A Variational Model for Dislocations in the Line Tension Limit

We study the interaction of a singularly-perturbed multiwell energy (with an anisotropic nonlocal regularizing term of H ^1/2 type) and a pinning condition. This functional arises in a phase field model for dislocations which was recently proposed by Koslowski, Cuitiño and Ortiz, but it is also of broader mathematical interest. In the context of the dislocation model we identify the Γ-limit of the energy in all scaling regimes for the number N _? of obstacles. The most interesting regime is N _? ≈|ln ?|/?, where ? is a nondimensional length scale related to the size of the crystal lattice. In this case the limiting model is of line tension type. One important feature of our model is that the set of energy wells is periodic, and hence not compact. Thus a key ingredient in the proof is a compactness estimate (up to a single translation) for finite energy sequences, which generalizes earlier results from Alberti, Bouchitté and Seppecher for the two-well problem with a H ^1/2 regularization.     

Image decomposition method for the analysis of a mixed dislocation in a general multilayer

The elastic solutions for a mixed dislocation in a general multilayer with N dissimilar anisotropic layers are obtained via a generalized image decomposition method. The original problem is decomposed into N homogeneous subproblems with strategically placed continuously distributed image (virtual) dislocations which satisfy the consistency conditions for degenerate N − M (M < N) layer problems. The image dislocations are used to satisfy the interface or free surface conditions, and represent the unknowns of the problem. The resulting singular Cauchy integral equations are transformed into non-singular Fredholm integral equations of the second kind using certain H- and I-integral transforms. The Fredholm integral equations are then solved via the classical Nyström method. The general decomposition and the elimination of all singular integrals yield an exact formulation of the problem; the approximation arises only in the Nyström method. The dislocation mixity and the number of layers dissimilar in thickness and elastic anisotropy can be handled without difficulty, constrained only by the number of linear algebraic equations in the Nyström method for large N. For the numerical study, image forces on a dislocation in two- and three-layer systems are calculated. The accuracy of the results is verified by checking the boundary conditions and by comparison with previous results. The dependence of the image force on the dislocation position and mixity, and on the layer thicknesses and elastic anisotropies, is also illustrated via numerical investigations.     

The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

A screw dislocation in a functionally graded material using the translation gauge theory of dislocations

The aim of this paper is to provide new results and insights for a screw dislocation in functionally graded media within the gauge theory of dislocations. We present the equations of motion for dislocations in inhomogeneous media. We specify the equations of motion for a screw dislocation in a functionally graded material. The material properties are assumed to vary exponentially along the x and y-directions. In the present work we give the analytical gauge field theoretic solution to the problem of a screw dislocation in inhomogeneous media. Using the dislocation gauge approach, rigorous analytical expressions for the elastic distortions, the force stresses, the dislocation density and the pseudomoment stresses are obtained depending on the moduli of gradation and an effective intrinsic length scale characteristic for the functionally graded material under consideration.     

High-temperature discrete dislocation plasticity

A framework for solving problems of dislocation-mediated plasticity coupled with point-defect diffusion is presented. The dislocations are modeled as line singularities embedded in a linear elastic medium while the point defects are represented by a concentration field as in continuum diffusion theory. Plastic flow arises due to the collective motion of a large number of dislocations. Both conservative (glide) and nonconservative (diffusion-mediated climb) motions are accounted for. Time scale separation is contingent upon the existence of quasi-equilibrium dislocation configurations. A variational principle is used to derive the coupled governing equations for point-defect diffusion and dislocation climb. Superposition is used to obtain the mechanical fields in terms of the infinite-medium discrete dislocation fields and an image field that enforces the boundary conditions while the point-defect concentration is obtained by solving the stress-dependent diffusion equations on the same finite-element grid. Core-level boundary conditions for the concentration field are avoided by invoking an approximate, yet robust kinetic law. Aspects of the formulation are general but its implementation in a simple plane strain model enables the modeling of high-temperature phenomena such as creep, recovery and relaxation in crystalline materials. With emphasis laid on lattice vacancies, the creep response of planar single crystals in simple tension emerges as a natural outcome in the simulations. A large number of boundary-value problem solutions are obtained which depict transitions from diffusional to power-law creep, in keeping with long-standing phenomenological theories of creep. In addition, some unique experimental aspects of creep in small scale specimens are also reproduced in the simulations.     

