Effect of loading direction and initial imperfections on the development of dynamic shear bands in a FCC single crystal

Summary We study plane strain dynamic thermomechanical deformations of a FCC single crystal deformed at an average strain-rate of 1 000 s^–1 along the crystallographic direction [380] with the plane of deformation parallel to the plane (001) of the single crystal. Four different situations are studied; in the first two there is no initial imperfection assumed in the crystal and it is either compressed or pulled, and in the other two the crystal is compressed but either the initial temperature is nonuniform or a small region around the centroid of the cross-section is misoriented relative to the rest of the cross-section. In each case, all twelve slip systems are assumed to be potentially active, and the crystal material is presumed to exhibit strain hardening, strain-rate hardening, and thermal softening. These effects are modelled by using a simple combined isotropic-kinematic hardening expression for the critical resolved shear stress, proposed by Weng, and modified to incorporate the effect of thermal softening of the material. It is found that each one of the slip systems , and contributes essentially equally to the plastic deformations of the crystal and these slip systems become active soon after the load is applied. The same holds for the slip systems , and except that they are active in a region different from that of the previous one. The remaining four slip systems either stay inactive throughout the deformation process, or become active at late stages of the deformation.     

Load relaxation studies of germanium

A series of compressive load relaxation experiments were conducted on germanium single crystals in the temperature range 400 to 885° C. The curvature of the log-log data obtained from load relaxation tests changes from concave upward to concave downward as the test temperature increases at fixed stress level, or as the strain level increases at fixed temperature. At intermediate temperatures, 600° C, the transition from concave upward to concave downward curvature happens on a single relaxation curve. These observations are consistent with the two-branch rheological model proposed by Hart to explain the deformation behaviour of metals and were analysed in terms of this model. The transition from concave upward to concave downward curvature could be moved to higher temperature by doping germanium with gallium, which decreases the dislocation glide velocity relative to that in pure germanium. The transition could be shifted to lower temperature by compressing samples along [1 1] rather than [1 0] because the [1 1] orientation favours cross-slip while the [1 0] orientation does not. Dislocation dipoles and straight dislocations dominated the microstructure of samples which had concave upward log-log curves, while well-developed dislocation cell structures dominated the microstructure of samples which yielded concave downward curves. The observed changes in the curvature of the load relaxation curves and the dislocation structure both indicate the increased importance of dislocation climb with increasing temperature. When compared through the Orowan equation, the load relaxation results are in good agreement with published stress-dislocation velocity data.     

Thermomechanic behavior of a polymer upon its formation in a crystallization process: Theoretical principles,hypotheses, and mathematical model

A model of thermomechanic behavior of a polymer upon its formation in a crystallization process is proposed. Based on methods of nonequilibrium thermodynamics governing relationships are obtained which make it possible to establish the dependence of the final degree of crystallicity of the material on the history of the crystallization process and to explain the mechanism of formation of the remanent stresses in a polymer article.Notation u translation vector - v velocity vector - acceleration vector - absolute temperature - density - c specific heat capacity - deformation tensor - strain tensor - specific enthropy - U ^* internal energy - z specific free enthalpy - _i internal parameters of state - t time - q heat flux vector - matrix of heat conduction coefficients - W ^* energy dissipation - F vector of mass forces - the 4th rank tensor of elastic pliabilities - matrix of heat expansion coefficients - tensor of contribution of structural variations to deformation - function of equilibrium value ^* - p mean pressure - deviator of the tensor of deformations - spherical part of the deformation tensor - deviator of the tensor of stresses - K volume modulus - unity tensor - Q enthalpy of the crystallization process - Q _eq enthalpy of the equilibrium crystallization process - _g glass transition temperature - ^*() the curve obtained in the equilibrium crystallization process - f final degree of crystallicity Institute of Mechanics of Continuous Media of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Institute of Technical Chemistry of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 479–485, May–June, 1995.     

