Dislocation Dynamics Theory Research Articles

A rate–dependent cyclic cohesive zone model incorporating material viscoelasticity and fatigue damage accumulation effects

Numerical simulation and analysis of high–frequency fatigue crack growth rates (FCGRs) have long been a challenging task in vibration fatigue assessment. The frequency effect on FCGRs can be effectively modeled through strain rate hardening. Additionally, the cyclic cohesive zone model (CCZM) has demonstrated good accuracy in simulating quasi–static fatigue crack growth (FCG). In this study, a rate–dependent cyclic cohesive zone model (RCCZM) is proposed by integrating the strain rate hardening term from the Johnson–Cook (J–C) model and the dynamic fracture toughness model with the CCZM. A linear scaling method is employed to improve both the speed and accuracy of FCGR predictions. The model is validated by comparing the theoretical range of FCGRs for aluminum alloys, based on dislocation dynamics theory, with experimental data for titanium alloys, and the extent of influence of the strain rate correction parameter is thoroughly analyzed. The RCCZM model demonstrates high reliability and accuracy, confirming that the frequency effect on metal FCGRs is primarily due to the strain rate’s impact on strength, with minimal effect from fracture toughness.

Dynamic Constitutive Model and Numerical Simulation of S32760 Duplex Stainless Steel Based on Dislocation Theory

Background: To describe the complex mechanical behavior of S32760 duplex stain-less steel under high strain rate and high-temperature loading conditions. Objective: The constitutive model of S32760 duplex stainless steel suitable for high strain rate was constructed from the micro-scale. Methods: Based on the theory of dislocation dynamics, the effects of different strain rates and strains on the plastic deformation of ferrite and austenite were analyzed, and the thermal stress term and non-thermal stress term of ferrite and austenite phases were coupled. Results: The simulation results of the model show that the S32760 dual-phase constitutive model has a high degree of fit with the experimental data at high strain rates. Conclusion: Compared with the classical J-C model, the results show that the constitutive model of this patent has more accurate predictability than the J-C model in describing the mechanical behavior of duplex stainless steel in the high strain range of 5000s-1 to 10000s-1.

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.

Plastic deformation behavior of ultrafine grained CoCrFeNiMn high entropy alloy with nanoparticles

High strength ultrafine grained CoCrFeNiMn high entropy alloys (HEAs) containing TiO(C) nanoparticles were fabricated by thermomechanical consolidation of a mechanically milled powder at 950 and 1000 °C, respectively. The sample consolidated at 950 °C with an average grain size of 417 nm had a high yield strength (YS) of 1457 MPa, but exhibited necking immediately after yield drop, leading to a limited elongation to fracture of 1.3%. In contrast, the sample consolidated at 1000 °C with an average grain size of 489 nm had a clearly lower YS of 1236 MPa, but exhibited yield drop, Lüders deformation, work hardening and then necking, with a significant elongation to fracture of 10.0%. Our work shows that by introducing intragranular Ni–Ti nanoparticles in the coarser grained sample through quenching to liquid nitrogen and aging, the YS of the sample was brought back to 1449 MPa, and still kept a substantial elongation to fracture of 4.2%. The transition of the plastic deformation behavior with grain refinement is analyzed based on dislocation dynamics theory. Based on this analysis, it is proposed that to maintain the uniform deformation, the grain sizes should be larger than a critical value, above which maximum flow stress of the material is higher than the upper yield stress. Introducing intragranular Ni–Ti nanoparticles is an effective strategy to further enhance yield strength without losing work hardening ability due to the synergistic effect with intragranular TiO(C) nanoparticles to inhibiting dislocation annihilation at grain boundaries.