Dislocation core spreading at interfaces between metal films and amorphous substrates

Experiments with transmission electron microscopy have shown that in a strong electron beam the contrast of dislocations may gradually disappear at an incoherent interface between a metal thin film and an amorphous substrate. There are reasons to believe that this phenomenon is caused by radiation-induced dislocation core spreading at the interface. A quantitative model accounting for this effect will be necessary for a better understanding of dislocation structures and plastic deformation in metal thin films. As a first step toward this objective, we develop a number of mathematical solutions for dislocation core spreading at an incoherent interface. For simplicity, we consider screw dislocations, and consider the interface to be characterized by a shear adhesive strength, τ₀, below which no core spreading occurs, and above which spreading takes place in a viscous manner. We determine the final equilibrium core width and the rate of core spreading for single or planar arrays of dislocations in a homogeneous bulk material or at the interface between a thin film and a semi-infinite substrate where the film and substrate may have the same, or different, elastic constants. Some of our solutions are analytic and others are based on an implicit finite difference method with a Gauss-Chebyshev quadrature scheme. The phenomenon of dislocation core spreading is expected to have a dramatic effect on the strength of crystalline films deposited on amorphous substrates.     

Viscoplastic flow rule for dislocation-mediated plasticity from systematic coarse-graining

The plastic response of metals is determined by the collective, coarse-grained dynamics of dislocations, rather than by the dynamics of individual dislocations. The evolution equations at both levels are quite different, for example considering their dependence on the applied mechanical load. On the one hand, the relation between the configurational force and dislocation velocity for individual dislocations is linear. On the other hand, in phenomenological crystal plasticity models, the relation between load and plastic slip is highly non-linear and often taken of power-law form. In this work, it is shown that this difference is justified and a consequence of emergent effects. Previously, an expression for the macroscopic dislocation flux was derived by systematic coarse graining (Kooiman et al., 2015). This expression has been evaluated numerically in this paper. The resulting relation between dislocation flux and applied mechanical load is found to be of power-law form with an exponent 3.7, while the underlying Discrete Dislocation Dynamics has a linear flux–load relation.     

Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x _i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β _n , corresponding to ${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$ , and ${\beta_n \gg 1}$ , and the two critical regimes β _n ~ 1/n and β _n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter β _n .     

The effects of couple stresses on dislocation strain energy

The correspondence theorem which relates the solutions of displacement boundary value problems in classical and couple stress elasticity is formulated and applied to derive the solutions for edge and screw dislocations in an infinite medium. The effects of couple stresses on the dislocation strain energy is evaluated for both types of dislocations. It is shown that within a radius of influence of each dislocation in a metallic crystal with dislocation density of 10¹⁰ cm⁻², the strain energy contribution from couple stresses (excluding the core energy) is about 15% in the case of an edge dislocation, and about 11% in the case of a screw dislocation. It is then shown that couple stresses make large effect on the total work of tractions acting on the dislocation core surface.     

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.     

A dislocation mechanics-based crystallographic model of a B2-type intermetallic alloy

A crystallographic slip based model for cubic oriented NiAl single crystals is derived from an idealization of the dislocation network observed in the active slip systems, viz. {110} 〈110〉. The crystallographic model successfully accounts for the cyclic steady-state behaviour of crystals subjected to strain histories within the range ϵ_〈100〉 = ϵ_m ± 0.5%, for ϵ_m = 0 and 35%, at 750 and 850°C. It accurately predicts the flow stress dependence on temperature, strain rate and dislocation density arising from the lattice resistance to dislocation motion and from discrete obstacle resistance due to dislocation interactions. The kinematic and isotropic hardening modes associated with defect trails left behind by gliding dislocations and dislocation storage, respectively, are properly represented. The average distance that dislocations have to glide for their density to increase beyond the level needed to balance dynamic recovery processes was predicted to be approximately 260 times the random forest dislocation spacing. Measured dislocation densities at different mean strains were found to be consistent with the predictions of the theoretical model.     

Simulation for surface self-nanocrystallization under shot peening

Driven by high frequency and multi-directional shot peens, dislocations of various orientations proliferate into the metal, and accumulate in high density in the surface layer of a shallow depth. Migration, generation and annihilation of dislocations dictate the evolution of mobile dislocation density. Simulation for the experiment of pure iron under repeated shot peen flux of 800 times per square millimeter is carried out, and a dislocation density up to 2.17×10¹¹ mm⁻² is achieved. Dislocations of such density in the surface layer are shown to be capable of forming nano-grains whose size is about 10 nm. Molecular dynamics simulation verifies the formation of nano-grained metals at such dislocation density level. The dislocations are first regrouped to form subcrystallites, then combined to form stable nanocrystallized grains after sufficiently long time of relaxation. The project supported by the National Natural Science Foundation of China (10121202)     