Study on the α2/γ interfacial structures in a two-phase (α2?+?γ) alloy deformed at elevated temperature

The details of structure modification on ₂/ interface induced by deformation in a hot-deformed Ti-45Al-10Nb alloy were investigated by conventional and high-resolution transmission electron microscopy. A new type of dislocation ledge containing 1/3[111] partials was identified. The Burgers vectors of these dislocation ledges were determined to be 1/2[110] or 1/2101]. The formation mechanism of this new type of dislocation ledge is discussed. Also, two types of hot deformation induced ₂/ interfaces, coherent interfaces with high density of ledges and misoriented semi-coherent ₂/ interfacial boundaries were observed. For the misoriented semi-coherent ₂ interfaces, the density of dislocation ledges in these interfaces increases with the misoriented angle between the (111) and (0001) planes, and 1/3[111] partial dislocations were involved in all the dislocation ledges. The formation mechanism of these deformation-induced ₂/ interfaces were discussed related to the role of ₂/ interfaces adjusting the deformation as a dislocation sink absorbing the slipping dislocations in the phase. Moreover, misoriented semi-coherent ₂ interface related deformation twinning and structure transformation induced by deformation were analyzed and discussed related to the role of ₂/ interfaces as a dislocation source during deformation.     

Microstructural development of Ti-24Al-14Nb-3V-0.5Mo alloy during hot tensile deformation

Microstructure of the Ti-24Al-14Nb-3V-0.5Mo (at.%) alloy deformed from room temperature to 700°C was studied by the H-800 transmission electron microscope (TEM) equipped with a side entry goniometer stage capable of ±45° tilt on the X or Y axis operating under two-beam condition. Especially dislocation types and slip systems of the ₂-phase (D0₁₉) were analyzed in accordance with invisibility criterion. The results indicated that with increasing the deformation temperature from room temperature to 700°C, the slip of -type dislocations on prismatic planes in the ₂-phase was enhanced whereas that on basal plane (0001) with hexagonal networks morphology was suppressed during tensile deformation. Being immobile, the dislocation networks were thought to be mainly responsible for the low ductility at low temperature. When the test temperature was increased to 700°C, the -type dislocation glide on pyramidal planes was also observed, but the hexagonal dislocation networks formed by the slip of -type dislocations on basal planes were hardly seen. During the high temperature deformation microstructural refinement was also observed due to dynamic recovery and recrystallization. This, together with the precipitation of secondary ₂ phase and O phase during deformation all contributed to ductility of the present alloy at high temperature.     

Computer experiment on deformation of nanocrystals of the chromium-niobium system. Part 1. Atomic-structural rearrangements

Molecular-dynamic simulation of the process of stretching of chromium and niobium nanocrystals and chromium-niobium bicrystals was performed to a pair-potential approximation. Atomic mechanism of deformation and fracture is illustrated, which depends on the crystal orientation with respect to the acting force . In the case of the initial orientation (100) , the reorientation mechanism (100) (110) is in action. In a bicrystal, it is niobium that deforms first and then the relay-race transfer of deformation from Nb to Cr takes place. With the initial configuration (110) , the fragmentation of the crystal and grain-boundary rearrangements are observed. Niobium crystallite in a bicrystal deforms only within the limits of elastic deformation whereas the deformation of chromium is of brittle-ductile mode. With the initial configuration (111) , the mechanism of pure brittle fracture is observed. In all cases, fracture of a bicrystal occurs along the interphase boundary. The causes of such atomic-structural transformations are explained.Translated from Problemy Prochnosti, No. 1, pp. 5–23, January–February, 2005.     

Heat flux measurement from the temperature field of a heated surface. 2. Inhomogeneous flux

Based on the previously obtained analytical solutions of the three-dimensional space-time problem of recalculating boundary conditions, algorithms are developed and a computational experiment is carried out to reconstruct the heat flux with arbitrary spatial distribution of the temperature field of the heated surface.Notation T(t, ) temperature over the plate surface - q(t ) heat flux to the plate surface - ={x,y} transverse coordinate - t time - k thermal conductivity - a ² thermal diffusivity - L plate thickness - Jacobi theta-function [4] - Fourier parameter - Jacobi theta-function [4] - transversal Laplacian Institute of Atmospheric Optics, Siberian Division, Russian Academy of Sciences, Tomsk, Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 622–628, July–August, 1995.     

