Dislocation Distribution Function Research Articles

An unstressed screw dislocation pileup against two circular elastic inhomogeneities

We use complex variable techniques and the theory of singular integral equations to study the problem associated with an unstressed pileup of identical parallel screw dislocations against two circular elastic inhomogeneities. Each of the circular inhomogeneities is assumed to be harder than the surrounding matrix rendering them repulsive to the nearby dislocations. Consequently, the pileup of dislocations can assume an equilibrium configuration even in the absence of remote loading. The condition required for equilibrium is formulated in terms of a singular integral equation which is solved numerically to arrive at the dislocation distribution function.

Unconventional cracks with dislocation-free zones

Using the method of continuously distributed dislocations, we study the dislocation-free zones for unconventional internal cracks in an infinite isotropic elastic medium under uniform remote anti-plane shear stress. The unconventional cracks are represented by a symmetric lip crack with two cusp tips and a symmetric airfoil crack with a single cusp tip. For each case, the equilibrium condition is formulated in terms of a single singular integral equation in the image plane. The resulting singular integral equation is solved numerically to arrive at the dislocation distribution function and the total number of dislocations in the plastic zone, the condition on remote loading and the local mode III stress intensity factor at the cusp tip. In the absence of the dislocation-free zones, the results are valid for the Dugdale model of these unconventional cracks.

Screw dislocation pileups in a two-phase thin film

The method of continuously distributed dislocations is used to study the distribution of screw dislocations in a linear array piled up near the interface of a two-phase isotropic elastic thin film with equal thickness in each phase. The resulting singular integral equation is solved numerically using the Gauss–Chebyshev integration formula to arrive at the dislocation distribution function and the number of dislocations in the pileup.

Phase transformation and microstructure evolution of nanoimprinted NiCoCr medium entropy alloys

In this study, the mechanical and processing properties of nanoimprinted nickel-cobalt-chromium medium-entropy alloy (NiCoCr MEA) are investigated using molecular dynamics (MD) simulation. The effects of microstructure, loading velocity, the taper angle of mould, and processing temperature are analyzed by loading force, deformation characteristic, crystal evolution, dislocation distribution, displacement vector, and radial distribution function. Based on the results, both stacking faults and twinning structures appear in NiCoCr MEA during the imprinting process. The formation of stacking fault is affected by grain size, and it always propagates in large grains (grain size = 8.5 nm). Besides, the maximum loading force and the residual stress increase with an increase in loading velocity (200 m/s), caused by the shorter relaxation time for the atoms in NiCoCr MEA specimen with higher loading velocity. Moreover, the temperature is found to affect the microstructure, phase transformation, and dislocation evolutions of NiCoCr MEA specimen. Most of the atoms transform into HCP or amorphous phase at a temperature of 900 K during the equilibrium process. The maximum loading force decreases with an increase in temperature, and atoms strongly adhesion on the bottom of the mould at the higher temperature (900 K). By contrast, the taper angle of mould gives little effect to the mechanical characteristics. There is no obvious trend in the deformation behaviors of the specimens with different taper angles.

A new system of singular integral equations for a curvilinear crack in bonded materials

The modified complex potentials (MCPs) functions are used to develop a new system of singular integral equations (SIEs) for a curvilinear crack in the upper part of bonded materials subjected to shear mode stress with the help of continuity conditions for resultant force and displacement functions. The unknown dislocation distribution function is mapped into a square root singularity function by using curved length coordinate method and the traction along the crack as the right hand term. The Gaussian quadrature rules were used to obtain the numerical solution for a new system of SIEs in order to compute the nondimensional stress intensity factors (SIFs) for these problems. Our results agree with those of the previous works. The findings have revealed that the nondimensional SIFs depend on the elastic constant ratio, crack geometries and the position of the cracks.

Dislocation Distribution Functions of the Edges of Mode I Crack under Several Boundary Conditions

Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

Dislocation Distribution Function of the Mode III Dynamic Crack Subjected to Moving Unit Step Load from a Point

Dislocation distribution functions of the edges of mode III propagation crack subjected to Moving unit step load from a point was studied by the methods of the theory of complex variable functions.By the methods, the problems researched can be facilely transformed into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements, dynamic stress intensity factor and dislocation distribution function were obtained by the methods of the theory of self-similar functions.In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were attained.

