Application of disclination concept to solid structures

Progress in Materials Science 54 (2009) 740–769

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Application of disclination concept to solid structures Alexey E. Romanov a,b,*, Anna L. Kolesnikova c a

Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26 Polytechnicheskaya, St. Petersburg 194021, Russia University of Tartu, 142 Riia, Tartu 51014, Estonia c Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 61 Bolshoi V.O., St. Petersburg 199178, Russia b

a r t i c l e

i n f o

a b s t r a c t Disclinations together with dislocations represent a class of linear defects in solids. Disclinations are characterized by typical singularities and the property of multi-value of the fields of displacement and rotation associated with the defects. We present an introduction to and overview of recent achievements of the disclination approach in physics and mechanics of solid structures. In the development of F.R.N. Nabarro ideas, the use of the disclination approach in materials science is demonstrated. The following milestones of the disclination concept are given and discussed: (i) definitions and designations for Volterra dislocations, Frank (rotation) vector of a disclination, wedge and twist disclinations; (ii) geometry of disclinations in structure-less and crystalline solids; (iii) the properties of screened low-energy disclination configurations, e.g. loops, dipoles, defects at the vicinity of a free surface, including the methods and results of calculation of their elastic fields and energies. Then using the properties of screened disclinations a number of qualitative and quantitative models for the structure formation and evolution in plastically deformed materials, is considered. Disclination theory of grain boundaries and their junctions in conventional polycrystals is presented. The bands with misorientated crystal lattice in metals and other materials are described as a result of partial wedge disclination dipole motion. Disclination approach is applied to the study of work-hardening at large strains. For nanocrystals, disclination approach allows to explain the peculiarities of the flow stress dependence on the grain size. The contribution of disclinations to relaxation of mechanical stresses in lattice mismatched thin layers placed on the bulk substrate is examined and linked to the appearance of domain patterns. Finally,

* Corresponding author. Address: Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26 Polytechnicheskaya, St. Petersburg 194021, Russia. Tel.: +7 812 2927304; fax: +7 812 2941017. E-mail address: [email protected] (A.E. Romanov). 0079-6425/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2009.03.002

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disclination models for the structure and properties of nanoparticles are presented. These models treat the pentagonal symmetry of micro- and nanoparticles and nanorods of materials with FCC crystal structure and explain stability and relaxation phenomena in such pentagonal objects. Ó 2009 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental observation of disclinations in solid structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrical properties of disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Disclinations in crystalline solids versus disclinations in liquid crystals. . . . . . . . . . . . . . . . . . 3.3. Connection between disclinations and dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic fields and energies of isolated straight disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Approaches to analyze elastic properties of disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Disclinations in comparison to dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screened disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Self-screening: disclination dipoles, quadrupoles, and loops . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Screening by a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disclination modeling of solid structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Rotational structures in deformed solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Plasticity and work-hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Disclinations in bulk nanocrystalline materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Disclinations in thin strained layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Disclinations in small particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Disclinations were originally introduced in the mechanics of deformed solids as specific sources of internal stresses. Vito Volterra [1] considered rotational dislocations (now known as disclinations) along with the translational dislocations (now known as dislocations) and named them as distortions. Fig. 1 shows six Volterra distortions in an elastic cylinder. The strength of a dislocation is determined by Burgers vector b, which is equal to the translational displacement of the non-deformed surfaces of the cut bounded by a dislocation line. In a similar way, the strength of a disclination is related to an axial vector x (Frank vector), which defines the mutual rotation of the undeformed surfaces of the cut bounded by a disclination line. Volterra distortions are linear defects in structure-less continuum. There are no low bound restrictions on the strength of disclinations and dislocations in the continuum. In crystals, the periodic structure dictates the appearance new features in the properties of dislocations and disclinations. The magnitudes of dislocation Burgers vector and disclination Frank vector should obey crystal symmetry. Frank Nabarro was one of the firsts who clearly indicates such property of disclinations in crystals. Fig. 2 reproduces the pictures of wedge disclinations given in his monograph ‘‘Theory of crystal dislocations” [2]. Frank Nabarro was also the first who paid attention to disclinations as physical objects, which can play an important role in deformation phenomena in solids. He proposed that disclination motion can contribute to processes of muscle contractions [3]. These ideas on the role of disclinations in solid structures were acquired first in St. Petersburg [4–7] and then by other groups in the World, e.g. works [8–13]. In 1992, owing to Nabarro’s proposal, the ‘‘Disclinations in Crystalline Solids” chapter [14] was published in the seminal ‘‘Dislocations in Crystals” series, for which he served as editor [15].

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Fig. 1. Volterra distortions in an elastic cylinder: (a) initial undistorted state of a hollow cylinder; (b) screw dislocation with Burgers vector b parallel to cylinder axis l; (c, d) edge dislocations with Burgers vector b perpendicular to the line of defect l; (e) wedge disclination with Frank vector (rotation pseudo-vector) x parallel to the line of the defect l; (f, g) twist disclinations with Frank vector x perpendicular to the line of the defect l.

Fig. 2. Perfect disclinations in crystalline structure (after Nabarro [2]): (a) positive wedge disclination of strength p/2 (in Nabarro’s designation – positive screw disclination +1/4); (b) positive wedge disclination of strength p/3 (in Nabarro’s designation – positive screw disclination +1/6).

Disclinations are defects with a long-range distortion field, De Wit [16], Romanov and Vladimirov [14]. A single straight-linear disclination is a special and a strong source of internal elastic stresses and possesses an energy which is very large in comparison to the energy of standard lattice dislocation.

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This property of single disclinations was considered to be an obstacle for their application in the modeling the plasticity of deformed solids along with or as replacement of dislocations, Friedel [17]. However in the case of self-screened disclination configurations (disclination dipoles, disclination loops, etc.) the latent energy may be relatively small, Romanov [18]. The other possibility for diminishing the disclination energy relies on the screening of disclination elastic fields by traction-free surfaces [19,20]. In this case screened disclinations can be important in subsurface layers, shells, thin films, and small crystalline particles [21,22]. Disclination concept has been successfully applied in various branches of physics. It has been developed most intensively for liquid crystals, for example Bouligand [23] Kleman [24], and Kleman and Friedel [25], for which disclinations were first introduced by Frank [26], and magnetics, for example Kleman [24]. Disclinations were used in the description of bulk biological objects, Bouligand [27], and two-dimensional crystals – biological membranes [28], nanocons, nanotubes including fullerenes [29]. The disclination approach has been used in the analysis of diffraction phenomena, Nye [30], vortex structures in superconductors, Anthony [31] and Anthony et al. [32], and superfluid helium, Salomaa and Volovik [33], phase transitions, Nelson [34], Earth crust displacements, Chou and Malasky [35], time-space singularities, Duan and Zhao [36], etc. Present paper follows Nabarro’s ideas and deals with disclination behavior in solid structures. We apply disclination approach to defect structures in deformed poly- and nanocrystalline solids, and small particles. First, we present the examples of rotational defect structures, which are related to disclinations and can be often observed in crystalline materials. Then we discuss the properties of disclinations in solids. Finally, we review a number of disclination models developed for explanation of physical and mechanical materials properties. 2. Experimental observation of disclinations in solid structures The defect configurations, which we interpret in terms of disclinations, can be easily found in the structure of deformed or as grown crystalline materials. Figs. 3–5 present some examples of experimental observations of partial disclinations together with schematics explaining the disclination content of materials structures under consideration. The typical microband pattern observed in rolled copper single crystal (fcc lattice), Klemm et al. [11,13], is shown in Fig. 3a. Microbands are regions of the crystal with misoriented crystal lattice; they may give rise to a dark/bright contrast in the transmission electron microscopy (TEM) micrographs, as can be seen in Fig. 3a. The misorientation in microbands is up to a few degrees. In the material region shown, microbands form junctions of cell walls. Special analysis based on the experimental technique

Fig. 3. Microbands in deformed materials and their disclination interpretation: (a) TEM micrograph of a microband in copper single crystal rolled down to 70% thickness reduction at room temperature, Klemm et al. [13]; (b) schematics showing cell wall junctions with disclination quadrupole configuration.

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Fig. 4. Terminated microband equivalent to wedge disclination dipole: (a) TEM image of defect structure in restrained polycrystalline Mo specimen, Luft [38]; (b) schematics showing partial wedge disclination dipole.

Fig. 5. Hierarchical structure of deformation twins, Müllner and Romanov [8]: (a) TEM of primary twin in austenitic steel with the family of secondary (internal) twins; (b) schematics showing alternating sign disclination walls along the boundaries of the primary twin.

developed by Klemm et al. [11,13,37] proves the presence of disclination defects in the neighboring junctions. The absolute value of disclination strength in this case is approx. 1°. It follows from the schematics of the Fig. 3b that disclinations arrange themselves in a quadrupole configuration. When disclinations form a quadrupole their elastic fields are mainly localized in the region between disclinations and the energy depends only on geometrical parameters of the quadrupole. The presence of disclinations and related terminated boundaries of misorientation does not depend on the crystal lattice type. Examples supporting the above statement are given for nickel (fcc), molybdenum (bcc), tungsten (bcc), titanium (hcp), etc., Rybin [5] and Klemm et al. [37]. Here in Fig. 4a we provide the micrograph of terminated microband configuration observed in molybdenum [38]. Usually, such elongated microbands originate from grain boundaries and propagate into grain interior. When accepting disclination language such terminated microband is equivalent to partial wedge dislination dipole as shown in Fig. 4b. The observations of disclination related structures in deformed metals reveal one common feature: at the place of cell walls termination and in nearby material regions a non-uniform TEM contrast can be detected. Kolesnikova et al. [39] have proposed that such a diffraction contrast can be associated with the elastic distortions of crystal lattice caused by disclinations. Modeling the diffraction contrast can provide a tool for disclination identification in deformed materials.

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Fig. 6. Periodic ferroelectric domain structure in epitaxial PbTiO3 films on SrTiO3 substrate: (a) cross-section TEM micrograph, Foster et al. [21], showing inclined twined domains with alternating spontaneous polarization; (b) schematics for disclination content at film/substrate interface.

