun and Subramaniam, Anandh(2011) 'Image forces on edge dislocations: a revisit of the fundamental concept with special regard to nanocrystals', Philosophical Magazine, 91: 5, 730 — 750, First published on: 04 December 2010 (iFirst) To link to this Article: DOI: 10.1080/14786435.2010.529089 URL: http://dx.doi.org/10.1080/14786435.2010.529089 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. 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Philosophical Magazine Vol. 91, No. 5, 11 February 2011, 730–750 Image forces on edge dislocations: a revisit of the fundamental concept with special regard to nanocrystals Prasenjit Khanikar, Arun Kumar and Anandh Subramaniam* Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 (Received 3 April 2010; final version received 30 September 2010) Two conditions under which image forces become significant are when a dislocation is close to a surface (or interface) or when the dislocation is in a nanocrystal. This investigation pertains to the calculation of image forces under these circumstances. A simple edge dislocation is simulated using finite element method (FEM) by feeding-in the appropriate stress-free strains in idealised domains, corresponding to the introduction of an extra half-plane of atoms. Following basic validation of the new model, the energy of the system as a function of the position of the simulated dislocation is plotted and the gradient of the curve gives the image force. The reduction in energy of the system arises from two aspects: firstly, due to the position of the dislocation in the domain and, secondly, due to deformations to the domain (/surfaces). The second aspect becomes important when the dislocation is positioned near a free-surface or in nanocrystals and can be calculated using the current methodology without constructing fictitious images. It is to be noted that domain deformations have been ignored in the standard theories for the calculation of image forces and, hence, they give erroneous results (magnitude and/or direction) whenever image forces play an important role. An important point to be noted is that, under certain circumstances, where domain deformations occur in the presence of an edge dislocation, the ‘image’ can be negative (attractive), zero or even positive (repulsive). The current model is extended to calculate image forces based on the usual concept of an ‘image dislocation’. Keywords: dislocation stress fields; image forces; nanocrystals; finite element method 1. Introduction A dislocation near a free surface feels a force towards the surface, which is called the image force [1]. The term image force is used in literature because a hypothetical negative dislocation is assumed to exist on the other side of the free surface for the calculation of the force. If the material with a dislocation is bonded with another material of a lower elastic modulus, the dislocation would still feel an attractive force towards the interface, which would be lower than that for a free surface. *Corresponding author. Email: [email protected] ISSN 1478–6435 print/ISSN 1478–6443 online ß 2011 Taylor & Francis DOI: 10.1080/14786435.2010.529089 http://www.informaworld.com Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 731 If the material across the interface is elastically harder, then the force experienced by the dislocation could become repulsive in nature (depending on the exact configuration). This could lead to an equilibrium position of a dislocation within the crystal (instead of the free surface) [2]. Additionally, if the free surface is replaced by a surface with constraints, then the magnitude and nature of the force experienced would be altered. The term ’image force’ can be extended to these cases as well, keeping in mind the fact that the construction of an image dislocation may or may not be possible, as in the case of a free surface-bounded crystal (wherein, the vacuum part is replaced with the same material having a negative dislocation). These cases, with dislocations near an interface, can be analysed as dislocations in semi-infinite domains [2]. Head [3] has analysed the force experienced by a dislocation near a surface film by using an infinite set of images. When the dislocation is near an interface, the interface deformations cannot be ignored and should be taken into account for the calculation of the energy of the system [4,5]. In free-standing nanocrystals, more than one surface will be at comparable distances from the dislocation line, and net force experienced by the dislocation would then be a superposition of all the image forces. Additionally, with decreasing size of the crystallite, the deformations to the domain shape become increasingly important. This implies that for free-standing nanocrystals, the standard theoretical formulations used in the analysis of semi-infinite domains do not yield correct results. The image force experienced by a dislocation can be resolved into a glide component parallel to the slip plane and a climb component perpendicular to the slip plane. When the glide component of the image force exceeds the Peierls force [6], it can lead to depletion of dislocations from regions near the surface of large crystals [1,7] and, in the case of nanocrystals, these forces can lead to a completely dislocation-free crystal [8]. The climb component of the image force is expected to play a role at high temperatures when dislocation climb becomes feasible. The utility finite element method (FEM) at the nanometre length-scale is highlighted through the work of Benabbas et al. [9], Zhang and Bower [10], Rosenauer [11] and other researchers. FEM has also proved to be an important tool in understanding dislocations in materials and its interactions with other stress fields [12–15]. Belytschko and co-workers [16–19] have made significant contributions to the understanding of dislocations; especially in the vicinity of interfaces (free