Equilibrium Position Of Dislocation Research Articles

Interaction between a generalized screw dislocation in the matrix and an inhomogeneity containing an elliptic hole in piezoelectric–piezomagnetic composite materials

The paper deals with the interaction of a generalized screw dislocation and an elliptic inhomogeneity containing a confocal elliptic hole in a magneto-electro-elastic composite material. Exact solutions are derived for the case where the generalized screw dislocation is located in the matrix under a remote anti-plane shear stress field, an in-plane electric field, and a magnetic field. Based on the complex variable method, the complex potentials of both the matrix and the inhomogeneity are obtained in series, and analytic expressions for the generalized stress and strain field, the image force, the generalized stress intensity factor of the blunt crack tip, and the energy release rate are derived explicitly. The presented solutions include some previous solutions, such as pure elastic, piezoelectric, piezomagnetic, and circular inclusions. Typical numerical examples are presented and the influences of the dislocation position, the volume of inhomogeneity, and the elliptic hole on these physical quantities are discussed. The results show that the magneto-electro-elastic coupling effect has a great influence on the image force and the equilibrium position of dislocation, especially when the dislocation approaches the interface; the coupling effect makes the image force on the screw dislocation follow different variation laws in piezoelectric–piezomagnetic composite materials compared with elastic materials.

A pileup of edge dislocations against an inclined bimetallic interface

A pileup of edge dislocations against an arbitrarily inclined flat bimetallic interface is considered. Equilibrium positions of dislocations are determined for a given number of dislocations and specified material properties, assuming that the resolved shear stress along the pileup plane from a remotely applied loading is uniform and equal for all interface inclination angles. Numerical results are compared for pileups at 0°, 30°, 45°, and 60° relative to the interface normal. The overall dislocation distribution is mildly affected by the inclination of the interface, although there are some notable differences. While an inclined interface repels the first and last dislocation stronger than the orthogonal interface, for piled-up dislocations in-between this is not necessarily the case. Small differences in the pileup length and the proximity of the leading dislocation to differently inclined interfaces can considerably affect the interface stresses. The magnitude of interface stresses decreases with the increase of the shear moduli ratio G2/G1 due to stronger repulsion exerted on dislocations by stiffer interfaces. The disparity in Poisson’s ratio also affects the interface stresses. The back stress behind a trailing dislocation is evaluated and discussed.

Equilibrium position of screw dislocation near circular inhomogeneity at the tip of an elastic wedge

The image force on a screw dislocation at the distance $$\rho =\alpha a\,(\alpha >1)$$ from the tip of an isotropic wedge of shear modulus G with a circular inhomogeneity of radius a and shear modulus $$G'$$ at its end is determined by the method of image dislocations. If the wedge angle is $$\pi /n$$ , for an integer $$n\ge 1$$ , $$4n-1$$ image dislocations are needed to achieve the required traction-free boundary conditions and the interface continuity conditions. It is shown that for the positive values of the inhomogeneity parameter $$\kappa =(G'-G)/(G'+G)$$ , there is an equilibrium position of the dislocation along the plane of symmetry of the wedge, specified by the parameter $$\alpha _\mathrm{eq}=\left[ 2n\kappa +\left( 1+4n^2\kappa ^2\right) ^{1/2}\right] ^{1/2n}$$ . The equilibrium position is stable relative to disturbances of the dislocation position along the plane of symmetry, but unstable for other disturbances. If the edges of the wedge and the inhomogeneity are fixed, rather than traction free, the equilibrium position of the dislocation along the plane of symmetry is specified by $$\alpha _\mathrm{eq}=\left[ 1-4n\kappa /(2n-1)\right] ^{1/4n}$$ , provided that the inhomogeneity is softer than the matrix material ( $$\kappa <0$$ ). The equilibrium position is unstable relative to disturbances of the dislocation position along the plane of symmetry. In the case of homogeneous wedges ( $$\kappa =0$$ ), the image force in the plane of symmetry for a wedge with fixed edges depends on the wedge angle and is given by $$f=(2n-1)k/2\rho$$ (away from the tip), while for a wedge with free edges the image force is independent of the wedge angle ( $$f=-k/2\rho$$ ).

Equilibrium positions of misfit dislocations in a nanolayer embedded in a matrix

The formation of two misfit edge dislocations in the interfaces of a stressed thin layer embedded in a matrix has been theoretically investigated from an energy variation calculation as a function of the lattice mismatch between the layer and matrix. The equilibrium positions of the dislocations have been determined and the two configurations where the arm orientations of the dipole are normal or inclined to the interfaces have been discussed versus the nanostructure dimensions and lattice mismatch.

