Peach Koehler Force Research Articles

The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity

We consider finite plasticity based on the decomposition F = F e F p of the deformation gradient F into elastic and plastic distortions F e and F p ( det F p = 1 ) . Within this framework the macroscopic Burgers vector may be characterized by the tensor field G = F p Curl F p . We derive a natural convected rate for G associated with evolution of F p and as our main result show that, for a single-crystal, • temporal changes in G —as characterized by its convected time derivative—may be decomposed into temporal changes in distributions of screw and edge dislocations on the individual slip systems. We discuss defect energies dependent on the densities of these distributions and show that corresponding thermodynamic forces are macroscopic counterparts of classical Peach–Koehler forces.

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.

Mesoscale Theory of Grains and Cells: Crystal Plasticity and Coarsening

Crystals with spatial variations in their axes naturally evolve into cells or grains separated by sharp walls. At high temperatures, polycrystalline grains form from the melt and coarsen with time: the dislocations can both climb and glide. At low temperatures under shear the dislocations (which allow only glide) form into cell structures. We present here a mesoscale theory of dislocation motion. It provides a quantitative description of deformation and rotation, grounded in a microscopic order parameter field exhibiting the topologically conserved quantities. The topological current of the Nye dislocation density tensor is derived from a microscopic theory of glide driven by Peach-Koehler forces between dislocations using a simple closure approximation. The resulting theory is shown to form sharp dislocation walls in finite time, both with and without dislocation climb.

Quantitative analysis of dislocation pile-ups nucleated during nanoindentation in MgO

Incipient plasticity in magnesium oxide single crystals has been investigated by nanoindentation with a spherical indenter. For very low loads, the surface deformation can be limited to a single slip line, making modelling of the deformation easier. The recently developed nanoetching technique has been used to investigate the corresponding dislocation structure. Single pile-ups of a few dislocation half loops have been observed and fully characterised. A simple elastic model, based upon the calculation of the Peach Koehler force induced by the pile-ups and the spherical indenter stress field, has been used to determine the equilibrium positions of dislocations in these pile-ups. A comparison between the experimental and calculated structures has led to the determination of the lattice friction stress for edge and screw dislocations in magnesium oxide single crystals. This stress is found to be 65MPa for edge dislocations and 86MPa for screw dislocations.

A note on material forces in finite inelasticity

In this contribution we aim to elaborate material forces in the context of multiplicative elasto-plasticity, which is considered as a representative and general framework for finite inelasticity. The comparison of different representations of the balance of linear momentum enables us to identify relevant Eshelbian stress tensors and corresponding volume forces. These material, or rather configurational, forces incorporate dislocation density tensors due to the general incompatibility of the underlying intermediate configuration. As an interesting application, the celebrated Peach–Koehler force, driving single dislocations in the context of finite-deformation inelasticity, allows representation in terms of the derived configurational volume forces.

Dislocation equilibrium conditions revisited

If there is an equilibrium arrangement of a given collection of dislocations, each having a fixed size and shape, in an externally loaded or unloaded elastic body, the corresponding potential energy will be stationary with respect to infinitesimal perturbations of the dislocation positions. This leads to the dislocation equilibrium conditions: the Peach–Koehler forces along the dislocation line of each dislocation due to externally applied stress and the interaction of the dislocation with other dislocations and its own image field is a set of self-equilibrated forces. The earlier proof of this result presented in the literature was based on an incomplete expression for the elastic strain energy. This is modified here by using the elastic strain energy expression that accounts for all dislocation core energy.

The material force acting on a screw dislocation in the presence of a multi-layered circular inclusion

In this paper, we study the interaction of a screw dislocation with a multi-layered interphase between a circularly cylindrical inclusion and a matrix. The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different shear moduli. The number of layers may also be arbitrary. Continuity of traction and displacement across all interfaces is assumed. We extend Honein et al.’s solution of circularly cylindrical layered media in anti-plane elastostatics to the case where all the singularities reside inside the inclusion core. The solution to this heterogeneous problem is given explicitly, for arbitrary singularities, as a rapidly convergent Laurent series, whose coefficients are expressed in terms of those of the complex potential of a corresponding homogeneous problem with the same singularities. We then consider the two particular cases of a screw dislocation, where, in the first instance, the dislocation resides inside the matrix, while, in the second instance, it is located in the inclusion core. In both instances, the Peach–Koehler force acting on the dislocation is calculated explicitly as a rapidly convergent series. We present several examples, where the effect of the layers on the material force is examined.

Interaction between an edge dislocation and a circular inclusion with an inhomogeneously imperfect interface

This research presents an analytical study of the interaction problem of an edge dislocation with a circular inclusion with a circumferentially inhomogeneously imperfect interface. The interface, which is modeled as a spring (interphase) layer with vanishing thickness, is characterized by that in which there is a displacement jump across the interface in the same direction as the corresponding tractions, and the same degree of imperfection is realized in both the normal and tangential directions. Furthermore, the interface parameter is nonuniform along the interface. In order to arrive at an elementary form solution, we introduce a conformal mapping function. Then the stress field as well as the Peach–Koehler force acting on the edge dislocation can be obtained from the derived complex potentials. Calculations demonstrate that the nonuniform interface parameter has a significant influence on the stress field.

