Continuum Dislocation Dynamics Theory Research Articles

Total Lagrange implementation of a finite-deformation continuum dislocation dynamics model of mesoscale plasticity

We present a computational algorithm for solving the recently developed finite-deformation continuum dislocation dynamics theory of mesoscale plastic deformation of single crystals (Starkey et al., 2020). This CDD theory is based on a vector density representation of dislocations governed by curl-type transport-reaction equations subjected to the divergence-free constraint of the appropriate dislocation density. These density evolution equations are to be solved simultaneously with the finite-deformation crystal mechanics. Specifically, our algorithm aims to solve the referential form of the governing equations for a representative volume element (RVE) subject to remote uniform loading. The mechanical fields at the mesoscale are thus split into RVE-averages plus fluctuating components and treated using a strain-driven homogenization scheme. A virtual work-based total Lagrange formulation was used to discretize the governing mechanics equations. A first-order system least squares finite element formulation was used to solve the transport equations. The two schemes are coupled in a staggered fashion. As a part of the crystal mechanics discretization, we derive a consistent tangent modulus and show that the stress update for this model is both linear and global. This linear stress update comes at the cost of solving the dislocation transport equations at every time step to update the plastic distortion caused by dislocation motion. Several test problems are given, demonstrating the ability of the discretization scheme to solve the problem, including the expansion of dislocation loop-like bundles under constant velocity and driven by a mean deformation gradient, dynamic recovery of two oppositely oriented tilt boundaries in a single crystal, and a uniaxial tension test of a single crystal with one slip system activated. In most of these examples, the evolution behavior of the dislocations in the finite deformation regime is demonstrated.

Emergence and role of dipolar dislocation patterns in discrete and continuum formulations of plasticity

The plasticity transition at the yield strength of a crystal typically signifies the tendency of dislocation defects towards relatively unrestricted motion. For an isolated dislocation the motion is in the slip plane with velocity proportional to the shear stress, while due to the long range interaction dislocation ensembles move towards satisfying emergent collective elastoplastic modes. Such collective motions have been discussed in terms of the elusively defined backstress. In this paper, we develop a two-dimensional stochastic continuum dislocation dynamics theory that clarifies the role of backstress and demonstrates a precise agreement with the collective behavior of its discrete counterpart, as a function of applied load and with only three essential free parameters. The main ingredients of the continuum theory is the evolution equations of statistically stored and geometrically necessary dislocation densities, which are driven by the long-range internal stress, a stochastic flow stress term and, finally, two local diffusion-like terms. The agreement is shown primarily in terms of the patterning characteristics that include the formation of dipolar dislocation walls.

Leaving the Slip System ‐ Cross Slip in Continuum Dislocation Dynamics

AbstractDislocations are the main contributors to plastic deformation of crystalline materials. An important step towards the description of hardening behavior is the consideration of cross slip, as it drives the exchange of dislocations between slip systems, thus playing a major role in dislocation multiplication or annihilation. In density based continuum theories of dislocations, dislocations are usually considered closed curves within their slip planes. Recent discrete dislocation dynamics simulations, however, suggest that only a small fraction of dislocations is actually closed on a single slip system, due to cross slip and dislocation reactions. We therefore investigate how the kinematics of moving open curves can be considered in dislocation density based models. The assumption of open planar curves leads to modified evolution equations for the dislocation state variables. These extended evolution equations are presented for the theory of geometrically necessary dislocations (GND) and for the higher dimensional continuum dislocation dynamics theory (hdCDD). The resulting equations are checked for plausibility by numerical calculations using the finite volume method.

A mesoscale continuum approach of dislocation dynamics and the approximation by a Runge-Kutta discontinuous Galerkin method

We consider a mesoscale continuum model for the evolution of dislocation density in small-strain crystal plasticity. The model is based on the continuum dislocation dynamics theory and extended by a formulation for impenetrable grain boundaries. We introduce a fully coupled numerical method combining a conforming finite element approximation of elasto-plasticity with an implicit Runge-Kutta discontinuous Galerkin discretization of the dislocation microstructure which allows for 3d computations including multiple slip systems and dislocation interaction. In addition, a numerical representation of grain boundaries impenetrable to dislocation flux is considered within this framework. The formulation is applied to a tricrystal focusing on the analysis of dislocation stress interaction between different grains. The results are compared to discrete dislocation dynamics data from the literature.

Annihilation and sources in continuum dislocation dynamics

Continuum dislocation dynamics (CDD) aims at representing the evolution of systems of curved and connected dislocation lines in terms of density-like field variables. Here we discuss how the processes of dislocation multiplication and annihilation can be described within such a framework. We show that both processes are associated with changes in the volume density of dislocation loops: dislocation annihilation needs to be envisaged in terms of the merging of dislocation loops, while conversely dislocation multiplication is associated with the generation of new loops. Both findings point towards the importance of including the volume density of loops (or ’curvature density’) as an additional field variable into continuum models of dislocation density evolution. We explicitly show how this density is affected by loop mergers and loop generation. The equations which result for the lowest order CDD theory allow us, after spatial averaging and under the assumption of unidirectional deformation, to recover the classical theory of Kocks and Mecking for the early stages of work hardening.

