Pile-up Of Screw Dislocations Research Articles

Unusual indentation depth dependent hardness and creep behaviors in NbMoTaW films at the nanoscale

Herein, nanocrystalline NbMoTaW and (NbMoTaW)N thin films were prepared via magnetron sputtering and investigated using nanoindentation creep tests at various indentation loads (P). The hardness (H) of films increases with P, exhibiting an abnormal indentation size effect (ISE). At the same time, with decreasing indentation depth (h), the creep rate and the creep rate sensitivity (m) both increase. Under a critical h (hc) for each film, m sharply increases. The diffusion between the indenter/film interface dominates the creep mechanism at h lower than hc, whereas the dislocation–grain boundary interaction dominates the creep deformation at h higher than hc. The coble creep and lattice diffusion were both excluded as the main deformation mechanism. The escape of screw dislocations to the surface leads to the softening of the film, whereas the pile-up of screw dislocations leads to hardening, explaining the abnormal ISE on H. The effect of residual stress, grain boundary structures, substrate, and impurities are also discussed.

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.

Screw dislocation pileups against a bimaterial interface incorporating surface elasticity

Screw dislocation pileups against a bimaterial interface incorporating surface elasticity

Screw dislocation pileups at a coated circular inhomogeneity

Screw dislocation pileups at a coated circular inhomogeneity

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.

CONFIGURATIONAL FORCE EXERTED ON AN INHOMOGENEITY BY A PILEUP OF SCREW DISLOCATIONS

Configurational forces exerted by a screw dislocation pileup on a circular inhomogeneity or a bimaterial interface are determined for a given number of dislocations, the level of remotely applied stress, and a specified degree of material inhomogeneity. The interface shear stress is also evaluated and discussed.

CONFIGURATIONAL FORCE EXERTED ON AN INHOMOGENEITY BY A PILEUP OF SCREW DISLOCATIONS

Configurational forces exerted by a screw dislocation pileup on a circular inhomogeneity or a bimaterial interface are determined for a given number of dislocations, the level of remotely applied stress, and a specified degree of material inhomogeneity. The interface shear stress is also evaluated and discussed.

A pileup of screw dislocations against an inclined bimetallic interface

Pileups of screw dislocations against an inclined bimetallic interface are considered. It is assumed that differently oriented pileups are either under the same remote uniform loading, or under the same resolved shear stress along the pileup direction for any orientation of the interface. The distributions of dislocations and the lengths of pileups are substantially different for differently oriented interfaces, particularly in the case of the same remote loading. The interface stresses are also strongly depended on the pileup orientation. The maximum stress can be higher for a pileup along an inclined direction than along the direction orthogonal to the interface. The back stress behind a pileup is evaluated and discussed.

Asymptotic expressions for the nearest and furthest dislocations in a pile-up against a grain boundary

In 1965, Armstrong and Head explored the problem of a pile-up of screw dislocations against a grain boundary. They used numerical methods to determine the positions of the dislocations in the pile-up and they were able to fit approximate formulae for the locations of the first and last dislocations. These formulae were used to gain insights into the Hall–Petch relationship. More recently, Voskoboinikov et al. used asymptotic techniques to study the equivalent problem of a pile-up of a large number of screw dislocations against a bimetallic interface. In this paper, we extend the work of Voskoboinikov et al. to construct systematic asymptotic expressions for the formulae proposed by Armstrong and Head. The further extension of these techniques to more general pile-ups is also outlined. As a result of this work, we show that a pile-up against a grain boundary can become equivalent to a pile-up against a locked dislocation in the case where the mismatch across the boundary is small.