Twinning gives the other example of the appearance of crystal regions with misoriented lattice. As it was pointed by Romanov and Vladimirov [14] and Müllner [9] many features of deformation twins can be understood in the framework of the disclination approach. Even such complex phenomena as internal twinning (see Fig. 5) can be qualitatively and quantitatively understood [8]. Plate like structures, i.e. domains, with misoriented (twined) crystal lattice are often formed in epitaxial ferroelectric thin films, see for example Speck et al. [40] and Fig. 6. These domain structures originate from the relaxation of elastic stresses caused by lattice mismatch at the film/substrate interface. As it was shown by Romanov et al. [41–43] different types of domain patterns may be generated in ferroelectric epitaxial films depending on the character of phase transition in the material of the film and crystallographic orientation of the substrate. At the position of domain plates terminations characteristic nonuniformities of elastic distortions can be observed. Speck et al. [40] and Pertsev and Zembilgotov [44] have proved that the sources of these nouniformities are interfacial disclinations, as shown in Fig. 6b. Another important example of the experimental observation of partial disclinations gives crystalline structures with fivefold symmetry. These structures are observed in bulk of strongly deformed (nanocrystalline) materials, Liao et al. [45,46] and in so-called pentagonal nano- and micro-particles, Gryaznov et al. [22]. Such particles have the shape of decahedron or icosahedron, and are frequently found for fcc materials. Pentagonal Cu micro-particles can be fabricated in the process of electro-deposition, Vikarchuk et al. [47–49]. Their possible morphology is illustrated in Fig. 7a and b. HREM image of pentagonal Au nanoparticles is given in Fig. 7c, Koga and Sugawara [50]. Galligan [51] and de Wit [52] were the first who established the disclination nature of pentagonal particles. Placed in the starlike junction of five twin boundaries the positive wedge disclination (see Fig. 7d) guarantees the closure of angular mismatch and the continuity of the particle body. 3. Geometrical properties of disclinations 3.1. Definitions The disclination is a linear defect, which bounds the surface of a cut in a continuous body; the undeformed faces of the cut obtain the relative displacement [u] produced by mutual rotation by the angle x around a fixed axis. In this process material is inserted into the holes which arise and is taken out from the areas of overlap. It has been proved, see for example De Wit [52] and Kroupa and Lejcek [53], that this relative displacement can be visualized with the help of axial vector x, known as Frank vector, in the following form:

½u ¼ x  ðrx  rÞ

ð1Þ

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Fig. 7. Pentagonal symmetry in small particles: (a) icosahedral Cu microparticle; (b) pentagonal Cu micro-rod, Yasnikov and Vicarchuk [48,49]; (c) decahedral Au nanoparticle (HREM image), Koga and Sugawara [50]; (d) schematics of star-disclination in the node of twin boundaries.

where r is the radius-vector and rx is the position of the axis for x. The displacement defined by Eq. (1) corresponds to rigid body rotation. This explains why sometimes x is cited as the vector of rotation. The magnitude of the Frank vector has the meaning of the disclination strength in the same way as the magnitude of the Burgers vector characterizes the dislocation strength. By the definition used above, both dislocation Burgers vector and disclination Frank vector are given for the whole defect and therefore are the same for any part of defect line. This constitutes the rule for the conservation of Burger vector along the dislocation line and Frank vector along disclination line. In view of the connection between dislocations and disclinations in solids (this connection will be discussed below) the Burgers vector conservation in crystals with disclinations should be applied to the whole dislocation/disclination defect. This leads to the effect of the possibility of dislocation termination at disclination line, emission of dislocations during disclination motion etc. Such effects are addressed in details in Refs. [14,16,25]. The elasticity of a body containing a disclination (or any Volterra distorsion) does not depend on the position and the shape of the surface of the cut, but is determined by the value, direction and the position of the disclination Frank vector, the shape of the defect line and the boundary condition, e.g. Mura [54]. Fig. 1 gives the examples of straight-linear dislocations and disclinations in a hollow cylinder with inner and outer radii R0 and R, respectively. With regard to the angle between the vector x and the line vector l, analogously to the screw (Fig. 1b) and edge dislocations (Figs. 1c and d) two

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fundamental types of disclinations can be defined: wedge disclinations with x || l (Fig. 1e) and twist disclinations with x\l (Fig. 1f and g). From the view-point of the elastic fields generated by defects, edge dislocations shown in Fig. 1c and d are equivalent. The same is true for twist disclinations shown in Fig. 1f and d. The difference is in the position of the cut surface used for defect introduction. In the first case the cut surface is normal either to Burgers or Frank vector (Fig. 1c and f), in the second case edge dislocation Burger vector or twist disclination Frank vector are in the plane of the cut surface. A note can be given concerning the importance of a hollow cylinder in Volterra procedure used for dislocation and disclination introduction shown in Fig. 1. In general, a hollow core prevents the appearance of a singularity in the displacement field at the defect line arising as a result of the cut surface mutual motion. Wedge disclination with Frank vector that coincides with disclination line (Fig. 1e), shows no such singularity. However it represents a degenerated case, since wedge disclination with Frank vector shifted with respect of defect disclination line already posses a singularity in displacement field. As it will be shown below wedge disclination shown in Fig. 1a still posses a logarithmic singularity in elastic strain and stress fields. Therefore the use of a hollow core is necessary to avoid such strain and stress behavior. 3.2. Disclinations in crystalline solids versus disclinations in liquid crystals The presence of crystal lattice symmetry leads to the existence of perfect and partial dislocations and disclinations, Romanov and Vladimirov [14], Kroupa and Lejcek [53]. The characteristic vectors b or x of perfect dislocations or disclinations, respectively, must be compatible with the available symmetry elements. That means, that the module of b has to be a translation period of the crystal lattice, while that of x has to correspond to the multiplicity n of a symmetry axis of the lattice: x = 2p/n (n = 2, 3, 4, 6). Depending on the physical properties of the condensed media under consideration the formation of disclinations of two types is possible: Frank disclinations, which are singularities of the field of spin (free) rotations, and Volterra disclinations, which are singularities of the field of orbital rotations connected with the non-uniform displacement field. Frank disclinations are peculiar to magnetics, liquid and ‘‘plastic” crystals, Friedel [55], Kleman [24], Kroupa and Lejcek [53], Friedel and Kleman [25]. In some cases they have no upper limits to their strength, Harris [56]. They do not have any connection with plastic flow and do not have a translation dislocation analogue. The origin of the difference between two types of disclinations is clear from the consideration of Fig. 8. 3.3. Connection between disclinations and dislocations Wedge disclination can be most easily imagined as an additional wedge of the material inserted in or taken out from the elastic body. Fig. 9 gives an example of so-called negative wedge disclination where the wedge of angle x is inserted into the region between the cut faces. As a result, near the negative disclination line (i.e. near the vertex of the wedge) the material is under compression. For positive wedge disclinations the wedge has to be taken out and the material near the disclination line is under tension. The physical distinction between positive and negative wedge disclinations is related to the fact that x is a pseudo-vector, Romanov and Vladimirov [14]. There exists a direct interrelationship between Volterra disclinations and dislocations. Any disclination can be represented in the form of a superposition of dislocations. Vise verse any dislocation defect admits an equivalent representation through a set of disclinations. As a characteristic example, a specific disclination dipole (Fig. 10a) may be considered. Romanov and Vladimirov [14] define this configuration as a single-line two-rotation-axes disclination dipole. In such a defect the disclination lines coincide and the Frank vectors x and x have a mutual displacement by a vector t. In the result, the single-line dipole appears to be a natural equivalent for a dislocation of Burgers vector b = t  x. The other important disclination dipole configurations are two-lines two-rotation-axes dipole of wedge disclinations (Fig. 10b), which is similar to the terminated edge dislocation wall and two-lines one-rotational-axis dipole (Fig. 10c), which demonstrates the properties of a dislocation quadrupole. One may note that in far field two-axis wedge disclination dipole is equivalent to an edge superdislocation having an effective Burgers vector of magnitude Beff = 2ax (2a is a dipole arm). The presence of different types of disclination dipoles is possible due to the existence of three sets of parameters which

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Fig. 8. Difference between disclinations in conventional and liquid crystals, Friedel [55]: (a) disordered state of a ‘‘plastic” crystal; (b) orientational ordering; (c) positive disclination (Volterra dislocation) of strength x = p/2 in the mass-point lattice; (d) positive disclination (Frank disclination) in the orientation position of molecules with x = p.

Fig. 9. Relation between wedge disclinations and terminated edge dislocation walls: (a) negative wedge disclination with strength x; (b) wedge of angle x, which has to be inserted to create the disclination; (c) a set of half-planes modeling the wedge; (d) terminated wall of equidistant edge dislocations.

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Fig. 10. Disclination dipoles and their dislocation analogues: (a) single-line, two-rotation-axes dipole of twist disclinations; (b) two-lines, two-axes wedge disclination dipole; (c) one-axis wedge disclination dipole.

