本工作提出了一个开源的基于位错密度的晶体塑性建模框架$ρ$-CP。采用Kocks型热激活流动来解释温度和应变率对晶体剪切率的影响。滑移系级别的可动和不可动位错密度，以及滑移系级别的背应力，被用作表示塑性变形期间亚结构演化的内状态变量。提出了一种全隐式数值积分方案，用于有限变形塑性模型的时间积分。该框架已实现并集成到开源有限元求解器Multiphysics Object-Oriented Simulation Environment (MOOSE)中。展示了该模型的示例应用，包括预测单晶和多晶hcp镁的各向异性力学响应、多晶fcc OFHC铜的应变率效应和循环变形，以及温度和应变率对多晶bcc钽热-力学变形的影响。演示了对真实Voronoi镶嵌微结构和Electron Back Scatter Diffraction (EBSD)微结构的模拟，以突出该模型预测塑性变形期间大变形和取向差发展的能力。
英文摘要：
This work presents an open source, dislocation density based crystal plasticity modeling framework, $ρ$-CP. A Kocks-type thermally activated flow is used for accounting for the temperature and strain rate effects on the crystallographic shearing rate. Slip system-level mobile and immobile dislocation densities, as well slip system-level backstress, are used as internal state variables for representing the substructure evolution during plastic deformation. A fully implicit numerical integration scheme is presented for the time integration of the finite deformation plasticity model. The framework is implemented and integrated with the open source finite element solver, Multiphysics Object-Oriented Simulation Environment (MOOSE). Example applications of the model are demonstrated for predicting the anisotropic mechanical response of single and polycrystalline hcp magnesium, strain rate effects and cyclic deformation of polycrystalline fcc OFHC copper, and temperature and strain rate effects on the thermo-mechanical deformation of polycrystalline bcc tantanlum. Simulations of realistic Voronoi-tessellated microstructures as well as Electron Back Scatter Diffraction (EBSD) microstructures are demonstrated to highlight the model's ability to predict large deformation and misorientation development during plastic deformation.
We study three structurally different states of nanocrystalline 316 steel and show that the state, where boundaries containing excess concentration of alloying elements are combined with mobile dislocations in grain interiors, allows maintaining extraordinarily high strength and remarkably enhanced plasticity. Underlying mechanisms featuring interaction between the segregations and mobile dislocations are discussed.
The importance of accurate simulation of the plastic deformation of ductile metals to the design of structures and components is well-known. Many techniques exist that address the length scales relevant to deformation pro- cesses, including dislocation dynamics (DD), which models the interaction and evolution of discrete dislocation line segments, and crystal plasticity (CP), which incorporates the crystalline nature and restricted motion of dis- locations into a higher scale continuous field framework. While these two methods are conceptually related, there have been only nominal efforts focu…
We report the first ab initio density-functional study of <111> screw dislocations cores in the bcc transition metals Mo and Ta. Our results suggest a new picture of bcc plasticity with symmetric and compact dislocation cores, contrary to the presently accepted picture based on continuum and interatomic potentials. Core energy scales in this new picture are in much better agreement with the Peierls energy barriers to dislocation motion suggested by experiments.
A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to model effects of geometrically-necessary dislocations (GND) only in work-hardening.
