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In this paper, a geometric field theory of dislocation dynamics and finite plasticity in single crystals is formulated. Starting from the multiplicative decomposition of the deformation gradient into elastic and plastic parts, we use Cartan’s moving frames to describe the distorted lattice structure via differential 1-forms. In this theory, the primary fields are the dislocation fields, defined as a collection of differential 2-forms. The defect content of the lattice structure is then determined by the superposition of the dislocation fields. All these differential forms constitute the internal variables of the system. The evolution equations for the internal variables are derived starting from the kinematics of the dislocation 2-forms, which is expressed using the notions of flow and of Lie derivative. This is then coupled with the rate of change of the lattice structure through Orowan’s equation. The governing equations are derived using a two-potential approach to a variational principle of the Lagrange–d’Alembert type. As in the nonlinear setting the lattice structure evolves in time, the dynamics of dislocations on slip systems is formulated by enforcing some constraints in the variational principle. Using the Lagrange multipliers associated with these constraints, one obtains the forces that the lattice exerts on the dislocation fields in order to keep them gliding on some given crystallographic planes. Moreover, the geometric formulation allows one to investigate the integrability—and hence the existence—of glide surfaces, and how the glide motion is affected by it. Lastly, a linear theory for small dislocation densities is derived, allowing one to identify the nonlinear effects that do not appear in the linearized setting. [ABSTRACT FROM AUTHOR]

Over the past decades, discrete dislocation dynamics simulations have been shown to reliably predict the evolution of dislocation microstructures for micrometer-sized metallic samples. Such simulations provide insight into the governing deformation mechanisms and the interplay between different physical phenomena such as dislocation reactions or cross-slip. This work is focused on a detailed analysis of the influence of the cross-slip on the evolution of dislocation systems. A tailored data mining strategy using the 'discrete-to-continuous (D2C) framework' allows to quantify differences and to quantitatively compare dislocation structures. We analyze the quantitative effects of the cross-slip on the microstructure in the course of a tensile test and a subsequent relaxation to present the role of cross-slip in the microstructure evolution. The precision of the extracted quantitative information using D2C strongly depends on the resolution of the domain averaging. We also analyze how the resolution of the averaging influences the distribution of total dislocation density and curvature fields of the specimen. Our analyzes are important approaches for interpreting the resulting structures calculated by dislocation dynamics simulations. [ABSTRACT FROM AUTHOR]

Abstract Cellular patterns formed by self-organization of dislocations are a most conspicuous feature of dislocation microstructure evolution during plastic deformation. To elucidate the physical mechanisms underlying dislocation cell structure formation, we use a minimal model for the evolution of dislocation densities under load. By considering only two slip systems in a plane strain setting, we arrive at a model which is amenable to analytical stability analysis and numerical simulation. We use this model to establish analytical stability criteria for cell structures to emerge, to investigate the dynamics of the patterning process and establish the mechanism of pattern wavelength selection. This analysis demonstrates an intimate relationship between hardening and cell structure formation, which appears as an almost inevitable corollary to dislocation dominated strain hardening. Specific mechanisms such as cross slip, by contrast, turn out to be incidental to the formation of cellular patterns.

With decreasing system sizes, the mechanical properties and dominant deformation mechanisms of metals change. For larger scales, bulk behavior is observed that is characterized by a preservation and significant increase of dislocation content during deformation whereas at the submicron scale very localized dislocation activity as well as dislocation starvation is observed. In the transition regime it is not clear how the dislocation content is built up. This dislocation storage regime and its underlying physical mechanisms are still an open field of research. In this paper, the microstructure evolution of single crystalline copper micropillars with a $\langle 1\,1\,0\rangle $ crystal orientation and varying sizes between $1$ and 10 $\mu \mathrm{m}$ is analysed under compression loading. Experimental in situ HR-EBSD measurements as well as 3d continuum dislocation dynamics simulations are presented. The experimental results provide insights into the material deformation and evolution of dislocation structures during continuous loading. This is complemented by the simulation of the dislocation density evolution considering dislocation dynamics, interactions, and reactions of the individual slip systems providing direct access to these quantities. Results are presented that show, how the plastic deformation of the material takes place and how the different slip systems are involved. A central finding is, that an increasing amount of GND density is stored in the system during loading that is located dominantly on the slip systems that are not mainly responsible for the production of plastic slip. This might be a characteristic feature of the considered size regime that has direct impact on further dislocation network formation and the corresponding contribution to plastic hardening.

