Michael Zaiser

Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics

The problem of the collective behavior of straight parallel edge dislocations is investigated. Starting from the equation of motion of individual dislocations a continuum description is derived. It is shown that the influence of the short range dislocation-dislocation interactions on the dislocation dynamics can be well described by a local back stress which scales like the square root of dislocation density plus a non-local diffusion-like term. The value of the corresponding diffusion coefficient is determined numerically, and implications for size effects in plasticity are discussed.

Scale invariance in plastic flow of crystalline solids

From the traditional viewpoint of continuum plasticity, plastic deformation of crystalline solids is, at least in the absence of so-called plastic instabilities, envisaged as a smooth and quasi-laminar flow process. Recent theoretical and experimental investigations, however, demonstrate that crystal plasticity is characterized by large intrinsic spatio-temporal fluctuations with scale-invariant characteristics: In time, deformation proceeds through intermittent bursts with power-law size distributions; in space, deformation patterns and deformation-induced surface morphology are characterized by long-range correlations, self-similarity and/or self-affine roughness. We discuss this scale-invariant behaviour in terms of robust scaling associated with a non-equilibrium critical point (‘yielding transition’). Contents PAGE 1. Introduction 186  1.1. Continuum mechanics of crystal plasticity 187  1.2. Crystal plasticity on the dislocation level: yield stress and depinning transition 191 2. Experimental investigation of fluctuation phenomena in plastic flow 197  2.1. Acoustic emission measurements 197   2.1.1. Experimental methodology 197   2.1.2. Acoustic emission in single- and polycrystals of ice 198   2.1.3. Acoustic emission in metals and alloys 201  2.2. Deformation-induced surface patterns 202   2.2.1. Slip-line patterns 202   2.2.2. Slip-line kinematography 203   2.2.3. Surface roughening in single- and polycrystals 205  2.3. Deformation of micron-size samples 209 3. Theoretical approaches 212  3.1. Dislocation dynamics 213   3.1.1. Simulation methods 213   3.1.2. Relaxation and creep of two-dimensional dislocation systems 218   3.1.3. Stepwise deformation curves and critical behaviour at yield 220  3.2. Models of microstrain evolution 224   3.2.1. Constitutive equations 224   3.2.2. Avalanche dynamics and surface morphology evolution 227  3.3. Phase-field models 233 4. Discussion and conclusions 236  4.1. Why has it not been seen before? 237  4.2. Open questions, doubts and prospects 240 Acknowledgements 243 References 243

Anticrack Nucleation as Triggering Mechanism for Snow Slab Avalanches

Snow slab avalanches are believed to begin by the gravity-driven shear failure of weak layers in stratified snow. The critical crack length for shear crack propagation along such layers should increase without bound as the slope decreases. However, recent experiments show that the critical length of artificially introduced cracks remains constant or, if anything, slightly decreases with decreasing slope. This surprising observation can be understood in terms of volumetric collapse of the weak layer during failure, resulting in the formation and propagation of mixed-mode anticracks, which are driven simultaneously by slope-parallel and slope-normal components of gravity. Such fractures may propagate even if crack-face friction impedes downhill sliding of the snowpack, indicating a scenario in which two separate conditions have to be met for slab avalanche release.

A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation

We propose a dislocation density measure which is able to account for the evolution of systems of three-dimensional curved dislocations. The definition and evolution equation of this measure arise as direct generalizations of the definition and kinematic evolution equation of the classical dislocation density tensor. The evolution of this measure allows us to determine the plastic distortion rate in a natural fashion and therefore yields a kinematically closed dislocation-based theory of plasticity. A self-consistent theory is built upon the measure which accounts for both the long-range interactions of dislocations and their short-range self-interaction which is incorporated via a line-tension approximation. A two-dimensional kinematic example illustrates the definitions and their relations to the classical theory.

Self-affine surface morphology of plastically deformed metals.

We analyze the surface morphology of metals after plastic deformation over a range of scales from 10 nm to 2 mm using atomic force microscopy and scanning white-light interferometry. We demonstrate that an initially smooth surface during deformation develops self-affine roughness over almost 4 orders of magnitude in scale. The Hurst exponent H of one-dimensional surface profiles initially decreases with increasing strain and then stabilizes at H approximately 0.75. We show that the profiles can be mathematically modeled as graphs of a fractional Brownian motion. Our findings can be understood in terms of a fractal distribution of plastic strain within the deformed samples.

Strain bursts in plastically deforming molybdenum micro- and nanopillars

Plastic deformation of micron and sub-micron scale specimens is characterized by intermittent sequences of large strain bursts (dislocation avalanches) which are separated by regions of near-elastic loading. In the present investigation we perform a statistical characterization of strain bursts observed in stress-controlled compressive deformation of monocrystalline molybdenum micropillars. We characterize the bursts in terms of the associated elongation increments and peak deformation rates, and demonstrate that these quantities follow power-law distributions that do not depend on specimen orientation or stress rate. We also investigate the statistics of stress increments in between the bursts, which are found to be Weibull distributed and exhibit a characteristic size effect. We discuss our findings in view of observations of deformation bursts in other materials, such as face-centred cubic and hexagonal metals.

Fractal analysis of deformation-induced dislocation patterns

The paper reports extensive analyses of the fractal geometry of cellular dislocation structures observed in Cu deformed in multiple-slip orientation. Several methods presented for the determination of fractal dimensions are shown to give consistent results. Criteria are formulated which allow the distinguishing of fractal from non-fractal patterns, and implications of fractal dislocation patterning for quantitative metallography are discussed in detail. For an interpretation of the findings a theoretical model is outlined according to which dislocation cell formation is associated to a noise-induced structural transition far from equilibrium. This allows relating the observed fractal dimensions to the stochastic properties of deformation by collective dislocation glide.

Depinning transition of dislocation assemblies: Pileups and low-angle grain boundaries

We investigate the depinning transition occurring in dislocation assemblies. In particular, we consider the cases of regularly spaced pileups and low-angle grain boundaries interacting with a disordered stress landscape provided by solute atoms, or by other immobile dislocations present in nonactive slip systems. Using linear elasticity, we compute the stress originated by small deformations of these assemblies and the corresponding energy cost in two and three dimensions. Contrary to the case of isolated dislocation lines, which are usually approximated as elastic strings with an effective line tension, the deformations of a dislocation assembly cannot be described by local elastic interactions with a constant tension or stiffness. A nonlocal elastic kernel results as a consequence of long-range interactions between dislocations. In light of this result, we revise statistical depinning theories of dislocation assemblies and compare the theoretical results with numerical simulations and experimental data.

Statistical modelling of dislocation systems

Abstract The paper discusses fluctuation phenomena in plastic flow. It is demonstrated that dislocation motion in materials with high dislocation mobility is characterized by large velocity fluctuations that lead to an intrinsically intermittent dynamics on microscopic and mesoscopic scales even if macroscopic deformation is stable and homogeneous. The large fluctuations indicate that the dislocation system in a deforming crystal is close to a critical point (‘yielding transition’). For motion of isolated dislocations through weak obstacle fields, this critical point corresponds to the depinning transition of an elastic manifold. A phenomenological approach to modelling the influence of collective, intermittent dislocation motions on dislocation microstructure evolution is discussed. Applications include the development of misorientations in dislocation cell structures and the interplay of strain hardening and dislocation patterning under deformation conditions where fractal dislocation arrangements are formed.

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.