surfaces and bimaterial interfaces). They have used techniques, such as enrichment of finite element space and J-integrals, to develop methodologies applicable to non-linear and anisotropic materials. Their contributions include calculating stress fields using XFEM along with Peach–Koehler forces. Schall et al. [20] have noted that, even on a scale of a few lattice vectors, the dislocation behaviour is well described by a continuum approach. In systems with complexity in: (i) distribution of dislocations or other internal stress fields, (ii) external loading and boundary conditions, (iii) geometry of the domain or (iv) material distribution, FEM becomes an indispensable tool. However, any model based on linear elasticity is not expected to be valid when the dislocation is very near an interface and atomistic models taking in the core structure can yield good results [21,22]. Researchers have used different computational methods to calculate image stress fields. Xin et al. [23] have used boundary element method for the investigation of dislocation–boundary interactions. Weinberger and Cai [24] have developed a Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 732 P. Khanikar et al. numerical method to compute image stress fields of defects in an elastic cylinder. Investigators studying dislocation dynamics (e.g. Van der Giessen and A. Needleman [25], Zaiser et al. [26], Weygand and coworkers [27]) have used alternatives to FEM [28] and also a hybrid of FEM and dislocation dynamics methodologies, to take into account image forces [29, 30]. Yasin et al. [31] have considered size and boundary effects in their 3D discrete dislocation model to understand the deformation of single crystal metals. Zbib and de la Rubia [32] have developed a multiscale model of plasticity wherein the effect of the interaction of interfaces with dislocations and point defects is taken into account. Verdier et al. [33] have reviewed the methods and techniques used for the simulation of dislocation dynamics on a mesoscopic scale, including the effects of image forces. Moriarty et al. [34] have covered the broader area of atomistic simulations of dislocations and defects in their review. Due to limitations of space, the review of literature presented here is but a sampling of the vast literature on the subject of dislocations. The reader can consult these references as a starting point to a detailed study. The manuscript is intended to make the following fundamental contributions to our understanding of image forces, especially with reference to nanocrystals and their computation using FEM: (i) A dislocation near a free surface deforms it and, hence, alters the energy state of the system along with the image force experienced by the dislocation. This effect is not taken into account in the standard formulations and will be a topic of focus in this work. (ii) In nanocrystals, the surface and domain deformations cannot be ignored, which will lead to an altered state of stress and energy of the domain and, hence, the image forces (which could be drastically altered). The current work considers these factors. (iii) In a fixed boundary scenario, it is easy to visualise the direction of the image force. In nanosized domains, which can bend or buckle, this cannot be taken for granted and the direction of the image force is a function of the position of the dislocation in the domain (i.e. it has to be calculated for a given position). The methodology developed is well suited to address this issue. (iv) In nanocrystals, due to proximity of more than one surface, the net image force, to a first approximation, is a superposition of multiple images. The current work proposes a methodology wherein multiple images need not be considered (i.e. no fictitious images need to be constructed). (v) The work shows a systematic comparison between the concept of an image dislocation for the calculation of image forces and the use of a single dislocation. In the context of nanocrystals, this also brings out the limitations of the standard equation for the calculation of the image force. (vi) This work brings out the concept of a positive image (via FEM), which brings about the convergence of the two and single dislocations models presented. (vii) The current methodology is easy to implement and extend to other configurations (e.g. in the presence of coherent or semi-coherent precipitates in a free standing film). (viii) There are other minor contributions, including finding a bimaterial configuration, which can yield a nearly zero image force over most positions of the dislocation in the domain (implying regions of stability even with low Peierls stress). The current work, which uses methods different from the standard discrete dislocation plasticity techniques, aims at the following tasks: (i) simulation of an edge dislocation in idealised domains using FEM, (ii) calculation of the glide and climb components of the image force, using a single dislocation or by using two Philosophical Magazine 733 Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 dislocations (one of which is the ‘image dislocation’) and compare the results of the simulations with that of the standard equations, (iii) determination of the image force experienced by an edge dislocation near a bi-material interface or near a constrained boundary, (iv) utilise this methodology to analyse the case of dislocations in nanocrystals, wherein domain deformations lead to a lowered net energy of the system and hence affects the image force. Additionally, efforts will be made to keep the procedures intuitive and simple; with suitable illustrations of the models considered. Some of the configurations considered are similar to those investigated by Sasaki et al. [13] and Belytschko and Gracie [18]. 