Peach Koehler Force Research Articles

Dislocations in nonlocal simplified strain gradient elasticity: Eringen meets Aifantis

The nonlocal strain gradient elasticity theory is used to address mechanical problems at small scales where size effects and regularization cannot be neglected. In this work, dislocations are investigated in the framework of nonlocal simplified first strain gradient elasticity. It is shown that nonlocal simplified strain gradient elasticity is the unification of the theories of Eringen’s nonlocal elasticity of Helmholtz type and simplified first strain gradient elasticity. Nonlocal simplified strain gradient elasticity contains two characteristic lengths, namely the characteristic length of nonlocal elasticity of Helmholtz type and the characteristic length of simplified first strain gradient elasticity. The advantage of nonlocal simplified first strain gradient elasticity is that the displacement, elastic distortion, plastic distortion, total stress, Cauchy stress and double stress fields of screw and edge dislocations which are calculated here are nonsingular and finite everywhere. Moreover, the Peach-Koehler force of two screw dislocations and two edge dislocations is derived and it is shown that the Peach-Koehler force is also nonsingular. Numerical examples for all dislocation fields of screw and edge dislocations in aluminum are given.

Couple-stress elasticity of intrinsic and extrinsic dislocations in three-dimensional multilayered materials

This work investigates the influence of couple stresses on intrinsic and extrinsic dislocations within three-dimensional heterogeneous multilayered structures. The materials consist of orthotropic and dissimilar layers with different elastic properties, incorporating the microstructural lengths from a special version of the anisotropic couple-stress elasticity. Double Fourier series expansions are used to describe extended displacements and tractions, considering rotation and couple-stress components and accounting for arbitrary displacement and rotation discontinuities associated with dislocation and disclination loops as well as periodic dislocation and disclination networks at semi-coherent interfaces. The dual-variable and position technique is used to recursively determine the field responses at any point in the multilayered laminate. The application examples focus on dislocation problems with couple stresses, revealing the effects of the core-spreading dislocation technique, free surfaces, anisotropic elasticity, and couple stresses on the field solutions. The analysis also uncovers pressure and von Mises stress amplitudes, non-zero rotation, and non-symmetric stress profiles due to couple stresses in copper/niobium bimaterials. Changes in the sign and magnitude of the Peach-Koehler forces are observed when couple stresses are introduced, which deviate from the classical theory of dislocations. The investigation finds a significant 29% reduction in energy per unit area at a semi-coherent interface between copper and niobium influenced by couple stresses. These results deepen our understanding of how couple stresses affect dislocation behavior in multilayered materials, and provide prospects for future discrete dislocation and disclination dynamics simulations with engineering applications in various nano- and micro-structured materials.

Applications of the Peach-Koehler force in liquid crystals

ABSTRACT In solids, external stress induces the Peach-Koehler force, which drives dislocations to move. Similarly, in liquid crystals, an external angular stress creates an analogous force, which drives disclinations to move. In this work, we develop a method to calculate the relevant angular stress either analytically or numerically, and hence to determine the force on a disclination. We demonstrate this method by applying the Peach-Koehler force theory to four problems: (a) Single disclination in a liquid crystal cell between two uniform in-plane alignments perpendicular to each other. (b) Array of disclinations in a liquid crystal cell with patterned substrates. (c) Pair of disclinations in a long capillary tube with homeotropic anchoring. (d) Radial hedgehog or disclination loop inside a sphere with homeotropic anchoring, and its response to an applied magnetic field. In all of these problems, the Peach-Koehler force theory predicts the equilibrium defect structure, and the predictions are consistent with the results of minimising the total free energy.

Shock wave induced migration of an edge dislocation dipole in alpha-Fe

The mobility of a 1/2〈111〉{110} edge dipole in alpha-iron has been studied using molecular dynamics simulations. Collision cascades generated by keV recoils have been shown to induce the migration of dislocations. In order to elucidate the origin of the motion of these dislocations, and separate the production of defects from temperature or pressure effects, a stressed region of different shapes (sphere and cylinder) is simulated close to the edge dipole. We observe that the generated shock wave triggers the movement of the dislocations even when no defects are produced. The shape of the distorted region and the character of the dislocations influence the way the dislocations move, due to the change in Peach-Koehler force direction and to the fact that the shock waves arrive to the different parts of the dislocations at different times.