Fig. 11. Disclination loops: (a) circular pure twist disclination loop; (b) rectangular pure wedge disclination loop. For pure defects, the rotation vector in the geometric centre of the loop.

are characteristic of a disclination defect: the value and the direction of the Frank vector x, the position rx of the Frank vector and the position of a disclination line. In disclination loop configurations the defect line forms closed circuits in the material, as shown in Fig. 11. It is common to distinguish twist disclination loops when the Frank vector is normal to the plane of the loop and wedge disclination loops when the Frank vector lies in the plane of the loop. The use of the term ‘‘wedge” for the loop with Frank vector being parallel to the loop plane is not very precise because the disclination is not of wedge character for any portion of closed disclination line. Somewhere disclination is of pure wedge or twist character but for the main part of the wedge disclination loop defect line it is of mixed wedge/twist character (the case is similar to a shear dislocation loop where dislocation line change its property from edge to screw dislocation character). It was also proposed to use the term tilt disclination loop but the designation wedge disclination loop became more common. As it was argued by Pertsev et al. [57], special attention should be devoted to rectangular disclination loops (RDLs) (Fig. 10b). The significance of RDLs is clear because they can easily be transformed into other disclination defects: an angular disclination, a disclination dipole, a [-shaped defect or a linear disclination. As a result any disclination defect can be built up with given accuracy as a set of rectangular loops having a common rotation vector. It was illustrated in Figs. 9d and 10b that for the case of a wedge disclination, an equivalent dislocation configuration can be introduced, i.e. straight wedge disclinations can be modeled as terminated dislocation walls. Such models give rise to the idea of partial disclinations, i.e. linear rotational defects with smaller vectors x but which are also associated with boundaries of misorientation. It is well

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documented that boundaries of misorientation observed in crystals cover a wide range of angles regardless of the crystal lattice type, Rybin [5]. So it is not crystallographic considerations that determine the nature of the rotations. Therefore the conclusion can be made that disclinations in crystals are partial, i.e. imperfect; they inevitably entail surface defects such as subgrain, grain or twin boundaries. The strength of partial disclinations x can vary over a wide range. The disclination line characteristic scale also may have a spectrum of length values. One may expect however that typical partial disclinations will have their segment sizes comparable to the elements of material structure, i.e. grain, subgrains or fragments, Klimanek et al. [12]. 4. Elastic fields and energies of isolated straight disclinations 4.1. Approaches to analyze elastic properties of disclinations To evaluate the elastic fields and energies of disclinations one can use linear or nonlinear theory of elasticity or apply the technique of computer simulation. The last two methods are more precise. However more complex problems may be investigated exactly in the framework of linear elasticity. Therefore one needs to determine the limits and the accuracy of the above three methods when analyzing the properties of disclinations. If we take into account geometrical nonlinearity (i.e. tensor of finite strains instead tensor of small strains) and materials nonlinearity (e.g. Murnaghan’s law instead Hooke’s law), we can obtain the following expression for the elastic energy of a wedge disclination per unit length of a cylinder with free surface, Vladimirov et al. [58]:

"

Ex c 

#

plðk þ lÞ 2 2 kþl kþl 3m2  m1 3 3 2 j R þ pl þ ða2  a1 Þ þ a jR 2ðk þ 2lÞ 3l l 2ðk þ 2lÞ2

ð2Þ

where j ¼ 2xp  1 characterizes the power of the disclination, R is the radius of a cylinder; a, a1, a2 are cumbersome combinations of the linear elastic modules k and l (Lame constants) and the second-order elastic constants m1, m2, m3; index ‘‘c” denotes ‘‘cylinder”. It is also assumed that the disclination is singular, i.e. inner cylinder radius is R0 = 0. In Eq. (2) the first term gives the well-known solution of isotropic linear elasticity. Usually it is written in the following form, Romanov and Vladimirov [14]:

Ex c ¼

1 D x2 R2 8

ð3Þ

where D = G/[2p(1  m)]; G = l is shear modulus, m is Poisson ratio. The analysis shows that for a not very large strength of defects (for example typically for partial disclinations of mesoscopic level x 6 10° and the second-order constants typical for real solids the stresses and energy of isolated disclinations are described by linear elasticity with an accuracy 610%. However, there is a set of qualitatively new nonlinear effects for disclinations. As follows from Eq. (2), there is a difference in energies of positive and negative disclinations. Unlike the case in the linear theory, the mean dilatation d of the body containing a wedge disclination is not equal to zero. This typical second-order effect is evidently the main motive for the application of nonlinear elasticity theory in elastostatic problems involving disclinations. In the second-order theory the dilatation d does not depend on the sign of x; a typical value for x  10° is d  103 and for perfect disclinations with x ¼ p3 the dilatation is d  102 to 3  102. According to current ideas about the structure of amorphous and nanocrystalline materials this result can be taken as a rough estimate of the difference in densities of the amorphous and bulk crystalline states of the material, Nazarov et al. [59]. Zubov [60] has developed a set of nonlinear elasticity solutions for disclinations in materials with Mooney–Rivlin constitutive relationships originally devised for rubber-like materials, which can undergo large elastic shears but may be considered as incompressible. It is useful to note here that for homogeneous elastically isotropic incompressible bodies wedge disclinations (and also screw dislocations) belong to the category of universal solutions of elasticity, Trusdell and Noll [61]. The strain fields of these solutions are such that the stress distribution satisfying the condition of static equilibrium may be found for every constitutive relationship compatible with the basic requirements of isotropy

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and incompressibility. Seeger and Romanov [62] have analyzed the interaction between the above universal solutions in the framework of nonlinear elasticity. For Hooke’s bodies there is no interaction between coaxial wedge disclination and screw dislocation. The combination of these defects constitutes the problem of its own in the theory of dispirations, Harris [56]. þd of As another example of nonlinear analysis we give here the expression for the total energy Ex c combined dislocated and disclinated cylinder made of neo-Hookean material. The strain energy function for such material has the simple form, Lur’e [63]:

wnH ¼ NðI1  3Þ

ð4Þ

where I1 is the first strain invariant, and N is the elastic coefficient, which can be related to shear modulus, formally for incompressible materials (I3 = 1, where I3 is the third strain invariant) N = G/2. The þd obtained by Seeger and Romanov [62] is as follows: result for Ex c

Ecxþd ¼

N 2 R 4R2 R20 2 R b ln þ Npj2 R2  R20  2 ln 2p R0 R0 R  R20 " !# 2 2 N 2 j R þ R0 R R ln  b j 1 ln 1þ 2 2 2p R R 2 0 0 R  R0

!

ð5Þ

where b is the magnitude of dislocation Burgers vector of; R0 is an internal radius of cylinder. Other designations are identical to designations in Eqs. (2)–(4). In the Eq. (5) the first and second terms correspond to the self-energies of screw dislocation and wedge disclination in a hollow cylinder, respectively. The third term represents the interaction energy, which vanishes in the approximation of small strains. There is another way to take into account nonlinearity when investigating the properties of disclinations – computer simulation, e.g. Doyama and Cotterill [64] and Zhigilei et al. [65]. The idea of the molecular dynamics technique consists in the construction of an initial atomic configuration for the disclination, introduction of the interaction potential between atoms and performing the relaxation procedure. It has been shown by Doyama and Cotterill [64] and Zhigilei et al. [65] that for potentials characterizing metals computer simulation of disclinations gives good agreement with the linear theory of elasticity (for defect energies and stresses). At the same time, the results of the nonlinear approach are also confirmed. The energies of positive and negative disclinations are distinguished and nonzero total dilatation exists. With the aid of computer simulations new effects for disclination elastic field screening have been predicted. These effects include the amorphization of disclination core, Mikhailin and Romanov [66], when new disclinations are generated near the initial defect, and the appearance of a dislocation cloud formed by defects nucleated at the surface of crystallite and propagated to the disclination core, Zhigilei et al. [65]. Recently the effect of disclination core amorphization has been studied in more details when modeling crack nucleation in nanorods. Formation of dislocation clouds near wedge disclinations was also confirmed in dislocation dynamics calculation by Perevezentsev and Sarafanov [67]. Both these processes, i.e. core amorphization and dislocation cloud formation, lead to the diminishing of the energy of disclinated crystallite in expense of increasing the number of defects. 4.2. Disclinations in comparison to dislocations The elastic fields of straight disclinations in an infinite isotropic elastic continuum may be found in analytical form from the general relations of the theory of defects, e.g. De Wit [52] and Mura [54]. For example, a positive wedge disclination with strength x and line oriented along z-axis has following stresses in Cartesian co-ordinates [52]:



rxx ¼ Dx ln r þ ryy

y2 r2



  x2 ¼ Dx ln r þ 2 r

ð6aÞ ð6bÞ

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Table 1 Characteristic disclination and dislocation terms in stress and elastic energy dependencies on distance from a defect line and screening length.

Stresses

Disclination

Dislocation

 Gx ln Rrs

 G br

2

 Gx

Energy

LR2s

2

 Gb L ln RRcs

G is a shear modulus, x is a value of Frank vector of disclination, b is a value of Burgers vector of dislocation, L is dislocation or disclination length, Rs is a screening length, Rc is a core radius of dislocation.

rzz ¼ Dxmð2 ln r þ 1Þ rxy

xy ¼ Dx 2 r

ð6cÞ ð6dÞ

where r2 = x2 + y2 and other designations are the same as in Eq. (3). Typical for wedge disclinations is the logarithmic divergence of stresses in the defect core and at large distances r from the defect line. In addition, twist disclinations possess also the linear divergence along their lines [52]. One may note that co-ordinate dependence in logarithm terms of Eqs. (6a)-(6c) is not normalized. This means that the stresses of disclinations in infinite media are defined with the accuracy of a constant term, which can be determined in course of the solution of corresponding elasticity boundary value problem. In Table 1 characteristic dependencies for stresses and elastic energies of disclinations are given in comparison with those for dislocations; lines of defects are along z-axis of cylindrical co-ordinate system (r, /, z). It is obvious that straight disclinations demonstrate stress divergence not only in the defect core but also at large distances. Their energy is proportional to the square of the characteristic screening length. Therefore disclinations can exist in an elastic continuum only in a screened state, when their energy is diminished. 5. Screened disclinations 5.1. Self-screening: disclination dipoles, quadrupoles, and loops The logarithmic divergence of the long-range stresses (see Eq. (6) and Table 1) can be disposed of by joining straight disclinations in dipole and other multipole configurations. Their stresses can be found by a superposition of contributions from individual disclinations. One-axis dipoles and quadrupoles of wedge disclinations are self-screened defect configurations. Dipole energy depends only on the distance between the disclinations inside the configuration. As it was shown by Romanov and Vladimirov [14], for the dipole having axes of rotation shifted by the distances t1 and t2 for each disclination, the energy is equal to:

  Rs t 2 þ t 21 Rs Exx ¼ Dx2 3a2 þ 2aðt 2  t 1 Þ þ ð2aða þ t2  t 1 Þ  t2 t 1 Þ ln þ 2 ln 2a 2 Rc

ð7Þ

where 2a is the dipole arm; Rc is a core radius of the dipole disclinations (which is similar to the inner radius R0 used for disclinations in the hollow cylinder); Rs is an external screening parameter, e.g. the body size. From Eq. (7) for t1 = t2 = 0 one can derive the energy of a two-axes dipole shown in Fig. 10b with unshifted axes of rotation:

  Rs Exx ¼ Dx2 a2 2 ln þ 3 : 2a

ð8Þ

It is easy to recognize in Eq. (8) the well-known relationship, e.g. Hirth and Lothe [68] and Table 1, for the energy of an edge dislocation. We already mentioned that such a dipole in a far field is equivalent to a superdislocation with an effective Burgers vector of magnitude Beff = 2ax.