2. Theoretical background The x component of the stress field of an edge dislocation in an infinite isotropic medium [35] has been modified by Sasaki et al. [13] for a finite cylindrical domain of radius r2: Gb y r2 2 3r2 2 3 1 2 x þ 1 2 y , ð1Þ x ¼ 2ð1 Þ ðx2 þ y2 Þ2 r2 r2 where G is the shear modulus, b is the modulus of the Burgers vector and is the Poisson’s ratio. When r2 ! 1, the equation for the infinite domain case is obtained. The energy of an edge dislocation per unit length (Edl) is given by [1] Gb2 2 þ ln , ð2Þ Edl ¼ r0 4ð1 Þ where is the radius of a cylindrical crystal and r0 is the core radius (usually assumed to be of the order of b [36]). The first term in the equation is due to the contribution from the core of the dislocation, which is estimated to be a fraction of the total energy [1]. The image force (Fimage) experienced by the edge dislocation at a distance d from the surface of a semi-infinite domain is given by [1] Fimage ¼ Gb2 : 4ð1 Þd ð3Þ For a finite domain of length L (Figure 1), the image force experienced by the edge dislocation (towards the vertical surfaces) can be computed as a superimposition of two image forces (by assuming two image dislocations) as follows: Gb2 1 1 Gb2 2x , ð4Þ ¼ Fimage ¼ 4ð1 Þ d L d ð1 Þ L2 4x2 where x is the position of the dislocation from the centre of the domain (Figure 1). It is seen from Equation (4) that the dislocation does not experience any force at the centre of the domain (x ¼ 0), which is symmetrical position with respect to the two surfaces. 734 P. Khanikar et al. Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Figure 1. Schematic illustration showing an edge dislocation in a finite domain experiencing two image forces. The ‘image dislocations’ are of opposite sign and lie on the extension of the slip plane. 3. Simulation methodology An p edge ﬃﬃﬃﬃﬃﬃﬃdislocation is simulated in aluminium (a0 ¼ 4.04 Å; slip system: h110i{111}, b ¼ 2a0 /2 ¼ 2.86 Å, G ¼ 26.18 GPa, ¼ 0.34837) by imposing the stress-free strain corresponding to the introduction of an extra half-plane of atoms. Stress-free strains are also called eigen strains (or Eshelby strains) and the point to be noted is that on the imposition of these strains, stresses would develop in the crystal (stress field of the edge dislocation in the present case). The term stress-free strain implies that if the body were unconstrained, then no stresses would develop in the body (e.g. thermal expansion or phase transformation of an unconstrained crystal). Use of anisotropic material properties is not expected to alter the methodologies developed in the current work. However, this allows the use of simple theoretical equations for comparison (Equations (1)–(4)). The value of G and are calculated by Voigt averaging single crystal data [35]. It is assumed that material properties of bulk crystals are valid at the length-scales used in the simulations. The structure and the energy of the core of the dislocation are ignored in the model. The energy of the core is not expected to play an important role in the calculation of image forces, except when the dislocation is at a distance of a few Burgers vectors from the an interface (i.e. the core energy is not expected to be a strong function of the position of a dislocation from the interface and, hence, would be nearly a constant additional term to the total energy of the system). At this point, it should be noted that the current model is amenable to modifications, to include a core model, as prescribed by Stigh [12], by altering the shape of the region where strains are imposed to a wedge shape. It is assumed that the dislocation has not split into partials. Isotropic, plane-strain conditions are assumed and the stress-free (eigen) strain ("T) introduced as thermal strains into a region (of width b) as shown in Figure 2 is "T ¼ ða0 þ bÞ a0 1 ¼ , 2 a0 ð5Þ where a0 (¼2b) is the face diagonal of the cubic unit cell. The strain is imposed only in a direction along the face diagonal of the cube (x-direction in Figure 2), as this corresponds to the insertion of an extra half-plane of atoms (to simulate the stress fields of an edge dislocation). This insertion implies that the additional half-plane is Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 735 Figure 2. Symmetrical half of a circular domain showing the regions where stress-free strains are imposed to simulate an edge dislocation. Two sets of boundary conditions are used to study the effect of deformations to the domain shape on the energy of the system: (a) less constrained system, and (b) more constrained system to restrict parts of circumference from deforming. The strain imposed in this case and future cases for the simulation of the edge dislocation are according to Equation (5). added along the face diagonal making the total length (a0 þ b) and, hence, the strain is calculated to be ½ as in Equation (5). In the absence of symmetry, where a full domain will have to be considered (instead of the symmetric half), this region where thermal strains are imposed to simulate an edge dislocation, will be of width 2b. Circular and rectangular domains were meshed with bilinear quadrilateral elements with mesh size b b (Figures 2 and 3). In the current simulations, L is taken to be 500b. Two sets of boundary conditions imposed are also shown in Figure 2 (the second set of boundary conditions is used to study the effect of deformations to the domain shape on the energy of the system). This simple geometry has been considered to make a comparison with the standard theoretical Equations (1) and (2) and to validate the FEM model of the edge dislocation. The numerical model was implemented using the Abaqus/standard (Version 6.7, 2007) FEM software. As the standard equations for stress fields and energy are for circular domains (cylindrical in three dimensions), the circular domain is used for the comparison of the FEM calculations with the theoretical equations [1,13] and to validate the finite element model. The size of the circular domain is varied to get a plot of the energy of the