Continuum dynamics of the formation,migration and dissociation of self-locked dislocation structures on parallel slip planes

In continuum models of dislocations, proper formulations of short-range elastic interactions of dislocations are crucial for capturing various types of dislocation patterns formed in crystalline materials. In this article, the continuum dynamics of straight dislocations distributed on two parallel slip planes is modelled through upscaling the underlying discrete dislocation dynamics. Two continuum velocity field quantities are introduced to facilitate the discrete-to-continuum transition. The first one is the local migration velocity of dislocation ensembles which is found fully independent of the short-range dislocation correlations. The second one is the decoupling velocity of dislocation pairs controlled by a threshold stress value, which is proposed to be the effective flow stress for single slip systems. Compared to the almost ubiquitously adopted Taylor relationship, the derived flow stress formula exhibits two features that are more consistent with the underlying discrete dislocation dynamics: (i) the flow stress increases with the in-plane component of the dislocation density only up to a certain value, hence the derived formula admits a minimum inter-dislocation distance within slip planes; (ii) the flow stress smoothly transits to zero when all dislocations become geometrically necessary dislocations. A regime under which inhomogeneities in dislocation density grow is identified, and is further validated through comparison with discrete dislocation dynamical simulation results. Based on the findings in this article and in our previous works, a general strategy for incorporating short-range dislocation correlations into continuum models of dislocations is proposed.     

A non-singular continuum theory of dislocations

We develop a non-singular, self-consistent framework for computing the stress field and the total elastic energy of a general dislocation microstructure. The expressions are self-consistent in that the driving force defined as the negative derivative of the total energy with respect to the dislocation position, is equal to the force produced by stress, through the Peach-Koehler formula. The singularity intrinsic to the classical continuum theory is removed here by spreading the Burgers vector isotropically about every point on the dislocation line using a spreading function characterized by a single parameter a, the spreading radius. A particular form of the spreading function chosen here leads to simple analytic formulations for stress produced by straight dislocation segments, segment self and interaction energies, and forces on the segments. For any value a>0, the total energy and the stress remain finite everywhere, including on the dislocation lines themselves. Furthermore, the well-known singular expressions are recovered for a=0. The value of the spreading radius a can be selected for numerical convenience, to reduce the stiffness of the dislocation equations of motion. Alternatively, a can be chosen to match the atomistic and continuum energies of dislocation configurations.     

Analysis of dislocation nucleation from a crystal surface based on the Peierls-Nabarro dislocation model

Dislocation nucleation from a stressed crystal surface is analyzed based on the Peierls-Nabarro dislocation model. The variational boundary integral approach is used to obtain the profiles of the embryonic dislocations in various three-dimensional nucleation configurations. The stress-dependent activation energies required to activate dislocations from their stable to unstable saddle point configurations are determined. Compared to previous analyses of this type of problem based on continuum elastic dislocation theory, the present analysis eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Moreover, atomic information can be incorporated to reveal the dependence of the nucleation process on the profile of the atomic interlayer potential as compared to continuum elastic dislocation theory in which only elastic constants and Burgers vector are relevant. Finally, the presented methodology can also be readily used to study dislocation nucleation from the surface heterogeneities such as cracks, steps, and quantum structures of electronic devices.     

Continuum framework for dislocation structure,energy and dynamics of dislocation arrays and low angle grain boundaries

We present a continuum framework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries that are allowed to be nonplanar or nonequilibrium. In our continuum framework, we define a dislocation density potential function on the dislocation array surface or grain boundary to describe the orientation dependent continuous distribution of dislocations in a very simple and accurate way. The continuum formulations incorporate both the long-range dislocation interaction and the local dislocation line energy, and are derived from the discrete dislocation model. The continuum framework recovers the classical Read–Shockley energy formula when the long-range elastic fields of the low angle grain boundaries are canceled out. Applications of our continuum framework in this paper are focused on dislocation structures on static planar and nonplanar low angle grain boundaries and misfitting interfaces. We present two methods under our continuum framework for this purpose, including the method based on the Frank׳s formula and the energy minimization method. We show that for any (planar or nonplanar) low angle grain boundary, the Frank׳s formula holds if and only if the long-range stress field in the continuum model is canceled out, and it does not necessarily hold for a total energy minimum dislocation structure.     