A theoretical modelling of strengthening mechanism in graphene-metal nanolayered composites

Graphene-metal nanolayered composites (GMNCs) are a new generation of nano-structural composites characterized by a very high density of graphene reinforced interfaces (GRI) between metal nanolayers. Compared to traditional graphene flake reinforced composites, GMNCs have much higher strength, toughness and ductility due to the excellent ability of GRI on constraining dislocation motion and crack propagation. Despite numerous experimental and numerical studies on the mechanical behavior of GMNCs, the underlying strengthening mechanism is still not fully understood due to the absence of appropriate theoretical model. This paper proposes a continuum mechanics based theoretical model to explain the strengthening mechanism in GMNCs. In this model, the metal matrix and the GRI are simulated as homogenous elastic medium of infinite extend and inextensible thin membrane of zero thickness, respectively. Using the theoretical model, two boundary value problems namely (i) A circular prismatic dislocation loop approaching to the GRI and (ii) A mixed mode I/II penny-shaped crack near the GRI are formulated to reveal the two key strengthening mechanism: dislocation blocking and crack shielding, respectively. The two problems are solved analytically by the Generalized Kelvin's Solution (GKS) based method for 3D elasticity and Fredholm integral integration technique. Exact closed form solution for the Peach-Koehler (P-K) force on the dislocation loop is obtained. An efficient numerical scheme is developed to solve the Fredholm integral integration for the crack problem with very high accuracy. It is shown that our theoretical model can well capture and explain the strengthening mechanism observed in experiments. Moreover, the dominant role of Poisson's ratio on the strengthening efficiency is also revealed by our model. This finding implies the exciting possibility that the strength of GMNCs can be tailored by controlling the Poisson's ratio of the metal matrix. The present theoretical modeling can provide valuable insights into the mechanics-based design of GMNCs.

Atomistic determination of Peierls barriers of dislocation glide in nickel

The Peierls barrier measures the lattice resistance to dislocation glide in crystalline solids. We use the nudged elastic band (NEB) method to calculate the Peierls barriers for screw and edge dislocation glide in a face-centered cubic (FCC) metal of Ni. The minimum energy paths (MEPs) across single or sequential Peierls barriers are determined under shear loading. The NEB results show the decreasing Peierls barrier with increasing shear stress, giving the Peierls stress at which the Peierls barrier vanishes. The effects of boundary condition and system size on Peierls barriers are studied by comparing strain- and stress-controlled NEB results. Furthermore, the free-end NEB methods are applied to determine MEPs with improved computational efficiency. The NEB results are also used to evaluate the energetic driving force of dislocation glide, which is consistent with that determined from the Peach-Koehler force. The accuracy of the present NEB results based on an empirical interatomic potential is assessed by comparison with a machine-learning potential. This work demonstrates the robust and efficient quantification of Peierls barriers to dislocation glide in an FCC metal, and it lays a solid foundation for the atomistic determination of Peierls barriers in compositionally complex alloys with the FCC structure in future studies.

Dislocation in a Strained Layer Embedded in a Semi-Infinite Matrix

Abstract The misfit stress in a thin layer embedded in a semi-infinite matrix has been first determined near the free surface of the structure, using the virtual dislocation formalism. From a Peach–Koehler force analysis, the different equilibrium positions (unstable and stable) of an edge dislocation gliding in a plane of the layer inclined with respect to the upper interface and emerging at the point of intersection of the upper interface and this free surface have been then characterized with respect to the lattice mismatch and the inclination angle of the gliding plane. It has been found that the dislocation may exhibit stable equilibrium position near the interface and/or near the free surface. A diagram of the position stability has been then determined versus the misfit parameter and the inclination angle. The energy variation due to the introduction of an edge dislocation from the free surface until the matrix layer interface has been finally determined, when the dislocation is gliding in the plane inclined with respect to the interface horizontal axis. A critical thickness of the layer beyond which the formation of the dislocation in the interfaces is energetically favorable has been finally determined as well as its position with respect to the free surface in the lower interface.