dislocation as a function of the domain size. The rectangular domain is used for image force calculations. For a comparison of the energy obtained from the finite element model, with the theoretical equation for energy (Equation (2)), the core contribution term in the equation was ignored. The energy of the system (per unit depth) is calculated from the model and then plotted as a function of the position of the dislocation. The direction of depth is parallel to the dislocation line, which is perpendicular to the xy-plane (Figure 3). Hence, the energy computed is the energy of the dislocation per unit length of the Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 736 P. Khanikar et al. Figure 3. Rectangular domain showing regions where stress-free strains are imposed, along with the boundary conditions used, for the simulation of an edge dislocation at a distance x from the centre of the domain along the x-direction. Figure 4. Domain, boundary conditions and the regions where stress-free strains are imposed to simulate a positive and a negative edge dislocation. This configuration is consistent with the theoretical concept of an ‘image dislocation’ and is used for the calculation of glide component of the image force. dislocation line. The image force (per unit depth) experienced by the dislocation is the slope of the energy versus distance plot (negative slopes correspond to attractive forces towards the boundary). By positioning the dislocation at various points along the x-direction, the glide component of the image force is computed. To make a direct comparison with the theoretical notion of an image of a dislocation, the configuration shown in Figure 4 is considered. It should be noted that the theoretical image dislocation annuls surface stresses in the case of a screw dislocation, but does not annul the xy (shear) stresses in the case of the edge dislocation [35]. Furthermore, the image force calculated using the image (edge) dislocation gives a correct result; but, the stresses have to be modified to annul the residual shear stresses. More on this aspect will be considered along with the results based on this model. As shown in the figure, the domain is an extended Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 737 Figure 5. FEM model used for the calculation of the climb component of the image force: (a) single-dislocation model, and (b) two-dislocation model (one of the dislocations being an ‘image dislocation’). Symmetrical half of the domain used in the FEM analysis is shown in both the figures. one containing a simulated edge dislocation of the opposite sign. The energy of these systems as the dislocations are brought towards each other (symmetrically) is plotted as a function of their position and the slope of this curve gives an alternate calculation of the image force; which is consistent with the concept of an image dislocation. The methodologies adopted for calculating the glide component of the image force can be repeated to determine the climb component of the image force. This is done using the models shown in Figure 5. The distance y (distance from the centre of the domain) is varied to position the dislocation at various points along the y-axis. As mentioned in Section 1, the concept of image force experienced by a dislocation can be applied to three distinct cases for the dislocation in the vicinity of: (i) a free-surface, (ii) an interface with another material, and (iii) a constrained surface. In the case of the interface with a harder material and the case of a constrained surface, the image force could be repulsive (this depends on the exact configuration). The case of a constrained surface is illustrated here with an example, wherein the nodes on the boundary are locked in the x-direction. In this model, the image developed is a positive dislocation and, hence, the force is repulsive in nature. It is worthwhile to note that the concept of an ‘image dislocation’ can directly be extended to the case of crystal bound by constrained surface (of the kind described), by making the image a positive one. To investigate these additional cases, the configurations shown in Figure 6 are used. Without loss in generality, the harder material can be made a softer one, which would lead to purely attractive force on the edge dislocation. Nickel is chosen as the harder material for the simulations (G ¼ 94.66 GPa, ¼ 0.2767). Aluminium will now play the role of the softer material. 738 P. Khanikar et al. Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Figure 6. Domains used for the simulation of image forces experienced by an edge dislocation in the vicinity of: (a) an interface with a material of higher elastic modulus, and (b) a surface locked in x-direction (constrained surface). Figure 7. FEM models which highlight the power of the new methodology developed in the current investigation: (a) inclined slip plane of length ‘s’ in a rectangular domain of aluminium with length (L) to breadth/height (h) ratio of 1:3, (b) a thin plate of aluminium with a horizontal slip plane (b ¼ 2.86 Å), and (c) an interfacial misfit edge dislocation in an heteroepitaxial system (Si/SiGe, b ¼ 3.84 Å). Having used the aforementioned models to validate the applicability of the new methodology; three further models (Figure 7) are considered, which illustrate those cases wherein the analytical formulation clearly breaks down (with respect to the direction and magnitude of the image force). These models will be used to highlight Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 739 Figure 8. Plot of the FEM-simulated x stress contours and its comparison with the theoretical equation [13] in a symmetrical half of a circular domain: a close match is seen between the two values. the power of the current method. The objectives behind the choice of these cases are outlined next. In the first model (Figure 7a), the slip plane is inclined, which