Rheological behaviour and modelling of semi-solid Sn-15% Pb alloy

The rheological behaviour and modelling of a semi-solid, Sn-15% Pb alloy characterized by a special coaxial-cylinder rheometer over a wide range of process conditions is reported. In particular, the effect of shear rate , volume fraction of solid (f _s), and cooling rate on the apparent viscosity () of the semi-solid Sn-15% Pb alloy under isothermal and various cooling conditions was studied. Based on the experimental data, the shear rate used in preparing the semi-solid alloy as well as the volume fraction of solid have the most dominant effects on the rheological properties of the semi-solid Sn-15% Pb alloy. A viscosity model expressed as is proposed in which f _s ^* is the critical solid fraction at which the apparent viscosity goes to infinity, (F _s) corresponds to the asymptotic viscosity at infinity shear rate, and characterizes the transition shear rate between the power-law and Newtonian regions. finally, measurements with a differential scanning calorimeter were made and used to correlate the temperature and volume fraction of solid which, in turn, was corroborated with available data from the literature.     

Structural features of Σ = 19, [110] GaAs tilt grain boundaries

= 19, [110] tilt grain boundaries have been observed to facet parallel to particular planes; the facets lie along _A/ _B, ( )_A/ )_B and ( )_A/( )_B. The structural unit of the = 19 ( )_A/( )_B [110] boundaries consists of 5- and 7-member rings, which are similar to the core structure of a/2[110] edge dislocations. The polarities in each grain on either side of the boundaries has been confirmed by CBED methods; a lower number of anti-site type cross-boundary bonds occur along the boundaries compared to when the polarity of one grain is reversed. The presence of 7-member rings and anti-site cross-boundary bonds results in a more open atomic structure at the boundary, shortening the distance between the first and the second {331} atomic planes from the boundary plane by 40%.     

A micromechanical method for predicting the precipitation hardening response of particle strengthened alloys hardened by ordered precipitates

Summary A micromechanical method was developed for predicting the precipitation hardening response of particle strengthened alloys hardened by ordered precipitates based on the microstructure, composition, and heat treatment, and utilizing a minimum number of experimental tests to evaluate the microstructural constants of the overall model. The overall approach was based on incorporating the dislocation particle interaction mechanics, particle growth and coarsening theory, thermodynamics, and particle strengthening mechanisms applicable to precipitation hardened alloys as part of the overall micromechanical method. The method/model evaluates, from a minimum number of experimental tensile tests, microstructural constants necessary in determining the precipitation srengthening response of a particle strengthened alloy. The materials that were used as vehicles to demonstrate and evaluate the model were precipitation hardenable aluminium-lithium-zirconium and nickel-aluminum alloys. Utilizing these demonstration alloys, the method used a total of four tensile tests to evaluate the necessary microstructural constants and thus predict the variation in strength as a function of aging time, aging temperature, and composition, for the underaged, the peak-aged, and the overaged conditions. Predictions of the precipitation strengthening response were made incorporating the Wagner particle distribution model to evaluate the size distributions of particles in the microstructures. The predicted variation of strength with aging practice and composition using the Wagner distribution model compared well with the corresponding experimental yield strength results.Notation b Burgers vector - average particle size diameter for a particle distribution - d _loop particle looping diameter for dislocation bypassing by Orowan looping - f _v volume fraction of precipitates - h() Wagner particle size distribution function - n _total total number of precipitate particles per unit area on a given microstructural plane - average particle size radius for a particle distribution - average planar particle size radius on a given microstructural plane - t aging time, in hours - average planar particle cross sectional area - G _t total shear modulus of the material - K _c particle growth rate constant - texture or Taylor grain orientation factor - N _v total number of precipitate particles per unit volume - Q _A activation energy for diffusion - R universal gas constant - T aging temperature - the interparticle separation or spacing - _y yield strength - _q as-quenched strength - _i intrinsic lattice strength - _c critical resolved shear strength - _loop critical resolved shear strength for dislocation particle bypassing via. Orowan looping - _particle total critical resolved shear strength for particle strengthening - _shear critical resolved shear strength for dislocation particle shearing, in underaged state     

On the motion of a curved and twisted rod

Summary In this paper, the equations of motion of a curved and twisted rod are derived from the basic principles of dynamics. The set of equations gives the extensional, flexural and torsional motions of the rod. The coupling among these types of motion due to the curvature and tortuosity of the rod is shown explicitly in the case of a helical spring. By successive simplification of the equations, the equations of motion of a circular ring and those of a straight rod are obtained. In this respect, the derived equations can be considered as a generalization of the elementary theories of rod in extensional, torsional and flexural motion.The dispersion relation of a helical spring is calculated for the two lower frequency modes. It is shown that the frequency-wave length relationship is not monotonically decreasing as in the cases of uncoupled flexural or torsional motion. Finally, frequencies are calculated based on the approximate frequency expression ofLove to show that Love's frequency expression for a helical rod is accurate.