Cellular patterns formed by self-organization of dislocations are a most conspicuous feature of dislocation microstructure evolution during plastic deformation. To elucidate the physical mechanisms underlying dislocation cell structure formation, we use a minimal model for the evolution of dislocation densities under load. By considering only two slip systems in a plane strain setting, we arrive at a model which is amenable to analytical stability analysis and numerical simulation. We use this model to establish analytical stability criteria for cell structures to emerge, to investigate the dynamics of the patterning process and establish the mechanism of pattern wavelength selection. This analysis demonstrates an intimate relationship between hardening and cell structure formation, which appears as an almost inevitable corollary to dislocation dominated strain hardening. Specific mechanisms such as cross slip, by contrast, turn out to be incidental to the formation of cellular patterns. [ABSTRACT FROM AUTHOR]

Abstract We explore new ways of regulating defect behavior in systems of conservation laws. Contrary to usual regularization schemes (such as a vanishing viscosity limit), which attempt to control defects by making them smoother, our schemes result in defects which are more singular, and we thus refer to such schemes as “irregularizations”. In particular, we seek to produce delta shock defects which satisfy a condition of stationarity. We are motivated to pursue such exotic defects by a physical example arising from dislocation dynamics in materials physics, which we describe.

Abstract 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.

The newly proposed microstructural constitutive model for polycrystal viscoplasticity in cold and warm regimes (Motaman and Prahl, 2019), is implemented as a microstructural solver via user-defined material subroutine in a finite element (FE) software. Addition of the microstructural solver to the built-in thermal and mechanical solvers of a standard FE package enabled coupled thermo-micro-mechanical or thermal-microstructural-mechanical (TMM) simulation of cold and warm metal forming processes. The microstructural solver, which incrementally calculates the evolution of microstructural state variables (MSVs) and their correlation to the thermal and mechanical variables, is implemented based on the constitutive theory of isotropic hypoelasto-viscoplastic (HEVP) finite (large) strain/deformation. The numerical integration and self-consistent algorithmic procedure of the FE implementation are explained in detail. Then, the viability of this approach is shown for (TMM-) FE simulation of an industrial multistep warm forging. [ABSTRACT FROM AUTHOR]

Abstract The initial γ / γ ′ microstructure (e.g. γ ′ shape, size, volume fraction) may dramatically affect creep behaviour of single crystal Nickel-based superalloys. However, it is difficult to have accurate control on each of these initial aspects and to understand the role of each by experimental methods. In the present work, a novel coupled model of phase-field and continuum dislocation dynamics is developed for the co-evolution of γ / γ ′ and dislocation microstructures, such that the creep deformation of single crystal Nickel-based superalloys can be described in a physical way. The creep curve can be directly obtained by averaging dislocation activity, without any phenomenological effort that is needed for traditional constitutive models. With the coupled model, we study the effect of initial γ / γ ′ microstructure on creep of single crystal Nickel-based superalloys. The role of initial γ ′ shape is studied by comparing simulations with different initial γ ′ shapes but with the same γ ′ size and volume fraction. The role of initial γ ′ size and volume fraction are studied in the same methodology. Simulation results show that as the γ / γ ′ misfit magnitude increases, the γ ′ shape transits from circular to cubic and the dislocation microstructure symmetry shifts. The cubic γ ′ shape corresponding to γ / γ ′ misfit of − 0.003 is the most beneficial for creep resistance, compare with γ / γ ′ misfit of − 0.0015 and 0. Bigger γ ′ size and higher γ ′ volume fraction also result in better creep resistance, especially the benefit of high γ ′ volume fraction to creep resistance is dramatic. The simulation results may bring inspirations for design of new single crystal Nickel-based superalloys with excellent creep resistance. Highlights • Accurate "control variable" on different aspects of initial γ / γ ′ microstructure. • A phase-field model applicable for both creep microstructure and property. • A comprehensive understanding on the role of initial γ / γ ′ microstructure in creep. [ABSTRACT FROM AUTHOR]

