Phase-field Model Of Dislocations Research Articles

Three-dimensional phase-field model of dislocations for a heterogeneous face-centered cubic crystal

A central aim of current materials studies is to develop a predictive modeling that incorporates dislocation-based plastic activity and microstructural evolution. Phase-field method has emerged as a powerful tool for addressing this issue, providing us with a versatile variational framework able to describe the movement of dislocations in interaction with underlying microstructures. In this article, a three-dimensional phase-field model of dislocations (PFMD) is developed with a discretization scheme that explicitly captures the face-centered cubic (FCC) geometry. Within this framework, continuous fields are discretized in a way that allows to consider strongly heterogeneous materials and sharp interfaces (free surfaces, stiffer precipitates, pores...) without generating numerical artifacts. The PFMD exposed in this work reproduces dislocation activity in FCC geometry, their reactions, and a particular attention is devoted to the dislocation core behaviors in order to remove effects present in prior generic PFMDs that can appear to be spurious for micron-scale applications. This allows us to rigorously reproduce the dislocation’s velocity with respect to experimental friction coefficients. The model is discussed and illustrated by applications standing at different space-scales that show how dislocations operate with microstructural heterogeneities such as free-surfaces (cylindrical nanopillar) and voids (pore under isostatic pressure).

Developing micro-scale crystal plasticity model based on phase field theory for modeling dislocations in heteroepitaxial structures

The capability of conventional crystal plasticity theory is extended in this paper to model a single dislocation behavior in heteroepitaxial structures. The plastic slip associated with each slip system is described by a continuous smooth field independently, which is the phase field interpolating between a slipped and an unslipped region. A dislocation is identified with location where the phase field value changes smoothly, to represent a smeared dislocation. Under a thermodynamically consistent framework that distinguishes between stored energy and dissipated energy during plastic deformation, the coupled balance equations of plastic slip evolution and quasi-static stress equilibrium are derived by using the principle of virtual power. With numerical implementation by finite element method, it is flexible to deal with material anisotropy and elastic modulus mismatch in heteroepitaxial structures at micro-scale, which are advantages of the proposed model compared to Khachaturyan-type phase field model of dislocations. Another advantage is that it is straightforward to handle the interaction and co-evolution of several types of material microstructures, such as dislocations and interfaces, within a unified continuum mechanics framework. Several examples are presented to illustrate the applicability and accuracy of the new method in modeling dislocations in complex heteroepitaxial structures.

High-resolution electron microscopy of dislocation ribbons in a CMSX-4 superalloy single crystal

High-resolution scanning transmission electron microscopy (STEM) has been used to study the structure of dislocations in single crystal superalloy samples that have been subjected to conditions that favour the primary creep regime. The study has revealed the detailed structure of extended a2〈112〉 dislocations as they shear the γ′ precipitates during creep. These dislocations dissociate in a manner that is consistent with predictions made using the phase-field model of dislocations and also suggests the importance of the reordering process during their movement. The shearing done by the a〈112〉 dislocations was also found to distort the γ/γ′ interface, changing its appearance from linear to a “saw tooth” pattern. Another important observation was the segregation of alloying elements with a high atomic mass to the stacking faults, presumably to reduce their energies during shear. Numerous a2〈110〉 dissociated dislocations were also observed in the γ channels of the superalloy. The high resolution provided by the STEM imaging enables one to study the high-energy faults that are usually difficult to observe in conventional weak-beam TEM, such as complex intrinsic and extrinsic stacking faults in the γ′ and intrinsic stacking faults in the γ, and to make estimates of their energies.

Shearing of γ′ precipitates in Ni-base superalloys: a phase field study incorporating the effective γ-surface

An extended phase field model of dislocations in Ni-base superalloys is presented. It incorporates the recently developed effective γ-surfaces for both matrix and precipitate phases, obtained from atomistic simulations. These novel γ-surfaces feature extrinsic stacking faults as additional local minima. Thus, they offer an increased number of available dislocation dissociation pathways within the phase field system. The new model has been used to simulate a variety of mechanisms for γ′ precipitate shearing proposed in literature. A critical assessment is made based on the modelling observations.