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Zur Bewegung eines gekrümmten und verdrillten Stabes

Zusammenfassung In dieser Arbeit werden die Bewegungsgleichungen eines gekrümmten, verdrillten Stabes von den Grundgleichungen der Dynamik hergeleitet. Dieses System von Gleichungen legt die Bewegung des Stabes durch Längsdehnung, Biegung und Torsion fest. Die Kopplung dieser Bewegunsarten, durch Krümmung und Verdrillung, wird für den Fall einer Spiralfeder explizit gezeigt. Durch Vereinfachung der Gleichungen werden die Bewegungsgleichungen des Kreisringes und die des geraden Stabes erhalten. In dieser Hinsicht können die hergeleiteten Gleichungen als Verallgemeinerung der elementaren Theorein der Bewegung eines Stabes betrachtet werden.Die Dispersionsgleichung der Spiralfeder wird für die beiden niedrigsten Frequenzen berechnet. Es wird gezeigt, daß das Verhältnis Frequenz-Wellenlänge nicht wie in den Fällen der ungekoppelten Biege-oder Torsionsbewegung monoton abnimmt. Abschließend werden die Frequenzen nach dem Loveschen Näherungsausdruck berechnet, um zu zeigen, daß die Genauigkeit dieses Ausdruckes für Spiralfedern gut ist.

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Nomenclature position vector of a point on axis of curved rod - position vector relating any pointB in a plane perpendicular to the axis of the rod to pointP where plane cuts axis of rod - R modulus of - linear momentum vector - density - axial force vector = - axial force vector at origin - transverse shear force vector - transverse shear force vector at origin - moment vector - moment vector at origin - s distance measured along axis of rod - t time - a radius of gyration of rod for circular sections - l total length of spring along axis - A cross-sectional area of rod - s/a — normalized coordinates measured along axis of rod - applied load vector - angular momentum vector - trihedral of space curve-unit vectors in tangential, normal and binormal directions - _t axial rotation - displacement vector - normalized displacement vector= - ₀,₀ curvature and tortuosity of rod - , normalized curvature and tortuosity of rod - , circular frequency and normalized frequency respectively - E, G Young's modulus and shear modulus - v Poisson's ratio - Wave number - pitch angle of helical spring - d wire diameter of helical spring - R coil radius of helical spring - N number of turns of helical spring With 5 Figures     

A comparison of strengthening mechanisms in rolled and axisymmetrically deformed Ti-20Y composites

Two Ti-20%Y metal-metal composites were deformation processed: one axisymmetrically and the other by rolling. The microstructures, preferred crystallographic orientations, and tensile strengths of each were measured periodically as the deformation progressed. The axisymmetrically deformed Ti matrix developed a [10 0] fiber texture, and the rolled composite acquired a texture with the <0001> tilted 31° from the sheet normal toward the transverse direction with [10 0] parallel to the rolling direction. The orientations of the {10 0}<11 0> and (0002)<11 0> slip systems in Ti with these textures were used in conjunction with the maximum possible slip distances for dislocation travel in each specimen to demonstrate that the axisymmetrically deformed material should be stronger than the rolled material for composites of equal phase thickness and spacing. The strengths of the two composites measured in this study were compared at similar microstructural phase sizes and spacings, and the axisymmetrically deformed composite was indeed found to be somewhat stronger, although the difference in strengths was not large.     

The effect of particles on the critical strain associated with the Portevin-LeChatelier effect in aluminium alloys

The effect of particles on the critical strain, _c, associated with the Portevin-LeChatelier (PL) effect of aluminium alloys is studied using Al-Mg-Ni and Al-Si alloys. Al-Mg-Ni and Al-Si alloy matrixes are composed of Al₃Ni and Si particles, respectively. Tensile tests were performed in the temperature range 223–273 K in which the critical strain decreases with increasing temperature, and strain rates between 10^–5 and 10^–2 s^–1 were chosen. According to the apparent activation energies, Q, Mg and Si solute atoms are responsible for the flow instability in Al-Mg-Ni and Al-Si alloys, respectively. The experimental results also show that the critical strain decreases with decreasing particle spacing, d _p. Since the particle spacing is small compared to the corresponding grain size, the decrease in critical strain should be ascribed to the effect of particles. Considering that the dislocation density is increased by the particles, a modified model showing the critical strain, _c, as a function of particle spacing, d _p, is proposed as T ^–1 exp (–Q/kT), in which , T and k are the strain rate, temperature and Boltzmann constant, respectively. Linear fit of the plots of In _c versus In d _p and In _c versus In d _p indicates that this equation is appropriate to rationalize the particle effect on the critical strain.     