Microcrystals deform very differently from their macroscopic counterparts, displaying a size-dependent yield stress and intermittent plastic strain bursts. Here we demonstrate that size effects and strain bursts may be just two different signatures of fundamental collective phenomena that govern the dynamics of dislocation systems on all scales. We confirm this conjecture by demonstrating agreement between the results of 2D and 3D dislocation dynamics simulations and experimental results on compressed micropillars. Our results indicate that many peculiarities of small-sample plasticity may ari…
The thermodynamic theory of dislocation-enabled plasticity is based on two unconventional hypotheses. The first of these is that a system of dislocations, driven by external forces and irreversibly exchanging heat with its environment, must be characterized by a thermodynamically defined effective temperature that is not the same as the ordinary temperature. The second hypothesis is that the overwhelmingly dominant mechanism controlling plastic deformation is thermally activated depinning of entangled pairs of dislocations. This paper consists of a systematic reformulation of this theory follo…
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 …
It is well known that diamond does not deform plastically at room temperature and usually fails in catastrophic brittle fracture. Here we demonstrate room-temperature dislocation plasticity in sub-micrometer sized diamond pillars by in-situ mechanical testing in the transmission electron microscope. We document in unprecedented details of spatio-temporal features of the dislocations introduced by the confinement-free compression, including dislocation generation and propagation. Atom-resolved observations with tomographic reconstructions show unequivocally that mixed-type dislocations with Bur…
Here we present a model to study the micro-plastic regime of a stress-strain curve. In this model an explicit dislocation population represents the mobile dislocation content and an internal shear-stress field represents a mean-field description of the immobile dislocation content. The mobile dislocations are constrained to a simple dipolar mat geometry and modelled via a dislocation dynamics algorithm, whilst the shear-stress field is chosen to be a sinusoidal function of distance along the mat direction. The latter, defined by a periodic length and a shear-stress amplitude, represents a pre-…
The statistics and origin of the first discrete plastic event in a one dislocation dynamics simulation are studied. This is done via a linear stability analysis of the evolving dislocation configuration up to the onset of irreversible plasticity. It is found, via a fold catastrophe, the dislocation configuration prior to loading directly determines the stress at which the plastic event occurs and that between one and two trigger dislocations are involved. The resulting irreversible plastic strain arising from the instability is found to be highly correlated with these triggering dislocations.
This work proposes a dislocation density-based strain gradient $J_2$ plasticity framework that models the strength contribution due to Geometrically Necessary Dislocations (GNDs) using a lower order, Taylor hardening backstress model. An anisotropy factor is introduced to phenomenologically represent the differential hardening between grains in this $J_2$ plasticity framework. An implicit numerical algorithm is implemented for the time integration of the finite deformation plasticity model. The framework is first used to predict directional hardening due to the GND-induced backstress during cy…
In an attempt to bridge the gap between atomistic and continuum plasticity simulations of hydrogen in iron, we present three dimensional discrete dislocation plasticity simulations incorporating the hydrogen elastic stress and a hydrogen dependent dislocation mobility law. The hydrogen induced stress is incorporated following the formulation derived by Gu and El-Awady (2018) which here we extend to a finite boundary value problem, a microcantilever beam, via the superposition principle. The hydrogen dependent mobility law is based on first principle calculations by Katzarov et al. (2017) and w…
Dislocation core properties of Al with and without H impurities are studied using the Peierls-Nabarro model with parameters determined by ab initio calculations. We find that H not only facilitates dislocation emission from the crack tip but also enhances dislocation mobility dramatically, leading to macroscopically softening and thinning of the material ahead of the crack tip. We observe strong binding between H and dislocation cores, with the binding energy depending on dislocation character. This dependence can directly affect the mechanical properties of Al by inhibiting dislocation cross-…
In the present work, we propose a novel model coupling phase-field, dislocation density based plasticity and damage. The dislocation density governing equations are constructed based on evolutions of mobile and immobile dislocations. Mechanisms including dislocation multiplication, annihilation, mobile-immobile transfer due to dislocation interactions and block of interfaces are incorporated in the model. Especially, the "swallow-gap" problem surrounding the coarsened second phase, which often appears in dislocation and phase-field coupled simulations, is solved in the present model. Moreover,…
The Discrete Dislocation (DD) analysis and its computional modeling have been advanced significantly over the past decade. This progress has been further magnified by the idea to couple DD with continuum mechanics analysis in association with computational algorithms such as finite elements. This coupling may pave the way to better understanding of the local response of materials at the micro scale and globally at the macroscale, increasing the potential for future applications in material, mechanical, structural and process engineering analyses.