Prediction of Mechanical Behaviour in Ni-Base Superalloys Using the Phase Field Model of Dislocations

The “Phase-Field Model of Dislocations” (PFMD) was used to simulate shearing of gamma-prime precipitate arrays in single crystal turbine blade superalloys. The focus of the work has been on the cutting of the L12 ordered precipitates by a<112>{111} dislocation ribbons during Primary Creep. The Phase Field Model presented incorporates specially developed Generalised Stacking Fault Energy (–surface) data obtained from atomistic simulations. The topography of this surface determines the shearing mechanisms observed in the model. The merit of the new –surface, is that it accounts for the formation of extrinsic stacking faults, making the model more relevant to creep deformation of superalloys at elevated temperatures.

Shearing of γ′ precipitates by a〈112〉 dislocation ribbons in Ni-base superalloys: A phase field approach

Abstract The phase field model of dislocations has been used to study the propagation of dislocation ribbons with an overall Burgers vector of a〈1 1 2〉 through a simulated Ni-base superalloy. The driving force for dislocation dissociation reactions and formation of planar faults is incorporated into the free energy functional using periodic functions specially fitted to ab initio γ-surface data. The model shows that the mechanism of cutting of the γ′ precipitates by these ribbons exhibits significant dependence on stress magnitude, orientation and precipitate shape. In the case of mixed screw–edge ribbons a change of shearing mode is observed, from stacking fault shear to anti-phase boundary shear, when the applied stress approaches the yield of the material. This transition is absent in pure edge ribbons.

Scaling laws in plastic deformation

In crystalline materials, plastic deformation is driven by the non-local long-range interaction of dislocations. This interaction is responsible for a series of features that obey scaling laws, such as the formation of fractal cellular structures, the intermittent plastic flow with scale-free avalanches following a power law and the Hall–Petch relation in which the yield stress depends on the sample size following a power law. A phase-field model of dislocations is described. The present theory is able to reproduce the jerky character of dislocation motion as well as size dependence in small-scale plasticity.

Incorporation of γ-surface to phase field model of dislocations: simulating dislocation dissociation in fcc crystals

Phase field method has become an attractive alternative to study dislocations of arbitrary configurations. Recently the model has been generalized for dislocation nodal reactions and network formation. Within the same framework, we study dissociation of various dislocation configurations into Shockley partials by incorporating directly γ-surface data from ab initio calculations into the crystalline energy of the phase field model. The model is first validated against the generalized P–N model for dissociation of straight edge and screw dislocations in Al and Pd using the same sets of γ-surface data and elastic constants. Then dissociation of dislocation nodes and a bent dislocation gliding on two intersecting {1 1 1} planes are investigated. The former leads to either an extended or a contracted node, depending on the directions of the sense vectors of the unit dislocations that join the node, while the latter leads to a stair-rod dislocation at the intersection of the two planes. With reliable ab initio γ-surface data being calculated routinely by first principles, the extended phase field model could provide a flexible and realistic treatment of dissociation of dislocations of arbitrary configuration, interactions among extended dislocations, and creation and annihilation of various planar faults.

Phase field model of dislocation networks

In phase field model of dislocations, the short-range, non-linear core–core interactions are characterized through the crystalline energy and the gradient energy. In this article we extend and generalize the approximations of these energies employed in previous phase field models to account for dislocation reactions leading to network formation. In order to characterize dislocation activities involving all slip planes, we suggest the crystalline energy to be a function of a general plastic strain tensor produced by arbitrary linear combinations of simple shears associated with each slip system. For the four {111} slip planes in an fcc crystal, a particular form of such a crystalline energy is formulated by simple linear superposition of the interplanar potential of each individual slip plane. A more detail and general form of the gradient energy is derived from the consideration of the total Burgers vector dependence of dislocation line energy. Examples of applications are presented for interactions between two dislocation loops expanding on either a single slip plane or two intersecting slip planes, as well as for more complex reactions taking place in dislocation networks. It is shown that the generalized expressions are able to handle self-consistently reactions among dislocations of all slip systems in accord with Frank’s rule. These extensions are necessary steps toward advanced applications of the phase field method to dislocation substructure formation and coarsening.

Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations

The first Phase Field model of evolution of a multi-dislocation system in elastically anisotropic crystal under applied stress is formulated. The model is a modification and extension of our Phase Field Microelasticity approach to the theory of coherent phase transformations. The long-range strain-induced interaction of individual dislocations is calculated exactly and is explicitly incorporated in the Phase Field formalism. It also automatically takes into account the effects of “short-range interactions”, such as multiplication and annihilation of dislocations and a formation of various metastable microstructures involving dislocations and defects. The proposed 3-dimensional Phase Field model of dislocations does not impose a priori constraints on possible dislocation structures or their evolution paths. Examples of simulation of the FCC 3D system under applied stress are considered.