implies proximity of the free surface. However, the domain is longer than this length, implying that the self energy of the dislocation will be high. Additionally, this configuration implies only a small bending of the domain. In the second model (Figure 7b), the bending is larger, leading to considerable deviation in energy (and stress plots), compared to a constrained domain. When the models shown in Figures 3 and 7a,b are put together, a broad range of domain deformations are covered. These represent cases wherein Equation (4) becomes progressively inapplicable. The third model (Figure 7c) corresponds to an interfacial misfit edge dislocation in a heteroepitaxial system (Si/Ge0.5Si0.5). As discussed later, the misfit edge dislocation represents a considerable deviation from the conventional edge dislocation and, hence, there are no available analytical expressions to compute the image force experienced by such a dislocation. The details of the simulation of the epitaxial system (model Figure 7c) along with the material properties are given elsewhere [14]. 4. Results and discussion Figure 8 shows the plot of x stress contours in a circular domain of 125b radius, determined from the current FEM simulations, along with those calculated using the theoretical Equation (1). A comparison of the two sets of contours in Figure 8 shows a close match. Unlike the case of the infinite domain, wherein the compressive and tensile regions are separated by a single horizontal zero stress line, the finite circular domain shows a stress reversal. A comparison of the variation in the energy of the dislocation (per unit length) in a circular domain, as a function of the domain size, determined from the FEM calculations, with that from the theoretical Equation (2) is shown in Figure 9 (these simulations correspond to the Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 740 P. Khanikar et al. Figure 9. Energy of the edge dislocation (per unit length of the dislocation line) as a function of the circular domain size: FEM-calculated values and its comparison with the theoretical values [1]. configurations shown in Figures 2a and b). Similar trends, along with a good match are observed between the theoretical results and the FEM computed values. It is also seen that the energy of the system increases with additional constraints to the deformation of the domain, with the energy of the more constrained domain being closer to the theoretical values. The FEM determined values are expected to be lower than the theoretical values, as relaxations are allowed in the FEM model (i.e. the surface is allowed to deform, except at points on which it has been constrained (Figure 2)). The shape of the deformed and the undeformed domains have been compared in Figure 10. It should be noted that slight geometrical relaxations could lead to large energetic relaxations. Figure 11 shows the plot of the FEM calculated energy values of the system as the simulated edge dislocation is positioned at various points along the x-axes. Two configurations of the system with edge dislocation(s) have been used to determine the energy as shown in Figures 3 and 4. These two configurations represent domains in which one domain is double the size of the other and a comparison in the energy values is made possible by taking half the energy determined from the double-sized domains. The image forces (glide component) calculated from these curves are also shown in Figure 11, along with the plot of the theoretical equation (Equations (3) and (4)). Positive image force values are indicative of an attraction towards the surface. Figure 12 shows the deformations to the free-surface, along with the x stress contours, when the edge dislocation is positioned at a distance of 20b from it. It is important to note that small geometric deformations can lead to considerable relaxation in energy and, hence, can have a profound effect on the image force experienced by the dislocation. It is to be noted that, in the model with twodislocations (Figure 4), these deformations would be restricted, compared to the Philosophical Magazine 741 Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Figure 10. Deformation to the shape of a circular domain (symmetrical half shown) of diameter 100b, due to the dislocation stress fields (plot of x contours shown on the deformed domain). The deformed shape has been shown exaggerated by a scale factor of six. Figure 11. Energy of an edge dislocation (per unit length of the dislocation line) and the image force (glide component) experienced by it as a function of its position along the x-axis from the centre of the rectangular domain. model with a single dislocation (Figure 3). This implies the energy of the system with a single dislocation would be lower, which is seen in Figure 11. It is seen from the plots in Figure 11 that there is a reasonable match between the theoretical plot of image force (Equation (4)) and the values calculated from the FEM simulation. However, it is seen that the values obtained from the simulation are higher than that from theory for positions of the dislocation away from the centre. This is as expected and is due to a steeper fall in the energy of the system as the dislocation moves away from the centre of the domain, which is consistent with the relaxations allowed by a free-standing crystal. Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 742 P. Khanikar et al. Figure 12. Deformations to a free surface due to the presence of a FEM-simulated edge dislocation at a distance of 20b from the surface. Plot of x contours shown on the deformed domain (the deformed shape has been shown exaggerated by a scale factor of six). An important point worthy of reiteration here is the fact that theoretical calculation of stress fields using an image edge dislocation need corrections [35]; unlike the stress fields computed using an image screw dislocation. However, the image force calculated from the image dislocation model turns out to be right [35]. This aspect further highlights the importance of single-dislocation model developed in the current work; wherein the construction of an image dislocation need not be considered and stress fields arise naturally from the model (hence, is an additional motivation for the current work). In light of these discussions, it becomes clear that the FEM model with an image dislocation has been considered for making an internal comparison and also to emphasise the importance of the single dislocation model (main theme of the current work). Figure 13 shows the plot of the climb component of the image force, along with theoretical plots of Equations (3) and (4), using the models shown in Figure 5. As in the plots in Figure 11, it is seen that the model with a single dislocation yields a lower energy for the system, compared to a model with two dislocations (one of the edge dislocations being an image dislocation as before). One difference to be noted between Figures 11 and 13 is the residual climb component of the image force present for the position of the dislocation at the centre of domain, using a single-dislocation model (in Figure 13). This is due to the asymmetry in the boundary conditions used (Figure 5a). This was necessitated by the requirement to leave a free-surface, towards which the dislocation can feel the image force. The lower surface could also have been left free (with the numerical requirement of locking just the central point on this line along the y-direction); which would lead to further relaxation of the system, by buckling of the domain maintaining the x ¼ 0 line mirror symmetry. In Figures 11 and 13, the theoretical plot of the image force obtained from Equation (3) has been included; which is valid only for semi-infinite domains. Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 743 Figure 13. Energy of an edge dislocation (per unit length of the dislocation line) and the image force (climb component) experienced by it as a function of its position along the y-axis from the centre of the rectangular domain. The inclusion of this plot is for the estimation of the distance of the dislocation from the surface, at which the domain seems like an infinite one, with regard to the image forces experienced. The rule of thumb, which can be evolved by analysing the plots, is that, when d/L 5 0.1, the finite domain behaves like an semi-infinite one within an error of 1%. At this point, it should be noted that the FEM simulations are expected to be valid for dislocation positions which are large compared to the core dimension (i.e. approximately, when d 4 5b). In the examples considered, the method has been used to calculate the image force along the x- and y-directions and can readily be extended to force calculations along any arbitrary direction. Figure 14 shows FEM calculations of image force (glide component) in the presence of an elastically harder material (according to the model in Figure 6a) of three thicknesses (t ¼ 5b, 50b and 100b). For t ¼ 5b, the edge dislocation feels an attractive force towards the interface in the validity regime of the simulations. For t ¼ 50b, there is an equilibrium position for the dislocation at a distance of about 20b from the interface. If the thickness of the harder material is made larger, the dislocation feels only a repulsive force from the interface. This is seen for the case of t ¼ 100b; wherein, it is additionally observed (Figure 14) that the repulsive force experienced by the dislocation is very small and nearly a constant, till the dislocation is at a distance of 50b from the interface. This implies that choosing a harder material of appropriate modulus and thickness, one can ‘taylor’ the image force to be nearly zero for most part of the domain. Head [2,37] had analysed the case of the image force experienced by a screw dislocation near a bi-material interface using an infinite set of images. It was pointed out in Head’s work that the equilibrium position of a screw dislocation in the presence of a harder material would be of the order of the thickness of the harder material, for the case of a semi-infinite domain. Additionally, Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 744 P. Khanikar et al. Figure 14. FEM calculations of image force (glide component) in the presence of an elastically harder material (according to the model in Figure 6a) of three thicknesses (t ¼ 5b, 50b and 100b). Head [2,38] had pointed out the difficulty in analysing the case of an edge dislocation near a bi-material interface. As seen from the example considered, the current methodology can avoid the use of images and can be used for finite domains, wherein the image forces can be considerably altered with respect to an infinite domain. A comparison of a single-dislocation model (model shown in Figure 6b) with a two-dislocations model, for the case of a dislocation approaching a constrained surface, is shown in Figure 15. The surface has been constrained in the x-direction. The model with two dislocations consists of two positive edge dislocations, unlike the case of a dislocation near a free-surface, wherein the image dislocation was a negative one. It is seen from the figure that a very good match is obtained between the models. This is to be expected, as the additional boundary condition used, acts like a ‘mirror’ (reflecting the domain on the left) and there are no free-surface deformations to differentiate the two models. It is interesting to note that the concept of an ‘image dislocation’ can be extended to the case of a constrained boundary, with a change in the sign of the image dislocation and with a reversal in the direction of the force experienced. The plot of state of stress ( x) for the models considered in Figure 7, are shown in Figure 16. Figures 7a and b also show the deformed shape of the domains in the