A multi-scale computational model of crystal plasticity at submicron-to-nanometer scales

Plastic flow in crystal at submicron-to-nanometer scales involves many new interesting problems. In this paper, a unified computational model which directly combines 3D discrete dislocation dynamics (DDD) and continuum mechanics is developed to investigate the plastic behaviors at these scales. In this model, the discrete dislocation plasticity in a finite crystal is solved under a completed continuum mechanics framework: (1) an initial internal stress field is introduced to represent the preexisting stationary dislocations in the crystal; (2) the external boundary condition is handled by finite element method spontaneously; and (3) the constitutive relationship is based on the finite deformation theory of crystal plasticity, but the discrete plastic strains induced by the slip of the newly nucleated or propagating dislocations are calculated by dislocation dynamics methodology instead of phenomenological evolution equations used in conventional crystal plasticity. These discrete plastic strains are then localized to the continuum material points by a Burgers vector density function proposed by us. Various processes, such as loop dislocation evolution, dislocation junction formation etc., are simulated to verify the reliability of this computational model. Specifically, a uniaxial compression test for micro-pillars of Cu is simulated by this model to investigate the ‘dislocation starvation hardening’ observed in the recent experiment.     

Study of size effects in thin films by means of a crystal plasticity theory based on DiFT

In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide given by Yefimov et al. [2004b. A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity simulations. J. Mech. Phys. Solids 52, 279-300]. In this dislocation field theory (DiFT) the transport equations for both the total dislocation density and geometrically necessary dislocation (GND) density on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested itself in the form of a back stress in addition to the external shear and the self-consistent internal stress. We here present the study of size effects in single crystalline thin films with symmetric double slip using the novel continuum theory. Two boundary value problems are analyzed: (1) stress relaxation in thin films on substrates subject to thermal loading, and (2) simple shear in constrained films. In these problems, earlier discrete dislocation simulations had shown that size effects are born out of layers of dislocations developing near constrained interfaces. These boundary layers depend on slip orientations and applied loading but are insensitive to the film thickness. We investigate the stress response to changes in controlled parameters in both problems. Comparisons with previous discrete dislocation simulations are discussed.     

Effect of source and obstacle strengths on yield stress: A discrete dislocation study

The combined effect of dislocation source strength τ_s, dislocation obstacle strength τ_obs, and obstacle spacing L_obs on the yield stress of single crystal metals is investigated analytically and numerically. A continuum theory of dislocation pileups emanating from a finite-strength source and impinging on asymmetric obstacles gives a closed-form expression for the yield stress. A 2d discrete dislocation model for a single-source/obstacle problem agrees well with the analytic model over a wide range of material parameters. Discrete dislocation simulations for a full tensile bar with statistically distributed sources and obstacles show that the distribution of obstacles plays a significant role in controlling the yield stress. Over a wide range of parameters, the simulations agree well with the analytic model using an effective obstacle spacing L_obs^* chosen to capture the strength-controlling statistically weaker pileup configurations. The analytic model can thus be used to guide the choice of source and obstacle parameters to obtain a desired yield stress. The model also shows how different combinations of internal source and obstacle parameters can generate the same macroscopic yield stress, and points to several internal length scales that could relate to size-dependent plasticity phenomena.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I

A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the positive-definite tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations.Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.     

An analysis of key characteristics of the Frank-Read source process in FCC metals

A well-known intragranular dislocation source, the Frank-Read (FR) source plays an important role in size-dependent dislocation multiplication in crystalline materials. Despite a number of studies in this topic, a systematic investigation of multiple aspects of the FR source in different materials is lacking. In this paper, we employ large scale quasistatic concurrent atomistic-continuum (CAC) simulations to model an edge dislocation bowing out from an FR source in Cu, Ni, and Al. First, a number of quantities that are important for the FR source process are quantified in the coarse-grained domain. Then, two key characteristics of the FR source, including the critical shear stress and critical dislocation configuration, are investigated. In all crystalline materials, the critical stresses and the aspect ratio of the dislocation half-loop height to the FR source length scale well with respect to the FR source length. In Al, the critical stress calculated by CAC simulations for a given FR source length agrees reasonably well with a continuum model that explicitly includes the dislocation core energy. Nevertheless, the predictions of the isotropic elastic theory do not accurately capture the FR source responses in Cu and Ni, which have a relatively large stacking fault width and elastic anisotropy. Our results highlight the significance of directly simulating the FR source activities using fully 3D models and shed light on developing more accurate continuum models.     

A model for the deformations and thermodynamics of liquids involving a dislocation kinetics

A model for the deformation and thermodynamics of liquids is developed that depends on dislocation kinetics. The approach uses concepts from statistical mechanics to model a stochastic evolution equation for a scalar dislocation density function. The dislocation density is used in an idealized model for the discrete discontinuous deformation due to dislocation motion and dislocation creation kinetics. The total deformation functional for a liquid is modelled as a continuum deformation of an idealized lattice structure plus the discontinuous deformation due to dislocation kinetics. This results in a thermodynamic model that has an elastic response from the continuum lattice structure and a fluid response from the dislocation kinetics.In the thermodynamics, a generalized internal energy functional is assumed to exist and to have a dependence on the functions of entropy, continuum lattice strain, scalar dislocation density, velocity, and mass density. The continuum lattice strain is termed the recoverable strain and its conjugate variable is the thermodynamic stress. The conjugate variable to the scalar dislocation density is the thermodynamic chemical potential for a dislocation configuration, somewhat analogous to Gibbs' treatment of chemical potential for various mass species.This model implies that a liquid and a crystalline solid have analogous deformation and thermodynamic responses. Their differences appear in the dislocation densities and in the dislocation chemical potentials. To illustrate the deformation response analogy, some solutions are developed for simple laminar shear flows. Also, using some concepts primarily from Kuhlmann-Wilsdorf's melting model, a definition for a specific dislocation creation heat equivalent is given. This thermodynamic formalism suggests that the melting process can be modelled as the consequence of a continuous change in the dislocation density function.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48.     

Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div-curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.     

A screw dislocation by nonlinear continuum mechanics

ＡＳＣＲＥＷＤＩＳＬＯＣＡＴＩＯＮＢＹＮＯＮＬＩＮＥＡＲＣＯＮＴＩＮＵＵＭＭＥＣＨＡＮＩＣＳＰａｎＫｅ－ｌｉｎ（潘客麟）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉｅｓ．ＴｏｎｇｊｉＵｎｉｖｅｒｓｉｔｙ,Ｓｈａｎｇｈａｉ）ＣｈｅｎＺ...     

Dislocation nucleation and work hardening in anti-plane constrained shear

The paper aims at studying the dislocation nucleation, the corresponding work hardening and the influence of resistance to the dislocation motion within the framework of continuum theory of dislocations. We consider an anti-plane constrained shear problem which admits an analytical solution. The interesting features of this solution are the energy and dissipation thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. The financial support by the DFG (German Science Foundation) within the Collaborative Research Center 526 (project D9) for K.C. Le is gratefully acknowledged     

A stress-gradient based criterion for dislocation nucleation in crystals

Dislocations are the main lattice defects responsible for the strength and ductility of crystalline solids. The generation of new dislocations is an essential aspect of crystal defect physics, but a fundamental understanding of the mechanical conditions which lead to dislocation nucleation has remained elusive. Here, we present a nucleation criterion motivated from continuum thermomechanical considerations of a crystalline solid undergoing deformation, and demonstrate the criterion's ability to correctly predict dislocation nucleation via direct atomistic simulations. We further demonstrate that the commonly held notion of a nucleation criterion based on the magnitude of local stress components is incorrect.     

Theoretical and numerical investigations on confined plasticity in micropillars

Multiscale dislocation dynamic simulations are systematically carried out to reveal the dislocation mechanism controlling the confined plasticity in coated micropillar. It is found that the operation of single arm source (SAS) controls the plasticity in coated micropillar and a modified operation stress equation of SAS is built based on the simulation results. The back stress induced by the coating contributes most to the operation stress and is found to linearly depend on the ‘trapped dislocation’ density. This linear relation is verified by comparing with the solution of the current higher-order crystal plasticity theory and is used to determine the material parameters in the continuum back stress model. Furthermore, based on the linear back stress model and considering the stochastic distribution of SAS, a theoretical model is established to predict the upper and lower bound of stress–strain curve in the coated micropillars, which agrees well with that obtained in the dislocation dynamic simulation.     

Constitutive analysis of finite deformation field dislocation mechanics

Driving forces for dislocation motion and nucleation in finite-deformation field dislocation mechanics are derived. The former establishes a rigorous analog of the Peach-Koehler force of classical elastic dislocation theory in a nonlinear, nonequilibrium field-theoretic context; the latter is a prediction of the theory. The structure of the stress response and permanent distortion are also derived. Sufficient boundary and initial conditions are indicated, and invariance under superposed rigid motions is discussed. Hyperelasticity and finite-deformation elastic theory of dislocations are shown to be special cases of the framework. Owing to the nonlocal nature of the theory, the results as well as the methods used to derive them appear to be novel.     

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations resulting from a piecewise quadratic Peierls potential; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles and lattice friction, resulting in hardening, path dependency and hysteresis. A chief advantage of the present theory is that it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. In particular, no numerical grid is required in calculations. The phase-field representation enables complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops. The theory also permits the consideration of obstacles of varying strengths and dislocation line-energy anisotropy. The theory predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect, whereby reverse yielding gradually eliminates the influence of previous loading; the evolution of the dislocation density under cycling loading, leading to characteristic ‘butterfly’ curves; and others.     

A Physical theory of asymmetric plasticity

Experiments have shown the strong rotation in plastic deformation, which is caused by the disclination, specific arrangement of dislocation and inhomogeneity of the gliding motion of the defects in the microscopic scale. Based on the microscopic mechanism of the rotational plastic deformation, the conservation equation satisfied by the defects motion (dislocation and disclination) has been developed in this paper. Then the diffusion motion of the defects are reduced based on the asymmetric theory of continuum mechanics. By utilizing the maximization procedure for the micro plastic work and a scale-invariance argument, various models of Cosserat-type plasticity are obtained in this manner.