The continuum dislocation dynamics framework for mesoscale plasticity is intended to capture the dislocation density evolution and the deformation of crystals when subjected to mechanical loading. It does so by solving a set of transport equations for dislocations concurrently with crystal mechanics equations, with the latter being cast in the form of an eigenstrain problem. Incorporating dislocation reactions in the dislocation transport equations is essential for making such continuum dislocation dynamics predictive. A formulation is proposed to incorporate dislocation reactions in the trans…
Despite decades of extensive research on mechanical properties of diamond, much remains to be understood in term of plastic deformation mechanisms due to the poor deformability at room temperature. In a recent work in Advanced Materials, it was claimed that room-temperature plasticity occurred in <001>-oriented single-crystal diamond nanopillars based on observation of unrecovered deformation inside scanning electron microscope. The plastic deformation was suggested to be mediated by a phase transition from sp3 carbon to an O8-carbon phase by molecular dynamics simulations. By comparison, our …
Plasticity modelling has long been based on phenomenological models based on ad-hoc assuption of constitutive relations, which are then fitted to limited data. Other work is based on the consideration of physical mechanisms which seek to establish a physical foundation of the observed plastic deformation behavior through identification of isolated defect processes ('mechanisms') which are observed either experimentally or in simulations and then serve to formulate so-called physically based models. Neither of these approaches is adequate to capture the complexity of plastic deformation which b…
Three-dimensional discrete dislocation dynamics simulations are used to study strain-controlled plastic deformation of face-centered cubic aluminium single crystals. After describing the rate and size dependence of the average stress-strain curves, we study the power-law distributed strain bursts and the average avalanche shapes, and find a universal power-law exponent $τ\approx 1.0$ for all imposed strain rates and system sizes, characterizing both the event sizes and their durations. We discuss the dependence of our results on loading rate and compare these with previous studies of strain-co…
A numerical strategy to simulate plastic deformation in Mg alloys including dislocation slip and twin propagation is presented. Dislocation slip is included through a crystal plasticity model which is solved using the finite element method while twin propagation is taken into account by means of a phase field model which is solved using a fast Fourier transform algorithm. The coupled crystal plasticity and phase field equations were solved using different discretizations of the simulation domain using the same time step for both of them. The numerical strategy was used to simulate the deformat…
We present a dislocation density-based strain hardening model for single crystal copper through a systematic coarse-graining analysis of more than 200 discrete dislocation dynamics (DDD) simulations of plastic deformation under uniaxial tension. The proposed constitutive model has two components: a generalized Taylor relation connecting resolved shear stresses to dislocation densities on individual slip systems, and a generalized Kocks-Mecking model for dislocation multiplication. The DDD data strongly suggests a logarithmic dependence of flow stress on the plastic shear strain rate on each sl…
Structural transitions are invariably affected by lattice distortions. If the body is to remain crack-free, the strain field cannot be arbitrary but has to satisfy the Saint-Venant compatibility constraint. Equivalently, an incompatibility constraint consistent with the actual dislocation network has to be satisfied in media with dislocations. This constraint can be incorporated into strain-based free energy functionals to study the influence of dislocations on phase stability. We provide a systematic analysis of this constraint in three dimensions and show how three incompatibility equations …
In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide due to Yefimov et al. (2004b). In this dislocation field theory (DiFT) the transport equations for both the total dislocation densities and geometrically necessary dislocation densities on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested…
A consistent, small scale description of plastic motion in a crystalline solid is presented based on a phase field description. By allowing for independent mass motion given by the phase field, and lattice distortion, the solid can remain in mechanical equilibrium on the timescale of plastic motion. Singular (incompatible) strains are determined by the phase field, to which smooth distortions are added to satisfy mechanical equilibrium. A numerical implementation of the model is presented, and used to study a benchmark problem: the motion of an edge dislocation dipole in a hexagonal lattice. T…
There is growing interest in promoting deformation twinning for plasticity in advanced materials, as highly organized twin boundaries are beneficial to better strength-ductility combination in contrast to disordered grain boundaries. Twinning deformation typically involves the kinetics of stacking faults, its interaction with dislocations, and dislocation - twin boundary interactions. While the latter has been intensively investigated, the dynamics of stacking faults has been less known. In this work, we report several new insights on the stacking fault behavior in twin induced plasticity from…
