Continuum Dislocation Research Articles

Threshold Stress Creep Behavior of Alloy 617 at Intermediate Temperatures

Creep of Alloy 617, a solid solution Ni-Cr-Mo alloy, was studied in the temperature range of 1023 K to 1273 K (750 °C to 1000 °C). Typical power-law creep behavior with a stress exponent of approximately 5 is observed at temperatures from 1073 K to 1273 K (800 °C to 1000 °C). Creep at 1023 K (750 °C), however, exhibits threshold stress behavior coinciding with the temperature at which a low volume fraction of ordered coherent γ′ precipitates forms. The threshold stress is determined experimentally to be around 70 MPa at 1023 K (750 °C) and is verified to be near zero at 1173 K (900 °C)—temperatures directly correlating to the formation and dissolution of γ′ precipitates, respectively. The γ′ precipitates provide an obstacle to continued dislocation motion and result in the presence of a threshold stress. TEM analysis of specimens crept at 1023 K (750 °C) to various strains, and modeling of stresses necessary for γ′ precipitate dislocation bypass, suggests that the climb of dislocations around the γ′ precipitates is the controlling factor for continued deformation at the end of primary creep and into the tertiary creep regime. As creep deformation proceeds at an applied stress of 121 MPa and the precipitates coarsen, the stress required for Orowan bowing is reached and this mechanism becomes active. At the minimum creep rate at an applied stress of 145 MPa, the finer precipitate size results in higher Orowan bowing stresses and the creep deformation is dominated by the climb of dislocations around the γ′ precipitates.

Simulations of Dislocation Structure and Response

Despite 5,000 years of metals in technology and 80 years since dislocations were postulated as the carriers of deformation, we still lack fundamental theories of dislocation substructure development and its relation to the stress-strain response in real materials. In this review, we focus on one type of tool being used to explore dislocation-based plasticity: the modeling and simulation of dislocation structures and properties. Before a discussion of the methods, we present a brief summary of aspects of dislocation theory that are critical to understanding plasticity. We then review three approaches to modeling and simulating dislocation properties at the mesoscale: discrete dislocation dynamics, phase-field dislocation methods, and continuous dislocation theory. We discuss the power and limitations of the methods when applied to real-world problems.

Analysis of dislocation pile-ups using a dislocation-based continuum theory

The increasing demand for materials with well-defined microstructure, accompanied by the advancing miniaturization of devices, is the reason for the growing interest in physically motivated, dislocation-based continuum theories of plasticity. In recent years, various advanced continuum theories have been introduced, which are able to described the motion of straight and curved dislocation lines. The focus of this paper is the question of how to include fundamental properties of discrete dislocations during their motion and interaction in a continuum dislocation dynamics (CDD) theory. In our CDD model, we obtain elastic interaction stresses for the bundles of dislocations by a mean-field stress, which represents long-range stress components, and a short range corrective stress component, which represents the gradients of the local dislocation density. The attracting and repelling behavior of bundles of straight dislocations of the same and opposite sign are analyzed. Furthermore, considering different dislocation pile-up systems, we show that the CDD formulation can solve various fundamental problems of micro-plasticity. To obtain a mesh size independent formulation (which is a prerequisite for further application of the theory to more complex situations), we propose a discretization dependent scaling of the short range interaction stress. CDD results are compared to analytical solutions and benchmark data obtained from discrete dislocation simulations.

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.

Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor

ABSTRACTIn the current paper we present a continuum theory of dislocations based on the second-order alignment tensor in conjunction with the classical dislocation density tensor (Kröner-Nye-tensor) and a scalar dislocation curvature measure. The second-order alignment tensor is a symmetric second order tensor characterizing the orientation distribution of dislocations in elliptic form. It is closely connected to total densities of screw and edge dislocations introduced in the literature. The scalar dislocation curvature density is a conserved quantity the integral of which represents the total number of dislocations in the system. The presented evolution equations of these dislocation density measures partly parallel earlier developed theories based on screw-edge decompositions but handle line length changes and segment reorientation consistently. We demonstrate that the presented equations allow predicting the evolution of a single dislocation loop in a non-trivial velocity field.

