The transient motion of a dislocation in a solid of general anisotropy

The transient motion of a dislocation starting from rest and moving in an arbitrary rectilinear motion in an anisotropic solid is analyzed by transform techniques and inversion according to the Cagniard-de Hoop technique.     

Distributional and regularized radiation fields of non-uniformly moving straight dislocations,and elastodynamic Tamm problem

This work introduces original explicit solutions for the elastic fields radiated by non-uniformly moving, straight, screw or edge dislocations in an isotropic medium, in the form of time-integral representations in which acceleration-dependent contributions are explicitly separated out. These solutions are obtained by applying an isotropic regularization procedure to distributional expressions of the elastodynamic fields built on the Green tensor of the Navier equation. The obtained regularized field expressions are singularity-free, and depend on the dislocation density rather than on the plastic eigenstrain. They cover non-uniform motion at arbitrary speeds, including faster-than-wave ones. A numerical method of computation is discussed, that rests on discretizing motion along an arbitrary path in the plane transverse to the dislocation, into a succession of time intervals of constant velocity vector over which time-integrated contributions can be obtained in closed form. As a simple illustration, it is applied to the elastodynamic equivalent of the Tamm problem, where fields induced by a dislocation accelerated from rest beyond the longitudinal wave speed, and thereafter put to rest again, are computed. As expected, the proposed expressions produce Mach cones, the dynamic build-up and decay of which is illustrated by means of full-field calculations.     

The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid

Ray theory is extended to consider the case of an accelerating disturbance which is producing waves in a rotating stratified fluid. Starting from the equations of motion, dispersion relations are derived for surface gravity waves, capillary waves, Rossby waves and internal-inertial waves. The wave system is studied in each case for the problem of a body starting impulsively from rest and for a body starting from rest and moving with constant acceleration.     

The transient motion of a nonuniformly moving dislocation

The solution to the transient subsonic motion of a nonuniformly moving screw dislocation starting from rest is obtained.

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Résumé On étudie le mouvement quelconque d'une dislocation se propageant à une vitesse nonuniforme subsonique à partir d'une position de repos.

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This research was conducted while the author held a visiting appointment at Brown University.     

Elastic precursor decay and radiation from nonuniformly moving dislocations

Precursor decay in plate impact experiments on single crystals is re-examined from the viewpoint of the elastodynamics of moving dislocations. Superposition of solutions for many dislocations set in motion by an incident plane wave is used to relate the decay of the wave amplitude at the front of the plane wave to the density and velocity of dislocations at the wavefront. The resulting precursor decay relation is the same as the one derived from an elastic/visco-plastic model of the material, except for a small correction due to differences between the effects of forward and backward propagating dislocations. Motivated by this added support for the validity of the precursor decay equation, the values used for the quantities in this equation are re-examined. Recent experimental results and the elastodynamics analysis are interpreted as indicating that the commonly-used values of dislocation velocity are probably satisfactory, but that the values used for dislocation density are several orders of magnitude too small near the lapped surfaces of the crystal. These large dislocation densities are identified as the probable dominant cause of the lower-than-predicted precursor amplitudes that are recorded in experiments. More accurate experimental data and inclusion of the non-linear elasticity effects are essential in determining whether or not the observed precursor decay in the bulk of the specimen can be explained by the motion of dislocations present initially. Calculations of energy radiated from screw and edge dislocations that start from rest and move thereafter at constant velocity confirm that dislocation drag forces due to continuum elasticity effects are small for dislocation velocities less than, say, 80% of the elastic shear wave speed. At supersonic speeds the continuum drag effects become so large that sustained supersonic motion of dislocations appears unlikely.     

Ideal gas flow behind a finite-amplitude shock wave

The equations of one-dimensional (with a plane of symmetry) adiabatic motion of an ideal gas are transformed to a form convenient for studying flows between a moving piston and a shock wave of variable intensity. The solution is found for the equations of a motion containing a shock wave which propagates through a quiescent gas with variable initial density and constant pressure. This solution contains four arbitrary constants and, in a particular case, gives an example of adiabatic shockless compression by a piston of a gas initially at rest.     

On the Equation of Motion and the Near-Field of a Nonuniformly Moving Dislocation

The O(1) terms in the near field asymptotic expansion of the radiated field from a nonuniformly moving dislocation are obtained by a singular asymptotic expansion. These terms enter the equation of motion of the dislocation and have not been given explicitly by Eshelby (1953). They depend on an integral over the history of the motion which physically represents the reaction of the motion to the radiation the dislocation has already emitted during its path.     

