On the transient dynamic antiplane contact problem in the presence of dry friction and slip

This article models the elastodynamic transient contact between two elastically similar half planes under antiplane loading and in the presence of friction. Contact is maintained along the positive real line under the presence of a certain remote contact pressure. An antiplane shear load is applied, which entails interfacial shear traction that opposes the frictional force entailed by the contact pressure. In order to balance the surface tractions, the surface must be allowed to slip. We derive the closed form solution of the interfacial traction due to a general antiplanar displacement distribution using a variant of the Wiener-Hopf technique. We also find closed-form expressions for the interfacial shear traction due to this remote antiplane load. In combination with the frictional force, this leads to an integral equation the solution to which is the distribution of relative slip. We quantify both this and the magnitude of the interfacial shear tractions under diverse loading, showing that transient loading leads to partial reverse slip of the contact surfaces. We show that the reverse slip tends to vanish over time, and that it is ameliorated if the friction coefficient is reduced

The role of the mobility law of dislocations in the plastic response of shock loaded pure metals

This article examines the role that the choice of a dislocation mobility law has in the study of plastic relaxation at shock fronts. Five different mobility laws, two of them phenomenological fits to data, and three more based on physical models of dislocation inertia, are tested by employing dynamic discrete dislocation plasticity (D3P) simulations of a shock loaded aluminium thin foil. It is found that inertial laws invariably entail very short acceleration times for dislocations changing their kinematic state. As long as the mobility laws describe the same regime of terminal speeds, all mobility laws predict the same degree of plastic relaxation at the shock front. This is used to show that the main factor affecting plastic relaxation at the shock front is in fact the speed of dislocations.The author acknowledges support by the EPSRC under the EPSRC Doctoral Prize Fellowship scheme. The author is indebted to D Dini, A P Sutton and D S Balint for their comments and useful discussions. The author reports no competing interests. This work did not involve any collection of human or animal data. This work does not have any experimental data

A stochastic study of the collective effect of random distributions of dislocations

The effect that random populations of dislocations have on a material is examined through stochastic integration of a random cloud of dislocations lying at some distance away from a material point. The problem is studied in one, two, and three dimensions. In 1D, the cloud consists of individual edge dislocations placed along the real line; in 2D, of edge dislocations and edge dipoles on the plane; in 3D, of dislocation loops. In all cases, the dislocation cloud is randomly distributed in space, associated to which several relevant physical parameters, including the material's slip geometry, the dislocation's sign, and its relative orientation, are also stochastically treated. A fully disordered population, i.e., one where the dislocation's signatures and orientations are entirely random, is first studied. It is shown that such disordered systems entail a strong indeterminacy in the collective stress fields, which here is solved by enforcing mass conservation locally. In 2D, this is achieved by modelling a cloud of edge dipoles instead of individual dislocations; in 3D, this is naturally guaranteed by the modelling of closed dislocation loops. The long-range fields of the dipoles in 2D and of the loops in 3D is modelled via their multipolar force expansions, which greatly simplifies the analytical treatment of the problem. The cloud's effect is then studied by performing the stochastic integration of the multipolar fields via Campbell's theorem. The local order, but not the magnitude of the dislocation density, is shown to be critical in contributing to the plastic relaxation of the material: fully disordered systems are shown to self-attenuate, leading to plastic neutrality; ordered and partially ordered systems, achieved when dislocation signatures are aligned, display a direct relationship between the dislocation density and the average stress shielding the material. We establish and generalise the conditions that a system of dislocations must fulfil to display Taylor's equation and the Hall-Petch relation, and offer adequate scaling laws related to this

How strong is the temperature increase due to a moving dislocation?

This article calculates the temperature increase resulting from the motion of a dislocation. The temperature rise is ascribed to two separate effects, both of which are calculated: the dissipative effect resulting from the energy lost by the dislocation as it overcomes the intrinsic lattice resistance to its motion; and the thermomechanical effect arising from the constrained changes in volume the dilatational field of a moving dislocation may entail. The dissipative effect is studied in an uncoupled continuum solid, whilst the thermomechanical effect is studied in a fully coupled thermo-elastodynamic continuum. Explicit solutions are provided, as well as asymptotic estimates of the temperature field in the immediacy of the dislocation core.Research funded with the support of Trinity College Cambridg

Generalised Kanzaki force field of extended defects in crystals, with applications to the modelling of edge dislocations

The Kanzaki forces and their associated multipolar moments are standard ways of representing point defects in an atomistically informed way in the continuum. In this article, the Kanzaki force approach is extended to other crystalline defects. The article shows how the resulting Kanzaki force fields are to be computed for any general extended defect by first computing the relaxed defect's structure and then defining an affine mapping between the said defect structure and the original perfect lattice. This methodology can be employed to compute the Kanzaki force field of any mass-conserving defect, including dislocations, grain and twin boundaries, or cracks. Particular focus is then placed on straight edge dislocation in face-centered cubic (fcc) and body-centered cubic (bcc) pure metals, which are studied along different crystallographic directions. The particular characteristics of these force fields are discussed, drawing a distinction between the slip Kanzaki force field associated with the Volterra disregistry that characterizes the dislocation, and the core Kanzaki force field associated with the specific topology of the dislocation's core. The resulting force fields can be employed to create elastic models of the dislocation that, unlike other regularization procedures, offer a geometrically true representation of the core and the elastic fields in its environs, capturing all three-dimensional effects associated with the core