DISLOCATION-BASED CONSTITUTIVE DESCRIPTION FOR MODELING OF THE 6008 ALUMINUM ALLOY BEHAVIOR

The behavior of the 6008 aluminum alloy under impact loading with different strain rates is experimentally studied. Quasi-static compressive stress–strain curves of these materials are obtained in this study by using an RPL100 material testing machine and a split Hopkinson pressure bar. As the strain rate increases, the yield stress and the peak stress of the materials are found to increase significantly. The adiabatic temperature rise generated during the impact process makes the materials soften, so that the rate of the increase in the flow stress gradually decreases with an increase in the strain rate. Based on the dislocation dynamics theory, a viscoplastic dynamic constitutive model for the 6008 aluminum alloy is constructed. The model accurately describes and predicts the mechanical behavior of this material under impact compression at room temperature.

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.

Multi-objective parameter identification and optimization for dislocation-dynamics-based constitutive modeling of Ti–6Al–4V alloy

Based on the plastic deformation mechanism and dislocation dynamics theory, a constitutive model to describe the mechanical behavior of the Ti–6Al–4V alloy over a wide range of strain rates and temperatures is proposed. The relationships between the constitutive model parameters and microstructural characteristics, as well as their physical significance, were described. The constitutive model contains 12 parameters. A new hybrid approach was developed to obtain the constitutive parameters from the existing experimental data. There were two main parts of the approach. First, to analyze the overall constitutive parameter sensitivity, the Latin hypercube sampling (LHS) and Spearman rank correlation (SRC) methods were applied. The effectiveness and accuracy of the parameter identification and sensitivity analysis results were improved significantly. Second, the parameter optimization was developed using an advanced genetic algorithm including the enhanced niche genetic algorithm, suspicious peak point judgment strategy, and local accuracy searching technology. Finally, the flow stress–plastic strain curve of the Ti–6Al–4V alloy over a wide range of strain rates and temperatures was reproduced through the proposed method. It was shown that the predicted results agreed reasonably well with the existing experimental data.

A bottom‐up continuum approach of crystal plasticity for the analysis of fcc microwires under torsion

AbstractThe microstructural evolution of face‐centered cubic microwires is studied using a physical motivated, homogenized continuum model of crystal plasticity. The dislocation configuration in the three‐dimensional space is thereby described via a Continuum Dislocation Dynamics (CDD) theory including a dislocation source term. The resulting spatial distribution of dislocation densities and strain components are shown for a relaxation problem with torsional loading.

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.

Global solution for a non-local eikonal equation modelling dislocation dynamics

In this paper, we study a non-local eikonal equation that arises from the theory of dislocation dynamics. For this equation, the global existence and uniqueness are available only for Lipschitz continuous viscosity solutions in some particular cases. Based on a new gradient entropy estimate, we prove the global existence of a continuous viscosity solution of this equation, considering non-signed velocity.

Microstructural comparison of the kinematics of discrete and continuum dislocations models

The Continuum Dislocation Dynamics (CDD) theory and the Discrete Dislocation Dynamics (DDD) method are compared based on concise mathematical formulations of the coarse graining of discrete data. A numerical tool for converting from a discrete to a continuum representation of a given dislocation configuration is developed, which allows to directly compare both simulation approaches based on continuum quantities (e.g. scalar density, geometrically necessary densities, mean curvature). Investigating the evolution of selected dislocation configurations within analytically given velocity fields for both DDD and CDD reveals that CDD contains a surprising number of important microstructural details.

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.

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.

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.

Defect-induced solid state amorphization of molecular crystals

We investigate the process of mechanically induced amorphization in small molecule organic crystals under extensive deformation. In this work, we develop a model that describes the amorphization of molecular crystals, in which the plastic response is calculated with a phase field dislocation dynamics theory in four materials: acetaminophen, sucrose, γ-indomethacin, and aspirin. The model is able to predict the fraction of amorphous material generated in single crystals for a given applied stress. Our results show that γ-indomethacin and sucrose demonstrate large volume fractions of amorphous material after sufficient plastic deformation, while smaller amorphous volume fractions are predicted in acetaminophen and aspirin, in agreement with experimental observation.