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For t1 = t2 = a, Eq. (7) gives the energy of the one-axis dipole shown in Fig. 10c:

  2a Ex0 ¼ Dx2 a2 2 ln  1 : Rc

ð9Þ

It is clear that in this case the disclination dipole energy has no dependence on external screening length Rs. The same property is peculiar to disclination quadrupoles and other higher multipoles. For quadrupole configuration given schematically in Fig. 3b, the self-energy obeys the following simple relation:

  a2 þ a2 a2 þ a2 Exq ¼ 2Dx2 a21 ln 2 2 1 þ a22 ln 2 2 1 a1 a2

ð10Þ

where 2a1 and 2a2 are the lengths of the two quadrupole sides. Disclination loops are defect configurations with maximal screening. It is convenient to investigate the properties of circular and rectangular disclination loops (RDLs), Romanov and Vladimirov [14]. Elastic energy of circular twist loop (CTL) shown in Fig. 11a and the energy of circular wedge disclination loop (CWL) with Frank vector laying in the plane of loop are following, see, for example Kolesnikova and Romanov [69]:

  Gx2 a3 p Jð2; 2; 0Þ r¼aRc 2  z¼0  Gx2 a3 p ¼ Jð2; 2; 0Þ r¼aRc 4ð1  mÞ

ECTL ¼ ECWL

ð11aÞ ð11bÞ

z¼0

where a is loop radius; Rc is core radius of disclination, J(m, n;p) are Lipschitz–Hankel integrals, Eason R1 jzj et al. [70]: Jðm; n; pÞ ¼ 0 J m ðjÞ J n ðj ar Þej a jp dj, Jm(j) is Bessel function. In approximation Rc  a, i.e. for large disclination loops, the term J(2, 2;0) is simplified and the expressions for disclination loop energies in Eqs. (11a) and (11b) acquire simpler forms:

  G 8a 8 xa3 ln  2 Rc 3   G 8a 8 ¼ xa3 ln  : 4ð1  mÞ Rc 3

ECTL ¼

ð12aÞ

ECWL

ð12bÞ

Pertsev et al. [57] have demonstrated that the elastic fields and energies of RDLs may be expressed through elementary functions. Their analysis is easy in comparison with the case of circular loops where special mathematical functions must be involved (see Eqs. (11a) and (11b)). For square twist and wedge disclination loops with the dimensions 2c  2c the corresponding energies are:

 8c  2:92 Rc p   2G 3 8c ¼ xc ln  2:80 : Rc p

ESTL ¼ ESWL

3G



xc3 ln

ð13aÞ ð13bÞ

STL and SWL denote square twist loop and square wedge loop correspondingly; the value for Poisson ratio was taken as m = 1/3. As it follows from the analysis, square loops have larger elastic energy than circular loops having the same area, as is obvious from general physical considerations. Nonetheless, in most applications this difference (which does not exceed 25%) is insignificant. What is important is the similar dependence on the loop size, proportional c3 or a3. The asymptotes of stresses for circular disclination loops coincide with those for square loops within a factor p/3. RDLs permit one to study the influence of the shape on the elastic fields and energies of disclination loops. In general, all characteristics and properties of RDLs are strongly influenced by their shape in contrast to those of dislocation loops. If the loop shape is changed, the sign of the energy of interaction between a disclination loop and a source of internal stresses may even be reversed.

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5.2. Screening by a free surface The solutions of elasticity boundary value problems for disclinations near free surfaces or interfaces in solids have been delivered for a number of geometries shown in Fig. 12: for straight disclinations with lines parallel to a free surface (Fig. 12a); for straight wedge disclinations perpendicular to the surface of a half-space; for straight disclinations, their dipoles and disclination loops in a plate of finite thickness (Fig. 12b); for disclinations in uniform and non-uniform cylinders (Fig. 12c); for a straight wedge disclination in an elastic spheroid (Fig. 12d); for disclinations near grain boundaries (not shown). The relevant references for the solutions of mentioned above boundary value problems can be found in the work by Romanov and Vladimirov [14]. Recently Kolesnikova and Romanov [69,71] have contributed to this field by applying the techniques of surface virtual defects for solving axial symmetric boundary value problems for disclinations and other defects. Here we provide the examples of the solutions for stresses for wedge disclination placed by a distance d from a planar free surface, as shown in Fig. 12a and for wedge disclination displaced by a distance R1 with respect to the cylinder axis, as shown in Fig. 12c. For a positive wedge disclination near free surface, the Airy stress function, which permits one to find the stresses in a state of plane strain, is [14]:

v¼

Dx 2 r þ r ln 2 þ r

ð14Þ

Fig. 12. External screening for wedge disclinations: (a) disclination parallel to the surface of the half-space; (b) disclination in the middle of the plate; (c) disclination shifted with respect to the cylinder axis; (d) disclination in spheroid.

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where r 2þ ¼ ðx þ dÞ2 þ y2 ; r2 ¼ ðx  dÞ2 þ y2 ; D ¼ G=½2pð1  mÞ, and the co-ordinate origin is at the free surface (Fig. 12a). The stresses of wedge disclination are found from Eq. (14):

rxx ¼ Dx ln

rþ y2 y2 y2  ðx  dÞ2 þ 2  2 þ 2xd r rþ r r4

!

rþ ðx þ dÞ2 ðx  dÞ2 ðx  dÞ2 ð2d  xÞ þ ð2d  3xÞy2 þ  þ 2d 2 2 r rþ r r 4   r ðx  dÞ rzz ¼ 2Dxm ln þ  d 2 r r ! 2 ðx þ dÞy ðx  dÞy d þ y 2  x2 : rxy ¼ Dx   2yd r 2þ r2 r 4

ð15aÞ !

ryy ¼ Dx ln

ð15bÞ ð15cÞ ð15dÞ

The Airy stress function for a wedge disclination in cylinder (Fig. 12c) is [14]:

  2 3  R2 ðx  R1 Þ2 þ y2 Dx 4 ðx2 þ y2 ÞðR21  R2 Þ5 2 2 þ : v¼ ðx  R1 Þ þ y ln  4 R2 ðxR1  R2 Þ2 þ y2 R1

ð16Þ

In the derivation of Eq. (16) it was assumed that the origin of the Cartesian co-ordinate system (x, y) coincides with the cylinder axis and the positive wedge disclination is placed at (R1, 0), R is a cylinder radius. For R1 = 0, i.e. for disclination laying along cylinder axis the Airy stress function is simplified to:

v¼

  Dx 2 r r 2 ; r ln  R 2 2

ð17Þ

and stresses acquire the following form [14]:



r R

rxx ¼ Dx ln þ

y2 r2



  r x2 ryy ¼ Dx ln þ 2 R r   r rzz ¼ Dxm 2 ln þ 1 R xy rxy ¼ Dx 2 : r

ð18aÞ ð18bÞ ð18cÞ ð18dÞ

Introduction of any type of surface means the appearance of external screening for disclinations. It is obvious that different surfaces differently influence the elastic properties of disclinations depending on their geometry and physical properties. Less trivial is the fact that external screening for disclinations is stronger than in the case of dislocations. If for dislocations instead of the stress dependence r / 1r at large distances r  d (d is the distance from a defect line to the surface of a half-space) there appears r / from r12 to r13 , disclinations are characterized instead of the stress divergence r / ln r by the relations r / from r12 to r13 depending on the type of disclination and the stress tensor component considered, Romanov and Vladimirov [14]. The energy per unit length of a wedge disclination is proportional to the square of the typical screening parameter in the problem: to the distance to a plane free surface d, to the half-thickness of a plate h, to the radius R of a cylinder. In dependence on the body shape the degree of screening diminishes in the following sequence: a half-space:

Ed ¼

1 2 D x2 d 2

ð19Þ

a plate:

Eh  0:182Dx2 h

2

ð20Þ

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a cylinder, see also Eq. (3):

Ex c ¼

1 ðR2  R2 Þ2 : D x2 1 2 8 R

ð21Þ

For elastic spheroid with disclination (see Fig. 12d) the energy is proportional to x2 and R3, and the stress state which is intermediate between plain strain and plain stress, Polonsky et al. [72]. The external screening of defect elastic fields in small particles, thin films or in subsurface layers leads to decreases of the disclination energy down to values which are comparable with the energy of an individual lattice dislocation. Grain and phase boundaries have very low screening ability for individual disclinations but they can cause additional configurational forces acting on disclination dipoles and loops. 6. Disclination modeling of solid structures Four main structure levels can be distinguished in experimental and theoretical analysis of plastic deformation in solids, Panin et al. [73], Romanov and Vladimirov [14]: (i) an atomic level – the scale of the lattice constant; (ii) a nano- or mesoscopic level – the scale of dislocation substructures; (iii) a structural level – the scale of grains in polycrystals; (iv) a macroscopic level – the scale of an average physical and mechanical properties (Fig. 13). Each level of solid structures requires the employment of special physical models and mathematical languages for understanding plasticity and deformation characteristics peculiar to the corresponding level. The nano-/mesoscopic level is conditioned by collective effects in the ensemble of defects of atomistic level. Its scale is

lmeso ¼

Gb

rl

ð22Þ

where b is the magnitude lattice dislocation Burgers vector, rl is the resistance to the motion of an individual dislocation (i.e. Peierls stress), G is the elastic modulus. Usually lmeso = 10 nm to 1:0 lm. It is useful to note that for nanocrystalline materials grain size can be of the order of lmeso. In such a case structural level can emerge for groups of grains. For each structure level the elementary linear defects: dislocations and disclinations may be introduced. Elementary defects for a crystal lattice are the usual translational lattice dislocations: the defects of the first order. For the mesoscopic level these are partial disclinations, dislocation arrays and dislocation pile-ups. In this section we provide the examples for the application of the disclination concept to the explanation of physical and mechanical properties of solid structures on mesoscopic level including conventional polycrystals, bulk nanostructured materials and small particles. It is necessary to note that the Sections 6.1 and 6.2 devoted to rotational structures in deformed solids and disclination plasticity and

Fig. 13. Structure scale levels in deformed materials: (a) atomic scale level with characteristic scale lmicro = 1 to 10 nm; (b) nano- or mesoscopic level with characteristic scale lmeso = 10 nm to 1 lm; (c) grain structure level with characteristic scale lsruct ¼ 1—100 lm; (d) macroscopic or structure-less level with characteristic scale lmacro 100 lm.