141 金属微視組織中に形成されるキンク変形の回位多重極子と連続分布転位モデルによる検討(材料力学・機械材料)

141 金属微視組織中に形成されるキンク変形の回位多重極子と連続分布転位モデルによる検討(材料力学・機械材料)

Mechanics and Dislocation Structures at the Micro-Scale: Insights on Dislocation Multiplication Mechanisms from Discrete Dislocation Dynamics Simulations

ABSTRACTThe plasticity of micro-pillar deformation has widely been studied by discrete dislocation dynamics simulations to explain the so-called size effect. In this study the role of glissile junctions forming during plastic deformation under various loading scenarios is in the center of interest. The activity of these naturally forming dislocation sources is followed in detail. Surprisingly these junctions are rather active sources and not just another obstacle as often assumed. Their relative contribution to the overall dislocation density for the simulated specimens reaches often values of 20% or even more. The formation of such a glissile junction is often correlated to stress drops or the end of a stress drop. It is therefore suggested – at least for the sample sizes considered – that this dislocation multiplication mechanism should be take into account in continuum models such as crystal plasticity of higher order dislocation continuum theories.

Computational Method of Rapidly Propagating Cracks Using Continuous Dislocations Model

The method of continuously distributed dislocations model is applied to the problem of rapidly propagating cracks. Weertman's solution of a running edge dislocation considering the effect of inertia (1961) has been utilized. A semi-infinite length crack running in a strip with clamped sides was analyzed. The singular integral equations representing the boundary condition of this problem were solved numerically by the newly proposed method based on the boundary collocation. The stress intensity factor was calculated and it has been made clear that precise solutions can be obtained even when the number of the collocation points is small. The crack opening displacement was also calculated and a startling result has been obtained. In the neighborhood of the crack tip, the crack opening displacement exceeds the applied displacement between both sides of the strip.

Comparison of closure approximations for continuous dislocation dynamics

ABSTRACTWe discuss methods to describe the evolution of dislocation systems in terms of a limited number of continuous field variables while correctly representing the kinematics of systems of flexible and connected lines. We show that a satisfactory continuum representation may be obtained in terms of only four variables. We discuss the consequences of different approximations needed to formulate a closed set of equations for these variables and propose a benchmark problem to assess the performance of the resulting models. We demonstrate that best results are obtained by using the maximum entropy formalism to arrive at an optimal estimate for the dislocation orientation distribution based on its lowest-order angular moments.

The Higher Dimensional Continuum Dislocation Dynamics Theory: A Runge‐Kutta Discontinuous Galerkin Approach

AbstractThe reformulation in conservative form of the higher‐dimensional continuum dislocations dynamics(hdCDD) theory of Hochrainer (Ph.D. thesis, 2006) is presented together with a framework for elasto‐plasticity problem based on this theory. A Runge‐Kutta discontinuous Galerkin(RKDG) method is used for the evolution of hdCDD to obtain information of the micro‐structure which is coupled with a finite element method for the stress computation. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linking Micro‐ and Macroscopic Aspects of a Dislocation Based Continuum Model by Investigating the Dislocation Double Pileup

AbstractIn this contribution we discuss the problem of a dislocation double pileup at impenetrable boundaries which serves as a benchmark test to examine the continuum dislocation dynamics (CDD). We enhance a formulation for a numerical implementation and the mean‐field theory to account for the ‘long‐range’ stress field due to inhomogeneous plastic deformations by a suitable correction term. This allows us to handle the motion of dislocation densities at impenetrable boundaries and study the influence of dislocation interactions on a scale much smaller than the scale of the proposed mean‐field stress. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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)

