Field Dislocation Mechanics Research Articles

Emergent fault friction and supershear in a continuum model of geophysical rupture

Important physical observations in rupture dynamics such as static fault friction, short-slip, self-healing, and the supershear phenomenon in cracks are studied. A continuum model of rupture dynamics is developed using the field dislocation mechanics (FDM) theory. The energy density function in our model encodes accepted and simple physical facts related to rocks and granular materials under compression. We work within a 2-dimensional ansatz of FDM where the rupture front is allowed to move only in a horizontal fault layer sandwiched between elastic blocks. Damage via the degradation of elastic modulus is allowed to occur only in the fault layer, characterized by the amount of plastic slip. The theory dictates the evolution equation of the plastic shear strain to be a Hamilton–Jacobi (H-J) equation, resulting in the representation of a propagating rupture front. A Central-Upwind scheme is used to solve the H-J equation. The rupture propagation is fully coupled to elastodynamics in the whole domain, and our simulations recover static friction laws as emergent features of our continuum model, without putting in by hand any such discontinuous criteria in the formulation. Estimates of material parameters of cohesion and friction angle are deduced. Short-slip and slip-weakening (crack-like) behaviors are also reproduced as a function of the degree of damage behind the rupture front. The long-time behavior of a moving rupture front is probed, and it is deduced that equilibrium profiles under no shear stress are not traveling wave profiles under non-zero shear load in our model. However, it is shown that a traveling wave structure is likely attained in the limit of long times. Finally, a crack-like damage front is driven by an initial impact loading, and it is observed in our numerical simulations that an upper bound to the crack speed is the dilatational wave speed of the material unless the material is put under pre-stressed conditions, in which case supersonic motion can be obtained. Without pre-stress, intersonic (supershear) motion is recovered under appropriate conditions.

Coupling Phase Field Crystal and Field Dislocation Mechanics for a consistent description of dislocation structure and elasticity

This work addresses differences in predicted elastic fields created by dislocations either by the Phase Field Crystal (PFC) model, or by static Field Dislocation Mechanics (FDM). The PFC order parameter describes the topological content of the lattice, but it fails to correctly capture the elastic distortion. In contrast, static FDM correctly captures the latter but requires input about defect cores. The case of a dislocation dipole in two dimensional, isotropic, elastic medium is studied, and a weak coupling is introduced between the two models. The PFC model produces compact and stable dislocation cores, free of any singularity, i.e., diffuse. The PFC predicted dislocation density field (a measure of the topological defect content) is used as the source (input) for the static FDM problem. This coupling allows a critical analysis of the relative role played by configurational (from PFC) and elastic (from static FDM) fields in the theory, and of the consequences of the lack of elastic relaxation in the diffusive evolution of the PFC order parameter.

Atomistic to continuum mechanics description of crystal defects with dislocation density fields: Application to dislocations and grain boundaries

The atomic structure of crystal defects such as dislocations, grain or phase boundaries, control these defects’ properties: their mobility, ability to cross-slip, or solute segregation. These crystal defects can be conveniently studied by atomistic simulations and one then needs to transfer relevant information at the upper scale to model microstructures containing a large number of defects, e.g., a polycrystal. Here, we propose an atomistic to continuum mechanics crossover method that (i) represents the atomic structure of dislocations cores by an appropriate Nye dislocation density tensor field and (ii), captures quantitatively the short and long range mechanical fields of defects. For (i), we propose a modified and improved interpolation method based on the original work by Hartley and Mishin. For (ii), we use a field dislocation mechanics framework that rigorously calculates/evaluates the mechanical fields associated with any Nye dislocation density distribution. The transfer method relies on molecular static calculations using two energetic models — ab-initio for screw dislocation core simulations in tungsten, and EAM potential for low and large angle grain boundaries in copper. Our findings demonstrate the effectiveness of the proposed approach in reconstructing the Burgers vector, and continuous strain and rotation fields. The framework is further applied to analyze the elastic interactions between extrinsic edge dislocations and a low angle grain boundary in copper.