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work-hardening present only a very brief description of the subject with the relevant references to corresponding reviews and key-articles. On the contrary, Sections 6.3–6.5 deal with new results that are available only in specialized articles. Therefore these sections contain additional details on disclination modeling of plastic behavior of bulk nanocrystalline materials, relaxation behavior of thin strained layers, and disclinations in small particles with axes of pentagonal symmetry. 6.1. Rotational structures in deformed solids By definition partial disclinations are associated with terminated boundaries of misorientations in otherwise perfect crystals, Romanov and Vladimirov [14]. Wedge disclination, as we have shown above, can be considered as a mesoscopic model for terminated tilt boundary. Boundaries of misorienation are related to disclinations have a different nature. They can be low angle dislocation walls (Fig. 14a) with the misorientation angle /DW, high-angle grain boundaries (Fig. 14b) with misorientations /GB, or twin or misorientation band boundaries with misorientation /TB or /MB, respectively. The geometry of terminated boundaries of misorientation can also be different. For single terminated tilt boundaries the disclination strength is exactly equal to the angle of misorientation. In Fig. 14a the negative wedge disclination x with the strength

x ¼ uDW

ð23Þ

is shown. At the junction of several boundaries (Fig. 14b) the resulting disclination with the strength xJ can appear under the condition

xJ ¼

X i

xi ¼

X

uGBi –0 ði ¼ 1; 2; 3Þ

ð24Þ

i

i.e. in the case of so-called non-compensated junctions, e.g. Likhachev and Rybin [4] and Bollman [74]. Disclination dipole configurations can be associated with terminated lamellae with misoriented crystal lattice: misorientation and kink bands or twins (as shown in Fig. 14c). In the latter case two terminated boundaries of misorientation belong to the disclination dipole. The concept of screened disclinations enables one to describe properties of misorientation boundaries themselves. The most promising here is the application of the disclination model to high-angle grain boundaries where the dislocation description does not work, Li [75], Gertsman et al. [76]. Within the framework of this model and using a limited number of experimentally measured parameters the dependence Eg(/) of grain boundary energy on misorientation u was evaluated. For example, it was shown that calculated dependences Eg(/) for symmetrical equilibrium tilt boundaries in Al are in good agreement with experiment Valiev et al. [77]. More recently Shenderova et al. [78] applied the disclination models for the multi-scale analysis of grain boundary properties in polycrystalline diamond (Fig. 15) and Nazarov [79] provided a comprehensive review in this field. Disclination models open

Fig. 14. Disclination models for terminated tilt boundaries in structure of deformed materials: (a) single terminated dislocation wall with misorientation /DW; (b) triple junction of high-angle grain boundaries; uGBi ði ¼ 1; 2; 3Þ are misorientations at individual grain boundaries; (c) misorientation (kink) band with misorientation angle /MB.

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Fig. 15. Disclination model of grain boundaries: (a) example of disclination structure for R 29 high-angle grain boundary in diamond; (b) Dependence of energy on angle for h0 0 1i symmetrical tilt-grain boundaries in diamond: the results of disclination model (dashed and solids lines) are compared with results of computer simulation (empty and filled circles and black squares) diamond, Shenderova et al. [78].

the possibility to investigate the so-called nonequilibrium state of grain boundaries, Nazarov et al. [80]. For this purpose a random distribution of disclination dipoles should be considered. Due to the nonuniformity in the dipole arrangement the grain boundary contains an exess energy, which may be comparable with the grain boundary energy in the lowest state. Additionally, grain boundaries with so-called quaisiperiodic structure have been analyzed, Mikaelyan et al. [81]. 6.2. Plasticity and work-hardening When moving, disclinations contribute to the plastic deformation. To calculate the amount of strain e the geometry of disclination motion must be known in detail. In order to obtain an elementary estimate of strain rate one can apply the following relation, e.g. Romanov [18]:

de ¼ hxlr V dt

ð25Þ

where h is the density of screened disclination systems (dipoles) with the strength x, V is their velocity and lr is the characteristic size (the dipole arm 2a). The motion of the dipoles of wedge partial disclination, as it is illustrated in Fig. 16, is the main mechanism of rotational plastic deformation. The

Fig. 16. Disclination contribution to the plastic strain: (a) macroscopic change in the sample shape to due to the disclination dipole motion; (b) microscopic mechanism of disclination dipole motion related to redistribution of dislocations in front of the dipole; (c) nucleation of disclination dipoles at triple junctions of grain boundaries.

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terminated band of misorientation is modeled as a disclination dipole. Vladimirov and Romanov [7] proposed that the dipole movement occurs when edge dislocations travel from the bulk of the grain to the planes where partial disclinations are located (see Fig. 16b). The disclination approach permits a description of the details of the development of misorientation bands in crystals and other solids. It provides a connection of the parameters of a band (the thickness 2a and misorientation / = x) with the density of mobile dislocations q and applied stress re. The nucleation of disclination dipoles and corresponding misorientation bands usually takes place near stress concentrators, e.g. Rybin [5] and Romanov and Valdimirov [14]. Grain boundary junctions and other imperfections of grain boundary structure serve as such concentrators; the example is given in Fig. 16c. The models considering the wedge disclination dipole generation near triple junctions predict the value for critical external shear stress sg, above which an event of disclination nucleation takes place, Gutkin et al. [82]. The numerical estimates for sg give the values of 103G. These estimates correspond to the level of deforming stress at the stage III of r  e diagram for initially monocrystalline materials. The critical stress sg is shown to be strongly dependent on the geometry and strength of initial disclinations at grain boundary faults. The further development of disclination structure at an isolated grain boundary fault demonstrates two main regimes of misorientation band development: stable and unstable propagation. Gutkin et al. [82] have found that the transition between the above two regimes of misorientation band development is controlled by the other critical value for external stress sp. Twinning of deformation origin can be effectively modeled in the framework of disclination approach. For the first time, disclinations were proposed for the description of deformation twins by Armstrong [83] in 1968. Then the disclination approach was applied to the twinning phenomena in the works of Müllner and co-workers [8,9,84,85]. The essential part of all these applications was based on the representation of the front of the moving deformation twin as a wedge disclination dipole combined with Somigliana dislocation dipole. Various twinning phenomena including twin intersections and internal twinning can be understood in the framework of disclination models [14,9,85]. Work-hardening at the stage of rotational structure formation is related to the resistance to dislocation motion in the elastic fields of existing disclinations, Romanov and Vladimirov [14]. At the stage of active rotational deformation the hardening appears to be due to interaction between disclinations. The extensive analysis of the work-hardening mechanisms operating at large strains and their relation to disclinations is given by Seefeldt [86]. Typical for disclination hardening is the following dependence of the deforming stress, e.g. Romanov [18]:

r ¼ bGx

ð26Þ

where b is a geometry factor determined by the actual type of interacting disclinations. Consideration of the above relation helps to explain deviations from the well-known dependence of the deforming stress on the dislocation density q:

pffiffiffiffi

r ¼ aGb q:

ð27Þ n

1 2

For the disclination mechanisms of hardening one can obtain r  q with < n 6 1. Such ‘‘anomalies” are often observed from the onset of stage III of the r(e) dependence. In order to couple the material structure evolution with the mechanical properties, the corresponding constitutive relations have to be introduced. There exists a number of attempts to reach this goal in the framework of the dislocation approach, e.g. Estrin et al. [87]. Seefeldt [86] summarized the achievements of the disclination description in the formulation of such constitutive relations. In particular, it was demonstrated that disclination based models provide a very good agreement with experimentally observed plastic flow curves for Cu monocrystals being deformed by compression at room temperature with the prescribed strain rate up to large (70%) strain degrees, also see Klimanek et al. [12]. Other deformation phenomena, for example texture formation, Nazarov et al. [88] and Van Hautte [89], superplasticity, Perevezentsev et al. [90], and transition to ductile fracture Rybin et al. [5,91,92] have found their explanation in the framework of disclination approach. For later case various disclination models for crack nucleation have been advanced including crack nucleation at grain boundary junctions, Wu et al. [93–95].

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6.3. Disclinations in bulk nanocrystalline materials Use of disclination models in the analysis of mechanical properties of bulk nanocrystalline materials (NCMs) emerged during the last decade Romanov [96,97], Gutkin and Ovid’ko [98]. In general, nanostructured materials cover a wide class of solids, in which the constitutive elements have a characteristic size from 1 to 100 nm. Nanostructured materials include bulk nanocrystalline materials, nanoscale films and free standing nanosized clusters of atoms, i.e. nanoparticles. It was shown over the past two decades, e.g. Gleiter [99], Gryaznov and Trusov [100], and Valiev et al. [101], that bulk NCMs demonstrate a unique set of physical and mechanical properties. Their unique properties originate mainly from two factors: (i) size effects and (ii) presence of high density of interfaces in the material interior. In mono-phase nanocrystalline materials (nanocrystals) the grain boundaries form a three-dimensional network with linear junctions and point nodes. Usually these are triple junctions and fourfold nodes. The density of junctions and nodes drastically increases with decreasing grain size. In nanocrystals prepared by severe plastic deformation technique the refinement of grains takes place during material processing. Therefore both the properties of nanocrystals and the nanocrystalline material processing itself cannot be understood without experimental and theoretical analysis of grain boundary network and disclination defects being peculiar to such network. Valiev et al. [101] have proposed that in bulk nanocrystalline materials produced by extreme plastic deformation grain boundaries are in metastable, nonequilibrium state. Nonequilibrium grain boundaries generate long-range elastic stress fields, which are responsible for modified properties of nanocrystals, e.g. Nazarov et al. [102]. As we already mentioned, the structure of nonequilibrium grain boundaries has been modeled in terms of disordered arrays of grain boundary dislocations and/or disclinations. The other important sources of internal stresses in nanocrystals are junction disclinations, Nazarov et al. [59]. Junction disclinations, e.g. shown in Fig. 14b have been proved to be necessary elements of defect structure at large strains. They appear as a result of the incompatibility of plastic strain in neighboring misoriented grains, Rybin et al. [103]. Nazarov et al. [102] have estimated various mechanical quantities for nanocrystalline materials in the framework of disclination approach. These quantities included root mean square (rms) strain, dilatation and stored energy. It was found, for example, that disclination contribution to dilatation d for

Fig. 17. Modeling of new grain nucleation in the initially disclinated grain, Orlova et al. [106]: (a) diagram illustrating the nucleation of a new grain at one of the initial grain boundaries (g, q, and k are the parameters characterizing the asymmetry of the disclination quadrupole configuration in the initial grain); (b) a change in the energy of the initial grain as a result of the formation of a new grain at one of the initial grain boundaries (D = G/[2p(1  m)], G is a shear modulus, m is a Poisson ratio for a model system with the parameters b ¼ a ¼ 1 lm; d=l ¼ 2; q ¼ 0:1; g ¼ k ¼ 1; p ¼ xx =x).