On XFEM applications to dislocations and interfaces

A method for modelling dislocations in systems with arbitrary materials interfaces is described. The method is based on the extended finite element method (XFEM) where dislocations are modelled in the manner of the Volterra dislocation model. A method for calculating the Peach–Koehler force by J-integrals in this framework is studied. The method is compared to closed form solutions for interface problems and excellent accuracy is obtained. The convergence and accuracy of the method is studied in two problems where analytical solutions are available: an edge dislocation interacting with a free-surface and an edge dislocation interacting with a bimaterial interface. The applicability of the method to more complicated problems is illustrated by the modelling of slip misorientation of an edge dislocation with a glide plane intersecting a material interface and dislocations in a multi-material domain with non-parallel interfaces.     

Screw dislocation pile-ups in two phase materials of finite rigidity

The arrangement of discrete screw dislocations piled-up under the action of a uniform applied stress against the welded interface between different elastically isotropic half-spaces has been determined by representing the pile-up as a continuous distribution of infinitesimal dislocations. The dislocation slip plane is inclined at an arbitrary angle $\text{1}\text{2}\text{aπ}$ to the normal to the interface, assuming a to be a rational number. The singular integral equation expressing the condition for static equilibrium of the dislocations under a constant applied stress is solved by a method based on the Wiener-Hoph technique with the Mellin transform, and from this solution the mean density of dislocations and the stress field of the pile-up are determined.     

Determination of interaction forces between parallel dislocations by the evaluation of J integrals of plane elasticity

The Peach–Koehler expressions for the glide and climb components of the force exerted on a straight dislocation in an infinite isotropic medium by another straight dislocation are derived by evaluating the plane and antiplane strain versions of J integrals around the center of the dislocation. After expressing the elastic fields as the sums of elastic fields of each dislocation, the energy momentum tensor is decomposed into three parts. It is shown that only one part, involving mixed products from the two dislocation fields, makes a nonvanishing contribution to J integrals and the corresponding dislocation forces. Three examples are considered, with dislocations on parallel or intersecting slip planes. For two edge dislocations on orthogonal slip planes, there are two equilibrium configurations in which the glide and climb components of the dislocation force simultaneously vanish. The interactions between two different types of screw dislocations and a nearby circular void, as well as between parallel line forces in an infinite or semi-infinite medium, are then evaluated.     

High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: Geometrically necessary dislocation densities

Experimental studies on indentation into face-centered cubic (FCC) single crystals such as copper and aluminum were performed to reveal the spatially resolved variation in crystal lattice rotation induced due to wedge indentation. The crystal lattice curvature tensors of the indented crystals were calculated from the in-plane lattice rotation results as measured by electron backscatter diffraction (EBSD). Nye's dislocation density tensors for plane strain deformation of both crystals were determined from the lattice curvature tensors. The least L²-norm solutions to the geometrically necessary dislocation densities for the case in which three effective in-plane slip systems were activated in the single crystals associated with the indentation were determined. Results show the formation of lattice rotation discontinuities along with a very high density of geometrically necessary dislocations.     

Continuum and discrete models of dislocation pile-ups. I. Pile-up at a lock

A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip plane under the action of an external force in the direction of a locked dislocation in that plane is considered. As n→∞ there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away.     

Unsteady gravity-driven slender rivulets of a power-law fluid

Unsteady gravity-driven flow of a thin slender rivulet of a non-Newtonian power-law fluid on a plane inclined at an angle α to the horizontal is considered. Unsteady similarity solutions are obtained for both converging sessile rivulets (when 0 < α < π/2) in the case x < 0 with t < 0, and diverging pendent rivulets (when π/2 < α < π) in the case x > 0 with t > 0, where x denotes a coordinate measured down the plane and t denotes time. Numerical and asymptotic methods are used to show that for each value of the power-law index N there are two physically realisable solutions, with cross-sectional profiles that are ‘single-humped’ and ‘double-humped’, respectively. Each solution predicts that at any time t the rivulet widens or narrows according to |x | ^{(2N+1)/2(N+1)} and thickens or thins according to |x | ^N/(N+1) as it flows down the plane; moreover, at any station x, it widens or narrows according to |t | ^?N/2(N+1) and thickens or thins according to |t | ^?N/(N+1). The length of a truncated rivulet of fixed volume is found to behave according to |t | ^N/(2N+1).     

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The non-linear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0<t^*<1 and reach the steady-state values for t^*≥4.