Stability and Motion of Low Angle Dislocation Boundaries in Precipitation Hardened Crystals

The present study investigates stability and motion of low angle dislocation boundaries in an array of precipitates. The model considers discrete dislocations and precipitates that are treated as impenetrable particles. Peach-Koehler forces, which originate due to the combined effect of dislocation-dislocation interactions and the applied stress, act the individual dislocations on. Both, the dislocation glide and the dislocation climb at elevated temperatures are taken into account. Results of the numerical study suggest that a critical applied shear stress (CASS) always exists which separates stable and unstable low angle boundary configurations. Varying particle size, interparticle spacing and density of dislocations in the boundary cause changes of the CASS that are systematically investigated. It is shown that the CASSs can considerably differ from the standard Orowan stress controlling the equilibrium of an isolated dislocation in a given microstructure. This result underlines the importance of long-range dislocation interactions that influence the high temperature strength of the precipitation-hardened alloys.

Mechanism-based strain gradient crystal plasticity—I. Theory

We have been developing the theory of mechanism-based strain gradient plasticity (MSG) to model size-dependent plastic deformation at micron and submicron length scales. The core idea has been to incorporate the concept of geometrically necessary dislocations into the continuum plastic constitutive laws via the Taylor hardening relation. Here we extend this effort to develop a mechanism-based strain gradient theory of crystal plasticity. In this theory, an effective density of geometrically necessary dislocations for a specific slip plane is introduced via a continuum analog of the Peach–Koehler force in dislocation theory and is incorporated into the plastic constitutive laws via the Taylor relation.

Dislocation dynamics in Rayleigh-Bénard convection.

Theoretical results on the dynamics of dislocations in Rayleigh-Bénard convection are reported both for a Swift-Hohenberg model and the Oberbeck-Boussinesq equations. For intermediate Prandtl numbers the motion of dislocations is found to be driven by the superposition of two independent contributions: (i) the Peach-Koehler force and (ii) an advection force on the dislocation core by its self-generated mean flow. Their competition allows to explain the experimentally observed bound dislocation pairs.

Partial Dislocations and Stacking Faults in 4H-SiC PiN Diodes

Using plan-view transmission electron microscopy (TEM), we have identified stacking faults (SFs) in 4H-SiC PiN diodes subjected to both light and heavy electrical bias. Our observations suggest that the widely expanded SFs seen after heavy bias are faulted dislocation loops that have expanded in response to strain of the 4H-SiC film, while faulted screw or 60° threading dislocations do not give rise to widely expanded SFs. Theoretical calculations show that the expansion of SFs depends on the Peach-Koehler (PK) forces on the partial dislocations bounding the SFs, indicating that strain plays a critical role in SF expansion.

Partial dislocations and stacking faults in 4H-SiC PiN diodes

Using plan-view transmission electron microscopy (TEM), we have identified stacking faults (SFs) in 4H-SiC PiN diodes subjected to both light and heavy electrical bias. Our observations suggest that the widely expanded SFs seen after heavy bias are faulted dislocation loops that have expanded in response to strain of the 4H-SiC film, while faulted screw or 60° threading dislocations do not give rise to widely expanded SFs. Theoretical calculations show that the expansion of SFs depends on the Peach-Koehler (PK) forces on the partial dislocations bounding the SFs, indicating that strain plays a critical role in SF expansion.

Mechanism-based strain gradient crystal plasticity

AbstractTo model size dependent plastic deformation at micron and submicron length scales the theory of mechanism-based strain gradient plasticity (MSG) was developed. The MSG approach incorporates the concept of geometrically necessary dislocations into continuum plastic constitutive laws via Taylor hardening relation. This concept is extended here to develop a mechanism-based strain gradient theory for crystal plasticity (MSG-CP) based on the notions of dislocation density tensor and resolved density force corresponding to the Peach-Koehler force in dislocation theory. An effective density of geometrically necessary dislocations is defined on the basis of resolved density force for specific slip systems and is incorporated into the plastic constitutive laws via Taylor relation.

転位動力学シミュレーションのＳＴＩ半導体構造への応用

We have proposed a method to infer the initiation points and slip system of generated dislocations through the combination of FEM calculation and dislocation dynamics simulation. In order to reproduce accurate shape of dislocations observed by TEM, we adopted the Brown's core splitting in the same procedure that Schwarz did. We applied our method to dislocation motion in a shallow trench isolation (STI) structure. Both initiation points and slip systems of four kinds of generated dislocation could be identified. It was found that the observed dislocation loops had certain shapes, which would be determined by the balance between line tension and Peach-Koehler force originated from applied stress field. This indicates that the estimation of the resolved shear stress field would be also possible.