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ultra fine grain Al (grain size 100 nm) can be of the order of 104. One may expect that disclinations will become more important with grain size reduction. The energy stored by junction disclinations and other mesodefects accumulated at grain boundaries in the process of severe plastic deformation can be transformed in additional defects. Based on this idea the process of grain subdivision has been modeled [104–106]. It was demonstrated that in a probe grain with junction disclinations of varying relative strength (characterized by coefficients g, q, and k, see Fig. 17a) the appearance of a new grain leads to the diminishing of the stored energy. This diminishing is caused by the screening of the elastic fields of junction disclinations by the defects associated with a new grain. It was proposed by Gryaznov et al. [107] that disclination mechanisms of hardening play an important role for NCMs. In particular, deviations from Hall-Petch dependence well-known for nano grain size materials (see for example Meyers et al. [108]) can be explained in the framework of disclination models. To illustrate this statement we consider the dislocation–disclination relay mechanism of plastic deformation of NCMs recently developed by Kolesnikova et al. [109], Romanov et al. [110]. The relay mechanism model investigates the propagation of localized plastic shear in NCMs that is schematically shown in Fig. 18a. It is supposed that some boundaries have low resistance rgb to grain boundary shear: rgb < s, where s is external applied shear stress, but the other boundaries still can resist to shear at such applied stress. In some places localized grain boundary shear can be transferred to the suitable ‘‘weak” boundary of the next grain. In the other places the shear can not be easily transferred through the boundaries surrounding such a grain. These ‘‘hard” grains (HG) serve as the places for the realization of dislocation–disclination relay mechanism. For simplicity the model assumes rectangular

Fig. 18. Relay dislocation–disclination mechanism of plasticity in nanocrystalline materials (NCMs), Romanov et al. [110]: (a) schematics of a localized shear development in a NCM with hard grains (HGs) and weak grains (WGs); (b–e) propagation of a localized shear trough a ‘‘hard grain” in nanocrystalline materials. Critical deforming stress sc as a functions of grain size d for (f) model with unsplitted dislocations at hard grain, and for (g) model with splitted dislocations at hard grain. Dependencies sc (d = d1) are given for aspect ratios f = d1/d2 as indicated by corresponding curves. Used parameters: magnitude of Burgers vector 1=2 ”, G is b = 0.3 nm, Poisson ratio m = 0.3, dislocation core parameter a = 1. Plots are presented in Hall-Petch co-ordinates ‘‘sc  d NCM shear modulus.

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cross-section of the grain with the sides d1 and d2 (see Fig. 18b). The localized shear, which is modeled by the straight edge dislocation with Burgers vector +b, will be arrested at left boundary of the HG. The original plane of the shear coincides with the easy glide boundary plane, in which the applied stress s acts. Because of geometrical and structural factors the dislocation is not allowed to enter HG. Nevertheless the initial shear can be transmitted to the right boundary of the HG by formation of the quadrupole of wedge disclinations located in the grain boundary junctions (Fig. 18c) and having the strength x. Such possibility follows from the disclination mechanisms of plastic deformation outlined in the previous Section 6.2. To preserve the volume of the shear transmitted through HG by disclinations, the Burgers vector magnitude b and the strength of junction disclinations x must be related via the simple formula

x ¼ b=d2 :

ð28Þ

The disclination quadruple formation can proceed for example via co-operative sliding in the HG interior along the planes inclined to the plane of the initial localized shear, or it can be achieved by co-operative diffusion process along the nanograin boundaries. As a consequence the translation mode of deformation transforms into the rotation mode (T?R). In this case crystallography planes in the HG grain become inclined. The rotations of the crystal lattice in nanograins are confirmed by ‘‘in situ” observations of plastic deformation of metal NCM, Shan et al. [111], Sergueeva and Mukherjee [112]. After the rotation deformation propagation trough the HG, the localized shear continues in the initial plane or in the plane parallel to initial one (Fig. 18d). In this case at the grain boundary the edge dislocation with opposite Burgers vector b forms and remains. The dislocation (plastic shear) +b propagates further in NCM along the easy shear grain boundary. The rotation mode of the plastic deformation transfers the ‘‘relay” to the translation mode (R?T). The remained at the grain defects are disclination quadrupole and dislocation dipole (Fig. 18d). Their energy depends only on the grain geometry, i.e. grain size and aspect ratio, and the magnitude of the localized shear b. The decreasing of the stored elastic energy can be achieved with the help of the dissociation (splitted) of the boundary dislocations with their transformation into junction dislocation quadrupole b2 =  b2 (Fig. 18e). We may note that transformation from one defect configuration to the other involves dislocation climb and therefore can be controlled by grain boundary diffusion. The proposed dislocation–disclination relay mechanism allows us to find the critical deforming stress for its realization. The details of calculations are reported by Kolesnikova et al. [109], Romanov et al. [110]. The analysis leads to the following expressions for the deforming stress in Models without and with dislocation splitting for equiaxial HG with d1 = d2 = d:

  Gb 0:4ad ln 2pð1  mÞd b   Gb 0:54ad ¼ ln 4pð1  mÞd b

sð1Þ c ¼

ð29aÞ

sð2Þ c

ð29bÞ

where a is the dislocation core parameter. ð1Þ;ð2Þ

In Figs. 18f and g the dependences sc ðdÞ are shown for general case of nonequiaxial grains for various grain size aspect ratio f = d1/d2 in traditional co-ordinates of Hall–Petch relationship, see for example 1=2 ”. From comparison of two possibilities of described relay mechanisms Meyers et al. [108], ‘‘s  d (models in Fig. 18d and e) one can give the preference to the model with splitting grain boundary dislocations. As it was already mentioned the transition from one model configuration to the other requires grain boundary diffusion and therefore is strongly temperature sensitive. In Figs. 18f and g the critical grain size dc at which the deforming stress becomes diminishing function of the grain size, is demonstrated. The estimates show that for the localized shear (Burgers vector) magnitude b = 0.3 nm, dislocation core parameter a = 1, and grain aspect ratio f = 1 the critical grain size dc = 2 nm. This is an underestimated value since the small magnitude of the localized shear and equal axial shape of the probe grain are used. For the grain aspect ratio f = 3 critical size is d3c  6 nm, and for b = 1.0 nm and a = 1 critical sizes are dc  7 nm and d3c  20 nm correspondently. These values of the grain size are close to

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experimentally observed values for the onset of anomaly in Hall–Petch dependence of the deforming stress on grain size in NCM materials [108].

6.4. Disclinations in thin strained layers Disclination approach has demonstrated a very effective application in the explanation of the domain structures in ferroelectric nanoscale films, Speck et al. [40], Pertsev and Zembilgotov [44], Romanov et al. [42,43]. For the epitaxial growth of ferroelectric films, elastic strains develop as a result of intrinsic lattice distortion (e.g. tetragonal or rhombohedral) and mismatch between the materials of film and substrate. These strains were shown to be relieved by the formation of domain patterns within the film, see the example in Fig. 6. At present the following types of domain patterns in epitaxial ferroelectric films have been investigated. For (0 0 1) oriented tetragonal ferroelectric (FT) films there are two possibilities for domain patterning: a1/a2 with {1 1 0} domain boundaries, which are normal to film/substrate interface, and a/c with {1 0 1} boundaries, which are inclined with respect to the film/substrate interface [40,41,44,113]. It is useful to note that the individual domains in a1/a2 patterns domains are mechanically equivalent, while the a and c domains in the a/c pattern are not energetically equivalent [40,44]. This leads to the formation of equal size domains for a1/a2 patterns and unequal size domains for a/c patterns. Moreover, the free surface of the film with a1/a2 pattern is flat whereas the a/c pattern has a puckered surface [40,114]. For (1 1 1) oriented FT all three possible domain variants di (i = 1, 2, 3) are mechanically equivalent and domains always have equal width [43]. Possible domain boundaries in a di/dj pattern are of the (1 1 0) type. The patterns with inclined {1 1 0} boundaries have a flat film surface and the patterns with normal {1 1 0} boundaries have a ‘puckered’ surface. Finally, in (0 0 1) oriented rhombohedral ferroelectric (FR) films the situation is similar to the (1 1 1) oriented FT films. The difference is that there are four possible mechanically equivalent domain variants ri (i = 1, 2, 3, 4) and domain boundaries are along {0 0 1} or {1 0 1} faces (in pseudocubic designations). Again the patterns ri/rj with domain boundaries normal to the film/substrate interface cause puckering of the film free surface [115]. The geometry and energetic of possible domain patterns can be most easily described in the framework of the coherency defect approach. As it was proved by Romanov et al. [42,43] the coherency interfacial defects have the features of disclinations and Somigliana dislocations. For domain with plane parallel boundaries throughout the film thickness, the residual stress distribution of the various domain patterns can be described by arrays of shear Somigliana dislocations and wedge disclinations aligned along the film/substrate interface (see Fig. 19): (0 0 1) FT a1/a2 patterns. It was proved by Romanov et al. [41] that the strain state in a1/a2 patterns in FT films can be described by the chain (cross grid) of Somigliana screw dislocations having alternating ; characterizes the dissign and the strength xs = eT (Fig. 19a), where the tetragonality strain eT ¼ ca b tortion for FT phase with a and c being the tetragonal lattice parameters and b* being the effective substrate lattice parameter [116]. (0 0 1) FT a/c patterns. For a/c domain pattern the mesoscopic coherency defect structure includes the chain of alternating sign wedge disclinations of the strength xd  eT and the chain of Somigliana edge dislocations with the strength xe = eT [40]. The lines of both disclinations and Somigliana edge dislocations coincide, however the sign convention gives the opposite sign for the coinciding defects (see Fig. 19b). (0 0 1) FR ri/rj patterns. In the case of ri/rj domain patterns in FR films the mesoscopic defect description depends on the orientation of the domain boundaries with respect to film/substrate interface. For normal orientation of the domain walls (as shown in Fig. 19c) the necessary defects are wedge disclinations with the strength xdr = 2d and Somigliana screw dislocations with the strength xsr = 2d, where d is angular deficiency characterizing rhombohedral elementary cell. For inclined orientation of domain walls in (0 0 1) FR films only Somigliana screw dislocations with the strength xsr are required [42]. (1 1 1) FT di/dj patterns. The coherency mesoscopic defects in the case of (1 1 1) FT films are similar to those used for (0 0 1) FR system. As shown in Fig. 19d, only Somigliana screw dislocations are required