A phase field model of a crystalline material at the mesoscale is introduced to develop the necessary theoretical framework to study plastic flow due to dislocation motion. We first obtain the elastic stress from the phase field free energy and show that it obeys the stress strain relation of linear elasticity. Dislocations in a two dimensional hexagonal lattice are shown to be composite topological defects in the amplitude expansion of the phase field, with topological charges given by the Burgers vector. This allows us to introduce a formal relation between dislocation velocity and the evolu…
Crystal plasticity is the result of the motion and interaction of dislocations. There is, however, still a major gap between microscopic and mesoscopic simulations and continuum crystal plasticity models. Only recently a higher dimensional dislocation density tensor was defined which overcomes some drawbacks of earlier dislocation density measures. The evolution equation for this tensor can be considered as a continuum version of dislocation dynamics. We use this evolution equation to develop evolution equations for the total dislocation density and an average curvature which together govern a…
Due to recent successes of a statistical-based nonlocal continuum crystal plasticity theory for single-glide in explaining various aspects such as dislocation patterning and size-dependent plasticity, several attempts have been made to extend the theory to describe crystals with multiple slip systems using ad-hoc assumptions. We present here a mesoscale continuum theory of plasticity for multiple slip systems of parallel edge dislocations. We begin by constructing the Bogolyubov-Born-Green-Yvon-Kirkwood (BBGYK) integral equations relating different orders of dislocation correlation functions i…
Results of recent large-scale molecular dynamics simulations of dislocation-mediated solid plasticity are campared with predictions of the statistical thermodynamic theory of these phenomena. These computational and theoretical analyses are in substantial agreement with each other in both their descriptions of strain-rate dependent steady plastic flow and of a transient stress peak associated with initially small densities of dislocations. The comparisons between the numerical simulations and basic theory reveal inconsistencies in some conventional phenomenological descriptions of solid plasti…
Enabling plastic co-deformation of disparate phases in a laser rapid solidified Sr-modified Al-Si eutectic through partial-dislocation-mediated-plasticity in Si
Stressed dislocation pattern formation in crystal plasticity at finite deformation is demonstrated for the first time. Size effects are also demonstrated within the same mathematical model. The model involves two extra material parameters beyond the requirements of standard classical crystal plasticity theory. The dislocation microstructures shown are decoupled from deformation microstructures, and emerge without any consideration of latent hardening or constitutive assumptions related to cross-slip. Crystal orientation effects on the pattern formation and mechanical response are also demonstr…
Grain structure plays a key role in the mechanical properties of alloy materials. Engineering the grain structure requires a comprehensive understanding of the evolution of grain boundaries (GBs) when a material is subjected to various manufacturing processes. To this end, we present a computationally efficient framework to describe the co-evolution of bulk plasticity and GBs. We represent GBs as diffused geometrically necessary dislocations, whose evolution describes GB plasticity. Under this representation, the evolution of GBs and bulk plasticity is described in unison using the evolution e…
Electroplasticity is defined as the reduction in flow stress of a material undergoing deformation on passing an electrical pulse through it. The lowering of flow stress during electrical pulsing has been attributed to a combination of three mechanisms: softening due to Joule-heating of the material, de-pinning of dislocations from paramagnetic obstacles, and the electron-wind force acting on dislocations. However, there is no consensus in literature regarding the relative magnitudes of the reductions in flow stress resulting from each of these mechanisms. In this paper, we extend a dislocation…
Dislocation slip is a general deformation mode and governs the strength of metals. Via discrete dislocation dynamics and molecular dynamics simulations, we investigate the strain rate and dislocation density dependence of the strength of bulk copper single crystals using 192 simulations spanning over 10 orders of magnitude in strain rate and 9 orders of magnitude in dislocation density. Based on these large set of simulations and theoretical analysis, a new analytical relationship between material strength, dislocation density, strain rate and dislocation mobility is proposed, which is in exce…
Plastic deformation in microscale differs from the macroscopic plasticity in two respects: (i) the flow stress of small samples depends on their size (ii) the scatter of plasticity increases significantly. In this work we focus on the scatter of plasticity. We statistically characterize the deformation process of micropillars under tension using results from discrete dislocation dynamics (DDD) simulations. We then propose a stochastic microplasticity model which uses the extracted information from the above statistical characterization to make statistical predictions regarding the micropillar …
The critical dynamics of dislocation avalanches in plastic flow is examined using a phase field crystal (PFC) model. In the model, dislocations are naturally created, without any \textit{ad hoc} creation rules, by applying a shearing force to the perfectly periodic ground state. These dislocations diffuse, interact and annihilate with one another, forming avalanche events. By data collapsing the event energy probability density function for different shearing rates, a connection to interface depinning dynamics is confirmed. The relevant critical exponents agree with mean field theory predictio…