Different Behavior between Edge and Screw Dislocations at the .GAMMA./.GAMMA.' Interface of Ni-Based Superalloy: A Molecular Dynamics Study

Different behaviors between the edge and screw dislocations in the γ/γ'microstructure of Ni-based superalloy are investigated by using molecular dynamics simulations. The simulations are implemented with a slab cell of Ni matrix involving an apex of a cuboidal Ni3A1 precipitate, that mimics a part of the idealized γ/γ' microstructure with about 3 million atoms. Mode II and III type displacements are applied on the cell to nucleate the edge and screw dislocations, respectively, and proceed them toward the precipitate. The edge dislocation decreases its velocity at the γ' precipitate, showing dislocation pinning there, then penetrates it under the force from the follow-on dislocation.The screw dislocation runs through the precipitate without slowdown by shrinking the with between its Shockley partials. Detail investigation of the stress distribution suggests that the constriction is due to interactions between the stress field around the precipitate and the partials : the stress causes repulsive Peach-Koehler force on the leading partial and attractive on the trailing since their edge components have opposite Burgers vectors. In addition to the fact that the strain energy of a screw dislocation is smaller than that of an edge, the constricted screw dislocation makes smaller atomic step on the γ' precipitate than the extended edge dislocation. Thus the screw dislocation easily penetrates the precipitate while the edge is pinned.

Interaction between a semi-infinite crack and a straight dislocation in a decagonal quasicrystal

The interaction between a semi-infinite crack and a line dislocation in a decagonal quasicrystalline solid is investigated in detail by employing the complex variable method. The local stress intensity factors and energy release rate at the crack tip as well as the Peach–Koehler force acting on the line dislocation are presented. The crack tip shielding effect due to the line dislocation can be easily observed from the expressions of the local stress intensity factors. It is found that all the four components of the Burgers vector of the line dislocation can affect the local phonon and phason stress intensity factors. Besides, the problem of a line dislocation in a decagonal quasicrystalline half-space in which the boundary of the half-space is traction-free is also investigated and the Peach–Koehler force on the dislocation due to the presence of the free surface is derived.

A method for linking thermally activated dislocation mechanisms of yielding with continuum plasticity theory

This work combines classical continuum mechanics, the continuously dislocated continuum theory as developed by Kröner and Bilby with discrete dislocation theory to develop quantities that permit models involving interactions between individual dislocations to be incorporated into a description of multiaxial yielding of a material. Two quantities distinguish this approach from earlier efforts: firstly, a dislocation mobility tensor relating the velocity of a dislocation configuration to the net Peach–Koehler force on the configuration and, secondly, a vector quantity representing the dislocation content of the materials. The theory of thermally activated motion of dislocations past obstacles is employed to relate the dislocation velocity to stress by a stress-dependent mobility tensor whose components are determined by the nature of the interaction of the moving dislocation with the obstacle. An example is presented in which the obstacle is a forest dislocation that affects a gliding dislocation through mutual interaction of their stress fields. The development leads to a quantity that can be used as a plastic potential for the construction of an associated flow law.

Constitutive analysis of finite deformation field dislocation mechanics

Driving forces for dislocation motion and nucleation in finite-deformation field dislocation mechanics are derived. The former establishes a rigorous analog of the Peach–Koehler force of classical elastic dislocation theory in a nonlinear, nonequilibrium field-theoretic context; the latter is a prediction of the theory. The structure of the stress response and permanent distortion are also derived. Sufficient boundary and initial conditions are indicated, and invariance under superposed rigid motions is discussed. Hyperelasticity and finite-deformation elastic theory of dislocations are shown to be special cases of the framework. Owing to the nonlocal nature of the theory, the results as well as the methods used to derive them appear to be novel.

On the stress dependence of partial dislocation separation and deformation microstructure in austenitic stainless steels

In the austenitic stainless steels the separation of Shockley partial dislocations is known to play an important role in the plastic deformation and produces a variety of deformation microstructures depending on test and material conditions. Theoretical calculations have been carried out in an attempt to explain the origin of the deformation microstructures which include large stacking faults and twins. Force balance equations for the leading and trailing partials are established by considering the Peach–Koehler force from an applied stress field, repulsive force between leading and trailing partial dislocations, attractive force due to the stacking fault energy, and resistance (or damping) force to the glide of the partial dislocations. For a simple dislocation and stress arrangement, an expression for separation distance was derived from the force balance equations. The results indicate that the separation distance varies with the directional relationship between the applied stress and the Burgers vectors of glide dislocations. Also, the separation distance increases with the applied stress and can diverge when the applied stress exceeds a critical stress. The critical stress is readily achievable in the uniform strain range by strengthening measures like irradiation, lowering test temperature, and increasing strain or strain rate. Further, using a stress-based analysis, some predictions were attempted for the influence of radiation-induced defects on deformation microstructure in austenitic stainless steels.