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Fig. 19. Examples for mesoscopic defect content for different domain patterns in coherent tetragonal ferroelectric FT and rhombohedral ferroelectric FR films: (a) a1/a2 – pattern in (0 0 1) FT films; (b) a/c – pattern in (0 0 1) FT films; (c) ri/rj -pattern in (0 0 1) FR films; (d) di/dj – pattern in (1 1 1) FT films.

for (1 1 1) FT domain patterns with inclined domain walls [43]; the strength of these dislocations is pffiffiffiffiffiffiffiffi screw dislocahowever different xst ¼ 2 2=3eT . For normal domain walls in addition to Somigliana pffiffiffi tions a chain of alternating sign wedge disclinations of the strength xdt ¼ 2= 3eT has to be superimposed [43]. The approach yields predictions of both the far and near interface elastic field distribution in the ferroelectric film and pffiffiffi provides the basis for the dependence of the domain size l on the film thickness h: scaling law l  h. Peculiarities of domain formation in ferroelectrics include the variation in the domain shape. Ullrich et al. [117] have demonstrated that the changes in the domain shape near interfaces lead to the diminishing of elastic energy associated with the constrained phase transformation in the ferroelectric film. More recently domain structures resulting from twinning in epitaxial rhombohedral La1x MnxTiO3 (LSMO) (1 0 0) and (1 1 0) oriented films were successfully described by applying disclination and Somigliana dislocation defect approach, Farag et al. [118,119]. 6.5. Disclinations in small particles Important area for the application of the disclination approach can be found in the description of the properties of pentagonal rods (PRs) and pentagonal particles (PPs), the examples of which are shown in Fig. 20. The origin of pentagonal symmetry in PRs and PPs follows from the schematics given in Fig. 20.As shown in Fig. 20a PR is a polycrystal consisting of five FCC monocrystalline regions divided by five twin boundaries (TBs). Lateral faces of this multiple-twinned PNR are crystallographic planes of {1 0 0}-type, whereas cup faces are of {1 1 1}-type. The axis of fivefold symmetry is parallel to h1 1 0i-type direction. The internal structure of PRs can be understood from the schematics of Figs

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Fig. 20. A disclination model for pentagonal rods (PRs) and pentagonal particles (PPs): (a) general schematics with crystallographic FCC planes for a PR with a coaxial positive wedge disclination with Frank vector x; (b) top-view of five FCC twinned crystals with four twin boundaries (TBs) of {1 1 1}-type and small angular gap; (c) cross-section of the PR with disclination x, crystallographic planes in the cross-section are of {1 1 0}-type; (d) general schematics for PP with positive wedge disclinations with Frank vectors x.

20b and c, in which we follow considerations given De Wit [120]. In the top-view of Fig. 20b five undistorted parts of the PNR are aligned along four twin boundaries (TBs), which in FCC crystals are {1 1 1}-type planes (for details of twinning crystallography in FCC crystals see, for example [68]). Because of FCC crystal geometry there exists a small angular gap x preventing the formation of completely connected and undistorted PR. This angular gap however can be eliminated by mutual rotation of the gap faces with the formation of the fifth TB. The closing of the gap is equivalent to the introduction of the positive wedge disclination of the strength x [120,14] along the PR axis. The resulting configuration of the crystal lattice in the PR cross-section (which is the plane of the {1 1 0}-type) is shown in Fig. 20c where a triangle designates a wedge disclination. The introduction of disclination leads to the elastic distortion of crystal lattice in the bulk of the PR. In the continuum mechanics model, which is suitable for the calculation of elastic fields and energies, a PR can be described as an elastic cylinder of the radius RP with coaxial positive wedge disclination. Typically, for experimentally observed PNRs RP varies from 10 nm to 1 lm [22,47,121]. In case of PPs the standard morphology is icosahedron with {1 1 1}-type crystallographic facets only [22,122]. The icosahedron possesses six disclinations shown in Fig. 20d. It has been proven, e.g. Gryaznov et al. [22] that each fivefold axis, which appears at the junction of five twin boundaries in a PR or a PP and contains a positive wedge disclination of the strength:



1



x ¼ 2p  10 sin1 pffiffiffi  7 200 : 3

ð30Þ

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Fig. 21. Energy release DE for pentagonal rod as function of (a) dislocation loop radius a; (b) dislocation position d; (c) ratio of core and shell radii t = Rc/Rs. Rp is a radius of PNR, b is a value of dislocation Burgers vector, e* is a misfit parameter between core and shell crystal lattice, G is a shear modulus, Poisson ratio m = 0.3.

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Owing to the presence of disclinations, non-uniform elastic distortions are generated in the particle interior. Elastic stresses related to these distortions may then relax in various ways. Gryaznov et al. [123] proposed, that the principal relaxation channels in pentagonal particles are the creation of additional lattice dislocations, displacement of the pentagonal axis (which is equivalent to the motion of the disclination towards the surface of the nanoparticle), splitting of the pentagonal axis (i.e. formation of new partial disclinations) and the emergence of pores in the disclination core. These findings are in good agreement with the available experimental results, see for example observation of empty channel in the PR shown in Fig. 7b [49]. Recent theoretical investigations of stress relaxation in pentagonal particles, Kolesnikova and Romanov [124,125], Dorogin et al. [126] have demonstrated the threshold character of additional defect formation in pentagonal nanorods and nanoparticles and have predicted the formation of mismatched shell on PRs and PPs. Fig. 21 shows the series of energy curves obtained when modeling stress relaxation in PRs. These results indicate elastic energy release for PR with dislocation loop, straight dislocation and lattice mismatched surface layer. It is interesting that the last theoretical prediction (i.e. mismatched layer formation) was later on independently confirmed in experimental observations of decahedral and icosahedral copper nanoparticles coherently covered by silver [127]. 7. Conclusions Disclinations in solid structures exist only in the form of screened configurations with lower selfenergy. The energy of disclinations is determined by the parameters of external and internal screening. The configurations with internal screening include dipoles, quadrupoles and disclination loops. Effective external screening is achieved for disclinations in small materials volumes such as nanoparticles and nanoscale thin films. Partial disclinations, which are observed in plastically deformed materials, manifest themselves on the nano- and mesoscopic levels from 10 nm to1:0 lm. The models based on the properties of screened disclinations are useful in studies of structure, deformation, work-hardening and fracture of poly- and nanocrystals, amorphous solids, thin films, and micro- and nanoparticles. Acknowledgments The support of Russian Foundation of Basic Researches (Projects Nos. 05-08-65503a and 07-0100659a) is acknowledged. AER is grateful to Project 1.00101-0337 of Program Meede 1.1. (Estonia). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

Volterra V. Annales Ecole Normale Sup Paris 1907;24:401. Nabarro FRN. Theory of crystal dislocations. Oxford: Clarendon Press; 1967. Nabarro FRN. In: Argon AS, editor. Physics of strength and plasticity. Cambridge, MA: The MIT Press; 1969. p. 97. Likhachev VA, Rybin VV. Izv Akad Nauk SSSR Fiz 1973;37:2433. Rybin VV. Large plastic deformations and fracture of metals. Moscow: Metalurgia; 1986 [in Russian]. Vladimirov VI, Zukovski IM. Sov Phys Solid State 1975;17:773. Vladimirov VI, Romanov AE. Sov Phys Solid State 1978;20:1795. Müllner P, Romanov AE. Solid State Phen 2002;87:239. Müllner P, Pirouz P. Mater Sci Eng A 1977;233:139. Seefeldt M, Klimanek P. Model Simul Mater Sci Eng 1998;6:349. Klemm V, Klimanek P, Seefeldt M. Phys Stat Solid (a) 1999;175:569. Klimanek P, Klemm V, Romanov AE, Seefeldt M. Adv Eng Mater 2001;3:877. Klemm V, Klimanek P, Motylenko M. Mater Sci Eng A 2001;324:174. Romanov AE, Vladimirov VI. In: Nabarro FRN, editor. Dislocations in solids, vol. 9. Amsterdam: North-Holland; 1992. p. 191. Nabarro FRN, editor. Dislocations in Solids. vol. 1–12. Amsterdam: North-Holland/Elsevier; 1979–2004. De Wit R. Continual theory of disclinations. Moscow: Mir; 1977 [in Russian]. Friedel J. Dislocations. London: Pergamon; 1964. Romanov AE. Mater Sci Eng A 1993;164:58. Romanov AE, Vladimitrov VI. Phys State Solid A 1980;59:R159. Vladimirov VI, Kolesnikova AL, Romanov AE. Phys Met Metall 1985;60:58. Foster CM, Pompe W, Daykin AC, Speck JS. J Appl Phys 1996;79:1405.