A study of the hot and cold deformation of twin-roll cast magnesium alloy AZ31

Recent advances in twin-roll casting (TRC) technology of magnesium have demonstrated the feasibility of producing magnesium sheets in the range of widths needed for automotive applications. However, challenges in the areas of manufacturing, material processing and modelling need to be resolved in order to fully utilize magnesium alloys. Despite the limited formability of magnesium alloys at room temperature due to their hexagonal close-packed crystalline structure, studies have shown that the formability of magnesium alloys can be significantly improved by processing the material at elevated temperatures and by modifying their microstructure to increase ductility. Such improvements can potentially be achieved by processes such as superplastic forming along with manufacturing techniques such as TRC. In this work, we investigate the superplastic behaviour of twin-roll cast AZ31 through mechanical testing, microstructure characterization and computational modelling. Validated by the experimental results, a novel continuum dislocation dynamics-based constitutive model is developed and coupled with viscoplastic self-consistent model to simulate the deformation behaviour. The model integrates the main microstructural features such as dislocation densities, grain shape and grain orientations within a self-consistent viscoplasticity theory with internal variables. Simulations of the deformation process at room temperature show large activity of the basal and prismatic systems at the early stages of deformation and increasing activity of pyramidal systems due to twinning at the later stages. The predicted texture at room temperature is consistent with the experimental results. Using appropriate model parameters at high temperatures, the stress–strain relationship can be described accurately over the range of low strain rates.

From systems of discrete dislocations to a continuous field description: stresses and averaging aspects

Metal plasticity is governed by the motion of dislocations, and predicting the interactions and resulting collective motion of dislocations is a major task in understanding and modeling plastically deforming materials. This task has, despite all the efforts and advances of the last few decades, not yet been fully accomplished. The reason for this is that discrete models which describe the dislocation system with high accuracy are only computationally feasible for small systems, small strains, and high strain rates. Classical continuum models do not suffer from these restrictions but lack sufficiently detailed information about dislocation microstructure. In this paper we present the steps that are needed for averaging systems of discrete dislocations toward a continuous and hence more efficient representation. Our main emphasis lies on investigating the effects of averaging on the description of stress fields and dislocation interactions. We show how the evolution of continuous dislocation fields can then be appropriately described by a dislocation density-based model and validate our results by comparison with discrete dislocation dynamic simulations.

Role of molecule flexibility on the nucleation of dislocations in molecular crystals

We show that a molecule's flexibility described by changes to its conformation and orientation during deformation is vital for the proper representation of dislocation nucleation in molecular crystals. This is shown for the molecular crystal hexahydro-1,3,5-trinitro-s-triazine (RDX) by comparing direct atomistic simulations to two alternate forms of a continuum dislocation nucleation model for a crack tip loaded in pure shear. The atomistic simulations show the emission of partial dislocations. These are compared to continuum dislocation nucleation models based on generalized stacking fault (GSF) energy surfaces where the molecules are allowed to be either rigid or flexible. The rigid molecules are unable to represent the partial dislocations whereas the flexible molecules agree with the direct atomistic model to within 17% of the stress intensity factor for emission of the first partial dislocation and to within 1% for the second partial. This agreement first indicates that the molecule flexibility serves a critical role in the ductile behavior of the molecular crystal and, second, the continuum dislocation nucleation model represents the correct atomistic behavior, showing two partial dislocations connected by a stacking fault, when parameterized with GSF energy surfaces that account for the molecule flexibility.

Continuum dislocation dynamics: Towards a physical theory of crystal plasticity

The plastic deformation of metals is the result of the motion and interaction of dislocations, line defects of the crystalline structure. Continuum models of plasticity, however, remain largely phenomenological to date, usually do not consider dislocation motion, and fail when materials behavior becomes size dependent. In this work we present a novel plasticity theory based on systematic physical averages of the kinematics and dynamics of dislocation systems. We demonstrate that this theory can predict microstructure evolution and size effects in accordance with experiments and discrete dislocation simulations. The theory is based on only four internal variables per slip system and features physical boundary conditions, dislocation pile ups, dislocation curvature, dislocation multiplication and dislocation loss. The presented theory therefore marks a major step towards a physically based theory of crystal plasticity.