Abstract Past models of stress-strain response under cyclic loading mainly rely on macroscopic equations which consider microstructure evolution indirectly or simply discard microstructure information. Modern materials science, on the other hand, seeks quantitative descriptions for the relations between microstructure and loading response. In the present work, we show a promising mesoscale phase-field framework which can describe co-evolution of phase/grain and defect microstructures, reveal microstructure mechanisms and simultaneously predict deformation properties as a natural outcome of microstrucuture interactions. The energy functionals for phase/grain and defect microstructures are constructed, followed by functional variation which leads to governing equations. Applying the developed framework to high temperature cyclic loading of single crystal Nickel-based superalloys, the simulated results show that cyclic loading-microstructure-property relations can be principally revealed. In the short term perspective (in one cycle), dislocations move back and forth, leading to cyclic loops consistent with characteristics observed in experiments. The plastic strains are one order of magnitude smaller than total strains, which explains why the cyclic loops are very "thin". In the long term perspective, all γ / γ ′ microstructures exhibit directional coarsening similar to creep under zero cyclic loading ratio, with the extent of rafting slightly dependents on cyclic waveform, period, etc. The plastic strains are sensitive to cyclic loading conditions both in terms of curve shape and in terms of magnitude. [ABSTRACT FROM AUTHOR]

In single crystal Nickel-based superalloys subject to creep loading, the γ / γ ′ phase microstructure co-evolves with the system of dislocations under load. Computational modeling thus requires multiphysics approaches capable of describing and simulating both phase and defect microstructures within a common conceptual framework. To do so, we formulate a coupled continuum model of the evolution of phase and dislocation microstructures. The simulated γ / γ ′ phase microstructure accounts for concentration as well as crystallographic order parameters. Dislocation microstructure evolution is described in terms of dislocation densities and associated stress-driven dislocation fluxes. The creep strain curve is obtained as a natural by-product of the microstructure evolution equations. We perform simulations of γ / γ ′ evolution for different dislocation densities and establish the driving forces for microstructure evolution by analyzing in detail the changes in different contributions to the elastic energy and chemical free energy density, as well as the evolution of stress concentrations that may trigger the transition from dislocation flow in the γ channels towards shearing of the γ ′ precipitates. Our investigation reveals the mechanisms controlling the process of directional coarsening (rafting) and demonstrates that the kinetics of rafting significantly depends on characteristics of the dislocation microstructure. In addition to rafting under constant load, we investigate the effect of changes in loading conditions and explore the possibility of improving creep properties by pre-rafting along a different loading path. [ABSTRACT FROM AUTHOR]

Continuum dislocation dynamics (CDD) is a single crystal strain gradient plasticity theory based exclusively on the evolution of the dislocation state. Recently, we derived a constitutive theory for the average dislocation velocity in CDD in a phase field-type description for an infinite domain. In the current work, so-called rational thermodynamics is employed to obtain thermodynamically consistent boundary conditions for the dislocation density variables of CDD. We find that rational thermodynamics reproduces the bulk constitutive equations as obtained from irreversible thermodynamics. The boundary conditions we find display strong parallels to the microscopic traction conditions derived by Gurtin and Needleman (M.E. Gurtin and A. Needleman, J. Mech. Phys. Solids 53 (2005) 1-31) for strain gradient theories based on the Kröner-Nye tensor. [ABSTRACT FROM AUTHOR]

In this study, a dislocation density-based model is introduced to analyze slip transmission across grain boundaries in polycrystalline materials. The method applies a combination of the misorientation of neighboring grains and resolved shear stress on relative slip planes. This model is implemented into a continuum dislocation dynamics framework and extended to consider the physical interaction between mobile dislocations and grain boundaries. The model takes full account of the geometry of the grain boundary, the normal and direction of incoming and outgoing slip systems, and the extended stress field of the boundary and dislocation pileups at the boundary. The model predicts that slip transmission is easier across grain boundaries when the misorientation angle between the grains is small. The modeling results are verified with experimental nanoindentation results for polycrystalline copper samples. [ABSTRACT FROM AUTHOR]

Nickel-based superalloys play a major role in many technologically relevant high temperature applications. Understanding and predicting the evolution of the phase microstructure during high temperature creep together with the evolution of the dislocation microstructure is a challenge that up to date has not yet been fully accomplished. Our two-dimensional coupled phase-field/continuum dislocation dynamics model explains microstructural mechanisms which are important during the early stage of rafting in a single crystal system. It shows how γ / γ ′ phases and dislocations interact giving rise to realistic creep behavior; no phenomenological fit parameters are required. [ABSTRACT FROM AUTHOR]