XFEM for multiphysics analysis of edge dislocations with nonuniform misfit strain: A novel enrichment implementation

In this work, the eXtended finite element method (XFEM) is implemented for the multiphysics analysis of edge dislocations near a heterostructure (bi-material) interface with nonuniform misfit strain. A novel enrichment is defined to incorporate the singular behavior of the electric potential due to dislocation. Nonlinear material properties due to temperature are taken for a thermo-electro-elastic analysis. The constituting materials having different lattice parameters result in a mismatch–misfit​ strain near the interface. This nonuniform misfit strain distribution affects the dislocations near the bi-material interface. Volterra dislocation modeling is performed to obtain all the primary and secondary fields around dislocations. An expression of the Peach–Koehler force considering the combined effect of the multiple fields is proposed. The results are obtained when the misfit strain is present on one side of the interface and both sides of the interface. A comparative study is performed on the Peach–Koehler force obtained in different cases of the misfit consideration with the case when the misfit strain is absent. Also for the case when the misfit strain is taken on the same side of the interface, the Peach–Koehler force is obtained by varying the misfit percentages corresponding to the different amounts of misfit relaxation.

Modelling needle domains with dislocations: A dislocation pile-up model and comparison with synchrotron experiment

The shape and strain field of a needle domain in a barium titanate single crystal are modelled using a distribution of dislocations and line charges. The arrangement of these dislocations and charges is a result of the balance of modified Peach–Koehler forces acting among the dislocations and a lattice friction assumed to act at each dislocation site. Based on measurements of needle shape by synchrotron X-ray diffraction, dislocation pile-up theory is used to compute the distribution of discrete dislocations along the needle and hence estimate the lattice friction. It is found that the lattice friction in this model is proportional to the opening angle of a wedge-shape needle domain and consistent with the observed magnitude of stress required to mobilize needle domains. The microstrain distribution around an a-a needle domain tip, obtained from X-ray diffraction measurement, is further used to test the dislocation model, with a similar pattern and magnitude of strains identified in the model and the experiment.

Singularity-free theory and adaptive finite element computations of arbitrarily-shaped dislocation loop dynamics in 3D heterogeneous material structures

The long-standing problem of arbitrarily-shaped discrete dislocation loops in three-dimensional heterogeneous material structures is addressed by introducing novel singularity-free elastic field solutions as well as developing adaptive finite element computations for dislocation dynamics simulations. The first framework uses the Stroh formalism in combination with the biperiodic Fourier-transform and dual variable and position techniques to determine the finite-valued Peach–Koehler force acting on curved dislocation loops. On the other hand, the second versatile mixed-element method proposes to capture the driving forces through dissipative energy considerations with domain integrals by means of the virtual extension principle of the surfacial discontinuities. Excellent agreement between theoretical and numerical analyses is illustrated from simple circular shear dislocation loops to prismatic dislocations with complicated simply-connected contours in linear homogeneous isotropic solids and anisotropic elastic multimaterials, which also serves as improved benchmarks for dealing with more realistic boundary-value problems with evolving dislocations. Examples of sophisticated dislocation applications include the short-range core reaction between intersecting dislocation loops in interaction with a spherical cavity, as well as the Orowan dislocation-precipitate bypass mechanism in a compressed micropillar of polycrystalline copper. The latter multiscale investigation spans three orders of magnitude in size scale, and is thus enabled by computationally efficient and robust adaptive mesh generation procedures for explicit dislocation propagation, interaction, and coalescence in three-dimensional materials and material structures.

Screw dislocations in a holed wedge

A numerical code has been designed and implemented to find the stress at any material point in a holed wedge material due to any number of crystal or real screw dislocations in the wedge. The code implements the image dislocation method together with the collocation-point method to find stresses at any material point. We demonstrate that the combined analytical–numerical method (analytical with the image dislocations and numerical with the collocation-point method) is valid in that it produces stress fields giving rise to zero traction on free surfaces as well at an arbitrary number of collocation points on void surfaces. In addition, the method makes it possible to find the Peach–Koehler force (PKF) acting on a real screw dislocation at any point in the wedge, as well as to determine points of equilibrium, where the PKF is zero. Numerous and easy to interpret contour plots reveal the utility of this method which was carried out using the mathematical software MATLAB.

