A non-singular continuum theory of dislocations

Plastic deformation of crystalline solids depends to a high degree on the mechanisms related to the dislocation network. In order to accommodate plastic deformation and to reduce the crystal's energy, new dislocations are nucleated and pile up near the grain or phase boundaries, thereby giving rise to material strengthening. The nucleation and motion of dislocations is hence an essential mechanism to explain plastic yielding, work hardening as well as size and hysteresis effects in crystal plasticity and needs embedding into the constitutive framework of modeling materials with microstructure. An important aspect of modeling dislocation microstructures by a continuum approach lies in a sensible representation of those effects stemming from the characteristics of the discrete crystal lattice which, in particular, prohibits high local dislocation concentrations. Such a saturation behavior gives rise to numerous experimentally observed effects. In particular, experimental investigations hint at an essential size-effect of many properties of elasto-plastic crystals (e.g., the size-dependence of the indentation force during nano-indentation experiments, the grain-size dependence of the yield stress of Hall-Petch or other type, etc.)

Journal of Materials Research, 2011

Miniaturization of components and devices calls for an increased effort on physically motivated continuum theories, which can predict size-dependent plasticity by accounting for length scales associated with the dislocation microstructure. An important recent development has been the formulation of a Continuum Dislocation Dynamics theory (CDD) that provides a kinematically consistent continuum description of the dynamics of curved dislocation systems [T. Hochrainer, et al., Philos. Mag. 87, 1261]. In this work, we present a brief overview of dislocation-based continuum plasticity models. We illustrate the implementation of CDD by a numerical example, bending of a thin film, and compare with results obtained by three-dimensional discrete dislocation dynamics (DDD) simulation.

2005

In the context of recent proposals to use statistical mechanics methods for building a continuum theory of dislocation lines, mathematical modelling has to answer three essential questions: (i) What is the mathematical object representing the single dislocation as basic "particle"? (ii) What is the law of motion of this object? (iii) What is the mathematical nature of a dislocation density built of such objects? If a mathematically rigorous answer to these questions can be given, one may expect to derive the kinetic evolution equation for such a density solely from its definition and a conservation law. We present a method for deriving classical and non-classical dislocation density measures as well as their evolution equations from the properties of single dislocations, using the close connection between differential forms and geometrical objects such as dislocation lines. Several dislocation density measures are compared in view of their ability to represent vital aspects of the statics and dynamics of discrete dislocation configurations. A dislocation density measure which considers line directions and curvatures is defined as differential form, and it is shown that its evolution correctly represents the essential features of dislocation motion.

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2003

As a guide to constitutive specification, driving forces for dislocation velocity and nucleation rates are derived for a field theory of dislocation mechanics. A condition of closure for the theory in the form of a boundary condition for dislocation density is also derived. Kinematical features of dislocation evolution like initiation of bowing of a pinned screw segment, and initiation of cross-slip of a screw segment are discussed. An exact solution for the expansion of a polygonal loop as well as representation within the theory of dislocation level Schmid and non-Schmid behavior, and unloaded stress-free and steady microstructures are also discussed.

Philosophical Magazine, 2010

Crystal plasticity is governed by the motion of lattice dislocations. Although continuum theories of static dislocation assemblies date back to the 1950's, 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 microbending.

Physical Review B, 1990

A new methodology in computational micromechanics, dislocation dynamics (DD), is introduced. Dislocation dynamics is developed for examining the dynamic behavior of dislocation distributions in solid materials. Under conditions of externally applied stress, dislocations exhibit glide with a velocity proportional to a power of the applied stress 0. and climb motion with a velocity that is a function of the applied stress and temperature. These motions result from long-range force fields, comprising both externally applied stress and long-range interactions between individual dislocations. Short-range reactions are represented as discrete events. The DD methodology is to be differentiated from particle methods in statistical mechanics (e.g. , molecular dynamics and the Monte Carlo method) in two respects. First, DD is developed to study the dynamical behavior of "defects" in the solid. Generally, the density of defects is less than that of the particles that make up the solid. Second, the small number of dislocations allows for a complete dynamical representation of the evolution of dislocations in the material medium without the requirement of statistical averaging. The purpose of the DD methodology is to bridge the gap between experimentally observed phenomena and theoretical descriptions of dislocation aggregates, particularly the evolution of self-organized dislocation structures under temperature, stress, and irradiation conditions.

Journal of the Mechanics and Physics of Solids, 2016

Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i) to represent dislocation kinematics in terms of a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. The kinematic problem (i) was recently solved through the introduction of continuum dislocation dynamics (CDD), which provides kinematically consistent evolution equations of dislocation alignment tensors, presuming a given average dislocation velocity (Hochrainer (2015), Philos. Mag. 95 (12), 1321-1367). In the current paper we demonstrate how a free energy formulation may be used to solve the dynamic closure problem (ii) in CDD. We do so exemplarily for the lowest order CDD variant for curved dislocations in a single slip situation. In this case, a thermodynamically consistent average dislocation velocity is found to comprise five mesoscopic shear stress contributions. For a postulated free energy expression we identify among these stress contributions a back-stress term and a line-tension term, both of which have already been postulated for CDD. A new stress contribution occurs which is missing in earlier CDD models including the statistical continuum theory of straight parallel edge dislocations (Groma et al. (2003), Acta Mater. 51, 1271-1281). Furthermore, two entirely new stress contributions arise from the curvature of dislocations.

2005

In the context of recent proposals to use statistical mechanics methods for building a continuum theory of dislocation lines, mathematical modelling has to answer three essential questions: (i) What is the mathematical object representing the single dislocation as basic "particle"? (ii) What is the law of motion of this object? (iii) What is the mathematical nature of a dislocation density built of such objects? If a mathematically rigorous answer to these questions can be given, one may expect to derive the kinetic evolution equation for such a density solely from its definition and a conservation law. We present a method for deriving classical and non-classical dislocation density measures as well as their evolution equations from the properties of single dislocations, using the close connection between differential forms and geometrical objects such as dislocation lines. Several dislocation density measures are compared in view of their ability to represent vital aspect...

2009

Plastic deformation of crystalline solids depends to a high degree on the mechanisms related to the dislocation network. In order to accommodate plastic deformation and to reduce the crystal's energy, new dislocations are nucleated and pile up near the grain or phase boundaries, thereby giving rise to material strengthening. The nucleation and motion of dislocations is hence an essential mechanism to explain plastic yielding, work hardening as well as size and hysteresis effects in crystal plasticity and needs embedding into the constitutive framework of modeling materials with microstructure. An important aspect of modeling dislocation microstructures by a continuum approach lies in a sensible representation of those effects stemming from the characteristics of the discrete crystal lattice which, in particular, prohibits high local dislocation concentrations. Such a saturation behavior gives rise to numerous experimentally observed effects. In particular, experimental investigations hint at an essential size-effect of many properties of elasto-plastic crystals (e.g., the size-dependence of the indentation force during nano-indentation experiments, the grain-size dependence of the yield stress of Hall-Petch or other type, etc.)

Philosophical Magazine, 2007

We propose a dislocation density measure which is able to account for the evolution of systems of three-dimensionally curved dislocations. The definition and evolution equation of this measure arise as direct generalisations of the definition and kinematic evolution equation of the classical dislocation density tensor. The evolution of this measure allows 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 visualises the definitions and their relations to the classical theory.