The constant stress tensile creep behaviour of a superplastic zirconia-alumina composite

A detailed study was conducted to evaluate the constant stress tensile creep behaviour of a superplastic 3 mol% yttria-stabilized zirconia-20 wt% alumina composite. The comprehensive experimental results indicate that creep deformation may be expressed in the form exp(–585 000/8.3T), where is the steady-state creep rate, is the imposed stress, is the linear intercept grain size andT is the absolute temperature. Microstructural observations revealed that there is very little dislocation activity, or change in grain size or shape. A detailed analysis was conducted to evaluate the possible rate-controlling mechanisms in terms of the experimentally determined mechanical properties and the microstructural observations. Based on the maintenance of an equiaxed microstructure and the strong grain size and stress dependence, it is concluded that creep occurs by a grain-boundary sliding/grain rearrangement process.     

Study on the interaction between a dislocation and impurities in KCl:Sr<Superscript>2+</Superscript> single crystals by the Blaha effect—Part IV influence of heat treatment on dislocation density

Strain-rate cycling tests associated with ultrasonic oscillation were carried out at 80–239 K for two kinds of KCl:Sr²⁺ (0.05 mol.% in the melt) single crystals: one is a quenched specimen and the other an annealed one. In this study, it was found that the density of moving dislocation is not influenced by the heat treatment. Furthermore, the increase in forest dislocation density for the annealed specimen seemed to be remarkable under the compression test, compared with that for the quenched specimen. As a result, the strain-hardening rate increased and the extent of plastic deformation region became short at a given temperature by annealing the quenched specimens. The investigation concerning forest dislocation density was conducted on the basis of the which will represent the variation of the strain-rate sensitivity due to dislocation cuttings with shear strain.     

In situ nuclear magnetic resonance study of defect dynamics during deformation of materials

Nuclear magnetic resonance techniques can be used to monitor in situ the dynamical behaviour of point and line defects in materials during deformation. These techniques are non-destructive and non-invasive. We report here the atomic transport, in particular the enhanced diffusion during deformation by evaluating the spin lattice relaxation time in the rotating frame, T ₁,in pure NaCl single crystals as a function of temperature (from ambient to about 900K) and strain-rate (to 1.0s^–1) in situ during deformation. The strain-induced excess vacancy concentration increased with the strain-rate while in situ annealing of these excess defects is noted at high temperatures. Contributions due to phonons or paramagnetic impurities dominated at lower temperatures in the undeformed material. During deformation, however, the dislocation contribution became predominant at these low temperatures. The dislocation jump distances were noted to decrease with increase in temperature leading to a reduced contribution to the overall spin relaxation as temperature is increased. Similar tests with an improved pulse sequence (CUT-sequence), performed on ultra-pure NaCl and NaF single crystals revealed slightly different results; however, strain-enhanced vacancy concentrations were observed. The applicability of these techniques to metallic systems will be outlined taking thin aluminium foils as an example.     

The model ofd-pairing in CuO2 planes of HTSC and the collective modes

We develop here for the first time a 2D path integral model of d-pairing in HTSC, which is analogous to the 3D model created earlier by Brusov and Brusova. Within this model we calculate the entire collective mode spectrum for all superconductive states of CuO₂ planes arisen from their symmetry classification. There are four collective modes in each phase. We found two high frequency modes in each phase while two remains modes seem to be Goldstone or quasi-Goldstone ones (having vanishing energy at zero momentum). It turns out that the collective mode spectrum in the phases and is the same.     