Screw Dislocations in a Three-Phase Composite Cylinder Model With Interface Stress

A three-phase composite cylinder model is utilized to study the interaction between screw dislocations and nanoscale inclusions. The stress boundary condition at the interface between nanoscale inclusion and the matrix is modified by incorporating surface/interface stress. The explicit solution to this problem is derived by means of the complex variable method. The explicit expressions of image forces exerted on screw dislocations are obtained. The mobility and the equilibrium positions of the dislocation near one of the inclusions are discussed. The results show that, compared to the classical solution (without interface stress), more equilibrium positions of the screw dislocation may be available when the dislocation is close to the nanoscale inclusion due to consider interface stress. Also, the mobility of the dislocation in the matrix will become more complex than the classical case.

A moving screw dislocation near interfacial rigid lines in two dissimilar anisotropic media

This paper attempts to investigate the problem for the interaction between a uniformly moving screw dislocation and interface rigid lines in two dissimilar anisotropic materials. Integrating Riemann-Schwarz’s symmetry principle with the analysis singularity of complex functions, we present the general elastic solutions of this problem and the closed form solutions for interfaces containing one and two rigid lines. The expressions of stress intensity factors at the rigid line tips and image force acting on moving dislocation are derived explicitly. The results show that dislocation velocity has an antishielding effect on the rigid line tip and a larger dislocation velocity leads to the equilibrium position of dislocation closing with the rigid line. The presented solutions contain previously known results as the special cases.

A piezoelectric screw dislocation interacting with an elliptical inclusion containing electrically conductive interfacial rigid lines

The problem of the electroelastic interaction between a piezoelectric screw dislocation and an elliptical inclusion with electrically conductive interfacial rigid lines under antiplane shear and inplane electrical loads is dealt with. The screw dislocation (antiplane singularity) is located inside either the elliptical inclusion or the infinite matrix. An efficient complex variable method for multiply connected regions is developed, in terms of which explicit series solutions for the electroelastic fields in the matrix and the inclusion regions are derived. The image force acting on the piezoelectric screw dislocation is then calculated by using the generalized Peach–Koehler formula. The influence of the aspect ratio of the ellipse, rigid line geometry and material electroelastic dissimilarity on the image force is discussed. It is shown that the effect of the interfacial rigid line on the equilibrium position of the dislocation near an elliptical inclusion is significant. A stable equilibrium position of the screw dislocation in the matrix can exist when the dislocation near the soft elliptical inclusion and the interfacial rigid line. An important founding is that there exists a same critical value of the interfacial rigid line angle to change the direction of the image force acting on the dislocation regardless of the shape of the inclusion as the area of the elliptical inclusion taking a constant.

DISLOCATIONS IN UNIAXIAL LAMELLAR PHASES OF LIQUID CRYSTALS, POLYMERS AND AMPHIPHILIC SYSTEMS

Dislocations in soft condensed matter systems such as lamellar systems of polymers, liquid crystals and ternary mixtures of oil, water and surfactant (amphiphilic systems) are described in the framework of continuum elastic theory. These systems are the subject of studies of physics, chemistry and biology. They also find applications in industry. Here we will discuss in detail the influence of dislocations on the bulk and surface properties of these lamellar phases. Especially the latter properties have only been recently studied in detail. We will present the experimental evidence of the existence of screw and edge dislocations in the systems and study their static properties such as: energy, line tension and core structure. Next we will show how the surface influences the equilibrium position of dislocations in the system. We will give the theoretical predictions and present the experimental results on thin copolymer films, free standing films of liquid crystals and smectic droplets shapes. In semi-infinite lamellar systems characterized by small surface tension the dislocation is stabilized at a finite distance, heq, from the surface, due to the surface bending elastic constant, Ks (for zero surface tension heq≈Ks/2K, where K is the bulk bending elastic constant). For large surface tension the edge dislocations are strongly repelled by the surface and the equilibrium location for finite symmetric systems such as free standing liquid crystal films shifts towards the center of the system. The surface is deformed by dislocations. These deformations are known as edge profiles. They will be discussed for finite systems with small and large surface tension. Surface deformations induce elastic interactions between edge dislocations, which decay exponentially with distance with decay length proportional to [Formula: see text] where D is the size of the system normal to lamellas. Two screw dislocations in finite system interact with the logarithmic potential, which is proportional to the surface tension and inversly proportional to D. The surface induced elastic interactions will be compared to, well-known, bulk deformation induced interactions. A new phenonenon discussed in our paper is the fluctuations induced interactions between edge dislocations, which follows from the Helfrich mechanism for flexible objects. At suitable conditions, edge dislocations can undergo an unbinding transition. Also a single dislocation loop can undergo an unbinding transition. We will calculate the properties of the loop inside finite system and discuss in particular the unbinding transition in freely suspended smectic films. We shall also compute the equilibrium size of the loop contained between two hard walls. Finally we will discuss the dynamical bulk properties of dislocations such as: mobility (climb and glide), permeation, and helical instability of screw dislocations. Lubrication theory will also be discussed.