Energies and distributions of dislocations in stacked pile-ups

To understand the kinematic and thermodynamic effects of representing discrete dislocations as continuous distributions in their slip planes, we consider stacked double ended pile-ups of edge and screw dislocations and compute their distributions and microstructural energies, i.e., the elastic interaction energies of geometrically necessary dislocations (GNDs). In general, three kinds of representations of GNDs are used: discrete, semi-discrete, and, continuous representation. The discrete representations are closest to reality. Therefore, the corresponding solutions are considered exact. In the semi-discrete representation, the discrete dislocations are smeared out into continuous planar distributions within discrete slip planes. The solutions to problems formulated using different descriptions are different. We consider the errors in: dislocation distributions (number of dislocations), and, microstructural energies; when the discrete description is replaced by the semi-discrete one. Asymptotic expressions are derived for: number of dislocations, maximum slip, and, microstructural energy density. They provide a powerful insight into the behavior of the system, and are accurate for a wide range of parameters. Two characteristic lengths emerge from the analysis: the ratio of pile-up length to slip plane spacing (lambda), and, the ratio of slip plane spacing to the Burgers vector. For large enough lambda and large enough number of dislocations, both the discrete and semi-discrete solutions are well-approximated by asymptotic solutions. Results of a comprehensive numerical study are presented.

Matched asymptotic expansion in modelling of edge dislocation pile-ups

There are two distinct approaches to the description of dislocations in solids. Often discrete dislocation modelling is too time-consuming and computationally intensive, whereas the solution of the equivalent problem in a continuum approximation can be obtained relatively easily. Although such solutions can provide information about the macroscopic stress it cannot be applied at the scale of the separation between neighbouring dislocations. We have already provided robust proof of the interconnection between continuum and discrete approaches to dislocation description and suggested a methodology for analyzing pile-ups of screw and edge dislocations in a uniform material and a pile-up of screw dislocations near an interface in a bimetallic solid. Here it is developed further to derive the equilibrium distribution of n edge dislocations in a linear pile-up stressed against an interface in a bimetallic solid. As n → ∞ the dislocation positions are located with sufficient accuracy that the stress distribution can be evaluated by a simple computational procedure. The stress is determined from a lumped discretisation of superdislocations away from the interface. We present an example in which a hundred dislocations can be replaced by just four superdislocations with only 1 % error in the computation.

Continuum and discrete models of dislocation pile-ups. II. Pile-up of screw dislocations at the interface in a bimetallic solid

The methodology developed in the precursor to this paper is used to find the positions of n screw dislocations in a pile-up against an interface bonding two crystalline solids. The pile-up is caused by a constant applied stress and, as n →∞, the dislocations are located with sufficient accuracy to predict the large but finite stress distribution at the interface. Such a prediction is impossible using a conventional continuum dislocation density.

Stress function determination for dislocation configurations obtained from Kröner’s theory

Based on the work of Borisova, a more general method is proposed to determine the stress functions, from which the stress field is derived, of an edge dislocation pile-up and wall. For this purpose, Kröner’s theory of discrete dislocation in a continuum is applied to these two typical infinite and periodic configurations of edge dislocations. It is shown that the calculated stress functions can be used to easily determine the stress field of other configurations like screw dislocation pile-up or symmetrical tilt boundary.

Green's function for an anisotropic film-substrate embedded with a screw dislocation

In this paper, the Green's function for an elastically anisotropic dissimilar film-substrate embedded with a screw dislocation is obtained analytically in series form by the method of continuously distributed image dislocations. The film-substrate interface conditions are satisfied exactly a priori, and it remains to satisfy only the free surface condition. This leads to a Fredholm integral equation of the second kind with the image dislocation density as the unknown. The solution is expressed in the form of a Neumann series, whose terms are certain integral transforms of rational functions. The series can be used as Green's function for a broad range of thin film problems, e.g., mode III crack, screw dislocation pileup, and edge stress concentration problems. Numerical studies of the Green's function focus on a screw dislocation in various semi-infinite film-substrate systems. The solutions satisfy the free surface condition very well using only three terms of the series. The importance of the free surface, elastic anisotropy and material inhomogeneity is emphasized in the numerical investigations.