Dislocation-grain boundary interaction plays a key role in the plasticity of polycrystalline materials. Capturing the effect of discrete dislocations interacting with a grain boundary in continuum models is not yet achieved. To date several approaches exist, but they have shortcomings in capturing the influence of dislocation-dislocation interaction across a grain boundary and the parameters which control grain boundary yield are phenomenologically motivated. In this work we show that grain boundary yielding is not inherently connected to physical dislocation transmission and that a realistic model needs to incorporate the interaction of dislocations across grain boundaries to capture the true strain distribution in the individual grains. By comparing discrete dislocation dynamics simulations of a single crystal with an artificial grain boundary to continuum dislocation dynamics results, a clear influence on the strain profile from the elastic interaction of dislocations belonging to different grains is shown. Our results demonstrate that continuum models like gradient plasticity need to extend their grain boundary modeling to incorporate dislocation interactions because a single yield criterion is not sufficient. [ABSTRACT FROM AUTHOR]

Continuum dislocation dynamics (CDD) is a single crystal strain gradient plasticity theory based exclusively on the evolution of the dislocation state. Recently, we derived a constitutive theory for the average dislocation velocity in CDD in a phase field-type description for an infinite domain. In the current work, so-called rational thermodynamics is employed to obtain thermodynamically consistent boundary conditions for the dislocation density variables of CDD. We find that rational thermodynamics reproduces the bulk constitutive equations as obtained from irreversible thermodynamics. The boundary conditions we find display strong parallels to the microscopic traction conditions derived by Gurtin and Needleman (M.E. Gurtin and A. Needleman, J. Mech. Phys. Solids 53 (2005) 1–31) for strain gradient theories based on the Kröner–Nye tensor.

Microwires have become of increasing interest for the miniaturization of structural components. A profound understanding of the deformation behavior of microwires is important for the assessment of their applicability and lifetime in specific components. In particular, the deformation behavior under torsional loading and the associated microstructure evolution are of interest. The exact involvement of individual slip systems and their activities in the complex stress field under torsional loading are mostly unknown. In this paper, the microstructure evolution of single crystalline gold microwires under torsion have been analyzed for the high-symmetry crystal orientations 〈 1 0 0 〉 , 〈 1 1 0 〉 , and 〈 1 1 1 〉 using simulation and experimental results. It is shown that a classification of the slip systems can be derived a priori by theoretical considerations. It is found, that the slip system activity, stress relaxation mechanism, as well as screw and edge composition of the piled-up dislocation density depends on specific slip system groups. Furthermore, the misorientation and its rotational axes including the identification of the slip system activities are discussed. [Display omitted] [ABSTRACT FROM AUTHOR]

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. • Strain driven homogenization scheme for a dislocation dynamics model on the mesoscale. • Dynamic recovery of two oppositely oriented tilt boundaries. • 3D simulation of FCC crystal under uniaxial loading and single slip conditions. [ABSTRACT FROM AUTHOR]

Abstract: Continuum dislocation dynamics (CDD) with a novel constitutive law based on dislocation density evolution mechanisms was developed to investigate the deformation behaviors of single crystals. The dislocation density evolution law in this model is mechanism-based, with parameters predicted by lower-length scale models or measured from experiments, not an empirical law with parameters back-fitted from the flow curves. Applied on iron single crystal, this model was validated by experimental data and compared with traditional single crystal constitutive models using a Hutchinson-type hardening law or a dislocation-based hardening law. The CDD model demonstrated higher fidelity than other constitutive models when anisotropic single crystal deformation behaviors were investigated. The traditional Hutchinson type hardening laws and other constitutive laws based on a Kocks formulated dislocation density evolution law will only succeed in a limited number of loading directions. The main advantage of CDD is the novel physics-based dislocation density evolution laws in describing the meso-scale microstructure evolution. Another advantage of CDD is on cross-slip, which is very important when loading conditions activate only one primary slip system. In addition to the dislocation hardening, CDD also takes into consideration dislocation defect interactions. Irradiation hardening of iron single crystal was simulated with validation from experimental results. [Copyright &y& Elsevier]

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. • A vector-density based continuum dislocation dynamics model was established with cross slip and junction reactions. • Two algorithms of plastic distortion and stress update were formulated and compared. • 3D simulation of FCC crystal under uniaxial loading was performed to show mechanical behavior and microstructure evolution. [ABSTRACT FROM AUTHOR]