A multiscale coupled Monte Carlo model to characterize microstructure evolution during hot rolling of Mo-TRIP steel

A multiscale modelling framework has been proposed to characterize microstructure evolution during hot strip rolling of transformation-induced plasticity (TRIP) steel. The modelling methodology encompasses a continuum dislocation density evolution model coupled with a lumped parameter heat transfer model which has been seamlessly integrated with a mesoscale Monte Carlo (MC) simulation technique. The dislocation density model computes the evolution of dislocation density and subsequently constitutive flow stress behaviour has been predicted and successfully validated with the published data. A lumped-parameter transient heat transfer model has been developed to calculate the average strip temperature in the time domain. The heat transfer model incorporates the effect of plastic work for different strain rates in the energy conservation formulation. A coupled initial value problem solver has been developed to integrate the system of stiff ordinary differential equations in the time domain to predict dislocation density and temperature profiles simultaneously. The temporal evolution of microstructure during hot rolling of TRIP steel is simulated by the MC method incorporating thermal and dislocation density data from the continuum models. Simulated microstructural maps, kinetics of recrystallization and grain size evolution have been generated in a 200×200 lattice system at different strain rates and temperatures. The simulation code has been implemented in a high-performance grid computing network. The predicted temporal evolution of grain size, recrystallized fractions and flow stress have been validated with the published literature and found to be in good agreement, confirming the predictive capability of the integrated model.

Nonlinear continuum dislocation theory revisited

The present paper provides the definition of resultant Burgers’ vector and the related dislocation density tensor in terms of the plastic deformation, regarded as that creating dislocations without deforming the crystal lattice. Based on this kinematics the thermodynamic framework of the finite strain continuum dislocation theory is developed. The proposed theory is then applied to the problem of antiplane constrained shear admitting an analytical solution.

Analysis of dissociation of 〈c〉 and 〈c + a〉 dislocations to nucleate twins in Mg

A mechanism for twin nucleation in Mg is studied in which edge 〈c〉 and mixed 〈c + a〉 lattice dislocations dissociate into a stable twin, having at least the minimum 6-layer thickness formed by three glissile twinning dislocations, plus a residual stair rod dislocation. Continuum dislocation theory is used to compute the energy of the initial and final states of the proposed dissociation process, using the twin boundary energy computed by density functional theory. For the 〈c〉 dislocation, the proposed dissociation is energetically favorable. An alternative dissociation path into partials on two -type pyramidal planes is possible, as seen in an atomistic analysis, and the continuum analysis predicts this alternative path to be more favorable than the twin process. For the 〈c + a〉 dislocation, the continuum model also predicts that dissociation into the twinned structure is energetically favorable for 6-layer and thicker twins. In both 〈c〉 and 〈c + a〉 cases, the equilibrium twin length is predicted to increase with increasing applied resolved shear stress and grow unstably beyond a critical stress. Atomistic simulations of these processes are then performed. For 〈c〉, a twinned structure is stable under zero loading but with higher energy than the alternative dissociation on two planes. Under a positive applied strain of 4%, resolved on the twin plane, the twinning structure grows while under a negative applied strain of −3%, it reverts back to the alternative low-energy dissociated configuration on the pyramidal planes. For the mixed 〈c + a〉 dislocation, the atomistic models predict that the dissociation into twinning dislocations does not occur spontaneously at zero applied strain but there is a stable twinned region at finite applied loads. These results demonstrate that dislocation-assisted mechanisms for twinning in Mg, initiating from lattice dislocations with large Burgers vectors, are physically feasible, and therefore twin nucleation from grain boundaries is not necessarily the dominant mechanism of twinning in Mg.

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.