The low-angle [011] tilt boundary in germanium II. Theoretical analysis of observed configurations and stability

Abstract A theoretical analysis of grain-boundary stability and of the equilibrium positions of dislocations in all types of observed structures has been carried out using the method of Key and Saada (1976). It is concluded that for medium angles of tilt (θ = 2° to 5°), the symmetric position, i.e. the {022} grain-boundary plane (G.B.P.), is unstable. Furthermore the {211} G.B.P. is always the position of minimum energy. A superposition method has been developed to measure precisely the splitting separation of partial dislocations. The intrinsic-fault energy deduced from high-resolution observation, is equal to 100 erg/cm2. The extrinsic-fault energy has been estimated to be of the same order of magnitude as this or lower. Although the core structure of dislocations is not yet completely determined, all our results are consistent with qualitative arguments on core energy based on open bonds; it explains why the sessile edge dislocation is not split.

The Behaviour of an Edge Dislocation in Non‐Homogeneous Anisotropic Body with Cracks

AbstractThe forces of interaction between a stationary edge dislocation and a partially bonded bimetallic interface is discussed in the framework of the elastic continuum approximation. Here, each of two semi‐infinite media, bonded together, is assumed to be anisotropic. The method of solution is based on the reduction to a type of dual nonhomogeneous Hilbert problem for two functions. Some numerical calculations are carried out for the simple case where the direction of the Burgers vector of dislocation is perpendicular to the interface. It is assumed that both of two materials are orthotropic and have the same elastic coefficients. The effect of mismatch of the axes of elastic symmetry on the glide force and the equilibrium position of dislocation is discussed in detail.

Equilibrium of Linear Dislocation Arrays in Heterogeneous Materials

The analysis of linear dislocation arrays, piled up in the soft phase of a two-phase material, is extended. The equilibrium positions of dislocations are determined by numerical analysis. A result of the computation is the confirmation of the linear approximation L/n = (A/σ) (α+βK), 0&amp;lt;K≤1. Here, L is the length of the array, n is the number of dislocations, σ is the applied stress, α and β are constants for a fixed n, and A is the usual material constant for the dislocation field. The heterogeneity parameter K is defined as a function of the elastic constants of the neighboring phases, ranging from 0 to 1 for the present study. It is shown that for large arrays of either screw dislocations against a welded boundary or edge dislocations against a slipping boundary, α and β are practically constants for all large n's and α =̇ 2 and β =̇ 1.1. For edge arrays against a welded boundary, only a simple case is examined, i.e., when Poisson's ratio for the two neighboring phases is identical. The linear relation is good in certain ranges. By using the above relation, an inverse square root expression for the applied stress and the length of the array (the Hall-Petch type) is derived for heterogeneous materials.

On the method of eshelby, frank and nabarro for calculating the equilibrium positions of dislocations

Abstract The method of Eshelby et al. transforms the non-linear equations of equilibrium of a group of dislocations, free to move in one dimension in a stress field, into another set of non-linear equations, greater in number but of a simpler form. The examples considered by Eshelby et al. are special cases for which the transformed equations are linear. The complexity of the transformed equations is shown to depend on the form of the stress field. The method is extended to dislocations which may be of opposite signs and which are free to move in two dimensions.

Single and Double Pile-Up of Dislocations in MgO Crystals

A quantitative analysis has been made of elastic interactions in three types of pile-up groups of dislocations in MgO crystals; (i) single pile-up, (ii) double pile-up on two orthogonal slip planes, and (iii) double pile-up on two slip planes intersecting at 120°. The equilibrium positions of dislocations in these groups are given. It is found that the stress concentration factor is highest in type (iii) and lowest in type (i). The number of dislocations in the pile-up groups sufficient to raise the tip stress to 1000σ0 is 1000 in type (i), about 120, determined by extrapolation, in type (ii), and 62 in type (iii).

The positions of dislocations in arrays

Abstract The equilibrium positions of dislocations in a number of types of array have been found numerically. Arrays considered are: a. that generated by a dislocation source which is between two locked dislocations,b. a dislocation pile-up against a locked dislocation which iR not on the slip-plane,c. a group of positive dislocations attracted to a group of negative dislocations but prevented from coalescing by locked dislocations,d. a group of positive dislocations attracted by a group of negative dislocations on a parallel slip plane. A short table of the zeros of the first derivative of the Laguerre polynomials is given.