Static and dynamic multipolar field expansions of dislocations and cracks in solids

This article provides a comprehensive analysis of the way multipolar field expansions, of common use in the study of point defects in solids, may be extended to study the long range fields of dis- locations and cracks. The long range fields of such defects are of relevance in fields as disparate as dislocation dynamics, microcrack and fragmentation, or radiation damage studies. The article provides a general framework for the development of multipolar field expansions in the continuum; one that may be used for any generalised force distribution. The general framework is combined with the Burridge-Knopoff force representation of dislocations and cracks, both in the planar and in the three dimensional cases, to achieve their respective multipolar field expansions for generalised dislocation loop and crack geometries. It is shown that, despite its simplicity, the multipolar field expansions provide a very accurate measure of the far field of both planar and three dimensional dislocations and cracks, and that the accuracy increases as higher order terms (i.e., quadrupolar, octopolar, etc) are introduced into the expansion. The formulation is then extended to the elas- todynamic case. Both a spatial-temporal multipolar field expansion and a spatial multipolar field expansion, are developed. The spatial-temporal multipolar field expansion is seen to capture only the leading terms of the elastodynamic fields, whilst the spatial multipolar expansions are seen to be very accurate at capturing the long range field behaviour so long as the characteristic speed of the dislocation or crack are a fraction of the longitudinal speed of sound

Elastodynamic image forces on screw dislocations in the presence of phase boundaries

The elastodynamic image forces acting on straight screw dislocations in the presence of planar phase boundaries are derived. Two separate dislocations are studied: (i) the injected, non-moving screw dislocation and (ii) the injected (or pre-existing), generally non-uniformly moving screw dislocation. The image forces are derived for both the case of a rigid surface and of a planar interface between two homogeneous, isotropic phases. The case of a rigid interface is shown to be solvable employing Head's image dislocation construction. The case of the elastodynamic image force due to an interface is solved by deriving the reflected wave's contribution to the global solution across the interface. This entails obtaining the fundamental solution (Green's function) for a point unit force via Cagniard's method, and then applying the convolution theorem for a screw dislocation modelled as a force distribution. Complete, explicit formulae are provided when available. It is shown that the elastodynamic image forces are generally affected by retardation effects, and that those acting on the moving dislocations display a dynamic magnification that exceed the attraction (or repulsion) predicted in classical elastostatic calculations.The author was financially supported by the Master and Fellows of Trinity College, University of Cambridge under the author’s Title A Fellowship

Adiabatic shear banding and the micromechanics of plastic flow in metals

This article studies the conditions that dislocation generation and motion must fulfil to promote the development of adiabatic shear bands in crystalline metals. First, we derive a stability criterion for the formation of shear bands, by linearising the conservation equations of thermo-plasticity. We then apply this criterion on the micromechanics-based Orowan equation for plastic flow, introducing a number of increasingly sophisticated constitutive assumptions on the model. It is found that there are two crucial promoters of shear band formation: the unfettered generation of dislocations that may be found in stage I plasticity; and the softening of the elastic constants with increasing temperature. In turn, we show that limiting the speed of dislocations tends to inhibit the formation of dislocations, even when the temperature dependence of the dislocation’s drag is accounted for. This leads to the existence of an upper temperature limit above which shear band formation appears unlikely

Elastodynamic image forces on dislocations.

The elastodynamic image forces on edge and screw dislocations in the presence of a planar-free surface are derived. The explicit form of the elastodynamic fields of an injected, quiescent screw dislocation are also derived. The resulting image forces are affected by retardation effects: the dislocations experience no image force for a period of time defined by the arrival and reflection at the free surface of the dislocation fields. For the case of injected, stationary dislocations, it is shown that the elastodynamic image force tends asymptotically to the elastotatic prediction. For the case of injected, moving dislocations, it is shown that the elastodynamic image force on both the edge and the screw dislocations is magnified by inertial effects, and becomes increasingly divergent with time; this additional effect, missing in the elastostatic description, is shown to be substantial even for slow moving dislocations. Finally, it is shown that the elastodynamic image force of an edge dislocation moving towards the surface at the Rayleigh wave speed becomes repulsive, rather than attractive; this is suggestive of instabilities at the core of the dislocation, and likely resonances with the free surface.EPSRC via the EPSRC Doctoral Prize Fellowship progra

A dynamic discrete dislocation plasticity method for the simulation of plastic relaxation under shock loading

In this article, it is demonstrated that current methods of modelling plasticity as the collective motion of discrete dislocations, such as two-dimensional discrete dislocation plasticity (DDP), are unsuitable for the simulation of very high strain rate processes (106 s-1 or more) such as plastic relaxation during shock loading. Current DDP models treat dislocations quasi-statically, ignoring the time-dependent nature of the elastic fields of dislocations. It is shown that this assumption introduces unphysical artefacts into the system when simulating plasticity resulting from shock loading. This deficiency can be overcome only by formulating a fully time-dependent elastodynamic description of the elastic fields of discrete dislocations. Building on the work of Markenscoff & Clifton, the fundamental time-dependent solutions for the injection and non-uniform motion of straight edge dislocations are presented. The numerical implementation of these solutions for a single moving dislocation and for two annihilating dislocations in an infinite plane are presented. The application of these solutions in a two-dimensional model of timedependent plasticity during shock loading is outlined here and will be presented in detail elsewhere. © 2013 The Author(s) Published by the Royal Society. All rights reserved