An FFT based adaptive grid framework to represent non-singular dislocations

Non-singular theories aim at regularizing the unrealistic stress singularity around a dislocation line predicted by elasticity by spreading the Burgers vector around the core. The use of these approaches in discrete dislocation models or field dislocation mechanics requires very fine discretization around the dislocation lines. FFT solvers commonly used in these problems rely on a regular grid which enforces the use of very fine discretizations or very small domains. In this work a methodology is proposed to solve the boundary value problems associated with dislocation modeling using FFT with an adaptive grid which is refined around dislocation cores. The framework introduces a regular domain to compute Fourier derivatives mapped to a physical domain with more points concentrated in the areas of interest. The linear differential operators involved are obtained by the product of the Fourier derivatives in the regular space and the Jacobian of the map defining the transformation between computational and physical domains. The periodic boundary value problem involved is transformed into a linear system that can be efficiently solved using Krylov solvers. The method for adaptive FFT grids is fully general and can be applied to any other micromechanical problem. It is demonstrated that the method allows to use several grid points within the core region of 2D and 3D dislocations even using coarse discretizations, allowing to resolve the non-singular stress fields within this region and also strongly reducing the numerical noise. Moreover, the method allows to preserve the accuracy of the results in fine meshes by reducing the grid spacing up to four times.

From electron tomography of dislocations to field dislocation mechanics: application to olivine

We propose a new procedure to extract information from electron tomography and use them as an input in a field dislocation mechanics. Dislocation electron tomography is an experimental technique that provides three-dimensional (3D) information on dislocation lines and Burgers vectors within a thin foil. The characterized 3D dislocation lines are used to construct the spatial distribution of the equivalent Nye dislocation density tensor. The model dislocation lattice incompatibility equation and stress balance equation are solved with a spectral code based on fast Fourier transform algorithms. As an output of the model, one obtains the 3D distribution of mechanical fields, such as strains, rotations, stresses, resolved shear stresses (RSSs) and energy, inside the material. To assess the potential of the method, we consider two regions from a previously compressed olivine sample. Our results reveal significant local variations in local stress fields and RSSs in various slip systems, which can impact the strong plastic anisotropy of olivine and the activation of different dislocation slip systems. It also evidences the built-up of kinematic hardening down to the nanometre scale.

Interface-Dominated Plasticity and Kink Bands in Metallic Nanolaminates

The theoretical and computational framework of finite deformation mesoscale field dislocation mechanics (MFDM) is used to understand the salient aspects of kink-band formation in Cu-Nb nano-metallic laminates (NMLs). A conceptually minimal, plane-strain idealization of the three-dimensional geometry, including crystalline orientation, of additively manufactured NML is used to model NMLs. Importantly, the natural jump/interface condition of MFDM imposing continuity of (certain components) of plastic strain rates across interfaces allows theory-driven ‘communication’ of plastic flow across the laminate boundaries in our finite element implementation. Kink bands under layer parallel compression of NMLs in accord with experimental observations arise in our numerical simulations. The possible mechanisms for the formation and orientation of kink bands are discussed, within the scope of our idealized framework. We also report results corresponding to various parametric studies that provide preliminary insights and clear questions for future work on understanding the intricate underlying mechanisms for the formation of kink bands.

Characterization of a nanopipe dislocation in GaN by means of HR-EBSD and field dislocation mechanics analysis

A nanopipe threading screw dislocation in a GaN layer is studied. A recently developed high-angular resolution electron backscatter diffraction technique, relying on a global image registration of Kikuchi patterns, is used to assess elastic strain and rotation fields in the defected area. The characterization is complemented with predictions from a piezoelectric field dislocation mechanics model of a threading screw dislocation line. In plane elastic fields are obtained at the free surface, which arise from the cancellation of the dislocation bulk shear stress field. The experimental and simulated long range fields agree qualitatively well and correspond to a screw dislocation with Burgers vector magnitude 3c = 1.56 nm.

