A continuum-dislocation theory for modeling dislocation microstructures and size effects in crystal plasticity (original) (raw)
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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.)
Size-dependent energy in crystal plasticity and continuum dislocation models
2015
In the light of recent progress in coarsening the discrete dislocation mechanics, we consider two questions relevant for the development of a mesoscale, size-dependent plasticity: (i) can the phenomenological expression for size-dependent energy, as quadratic form of Nye’s dislocation density tensor, be justified from the point of view of dislocation mechanics and under what conditions? (ii) how can physical or phenomenological expressions for size-dependent energy be computed from dislocation mechanics in the general case of elastically anisotropic crystal? The analysis based on material and slip system symmetries implies the negative answer to the first question. However, the coarsening method developed in response to the second question, and based on the physical interpretation of the size-dependent energy as the coarsening error in dislocation interaction energy, introduces additional symmetries. The result is that the equivalence between the phenomenological and the physical expressions is possible, but only if the multiplicity of characteristic lengths associated with different slip systems, is sacrificed. Finally, we discuss the consequences of the assumption that a single length scale governs the plasticity of a crystal, and note that the plastic dissipation at interfaces has a strong dependence on the length scale embedded in the energy expression.
Journal of the Mechanics and …, 2010
"Starting from the standard coarsening of dislocation kinematics, we derive the sizedependent continuum crystal plasticity by systematic thermodynamic coarsening of dislocation mechanics. First, we observe that the energies computed from different kinematic descriptions are different. Then, we consider systems without boundary dissipation (relaxation) and derive the continuum approximation for the free energy of elastic–plastic crystals. The key elements are: the two-dimensional nature of dislocation pile-ups at interfaces, the localized nature of the coarsening error in energy, and, the orthogonal decomposition theorem for compatible and incompatible elastic strain fields. Once the energy landscape is defined, the boundary dissipation is estimated from the height of energy barriers. The characteristic lengths are the average slip plane spacing for each slip system. They may evolve through the double-cross slip mechanism. The theory features the slipdependent interface free energy and interface dissipation for penetrable interfaces. The main constitutive parameters are derived from elasticity. The exception is the dependence of interface energy on slip plane orientation, which is determined from numerical results. The theory requires no higher order boundary conditions. The only novel boundary conditions are kinematic, involving slip relaxation on the two sides of an interface."
Continuum Dislocation Theory and Related Size Effects in Crystal Plasticity
2010
This Chapter discusses the continuum dislocation theory and its applications in crystal plasticity. We aim at studying the dislocation nucleation and accumulation, the resulting work hardening and the influence of the resistance to dislocation motion. Among boundary-value problems we consider plane constrained shear, plane strain uniaxial extension and their combination for single and bi-crystals, which admit analytical solutions. The interesting features of these solutions are the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effects.
Size effects under homogeneous deformation of single crystals: A discrete dislocation analysis
Journal of The Mechanics and Physics of Solids, 2008
Mechanism-based discrete dislocation plasticity is used to investigate the effect of size on micron scale crystal plasticity under conditions of macroscopically homogeneous deformation. Long-range interactions among dislocations are naturally incorporated through elasticity. Constitutive rules are used which account for key short-range dislocation interactions. These include junction formation and dynamic source and obstacle creation. Two-dimensional calculations are carried out which can handle high dislocation densities and large strains up to 0.1. The focus is laid on the effect of dimensional constraints on plastic flow and hardening processes. Specimen dimensions ranging from hundreds of nanometers to tens of microns are considered. Our findings show a strong size-dependence of flow strength and work-hardening rate at the micron scale. Taylor-like hardening is shown to be insufficient as a rationale for the flow stress scaling with specimen dimensions. The predicted size effect is associated with the emergence, at sufficient resolution, of a signed dislocation density. Heuristic correlations between macroscopic flow stress and macroscopic measures of dislocation density are sought. Most accurate among those is a correlation based on two state variables: the total dislocation density and an effective, scale-dependent measure of signed density. r
Plasticity of crystals and interfaces: From discrete dislocations to size-dependent continuum theory
Theoretical and Applied Mechanics, 2010
In this communication, we summarize the current advances in size-dependent continuum plasticity of crystals, specifically, the rate-independent (quasistatic) formulation, on the basis of dislocation mechanics. A particular emphasis is placed on relaxation of slip at interfaces. This unsolved problem is the current frontier of research in plasticity of crystalline materials. We outline a framework for further investigation, based on the developed theory for the bulk crystal. The bulk theory is based on the concept of geometrically necessary dislocations, specifically, on configurations where dislocations pile-up against interfaces. The average spacing of slip planes provides a characteristic length for the theory. The physical interpretation of the free energy includes the error in elastic interaction energies resulting from coarse representation of dislocation density fields. Continuum kinematics is determined by the fact that dislocation pile-ups have singular distribution, which allows us to represent the dense dislocation field at the boundary as a superdislocation, i.e., the jump in the slip filed. Associated with this jump is a slip-dependent interface energy, which in turn, makes this formulation suitable for analysis of interface relaxation mechanisms.
Discrete Dislocation Plasticity Analysis of Size Effects in Single Crystals
The effect of loading conditions on the tensile stress versus strain response of micron-sized planar crystals with a single active slip system is investigated via finite and small deformation discrete dislocation plasticity analyses. When rotation of the tensile axis is prevented, lattice curvature is induced in the crystal in both the small and finite strain analyses with the build-up of geometrically necessary dislocations resulting in a hardening response. The hardening rate is higher in the small strain analyses and this is attributed to the assumption of linear kinematics in that analysis. On the other hand, when rotation of the tensile axis is permitted, no lattice curvature is induced in the crystal in the small strain analysis resulting in an ideally plastic response. However, the change in the geometry of the crystal induces bending moments in the crystal in the finite strain analyses giving rise to a mildly hardening tensile stress versus strain response.
Continuum dislocation theory accounting for redundant dislocations and Taylor hardening
International Journal of Engineering Science, 2016
This paper develops the phenomenological continuum dislocation theory accounting for the density of redundant dislocations and Taylor hardening for single crystals. As illustration, the problem of anti-plane constrained shear of single crystal deforming in single slip is solved within the proposed theory. The distribution of excess dislocations in the final state of equilibrium as well as the stress-strain curve exhibiting the Bauschinger translational work hardening and the size effect are found. Comparison with the stress-strain curve obtained from the continuum dislocation theory without the density of redundant dislocations and Taylor hardening is provided.
Physical Review B, 2008
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 in a grand canonical ensemble. Approximate pair correlation functions are obtained for single-slip systems with two types of dislocations and, subsequently, for general multiple-slip systems of both charges. The effect of the correlations manifests itself in the form of an entropic force in addition to the external stress and the self-consistent internal stress. Comparisons with a previous multiple-slip theory based on phenomenological considerations shall be discussed.
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.