presence of the dislocation. Table 1 gives a comparison of the magnitude and direction of the image force, between the analytical results (Equation (4)) and the current work. It is seen that when large domain deformations occur in the presence of the dislocation (leading to considerable relaxation of energy of the system), a major alteration to the image force takes place (both in magnitude and in direction). The important point to be noted is that ‘image’ can be negative (attractive but with altered direction and magnitude), zero (no image construction possible) or even positive (repulsive). From Figures 16a and b, it is seen that the compression–tension Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 745 Figure 15. Comparison of single- and two-dislocation finite element models for the image force experienced by a dislocation in the vicinity of a constrained surface. In the model with two dislocations, two positive edge dislocations are used for the determination of the image force. inversion symmetry present in the stress plot of a usual edge dislocation is lost. This is an additional point to be noted and will occur when there is buckling (additionally, geometrical configuration of the system can lead to this asymmetry, as in Figure 16a). Special attention should be focussed on the case of the misfit edge dislocation in a heteroepitaxial film. An interfacial edge dislocation represents a very different case as compared to a normal edge dislocation, in the following aspects: (i) the compression– tension inversion symmetry present in the stress plot of an usual edge dislocation is broken (firstly due to the asymmetry in the geometry), (ii) the material properties above (film) and below (substrate) the interface are different, (iii) the interaction of coherency stresses of the system with the dislocation stress fields significantly modifies the latter. Hence, there are no analytical expressions for comparison with the results of the current work (Table 1). The interfacial misfit dislocation feels a force towards the centre of the domain from the position indicated in Figure 7c (99.5b), as the energy of the system can be lowered by its position at the centre of the domain. As for the case considered in Figure 7b (climb force), this aspect is reflected by the negative sign for the image force in Table 1. This in not an ‘image-force’ in the usual sense of the phrase and arises primarily due to interaction of coherency stresses with the dislocation stress fields. Nonetheless, the dislocation feels a force which can be calculated from the current method and does not lead to the erroneous conclusion that the dislocation is attracted towards the free surface. When the dislocation is ‘sufficiently close’ the surface the image force will again be positive (attractive towards the surface). Even in the case of the thin plate with horizontal slip plane (Figure 16b), the applicability of the analytical formula (Table 1) is questionable, as in the analytical Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 746 P. Khanikar et al. Figure 16. x plot for the models considered in Figures 7a–c, respectively. x ¼ 20b (y ¼ 10b) in part (b) of the figure. The shape of the deformed domains in the presence of edge dislocation are to be noted (a) and (b). Dimensions correspond to the undistorted domains. model the domain is assumed to be infinite in the y-direction. A thin plate deviates considerably from this assumption. The error in direction (angles) calculated in Table 1 are with respect to the domain where distortions are not considered; wherein, the direction of the force would be along the x-axis (horizontal). Additionally, the angles calculated in Table 1 are approximate as the deformations are non-uniform. The cases illustrated in Figure 7 are just a few of the cases where the current model can be used. Further configurations include image forces experienced by dislocation loops (normal and misfit), dislocations in the presence of other dislocations, dislocations in epitaxial superlattices, etc. Although, on the one hand, the FEM models considered suffer from the following assumptions: (i) material properties of bulk crystals have been used for the nano-sized crystals, (ii) Voigt averaged material properties have been used to simplify the comparisons between theory and simulations, (iii) structure and energy of the core of the edge dislocation have been ignored in the simulation, (iv) dissociation of the dislocation into partials has not been considered, (v) certain idealised domain shapes have been used instead of real shapes of crystals, and (vi) calculations are for 300 K (room temperature). On the other hand, it offers the following advantages: (i) can be used in cases standard equations of stress fields and image forces are not valid (e.g. small domain sizes of any shape), (ii) the current model can account for surface deformations as a result of the internal stress fields, 2.95 103 (Error of 85%) (for x ¼ 40.5b, y ¼ 10b) Fx ¼ 8.61 102 (Glide) (Error of 89.5%) (for x ¼ 20.5b, y ¼ 10b) Fx ¼ 1.8 102 (Glide) (Error of 100%) (for x ¼ 0, y ¼ 14.5b) Fy ¼ 1.03 101 (Climb) (Error of 346%) None available for comparison Plate with horizontal slip plane along the length of the plate (Figure 7b) Epitaxial film with misfit edge dislocation (Figure 7c) Image force (Analytical) (N/m) (% Error) Inclined slip plane in a plate (Figure 7a) Model 4 3.5 4 180 (180 ) 9.0 103 0 2.53 101 (towards the centre) 5.28 102 (towards the centre) Error in direction 4.4 104 Image force (FEM) (N/m) Table 1. Comparison between analytical (Equation (4)) and results from the current work for the models shown in Figure 7. ‘Error in direction’ refers to the error in the analytical model with respect to the FEM calculations (see Figure 16). The negative signs in column 3 are to be noted, which implies a force away from the surface (details are explained in the text). Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 747 Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 748 P. Khanikar et al. which leads to a lowered state of energy and the deformed shape can be computed (this aspect is not considered in the standard theoretical formulations), (iii) it is very simple to implement and extend to three-dimensions and to complicated configurations (e.g. presence of multiple dislocations, splitting of the dislocation into partials, presence of coherent precipitates etc.) and material properties (e.g. multiphase materials, anisotropy etc.). Additionally, when domain deformations are small, an approximate estimate of the image force experienced by a pure screw dislocation can be determined by considering a multiplication factor of (1 ) edge screw ¼ Fimage [1]). The biggest advantage perhaps, is the simplicity of the (i.e. ð1 Þ Fimage model and its ease of implementation using any standard FEM software. A comparison of the current work with the state of the art techniques in dislocation dynamics is perhaps due at this stage; even though the two methods employ entirely different approaches and have different goals. As stated before, studies using discrete dislocation dynamics techniques have provided considerable insight into the mechanical behaviour of materials starting with dislocations and other defects at the microscale. The evolution of the system in the presence of internal and external stresses (and constraints) at multiple length scales has been successfully implemented using an elasto-viscoplastic continuum framework [32]. In these techniques, the well known solutions of the dislocation fields are imposed as in an infinite domain and the fields are corrected for (surface tractions [28]) the presence of surfaces (and interfaces) using the superposition principle [31]. Computational efficiency (using FEM) is obtained by replacing the defected inhomogeneous volume element with a homogenous body with eigenstress [32]. The aim of the current work is to simulate dislocation stress fields using eigen strains and further calculate image forces using a FEM model. In this model, a priori knowledge of dislocation stress fields either in a finite or infinite body is not required. Hence, image forces naturally arise out of the model and not as a correction to the infinite body solution. Though the model can handle complex material and geometric, configurations (including the presence of other defects in the material); it lacks any dynamics capability and is meant for static calculations. 5. Summary and conclusions An edge dislocation is simulated using stress-free strains corresponding to the insertion of an extra half-plane of atoms. The stress state and energy of the system is calculated using the finite element model and are validated by comparison with the standard theoretical equations (for cases where these expressions are valid). By varying the position of the simulated dislocation, the variation of energy of the system is calculated. The glide and climb components of the image force, experienced by the edge dislocation are determined from these plots (as a slope of the curves). It is seen that the image force experienced by the edge dislocation is a natural consequence of its asymmetric position in the domain and can be calculated using the FEM model without externally applying any forces or invoking the concept of fictitious images. Surface/domain-shape relaxations are also taken into account in the model, which are ignored in the standard theoretical formulations. An added benefit of the method is its computational simplicity and efficiency. Downloaded By: [University of Cincinnati] At: 20:01 21 March 2011 Philosophical Magazine 749 An alternate methodology for the determination of the image force is also developed based on the concept of an ‘image dislocation’; which is in accord with the theoretical notion of an ‘image force’. Good match is seen for the image force determined by both the methodologies with the standard theoretical formulations. However, it is seen that the model with a single dislocation, which takes into account the surface/domain deformations (leading to some relaxation in energy), is a better numerical model for dislocations in nanocrystals or for dislocations near a freesurface. These are the two cases where image forces become important, but the theoretical formulation breaks down (gives erroneous results). Further, an important point seen is that under certain circumstances where domain deformations occur the ’image’ can be negative (attractive but with altered magnitude and direction), zero (no image construction is possible) or even positive (repulsive). Hence, this aspect leads to a radical deviation from the conventional concept of an ‘image’. The numerical model for the calculation of image forces is extended for the cases of the edge dislocation near a bimaterial interface and near a constrained surface. It is to be noted that for these cases the traditional concept of an ‘image dislocation’ is no longer valid. For a dislocation near an interface with a harder material the equilibrium position of the dislocation is determined using the model (which is not the surface as in the case of a dislocation near a free-surface or near an interface bordering a softer material). The case of a dislocation near the type of constrained surface considered can be analysed using a ‘positive image dislocation’ rather than the usual ‘negative image dislocation’. The simplicity, versatility and power of the simulation methodologies are brought forth via the examples considered. The methodology developed in the current work can be extended to various complicated cases, like the presence of multiple edge dislocations, anisotropic material properties, systems with multiple materials, presence of coherent precipitates etc., without recourse to additional concepts. 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