768 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90]

A.E. Romanov, A.L. Kolesnikova / Progress in Materials Science 54 (2009) 740–769 Gryaznov VG, Heydenreich J, Kaprelov AM, Nepijko SA, Romanov AE, Urban J. Cryst Res Technol 1999;34:1091. Bouligand Y. In: Balian R, Kleman M, Poirier JP, editors. Physique des defauts. Amsterdam: North-Holland; 1981. p. 665. Kleman M. Points, lines and walls. New York: Wiley; 1983. Kleman M, Friedel J. Rev Modern Phys 2008;80:61. Frank FC. Disc Farad Soc 1958;25:19. Bouligand Y. In: Balian R, Kleman M, Poirier JP, editors. Physique des defauts. Amsterdam: North-Holland; 1981. p. 777. Sleytr UB, Messner P, Pum D, Saèra M, editors. Crystalline bacterial cell surface layers. Berlin: Springer-Verlag; 1988. Chattopadhyay S, Chen L-Ch, Chen K-H. Critical Rev Sol State Mater Sci 2006;31:15. Nye JF. Proc Roy Soc A 1983;387:105. Anthony K. Solid State Phen 2002;87:15. Anthony K, Essmann U, Seeger A, Trauble H. In: Kröner E, editor. Mechanics of generalized continua. Berlin: Springer; 1968. p. 355. Salomaa MN, Volovik GE. Rev Mod Phys 1987;59:533. Nelson DR. Defects and geometry in condensed matter physics. Cambridge: Cambridge University Press; 2002. Chou T-W, Malasky HJ. J Geophys Res B 1979;84:6083. Duan YS, Zhao L. Int J Modern Phys 2007;22:1355. Klemm V, Klimanek P, Motylenko M. Solid State Phen 2002;87:57. Luft A. Progr Mater Sci 1991;35:97. Kolesnikova AL, Klemm V, Klimanek P, Romanov AE. Phys State Solid (A) 2002;191:467. Speck JS, Daykin AC, Seifert A, Romanov AE, Pompe W. J Appl Phys 1995;78:1696. Romanov AE, Pompe W, Speck JS. J Appl Phys 1996;79:6170. Romanov AE, Levefre MJ, Speck JS, Pompe W, Streiffer SK, Foster CM. J Appl Phys 1998;83:2754. Romanov AE, Vojta A, Pompe W, Levefre MJ, Speck JS. Phys State Solid (A) 1999;172:225. Pertsev NA, Zembilgotov AG. J Appl Phys 1995;80:6170. Liao XZ, Huang JY, Zhu YT, Zhou F, Lavernia EJ. Phil Mag 2003;83:3065. Liao XZ, Zhao YH, Srinivasan SG, Zhua YT, Valiev RZ, Gunderov DV. Appl Phys Lett 2004;84:592. Vikarchuk AA, Volenko AP. Phys Solid State 2005;47:352. Yasnikov IS, Vikarchuk AA. Tech Phys Lett 2006;83:42. Yasnikov IS, Vikarchuk AA. Phys Solid State 2006;48:1433. Koga K, Sugawara K. Surf Sci 2003;529:23. Galligan JM. Scripta Met 1972;6:161. De Wit R. J Res Nat Bur Stand 1973;A77:607. Kroupa F, Lejcek L. Solid State Phen 2002;87:1. Mura T. Micromechanics of defects in solids. Boston: Martinus Nijhoff; 1987. Friedel J. In physiques des defauts. North Holland: Amsterdam; 1981. 1. Harris WF. Surf Defect Proper Solids 1974;3:57. Pertsev NA, Romanov AE, Vladimirov VI. Phil Mag A 1984;49:591. Vladimirov VI, Polonsky IA, Romanov AE. Sov Phys Techn Phys 1988;58:882. Nazarov AA, Romanov AE, Valiev RZ. Scripta Mater 1996;34:729. Zubov LM. Nonlinear theory of dislocations and disclinations in elastic bodies. Berlin: Springer; 1997. Truesdell C, Noll W. Handbuch der physik III/3. Berlin: Springer; 1965. Seeger A, Romanov AE; 1995 (unpublished). Lur’e AI. Nonlinear theory of elasticity. Amsterdam: North-Holland; 1990. Doyama M, Cotterill RMJ. Phil Mag A 1984;50:L7. Zhigilei LV, Mikhailin AI, Romanov AE. Phys Met Metall 1988;66:56. Mikhailin AI, Romanov AE. Sov Phys Solid State 1986;28:337. Sarafanov GF, Perevezentsev VN. Phys Solid State 2007;49:1867. Hirth JP, Lothe J. Theory of dislocations. New York: McGraw-Hill; 1982. Kolesnikova AL, Romanov AE. Phys Solid State 2003;45:1626. Eason G, Noble B, Sneddon IN. Phil Trans Roy Soc 1955;247:529–51. Kolesnikova AL, Romanov AE. J Appl Mech 2004;71:409. Polonsky IA, Romanov AE, Gryaznov VG, Kaprelov AM. Phil Mag A 1991;64:281. Panin VE, Likhachev VA, Grinyaev YuV. Structure levels of deformation in solids. Novosibirsk: Nauka; 1985. Bollmann W. Philos Mag A 1988;57:181. Li JCM. Surf Sci 1972;31:12. Gertsman VY, Nazarov AA, Romanov AE, Valiev RZ, Vladimirov VI. Phil Mag A 1989;59:1113. Valiev RZ, Vladimirov VI, Gertsman Vyu, Nazarov AA, Romanov AE. Phys Met Metallogr 1990;69(3):30. Shenderova OA, Brenner DW, Nazarov AA, Romanov AE. Phys Rev B 2000;57:3181. Nazarov AA. Solid State Phen 2002;87:193. Nazarov AA, Romanov AE, Valiev RZ. Acta Mater 1993;41:1033. Mikaelyan KN, Ovid’ko IA, Romanov AE. Mater Sci Eng A 2000;288:61. Gutkin MY, Mikaelyan KN, Romanov AE, Klimanek P. Phys State Solid (A) 2002;193:35. Armstrong RW. Science 1968;162:799. Müllner P, Romanov AE. Acta Materialia 2002;48:2323. Müllner P. Z Metall 2006;97:2005. Seefeldt M. Rev Adv Mater Sci 2001;2:44. Estrin Y, Toth LS, Molinari A, Brechet Y. Acta Mater 1998;46:5509. Nazarov AA, Enikeev NA, Romanov AE, Orlova TS, Alexandrov IV, Beyerlein IJ, et al. Acta Mat 2006;54:985. Van Hautte P, Kanjarla AK, Van Bael A, Zeefeldt M, Delanney L. Europe J Mech 2006;25:634. Perevesentsev VN, Rybin VV, Chuvildeev VN. Acta Met Mat 1992; 40: 887, 895, 907, 915.

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[91] Rybin VV, Zukovskii IM. Sov Phys Solid State 1978;20:1056. [92] Rybin VV, Zisman AA, Zukovskii IM. Probl Prochn 1982;12:10. [93] Wu MS. In: Klimanek P, Romanov AE, Seefeldt M, editors. Local lattice rotations and disclinations in microstructures of distorted crystaline materials. Switzerland: Scitec Publications Ltd; 2002. p. 277. [94] Zhou K, Nazarov AA, Wu MS. Phys Rev Lett 2007;98:035501. [95] Wu MS, Zhou K, Nazarov AA. Phys Rev B 2007;76:134105. [96] Romanov AE. Nanostruct Mat 1995;6:125. [97] Romanov AE. Adv Eng Mat 2003;5:301. [98] Yu Gutkin M, Ovid’ko IA. Plastic deformation in nanocrystalline materials. Berlin: Springer; 2004. [99] Gleiter H. Prog Mater Sci 1989;33:223. [100] Gryaznov VG, Trusov LI. Prog Mater Sci 1993;37:289. [101] Valiev RZ, Islamgaliev RK, Alexandrov IV. Prog Mater Sci 2000;45:103. [102] Nazarov AA, Romanov AE, Valiev RZ. Nanostruct Mat 1995;6:775. [103] Rybin VV, Zisman AA, Zolotorevsky NYu. Acta Metall Mater 1993;41:2211. [104] Enikeev NA, Orlova TS, Alexandrov IV, Romanov AE. Solid State Phen 2005;101–102:319. [105] Orlova TS, Nazarov AA, Enikeev NA, Alexandrov IV, Valiev RZ, Romanov AE. Phys Solid State 2005;47:845. [106] Orlova TS, Romanov AE, Nazarov AA, Enikeev NA, Alexandrov IV, Valiev RZ. Tech Phys Lett 2005;31:1015. [107] Gryaznov VG, Gutkin MY, Romanov AE, Trusov LI. J Mater Sci 1993;28:4359. [108] Meyers MA, Mishra A, Benson DJ. Prog Mater Sci 2006;51:427. [109] Kolesnikova AL, Ovid’ko IA, Romanov AE. Tech Phys Lett 2007;33:641. [110] Romanov AE, Kolesnikova AL, Ovid’ko IA, Aifantis EC. Mater Sci Eng A 2009;503:62. [111] Shan Zh, Stach EA, Wiezorek JMK, Knapp JA, Follstaedt DM, Mao SX. Science 2004;304:654. [112] Sergueeva AV, Mukherjee AK. Rev Adv Mater Sci 2006;13:1. [113] Pertsev NA, Zebilgotov AG. J Appl Phys 1996;80:6401. [114] Foster CS, Pompe W, Daykin AC, Speck JS. J Appl Phys 1996;79:1405. [115] Strifer SK, Parker CB, Romnov AE, Lefevre MJ, Zhao L, Speck JS, et al. J Appl Phys 1998;83:2742. [116] Speck JS, Strifer SK, Pompe W, Ramesh R. J Appl Phys 1994;76:477. [117] Ullrich A, Pompe W, Speck JS, Romanov AE. Solid State Phen 2002;87:245. [118] Farag N, Bobeth M, Pompe W, Romanov AE, Speck JS. J Appl Phys 2005;97:113516. [119] Farag N, Bobeth M, Pompe W, Romanov AE. Phil Mag 2007;87:823. [120] De Wit R. J Phys C 1972;5:529. [121] Tang Z, Kotov NA. Adv Mater 2005;17:951. [122] Howie A, Marks LD. Phil Mag A 1984;49:95. [123] Gryaznov VG, Kaprelov AM, Romanov AE, Polonsky IA. Phys State Solid (b) 1991;167:441. [124] Kolesnikova AL, Romanov AE. Tech Phys Lett 2007;33:886. [125] Kolesnikova AL, Romanov AE. Phys State Solid (RRL) 2007;1:271. [126] Dorogin LM, Kolesnikova AL, Romanov AE. Tech Phys Lett 2008;34:779. [127] Langlois C, Alloyeau D, Le Bouar Y, Loiseau A, Oikawa T, Mottet C, et al. Farad Disc 2008;138:375.