A one-dimensional field dislocation mechanics model using discontinuous Galerkin method

A numerical solution strategy for a one-dimensional field dislocation mechanics (FDM) model using the Discontinuous Galerkin (DG) method is developed. The FDM model is capable of simulating the dynamics of discrete, nonsingular dislocations using a partial differential equation involving a conservation law for the Burgers vector content with constitutive input for nucleation and velocity. Modeling of individual dislocation lines with an equilibrium compact core structure in the context of this continuum elastoplastic framework requires a non-convex stored energy density. Permanent deformation and stress redistribution caused by the dissipative transport of dislocations is modeled using thermodynamics-based constitutive laws. A DG method is employed to discretize the evolution equation of dislocation density yielding high orders of accuracy when the solution is smooth. The trade-offs of using a high order explicit Runge–Kutta time stepping and an implicit–explicit scheme are discussed. The developed numerical scheme is used to simulate the transport of a single screw dislocation wall in the case of a non-zero applied strain.

Mechanics of micropillar confined thin film plasticity

Micropillar compression experiments probing size effects in confined plasticity of metal thin films, including the indirect imposition of ‘canonical’ simple shearing boundary conditions, show dramatically different responses in compression and shear of the film. The Mesoscale Field Dislocation Mechanics (MFDM) model is confronted with this set of experimental observations and shown to be capable of modeling such behavior, without any ad-hoc modification to the basic structure of the theory (including boundary conditions), or the use of extra fitting parameters. This is a required theoretical advance in the current state-of-the art of strain gradient plasticity models. It is also shown that significantly different inhomogeneous fields can display qualitatively similar size effect trends in overall agreement with the experimental results. The (plastic) Swift and (elastic) Poynting finite deformation effects are also demonstrated.

A numerical study of reversible plasticity using continuum dislocation mechanics

In this contribution, an elasto-viscoplastic fast Fourier transform-based (EVPFFT) numerical implementation of the Mesoscale Field Dislocation Mechanics (MFDM) formulation, called MFDM-EVPFFT, is applied to study the reversible plastic behavior of periodic two-phase crystalline composites with an elasto-viscoplastic plastic matrix and a purely elastic second phase. Periodic laminate microstructures of this kind with different periods (i.e. sizes) are considered to examine the size dependence of the Bauschinger effect and hardening during cyclic loading. Comparisons with classic composite effects obtained with conventional crystal plasticity are discussed. Specifically, the MFDM-EVPFFT results shed light on the hardening mechanisms due to piling-up/unpiling-up of geometrically-necessary dislocations (GND) during reverse loading.

Situating the Vector Density Approach Among Contemporary Continuum Theories of Dislocation Dynamics

Abstract For the past century, dislocations have been understood to be the carriers of plastic deformation in crystalline solids. However, their collective behavior is still poorly understood. Progress in understanding the collective behavior of dislocations has primarily come in one of two modes: the simulation of systems of interacting discrete dislocations and the treatment of density measures of varying complexity that are considered as continuum fields. A summary of contemporary models of continuum dislocation dynamics is presented. These include, in order of complexity, the two-dimensional statistical theory of dislocations, the field dislocation mechanics treating the total Kröner–Nye tensor, vector density approaches that treat geometrically necessary dislocations on each slip system of a crystal, and high-order theories that examine the effect of dislocation curvature and distribution over orientation. Each of theories contain common themes, including statistical closure of the kinetic dislocation transport equations and treatment of dislocation reactions such as junction formation. An emphasis is placed on how these common themes rely on closure relations obtained by analysis of discrete dislocation dynamics experiments. The outlook of these various continuum theories of dislocation motion is then discussed.