We present a framework which unifies classical phenomenological $J_2$ and crystal plasticity theories with quantitative dislocation mechanics. The theory allows the computation of stress fields of arbitrary dislocation distributions and, coupled with minimally modified classical ($J_2$ and crystal plasticity) models for the plastic strain rate of statistical dislocations, results in a versatile model of finite deformation mesoscale plasticity. We demonstrate some capabilities of the framework by solving two outstanding challenge problems in mesoscale plasticity: 1) recover the experimentally o…
Because of the enormous range of time and space scales involved in dislocation dynamics, plastic modeling at macroscale requires a continuous formulation. In this paper, we present a rigorous formulation of the transition between the discrete, where plastic flow is resolved at the scale of individual dislocations, and the continuum, where dislocations are represented by densities. First, we focus on the underlying coarse-graining procedure and show that the emerging correlation-induced stresses are scale-dependent. Each of these stresses can be expanded into the sum of two components. The firs…
We develop a nonlinear, three-dimensional phase field model for crystal plasticity which accounts for the infinite and discrete symmetry group G of the underlying periodic lattice. This generates a complex energy landscape with countably-many G-related wells in strain space, whereon the material evolves by energy minimization under the loading through spontaneous slip processes inducing the creation and motion of dislocations without the need of auxiliary hypotheses. Multiple slips may be activated simultaneously, in domains separated by a priori unknown free boundaries. The wells visited by t…
Creep in single crystal Nickel-based superalloys has been a topic of interest since decades, and nowadays simulations are more and more able to complement experiments. In these alloys, the $γ/γ'$ phase microstructure co-evolves with the system of dislocations under load, and understanding the mutual interactions is essential for understanding the resulting creep properties. Predictive modeling thus requires multiphysics frameworks capable of modeling and simulating both the phase and defect microstructures. To do so, we formulate a coupled model of phase-field evolution and continuum dislocati…
The deformation behaviour of the intermetallic Al$_{2}$Cu-phase was investigated using atomistic simulations and micropillar compression, where slip on the unexpected {211} and {022} slip planes was revealed. Additionally, all possible slip systems for the intermetallic phases were further evaluated and a preference for the activation of slip systems based on their effective interplanar distances as well as the effective Burgers vector is proposed. The effective interplanar distance corresponds to the manually determined interplanar distance, whereas the effective Burgers vector takes a potent…
Along with high strength, plasticity is what makes metals so widely usable in our material world. Both strength and plasticity properties of a metal are defined by the motion of dislocations - line defects in the crystal lattice that divide areas of atomic planes displaced relative to each other by an interatomic distance. Here we present first fully dynamic atomistic simulations of single crystal plasticity in metal tantalum predicting that above certain maximum rate of straining - the ultimate limit - the dislocations can no longer relieve mechanical loads and another mechanism, twinning, co…
The spontaneous emergence of heterogeneous dislocation patterns is a conspicuous feature of plastic deformation and strain hardening of crystalline solids. Despite long-standing efforts in the materials science and physics of defect communities, there is no general consensus regarding the physical mechanism which leads to the formation of dislocation patterns. In order to establish the fundamental mechanism, we formulate an extremely simplified, minimal model to investigate the formation of patterns based on the continuum theory of fluxes of curved dislocations. We demonstrate that strain hard…
In Part I of this set of two papers, a model of mesoscopic plasticity is developed for studying initial-boundary value problems of small scale plasticity. Here we make qualitative, finite element method-based computational predictions of the theory. We demonstrate size effects and the development of strong inhomogeneity in simple shearing of plastically-constrained grains. Nonlocality in elastic straining leading to a strong Bauschinger effect is analyzed. Stability of the time dependent, spatially homogeneous, simple shearing solution of PMFDM is studied. Results from thermal cycling of small…
Coarse-grained, metallic materials undergo microstructure refinement during tribological loading. This in turn results in changing tribological properties, so understanding deformation under tribological load is mandatory when designing tribological systems. Single-trace experiments were conducted to understand the initiation of deformation mechanisms acting in various tribological systems. The main scope of this work was to investigate the influence of normal and friction forces as well as crystal orientations on the dominating deformation mechanism in a face-centred cubic concentrated solid …
Plastic deformation mediated by collective dislocation dynamics is investigated in the two-dimensional phase-field crystal model of sheared single crystals. We find that intermittent fluctuations in the dislocation population number accompany bursts in the plastic strain-rate fluctuations. Dislocation number fluctuations exhibit a power-law spectral density $1/f^2$ at high frequencies $f$. The probability distribution of number fluctuations becomes bimodal at low driving rates corresponding to a scenario where low density of defects alternate at irregular times with high population of defects.…