Internal stresses in a homogenized representation of dislocation microstructures

To develop a continuum theory based on the evolution of dislocation microstructures, two challenges have to be resolved: the correct representation of the kinematics of dislocation motion in terms of dislocation density and the formulation of a mobility law reflecting an effective description of the physical behavior of the discrete many-body problem.Kröner's classical continuum theory has inspired different approaches to model plasticity based on the motion of dislocations. Amongst them, the Continuum Dislocation Dynamics (CDD) theory was formulated as a generalization of the classical theory. The CDD theory allows for a continuous representation of the evolution of dislocation microstructures and is found to be kinematically complete.Here, a numerical formulation of the CDD theory is presented and constitutive laws for the incorporation of dislocation interactions are derived based on the representation of the dislocation microstructure in two dimensions. An error measure is introduced to analyze the constitutive law and the results are compared to discrete dislocation dynamics simulations. Important aspects for the implementation of a 3D theory are discussed.

A link between microstructure evolution and macroscopic response in elasto-plasticity: Formulation and numerical approximation of the higher-dimensional continuum dislocation dynamics theory

Micro-plasticity theories and models are suitable to explain and predict mechanical response of devices on length scales where the influence of the carrier of plastic deformation – the dislocations – cannot be neglected or completely averaged out. To consider these effects without resolving each single dislocation a large variety of continuum descriptions has been developed, amongst which the higher-dimensional continuum dislocation dynamics (hdCDD) theory by Hochrainer et al. (Phil. Mag. 87, pp. 1261–1282) takes a different, statistical approach and contains information that are usually only contained in discrete dislocation models. We present a concise formulation of hdCDD in a general single-crystal plasticity context together with a discontinuous Galerkin scheme for the numerical implementation which we evaluate by numerical examples: a thin film under tensile and shear loads. We study the influence of different realistic boundary conditions and demonstrate that dislocation fluxes and their lines' curvature are key features in small-scale plasticity.

Orientation-dependent Pattern Formation in a 1.5D Continuum Model of Curved Dislocations

ABSTRACTDislocation pattern formation is a phenomenon where during significant plastic deformation dislocations organize themselves into (meta)stable structures. Modeling such systems is a non-trivial task, because the number of interacting dislocations is high, bringing discrete simulation models to their computational limits. Continuum models, although more efficient, generally do not contain sufficient information for a physically detailed representation of such systems. In this paper we show how a continuum dislocation dynamics theory can be used to model idealized pattern formation. Furthermore, we show how discrete dislocation dynamics (DD) simulations can be used to provide physical input for our continuum model.

Numerical Implementation of Continuum Dislocation Dynamics with the Discontinuous-Galerkin Method.

ABSTRACTIn the current paper we modify the evolution equations of the simplified continuum dislocation dynamics theory presented in [T. Hochrainer, S. Sandfeld, M. Zaiser, P. Gumbsch, Continuum dislocation dynamics: Towards a physical theory of crystal plasticity. J. Mech. Phys. Solids. (in print)] to account for the nature of the so-called curvature density as a conserved quantity. The derived evolution equations define a dislocation flux based crystal plasticity law, which we present in a fully three-dimensional form. Because the total curvature is a conserved quantity in the theory the time integration of the equations benefit from using conservative numerical schemes. We present a discontinuous Galerkin implementation for integrating the time evolution of the dislocation state and show that this allows simulating the evolution of a single dislocation loop as well as of a distributed loop density on different slip systems.

A crystal plasticity framework based on the Continuum Dislocation Dynamics theory: relaxation of an idealized micropillar

AbstractThe motion and interaction of dislocation lines are the physical basis of the plastic deformation of metals. Although ‘discrete dislocation dynamic’ (DDD) simulations are able to predict the kinematics of dislocation microstructure (i.e. the motion of dislocations in a given velocity field) and therefore the plastic behavior of crystals in small length scales, the computational cost makes DDD less feasible for systems larger than a few micro meters. To overcome this problem, the Continuum Dislocation Dynamics (CDD) theory was developed. CDD describes the kinematics of dislocation microstructure based on statistical averages of internal properties of dislocation systems. In this paper we present a crystal plasticity framework based on the CDD theory. It consists of two separate parts: a classical 3D elastic boundary value problem and the evolution of dislocation microstructure within slip planes according to the CDD constitutional equations. We demonstrate the evolution of dislocation density in a micropillar with a single slip plane. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scaling theory of continuum dislocation dynamics in three dimensions: Self-organized fractal pattern formation

We focus on mesoscopic dislocation patterning via a continuum dislocation dynamics theory (CDD) in three dimensions (3D). We study three distinct physically motivated dynamics which consistently lead to fractal formation in 3D with rather similar morphologies, and therefore we suggest that this is a general feature of the 3D collective behavior of geometrically necessary dislocation (GND) ensembles. The striking self-similar features are measured in terms of correlation functions of physical observables, such as the GND density, the plastic distortion, and the crystalline orientation. Remarkably, all these correlation functions exhibit spatial power-law behaviors, sharing a single underlying universal critical exponent for each type of dynamics.

Continuum modeling of dislocation plasticity: Theory, numerical implementation, and validation by discrete dislocation simulations

Abstract

Atomistic simulations for the evolution of a U-shaped dislocation in fcc Al

Through massively parallel molecular dynamics simulations for the evolution of U-shaped dislocation in face-centered cubic aluminum, conventional and noncoplanar (in a helical form) evolutions of dislocation segments are revealed at the atomistic scale. The two different evolutions are closely related to the Frank-Read multiplication mechanism. The possibility of noncoplanar process is quantitatively analyzed using a combination of continuum dislocation dynamics theory and atomistic simulations. The cross-slip mechanism involving in the noncoplanar evolution is supported by examining its energy barrier and critical stress. It is suggested that the operations of two different evolutions are dictated by the strain rate and the crystal size.