Experimental and numerical investigation of key microstructural features influencing the localization of plastic deformation in Fe-TiB2 metal matrix composite

A new generation of iron-based matrix composite reinforced by TiB2 particles was deformed in tension to investigate at a mesoscopic scale the localization of plastic deformation in relation with characteristic microstructural features of the composite (in particular, ferrite grain boundaries and particle/matrix interfaces). Large electron back-scattered diffraction (EBSD) maps with improved angular resolution were acquired to evaluate statistically the evolution of the geometrically necessary dislocation (GND) density from the early stage of the deformation. GNDs were found to accumulate preferentially at matrix/particle interfaces with hot-spots located at the tips of elongated particles. Additional length-scale parameters derived from EBSD data evidenced the key influence of two main microstructural features: the particle morphology and the particle clustering. Finally, we present results of an advanced full-field micromechanical model that is best suited to capture these effects, based on an enhanced crystal plasticity elasto-viscoplastic Fast Fourier Transform (EVP-FFT) formulation coupled with a phenomenological continuum Mesoscale Field Dislocation Mechanics (MFDM) theory. By taking the experimental TiB2 particle distribution into account, the model describes qualitatively the observed effect of particle morphological features on the heterogenous distribution of GNDs.Graphical abstract

On the computational solution of vector-density based continuum dislocation dynamics models: A comparison of two plastic distortion and stress update algorithms

Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan's law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test was first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions. By considering dislocation interactions caused by junctions, a hardening rate similar to discrete dislocation dynamics simulation results was obtained. The simulations show that dislocations exhibit some self-organized structures as the strain is increased.

Spectral discrete dislocation dynamics with anisotropic short range interactions

Discrete dislocation dynamics models dislocation-mediated plastic deformation and microstructural evolution in solids whilst discretely representing and tracking all line defects present in the medium. Hybrid methodologies for evaluating mechanical fields using continuum approaches while retaining the discrete formalism for evaluating dislocation kinematics and interactions have greatly improved the numerical efficiency of the method. In the present work, we propose a spectrally-based continuum solution of the mechanical fields based on the field dislocation mechanics framework which produces highly accurate elastic fields near dislocation cores in both isotropic and anisotropic media. Application of the model to a few elementary cases (i.e. dislocation junction formation, dislocation migration) show that the model proposed predicts accurate elastic fields both on the short and long range.

Periodic smoothing splines for FFT-based solvers

The aim of this paper is to develop a periodic smoother based on splines for FFT-based solvers. Spurious oscillations in FFT-based methods are shown to be due to pseudo-spectral differentiation of discontinuous fields. An automatic smoother based on polynomial splines is developed, which permits to add smoothness to initial material properties. The method, which is applied in various problems including conductivity, elasticity and field dislocation mechanics, improves significantly the local fields and reduces spurious oscillations.

Revisiting the Application of Field Dislocation and Disclination Mechanics to Grain Boundaries

We review the mechanical theory of dislocation and disclination density fields and its application to grain boundary modeling. The theory accounts for the incompatibility of the elastic strain and curvature tensors due to the presence of dislocations and disclinations. The free energy density is assumed to be quadratic in elastic strain and curvature and has nonlocal character. The balance of loads in the body is described by higher-order equations using the work-conjugates of the strain and curvature tensors, i.e., the stress and couple-stress tensors. Conservation statements for the translational and rotational discontinuities provide a dynamic framework for dislocation and disclination motion in terms of transport relationships. Plasticity of the body is therefore viewed as being mediated by both dislocation and disclination motion. The driving forces for these motions are identified from the mechanical dissipation, which provides guidelines for the admissible constitutive relations. On this basis, the theory is expressed as a set of partial differential equations where the unknowns are the material displacement and the dislocation and disclination density fields. The theory is applied in cases where rotational defects matter in the structure and deformation of the body, such as grain boundaries in polycrystals and grain boundary-mediated plasticity. Characteristic examples are provided for the grain boundary structure in terms of periodic arrays of disclination dipoles and for grain boundary migration under applied shear.