Non-linear response of internal friction to tensile strain rate and frequency during plastic deformation of high-purity aluminium

The internal friction of high-purity aluminium during the process of plastic deformation was measured by a middle torsion pendulum on a modified tensile testing machine. The effects of tensile strain rate, , in the range of 0.73×10^–6 to 50×10^–6s^–1, and frequency of internal friction measurement, f, in the range of 0.38 to 2.6 Hz were studied. The results showed a non-linear dependence of internal friction, Q ^–1, on and f ^–1 or on (=2 f). The interrelationship between internal friction during the process of plastic deformation and dislocation motion, and the effect of non-linearity on the dynamic behaviour of dislocations are discussed.     

Interfacial Tension and Interface Delocalization Phase Boundary for Strongly Type-I Superconductors

We analytically determine the interface delocalization (or wetting) transition phase boundary in the limit of strongly type-I superconductors. In particular, within Ginzburg–Landau theory we derive an analytic expression for the reduced surface tension, _SC/N, of a type-I superconductor. We find that the truncated expansion (where is the Ginzburg–Landau parameter) is so accurate in the entire type-I regime that derivation of higher-order terms is unnecessary. We further derive an expression for the wall/superconductor interfacial tension which again proves accurate across a broad range of values. These expansions allow us to locate the low- interface delocalization phase boundary accurately, complementing previous numerical results for the wetting phase diagram.     

Behavior of a screw dislocation near a partially bonded bimetallic interface

This paper, based upon the isotropic elastic continuum approximation, deals with the forces of interaction between a stationary screw dislocation and a partially bonded bimetallic interface. The complex variable method is used throughout and the closed form solution is obtained. The result indicates that if the material on the other side of the interface is more rigid than that in which the dislocation is present, the bonded region will act as a barrier to the dislocation located in a certain region near the interface and that there is no stable equilibrium position of single dislocation line even in the aforementioned case since the free surface operates an attractive force on it.

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Zusammenfassung Diese Arbeit behandelt die Kraefte der Wechselwirkung zwischen einer stationaeren Schraubenverschiebung und einer teilweisen unfreien zwei-metallischen Zwischenflaeche. Sie beruht auf dem isotropischen, elastischen Kontinuumnaeherungswert. Die Methode der komplexen Veraenderlichen wurde benutzt und die Loesung der geschlossenen Form wurde erhalten.Das Ergebnis zeigt, dass, sollte das Material auf der anderen Seite der Zwischenflaeche haerter sein, als jenes in welchem die Verschiebung stattfindet, das unfreie Gebiet als Barriere zu der Verschiebung, die sich in einem bestimmten Gebiet in der Nahe der Zwischenflaeche befindet, auftritt. Ferner gibt es keine stabile Gleichgewichtsposition von einer einzelnen Verschiebungslinie, selbst in dem erwaehnten Falle nicht, da die frei Oberflaeche Zugkraft ausuebt.

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Résumé Se basant sur l'approximation d'un milieu continu élastique et isotrope, le mémoire est relatif aux forces d'interaction entre une dislocation-vis et l'interface de deux corps métalliques partiellement solidarisés. On utilise la méthode des variables complexes, et l'on obtient une solution particulière et limitée.Les résultats indiquent que si le mátériau qui se trouve de l'autre côte de l'interface est plus rigide que celui dans lequel se trouve la dislocation, la zone de liaison fera office de barrière pour toute dislocation située à une certaine distance de cet interface. Par ailleurs, il n'y a pas de position d'équilibre stable pour une ligne simple de dislocations se trouvent dans le cas susmentionné, puisque la surface libre opère une force d'attraction sur cette ligne.

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Notation a, h location of screw dislocation - b magnitude of Burgers vector of screw dislocation - force acting on the dislocation - F_y y component of - i –1 - i, i unit vectors in x and y directions - half length of bonded segment - L, L bonded segment and traction free one on real axis - S⁺, S^– upper and lower half-planes in z plane - u, v, w carte sian displacement components - x, y, t cartesian coordinates - X(z) (z+)^–1/2(z–)^–1/2 - z x+iy - h+ia - eeⁱ z– - tan^–1 [a/(+h)] - tan^–1 [a/(–h)] - µ^* shear modulus - - _* - _xt, _yt shear stress components - _xt ⁽ⁱ⁾ _yt(¹) shear stress components at the location of dislocation due to the interaction with the interface - (z) complex displacement potential - (z) potential as defined in text On leave from Department of Mechanical Engineering, Tohoku University, Sendai, Japan.