Transient dislocation emission from a crack tip under dynamic mode III loading

Summary  Transient dislocation emission from a crack tip under dynamic mode III loading is analyzed. By taking into account the dynamic interaction between the crack and dislocation, the governing equation for the dislocation motion is derived under the quasi-steady assumption. The behavior of dislocation emission is explored in detail by solving this equation numerically. A critical initial speed can be determined, which must be exceeded by dislocations to escape from the crack tip. The dislocation emission process is found to be completed in such a short time period that the applied load may be approximately treated as constant during dislocation emission. Based on this fact, an asymptotic criterion for transient dislocation emission is developed, from which the critical initial speed can be evaluated. In the case that the dislocation is emitted from rest, we recover the quasi-static criterion of dislocation emission. Received 22 November 2000; accepted for publication 20 March 2001     

On the dynamic behaviour of a beam with an accelerating mass

Summary The dynamic behaviour of an Euler beam traversed by a moving concentrated mass, is analyzed for the general case of a mass moving with a varying speed. The equation of motion in a matrix form is formulated using the Lagrangian approach and the assumed mode method. The dimensionless form of the equation enables the numerical results to be applicable for a wide range of system parameters. The possibility of the mass separating from the beam is analyzed by examining the contact forces between the mass and the beam during the motion.     

Effect of solutes on dislocation motion —a phase-field simulation

Based on recent advances in phase-field models for integrating phase and defect microstructures as well as dislocation dynamics, the interactions between diffusional solutes and moving dislocations under applied stresses are studied in three dimensions. A new functional form for describing the eigenstrains of dislocations is proposed, eliminating the dependence of the magnitude of the dislocation Burgers vector on the applied stress and providing correct stress fields of dislocations. A relationship between the velocity of the dislocation and the applied stress is obtained by theoretical analysis and numerical simulations. The operation of Frank–Read sources in the presence of diffusional solutes, the effect of chemical interactions in solid solution on the equilibrium distribution of Cottrell atmosphere, and the drag effect of Cottrell atmosphere on dislocation motion are examined. The results demonstrate that the phase-field model correctly describes the long-range elastic interactions and short-range chemical interactions that control dislocation motion.     

Tensile response of passivated films with climb-assisted dislocation glide

The tensile response of single crystal films passivated on two sides is analysed using climb enabled discrete dislocation plasticity. Plastic deformation is modelled through the motion of edge dislocations in an elastic solid with a lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation incorporated through a set of constitutive rules. The dislocation motion in the films is by glide-only or by climb-assisted glide whereas in the surface passivation layers dislocation motion occurs by glide-only and penalized by a friction stress. For realistic values of the friction stress, the size dependence of the flow strength of the oxidised films was mainly a geometrical effect resulting from the fact that the ratio of the oxide layer thickness to film thickness increases with decreasing film thickness. However, if the passivation layer was modelled as impenetrable, i.e. an infinite friction stress, the plastic hardening rate of the films increases with decreasing film thickness even for geometrically self-similar specimens. This size dependence is an intrinsic material size effect that occurs because the dislocation pile-up lengths become on the order of the film thickness. Counter-intuitively, the films have a higher flow strength when dislocation motion is driven by climb-assisted glide compared to the case when dislocation motion is glide-only. This occurs because dislocation climb breaks up the dislocation pile-ups that aid dislocations to penetrate the passivation layers. The results also show that the Bauschinger effect in passivated thin films is stronger when dislocation motion is climb-assisted compared to films wherein dislocation motion is by glide-only.     

Small‐Time Expansion of Wave Motion Generated by a Submerged Sphere

A nonlinear problem of motion of a solid sphere near a free surface of an infinitely deep fluid is considered. For the case of motion with a constant acceleration starting from rest, the solution is studied using a smalltime expansion. Expansion coefficients up to the fourth power inclusive are found for the free surface elevation and for the force acting on the sphere. The solutions for linear and nonlinear conditions on the free surface are compared.     

On the flows produced by sudden application of a constant pressure gradient or by impulsive motion of a boundary

Some properties of the time-dependent Navier-Stokes equations are discussed for flows impulsively started from rest by sudden application of a constant pressure gradient or by the impulsive motion of a boundary. Five illustrative examples are given. They are: unsteady flow in a circular cylinder moving parallel to its length, starting flow in a circular pipe, unsteady flow in a rotating cylinder, starting flow in a rectangular channel moving parallel to its length and unsteady flow in a channel of rectangular cross-section. It is found that the expressions of the quantities such as velocity, flux and skin friction are in series forms which may be rapidly convergent for large values of the time but slowly convergent for small values of the time or vice versa. It is shown that if their expressions can be found for one of large values of the time or small values of the time, these expressions can be used for the other.     