On the thermo-mechanical theory of field dislocations in transient heterogeneous temperature fields

A strong coupling between the field theory of dislocation mechanics and heat conduction is proposed. The novel model, called the Thermal Field Dislocation Mechanics (T-FDM) model, is designed to study the dynamics of dislocations during rapid or gradual temperature changes in a body having a heterogeneous temperature distribution; for example, such conditions occur in a heat-affected crystalline solid during an additive manufacturing process. Thermal strains are uniquely separated into compatible and incompatible components via the Stokes-Helmholtz decomposition and the curl of the incompatible part of thermal strains is directly related to the areal dislocation density tensor. A dislocation density evolution (including transport) law is developed and shown to be related to the evolution of the curl of incompatible thermal strains. This relationship demonstrates that dislocation generation, annihilation, motion and/or interactions with other defects can be triggered due to transient temperature changes, and conversely an evolving dislocation density induces temperature changes. The model development is completed with constitutive laws derived from energetic and dissipative considerations. The advantages and consequences of the assumptions of the T-FDM model under rapidly changing temperatures, both spatially and temporally, are discussed. The T-FDM model is intended for application at the length scale where individual dislocations can be characterized. At this level, local thermodynamic equilibrium is found to be a reasonable assumption even for high rates of change of temperature such as those occurring during an additive manufacturing process. Some illustrative examples are presented to demonstrate the applicability of the model and to better understand some of the novel concepts proposed in this work.

Field dislocation mechanics and phase field crystal models

A new formulation of the Phase Field Crystal model is presented that is consistent with the necessary microscopic independence between the phase field, reflecting the broken symmetry of the phase, and both mass density and elastic distortion. Although these quantities are related in equilibrium through a macroscopic equation of state, they are independent variables in the free energy, and can be independently varied in evaluating the dissipation functional that leads to the model governing equations. The equations obtained describe dislocation motion in an elastically stressed solid, and serve as an extension of the equations of plasticity to the Phase Field Crystal setting. Both finite and small deformation theories are considered, and the corresponding kinetic equations for the fields derived.

Finite element approximation of finite deformation dislocation mechanics

We develop and demonstrate the first general computational tool for finite deformation static and dynamic dislocation mechanics. A finite element formulation of finite deformation (Mesoscale) Field Dislocation Mechanics theory is presented. The model is a minimal enhancement of classical crystal/J2 plasticity that fundamentally accounts for polar/excess dislocations at the mesoscale. It has the ability to compute the static and dynamic finite deformation stress fields of arbitrary (evolving) dislocation distributions in finite bodies of arbitrary shape and elastic anisotropy under general boundary conditions. This capability is used to present a comparison of the static stress fields, at finite and small deformations, for screw and edge dislocations, revealing heretofore unexpected differences. The computational framework is verified against the sharply contrasting predictions of geometrically linear and nonlinear theories for the stress field of a spatially homogeneous dislocation distribution in the body, as well as against other exact results of the theory. Verification tests of the time-dependent numerics are also presented. Size effects in crystal and isotropic versions of the theory are shown to be a natural consequence of the model and are validated against available experimental data. With inertial effects incorporated, the development of an (asymmetric) propagating Mach cone is demonstrated in the finite deformation theory when a dislocation moves at speeds greater than the linear elastic shear wave speed of the material.

Particle interspacing effects on the mechanical behavior of a Fe\u2013TiB2 metal matrix composite using FFT-based mesoscopic field dislocation mechanics

This paper presents an application to metal matrix composites (MMCs) of an enhanced elasto-viscoplastic Fast Fourier Transform (EVP-FFT) formulation coupled with a phenomenological continuum Mesoscale Field Dislocation Mechanics (MFDM) theory. Contrary to conventional crystal plasticity, which only accounts for plastic flow and hardening induced by statistically stored dislocations (SSDs), MFDM-EVP-FFT also describes the evolution of polarized geometrically necessary dislocation (GND) density and its effect on both plastic flow and hardening. Numerical results for a Fe–TiB2 MMC made of a ferrite matrix (α-Fe) and elastic ceramic particles (TiB2) are presented. Full-field simulations are performed using synthetic periodic unit cells representative of the MMC, with single-crystalline and polycrystalline matrix, for different particle interspacing distances. A strong dependence of the predicted equivalent stress, cumulated plastic strain and GND density fields with particle interspacing distance is observed, in contrast with conventional crystal plasticity. Correlations between these mechanical fields and microstructural features, and their influence on local and global mechanical behavior are examined for the different MMC microstructures.