Stefan Sandfeld

Numerical implementation of a 3D continuum theory of dislocation dynamics and application to micro-bending

Crystal plasticity is governed by the motion of lattice dislocations. Although continuum theories of static dislocation assemblies date back to the 1950s, the line-like character of these defects posed serious problems for the development of a continuum theory of plasticity which is based on the averaged dynamics of dislocation systems. Only recently the geometrical problem of performing meaningful averages over systems of moving, oriented lines has been solved. Such averaging leads to the definition of a dislocation density tensor of second order along with its evolution equation. This tensor can be envisaged as the analogue of the classical dislocation density tensor in an extended space which includes the line orientation as an independent variable. In the current work, we discuss the numerical implementation of a continuum theory of dislocation evolution that is based on this dislocation density measure and apply this to some simple benchmark problems as well as to plane-strain micro-bending.

Scaling properties of dislocation simulations in the similitude regime

Dislocation systems exhibit well-known scaling properties such as the Taylor relationship between flow stress and dislocation density, and the ?law of similitude? linking the flow stress to the characteristic wavelength of dislocation patterns. Here, we discuss the origin of these properties, which can be related to generic invariance properties of the equations of evolution of discrete dislocation systems, and their implications for a wide class of models of dislocation microstructure evolution. We demonstrate that under certain conditions dislocation simulations carried out at different stress, dislocation density and strain rate can be considered to be equivalent, and we study the range of deformation conditions (?similitude regime?) over which this equivalence can be expected to hold. In addition, we discuss restrictions imposed by the stated invariance properties for density-based, non-local or stochastic models of dislocation microstructure evolution, and for dislocation patterns and size effects.

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.

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.

Universal features of amorphous plasticity

Plastic yield of amorphous solids occurs by power law distributed slip avalanches whose universality is still debated. Determination of the power law exponents from experiments and molecular dynamics simulations is hampered by limited statistical sampling. On the other hand, while existing elasto-plastic depinning models give precise exponent values, these models to date have been limited to a scalar approximation of plasticity which is difficult to reconcile with the statistical isotropy of amorphous materials. Here we introduce for the first time a fully tensorial mesoscale model for the elasto-plasticity of disordered media that can not only reproduce a wide variety of shear band patterns observed experimentally for different deformation modes, but also captures the avalanche dynamics of plastic flow in disordered materials. Slip avalanches are characterized by universal distributions which are quantitatively different from mean field predictions, both regarding the exponents and regarding the form of the scaling functions, and which are independent of system dimensionality (2D vs 3D), boundary and loading conditions, and uni-or biaxiality of the stress state. We also measure average avalanche shapes, which are equally universal and inconsistent with mean field predictions. Our results provide strong evidence that the universality class of plastic yield in amorphous materials is distinct from that of mean field depinning.Plastic yielding of amorphous solids occurs by power-law distributed deformation avalanches whose universality is still debated. Experiments and molecular dynamics simulations are hampered by limited statistical samples, and although existing stochastic models give precise exponents, they require strong assumptions about fixed deformation directions, at odds with the statistical isotropy of amorphous materials. Here, we introduce a fully tensorial, stochastic mesoscale model for amorphous plasticity that links the statistical physics of plastic yielding to engineering mechanics. It captures the complex shear patterning observed for a wide variety of deformation modes, as well as the avalanche dynamics of plastic flow. Avalanches are described by universal size exponents and scaling functions, avalanche shapes, and local stability distributions, independent of system dimensionality, boundary and loading conditions, and stress state. Our predictions consistently differ from those of mean-field depinning models, providing evidence that plastic yielding is a distinct type of critical phenomenon.

Pattern formation in a minimal model of continuum dislocation plasticity

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 hardening as embodied in a Taylor-type dislocation density dependence of the flow stress, in conjunction with the structure of the kinematic equations that govern dislocation motion under the action of external stresses, is already sufficient for the formation of dislocation patterns that are consistent with the principle of similitude.

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.

Continuum representation of systems of dislocation lines: A general method for deriving closed-form evolution equations

Abstract Plasticity is governed by the evolution of, in general anisotropic, systems of dislocations. We seek to faithfully represent this evolution in terms of density-like variables which average over the discrete dislocation microstructure. Starting from T. Hochrainers continuum theory of dislocations (CDD) ( Hochrainer, 2015 ), we introduce a methodology based on the ‘Maximum Information Entropy Principle’ (MIEP) for deriving closed-form evolution equations for dislocation density measures of different order. These equations provide an optimum representation of the kinematic properties of systems of curved and connected dislocation lines with the information contained in a given set of density measures. The performance of the derived equations is benchmarked against other models proposed in the literature, using discrete dislocation dynamics simulations as a reference. As a benchmark problem we study dislocations moving in a highly heterogeneous, persistent-slip-band like geometry. We demonstrate that excellent agreement with discrete simulations can be obtained in terms of a very small number of averaged dislocation fields containing information about the edge and screw components of the total and excess (geometrically necessary) dislocation densities. From these the full dislocation orientation distribution which emerges as dislocations move through a channel-wall structure can be faithfully reconstructed.

Avalanches, loading and finite size effects in 2D amorphous plasticity: results from a finite element model

Crystalline plasticity is strongly interlinked with dislocation mechanics and nowadays is relatively well understood. Concepts and physical models of plastic deformation in amorphous materials on the other hand - where the concept of linear lattice defects is not applicable - still are lagging behind. We introduce an eigenstrain-based finite element lattice model for simulations of shear band formation and strain avalanches. Our model allows us to study the influence of surfaces and finite size effects on the statistics of avalanches. We find that even with relatively complex loading conditions and open boundary conditions, critical exponents describing avalanche statistics are unchanged, which validates the use of simpler scalar lattice-based models to study these phenomena.

Deformation patterns and surface morphology in a minimal model of amorphous plasticity

We investigate a minimal model of the plastic deformation of amorphous materials. The material elements are assumed to exhibit ideally plastic behavior (J2 plasticity). Structural disorder is considered in terms of random variations of the local yield stresses. Using a finite element implementation of this simple model, we simulate the plane strain deformation of long thin rods loaded in tension. The resulting strain patterns are statistically characterized in terms of their spatial correlation functions. Studies of the corresponding surface morphology reveal a non-trivial Hurst exponent H ≈ 0.8, indicating the presence of long-range correlations in the deformation patterns. The simulated deformation patterns and surface morphology exhibit persistent features which emerge already at the very onset of plastic deformation, while subsequent evolution is characterized by growth in amplitude without major morphology changes. The findings are compared to experimental observations.