Dislocation Distribution Model of Mode I Dynamic Crack

Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

Collinear Zener–Stroh crack problem in plane elasticity

This paper investigates the collinear Zener–Stroh crack problem in plane elasticity. Two cracks in series are chosen as an example in analysis. Different to the Griffith crack problem, an initial displacement discontinuity exists in the Zener–Stroh crack problem, which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. From the traction free condition along the cracks, the dislocation distribution function as well as the complex potential with undetermined coefficients is suggested. The involved undetermined coefficients can be evaluated from the condition of the assumed initial displacement discontinuity. Finally, a closed form solution for the problem is obtained and the calculated stress intensity factors at crack tips are presented. A problem for two collinear Zener–Stroh cracks with different lengths is also studied.

Dislocation Distribution Function of the Surfaces of Mode III Dynamic Crack Subjected to Unit-Step Loads and Moving Increasing Loads

By the theory of complex functions, dislocation distribution function concerning mode dynamic crack propagation problem under the conditions of unit-step loads and moving increasing loads was studied respectively. Analytical solution representations are attained by the methods of self-similar functions. The problems investigated can be transformed into Riemann-Hilbert problems and their closed solutions are obtained rather simple by this approach.

Stress intensity factors and kink angle of a crack interacting with a circular inclusion under remote mechanical and thermal loadings

A problem of a circular elastic inhomogeneity interacting with a crack under uniform loadings (mechanical tension and heat flux at infinity) is solved. The singular integral equations for edge and temperature dislocation distribution functions are constructed and solved numerically, to obtain the stress intensity factors. The effects of the material property ratio on the stress intensity factor (SIF) are investigated. The computed SIFs are used to predict the kink angle of the crack when the crack grows.

Natural strong pinning sites in laser-ablatedYBa2Cu3O7−δthin films

At low temperatures dislocations are the dominant flux-pinning centers in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ deposited on (100) ${\mathrm{SrTiO}}_{3}$ substrates [B. Dam et al., Nature (London) 399, 439 (1999)]. Using a wet-chemical etching technique in combination with atomic force microscopy, both the length and the lateral dislocation distribution are determined in laser ablated ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ films. We find that (i) dislocations are induced in the first stages of film growth, i.e., close to the substrate-film interface, and persist all the way up to the film surface parallel to the c axis, resulting in a uniform length distribution, and (ii) the radial dislocation distribution function exhibits a universal behavior: it approaches zero at small distances, indicating short-range ordering of the defects. This self-organization of the growth-induced correlated disorder makes epitaxial films completely different from single crystals with randomly distributed columnar defects created by means of heavy-ion irradiation. Since the substrate temperature can be used to tune the dislocation density ${n}_{\mathrm{disl}}$ over almost two orders of magnitude (\ensuremath{\sim}1--100/\ensuremath{\mu}${\mathrm{m}}^{2}$), the mechanism by which dislocations are induced is closely related to the ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ nucleation and growth mechanism. We present evidence for preferential precipitation in the first monolayer and precipitate generated dislocations.

Distribution of dislocations at a mode I crack tip and their shielding effect

Distribution of dislocations at a finite mode I crack tip is formulated. Closed form solutions for the dislocation distribution function, the dislocation-free zone (DFZ), the local stress intensity factor and the crack tip stress field are obtained. The dislocation distribution has similar features to a mode III crack model. Under a given applied stress, there may exist different configurations of plastic zone and DFZ. Crack tip shielding by dislocations depends on both applied stresses and the configuration.

Influence of irradiation on the dislocation kinetics with allowance for the dislocation velocity distribution

A model of plastic deformation evolution and flow localisation processes in irradiated materials is proposed. This models takes into account the dislocation distribution function dependence on dislocation velocity in an ensemble. It is shown that the fraction of dislocations overcoming radiation defects with high velocities in the dynamical regime grows with increasing radiation hardening.

Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations

For describing the dislocation pattern formation, two different approaches are applied. In one of them the collective behavior of many individual dislocations is investigated by analytical or numerical methods, in the other one the dislocation system is described by continuous functions of the space coordinates. A method is proposed for establishing the link between the two length scales. Starting from the exact evaluation equation of the $N$-dislocation distribution function, a hierarchy of equations is derived for the one-, two-, three-, etc., particle distribution functions. Although the dislocations form a nonconservative system, the applied method is similar to the derivation of the so called BBGKY hierarchy which is frequently used in plasma physics. By assuming that the three-particle correlation function is negligible a closed set of equations can be obtained for the dislocation density and the two-particle correlation functions. The possible origin of appearance of dislocation patterns is also investigated.

Interaction of an antiplane shear crack with a transforming inclusion

The influence of a circular transforming inclusion on the stress field of an antiplane shear crack is investigated. Further, the analytical expressions for the stress field of an antiplane shear crack submitted to antiplane shear stresses due to the eigenstrain transformation of the constrained inclusion and an applied constant antiplane stress field at infinity are given by using an appropriate dislocation distribution function. The stress intensity factor K III is obtained analytically and its values vary with the size, position and stress-free strains of the transforming inclusion. The characteristics of this crack-inclusion interaction problem are analyzed and discussed.

Correlation between volume fraction of radiation-induced precipitates and toughness of Cr-Mo-V alloyed VVER pressure vessel steel

A model of plastic deformation evolution and flow localisation processes in irradiated materials is proposed. This models takes into account the dislocation distribution function dependence on dislocation velocity in an ensemble. It is shown that the fraction of dislocations overcoming radiation defects with high velocities in the dynamical regime grows with increasing radiation hardening.

Effect of grain boundary ledges on mode III fracture

Abstract Dislocations piled up between a finite crack tip and a grain boundary ledge with a dislocation-free zone are investigated by a continuous dislocation modelling method. The dislocation distribution function to simulate the crack and plastic zone is analytically derived. A closed form for the number of dislocations in the plastic zone is analytically derived. A closed form for the number of dislocations in the plastic zone and stress field is obtained. From the stress field, the stress intensity factor K m and stress concentration at the grain boundary are obtained. When there is no dislocation-free zone in front of crack tip, the stress intensity factor K m is zero, no matter whether the grain boundary exists or not. The effect of the grain boundary ledge on the fracture toughness and the effect of the crack on the strength of the grain boundary, such as described by the Hall-Petch equation, are discussed. Finally, our results can reduce to several special cases, and these are discussed.

Elastic interaction between screw dislocations and collinear edge and internal cracks

The elastic interaction between screw dislocation and collinear internal and surface cracks has been investigated by the dislocation modelling method. The dislocation distribution function to simulate the cracks, the stress field and the stress intensity factor at the crack tip are obtained. From the stress field, we have derived the force on the dislocation, the zero-force position and the critical stress intensity factor for dislocation emitted from the crack tip. The critical stress intensity factor for dislocation emitted from the crack tip is affected by the other crack. The dislocations inside the crack also play an important role in fracture. Finally, four special cases can be obtained from our results: 1. (1) the interaction between a dislocation and a surface crack; 2. (2) the interaction between a dislocation and an internal crack; 3. (3) the interaction between two dislocations with a surface crack; 4. (4) the interaction between a dislocation and an internal crack near a free surface.

Interaction of lateral cracks and plastic zones

A computational method is developed for analyzing interactions between near-surface lateral cracks and residual stresses due to localized plastic zones. The latter are approximated as either prismatic or shear dislocation dipole pairs. Two-dimensional plane-strain conditions are assumed. A complete linear elastic solution is obtained using half-space dislocations for the dipole pairs and for the dislocation distribution function that accounts for the stress-free crack surface boundary condition. The crack-tip stress intensity factors show that mode-II driving forces are important for both prismatic and shear zones. The tendency for crack deflection to the free surface derived from energy release rates is complex. The dipole representation results do not provide a simple rationalization for lateral crack chipping.