Near threshold fatigue curve based on the dislocation free zone model of fracture

Dislocation pileup along an inclined direction from a crack tip has been observed in metals during tensile deformation in an electron microscope. Dislocations are emitted from the crack tip, move through a dislocation free zone, and pile up in the plastic zone. The problem of inclined pileup of screw dislocations at a crack tip with a dislocation free zone has been solved by applying the Wiener–Hopf method. The result can be used to calculate the crack tip blunting and the local stress intensity K for the applied stress intensity K 3. According to the fatigue theory of Laird or Neumann, the crack propagation per cycle has the order of magnitude of the blunting. The blunting versus K 3 is then plotted. It has the shape of the near threshold fatigue curve. To plot this curve, the local stress intensity K has been assumed to be the critical value K g for the dislocation emission at the crack tip.

Solitary wave equation for steady-state propagation of an extended defect in a solid

Lattice models to describe the steady-state propagation of (i) a screw dislocation and (ii) a crack as a pileup of screw dislocations appear to be useful. This prompts us to exhibit a solitary wave equation which describes the continuum limit of the lattice model of Celli and Flytzanis for a screw dislocation. Some numerical consequences of this continuum equation are also presented.

Strain Induced Grain Boundary Premelting due to Heavy Pile-Up of Screw Dislocations in Deformed Copper Bicrystals

Strain Induced Grain Boundary Premelting due to Heavy Pile-Up of Screw Dislocations in Deformed Copper Bicrystals

Force between two parallel screw dislocations and application to linear screw dislocation pileups—Gauge theory results

An analytic expression for the force between two parallel screw dislocations, derived earlier on the basis of the gauge theory of dislocations, has been used to investigate the static distribution of a given numberN of parallel screw dislocations confined between two immobile dislocation obstacles. It is shown that in the limit of a continuous distribution of dislocations the equilibrium condition leads to a Fredholm integral equation of first type which does not admit any nontrivial solution. Implication of this result is discussed. For a finite number of dislocations, the ratio (η) of the obstacle separation to the core radius is an important parameter governing the nature of solution of the discrete equation. It is found that for a givenN, there is a critical valueη c below which there does not exist any solution.

Strain induced grain boundary premelting in bulk copper bicrystals

In bulk bicrystals strain induced grain boundary premelting (SIGBPM) occurs when heavy screw dislocation pileup can be held up to a certain high temperature, approximately 0.6T M, where T M is the melting point of bulk material in Kelvin. SIGBPM occurs at grain boundaries to which new twist component is added due to the rotation of both component crystals toward opposite direction about the axis perpendicular to the grain boundary plane. At the original grain boundary, grain boundary sliding takes place due to this relative rotation. In f.c.c. metals with relatively low stacking fault energies such as copper, nickel, brass(30Zn) and silver, dislocations dissociate into partials. Therefore high density tangled dislocations introduced during plastic deformation hardly loose. If these dislocations can be held to high temperatures, SIGBPM is promoted. Formation of static or dynamic recrystallized grains suppresses SIGBPM itself and the propagation of grain boundary cracks formed by SIGBPM.

Strain Induced Grain Boundary Premelting in Copper Bicrystals

In bulk bicrystals strain induced grain boundary premelting (SIGBPM) occurs when heavy screw dislocation pileup can be held up to a certain high temperature, approximately 0.6TM, where TM is the melting point of bulk material in Kelvin. SIGBPM occurs at grain boundaries to which new twist component is added due to the rotation of both component crystals toward opposite direction about the axis perpendicular to the grain boundary plane. At the original grain boundary, grain boundary sliding takes place due to this relative rotation. In f.c.c. metals with relatively low stacking fault energies such as copper, nickel, brass(30Zn) and silver, dislocations dissociate into partials. Therefore high density tangled dislocations introduced during plastic deformation hardly loose. If these dislocations can be held to high temperatures, SIGBPM is promoted. Formation of static or dynamic recrystallized grains suppresses SIGBPM itself and the propagation of grain boundary cracks formed by SIGBPM.