Causal Stroh formalism for uniformly-moving dislocations in anisotropic media: Somigliana dislocations and Mach cones

In this work, Stroh’s formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity–including supersonic regimes–without needing to consider subsonic and supersonic cases separately. As an illustration, the distortion field of a Somigliana dislocation of the Peierls–Nabarro–Eshelby-type with finite-width core is computed analytically, starting from the Green’s tensor of elastodynamics. To obtain the result in the form of a single compact expression, use of the modified Stroh formalism requires splitting the Green’s function into its reactive and radiative parts. In supersonic regimes, the solution obtained displays Mach cones, which are supported by Dirac measures in the Volterra limit. From these results, an explanation of Payton’s ‘backward’ Mach cones (Payton, 1995) is given in terms of slowness surfaces, and a simple criterion for their existence is derived. The findings are illustrated by full-field calculations from analytical formulas for a dislocation of finite width in iron, and by Huygens-type geometric constructions of Mach cones from ray surfaces.     

A wave propagation problem for non-uniform screw dislocation motion in a viscoelastic half-plane

The wave propagation problem for a largely arbitrary anti-plane displacement discontinuity imposed along a line perpendicular to the surface of a stress-free linearly viscoelastic half-plane is considered. The general Laplace transform solution is obtained and then inverted for the case of a screw dislocation moving at an arbitrary speed in a Maxwell material. It is shown that the material viscoelasticity alters the coefficient of the dislocation edge stress singularity and damps the surface displacements from the elastic values. The surface damping increases with time, distance from the dislocation path and dislocation speed, whether sub- or supersonic.     

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.     

Molecular dynamics study of solute strengthening in Al/Mg alloys

The strengthening of Al by Mg solute atoms is investigated using molecular dynamics (MD) studies of single dislocations moving through a field of randomly placed solutes. The MD method permits explicit treatment of “core” effects, dislocation pinning and deceleration, and dislocation unpinning by thermal activation, all under an applied load. Choice of an appropriate MD simulation cell size is assessed using analytic concepts developed by Labusch. The interaction energy of a single Mg atom with straight edge and screw dislocations is computed and compared with continuum models. Using the single Mg energies, a one-dimensional energy landscape for the motion of a straight edge dislocation through a random field of Mg solutes is computed. The minima in this landscape match well with those found in the MD simulations at zero temperature. The stress to unpin a straight edge dislocation trapped in a local energy minimum generated by the solutes is then predicted semi-analytically using the energy landscape, and good agreement is obtained with the MD results. At temperatures of 300 and 500 K, the thermally activated rate of unpinning vs. stress and temperature is calculated semi-analytically, and agreement with the full MD results is again obtained with the fitting of a single attempt frequency in a transition state model. The agreement of the semi-analytical models provides a basis for calculating yield stress vs. strain rate and temperature, resulting from statistical pinning, for the case of non-interacting dislocations on a single slip system, and for extending the analysis to study dynamic strain aging effects resulting from diffusion of Mg atoms around a pinned dislocation.     

Dynamics of ideal fibre-reinforced rigid-plastic beams

A mechanism is proposed by which discontinuities in slope can propagate along an ideal fibr-ereinforced beam which is inextensible in the direction of its axis. The equations of motion of the beam are formulated, including the dynamical conditions which must be satisfied at the discontinuity. Constitutive equations for a rigid-plastic fibre-reinforced beam are established, and it is shown that slope discontinuities may propagate in a strain-hardening material, but are stationary in a perfectlyplastic beam. The theory is illustrated by its application to the problem of a beam moving in a direction normal to its axis brought to rest by striking a rigid stop at its mid-point. It is shown that in the subsequent motion slope discontinuities travel outwards from the centre of the beam. A complete explicit solution is obtained for the case of a beam with linear strain-hardening.     

Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion

Two-dimensional unsteady separated flow past a semi-infinite plate with transverse motion is considered. The rolling-up of the separated shear-layer is modelled by a point vortex whose time-dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed along with a power law for the transverse motion. The effects of the motion of the plate on the starting vortex circulation and trajectory are presented. A suitable vortex shedding mechanism is introduced and a class of flows involving several vortices is presented. Finally, some possibilities for actively controlling the production of circulation by moving the plate are discussed.     

刃型位错和圆弧形刚性线夹杂的干涉效应

位错和夹杂的干涉效应对于理解材料的强化和韧化机理具有十分重要的意义。文中研究了晶体材料中刃型位错和多条共圆弧刚性线夹杂的干涉作用。利用Riemann—Schwarz反照原理和复势函数的奇性主部分析技术，得到了问题的一般解答；对于只含一条刚性线夹杂的情况，给出了复势函数的封闭形式解。由Peach-Koehler公式求出了作用在刃型位错上的位错力，并讨论了圆弧形刚性线夹杂对位错力的影响规律，发现弧形刚性线对刃型位错有很强的排斥作用。本文解答不但可作为格林函数获得任意分布位错的相应解答，而且可以用于研究刚性线夹杂和任意形状裂纹的干涉效应问题。