A phase field crystal theory of the kinematics of dislocation lines

We introduce a dislocation density tensor and derive its kinematic evolution law from a phase field description of crystal deformations in three dimensions. The phase field crystal (PFC) model is used to define the lattice distortion, including topological singularities, and the associated configurational stresses. We derive an exact expression for the velocity of dislocation line determined by the phase field evolution, and show that dislocation motion in the PFC is driven by a Peach–Koehler force. As is well known from earlier PFC model studies, the configurational stress is not divergence free for a general field configuration. Therefore, we also present a method (PFCMEq) to constrain the diffusive dynamics to mechanical equilibrium by adding an independent and integrable distortion so that the total resulting stress is divergence free. In the PFCMEq model, the far-field stress agrees very well with the predictions from continuum elasticity, while the near-field stress around the dislocation core is regularized by the smooth nature of the phase-field. We apply this framework to study the rate of shrinkage of an dislocation loop seeded in its glide plane.

Nonlinear thermo-elastic analysis of edge dislocations with Internal Heat Generation in Semiconductor Materials

In this work, the nonlinear thermo-elastic analysis of edge dislocations with internal heat generation is performed by the extended finite element method (XFEM) in semiconductor materials. The nonlinearity due to the temperature dependent electrical conductivity, thermal conductivity and thermal expansion coefficient is examined. The presence of an electric field in the direction of the dislocation line results in internal heat generation due to the electrical resistivity. The continuous Volterra model of dislocation is used for XFEM analysis of dislocations. The influence of Joule heat on thermal and elastic fields is observed. Different temperatures and electric field intensity values are taken for understanding the dislocation behavior. The Peach-Koehler (P–K) force for different dislocation configurations such as dislocation-dislocation interaction, dislocation-free surface interaction and dislocation-material interface interaction is evaluated. The results are found quite different with the amount of heat generation. The P–K force values are found more than that of pure elastic case. With the knowledge of the movability function, the obtained results can be used to predict the velocity of the dislocations.

Finite-deformation phase-field microelasticity with application to dislocation core and reaction modeling in fcc crystals

The purpose of the current work is the formulation of finite-deformation phase-field microelasticity (FDPFM) and its application to the modeling of (i) the dislocation core and (ii) dislocation interaction/reaction on intersecting glide planes in fcc crystals. The corresponding model formulation is carried out in the context of continuum thermodynamics; for simplicity, isothermal, quasi-static conditions are assumed. The resulting model rests in particular on generalized Ginzburg–Landau relations for dislocation slip-line phase field evolution depending on the Peach-Koehler (PK) force. Restricting attention to (periodic) bulk single crystal behavior, the algorithmic formulation and numerical implementation of FDPFM employs Green function, spectral, and Fourier methods generalized to geometric non-linearity. More specifically, algorithms for control of either the mean deformation gradient, or the mean Cauchy stress (i.e., PK force control) are developed. As in ”standard” linear PFM, the free energy model in FDPFM is based on elastic, crystal and gradient contributions. In order to model dislocation interaction and reactions on multiple intersecting glide planes, the ”standard” 2D stacking-fault-energy-based PFM crystal energy is generalized here to 3D crystallographic form for dissociation, partial fcc dislocations and stacking fault formation. Comparison of FDPFM and atomistic (MS) simulation results for higher-order local continuous and discrete deformation measures (e.g., dislocation tensor) in the core of a dissociated edge dislocation in Au show generally good agreement. In addition, results from FDPFM and MS are compared for dislocation reactions in Al on two intersecting glide planes. In the case of collinear reaction, for example, good qualitative agreement with MS results is obtained only when the dislocation slip on multiple glide planes is energetically coupled in the crystal energy.

Elasto-plastic evolution of single crystals driven by dislocation flow

This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the movement of dislocations, that is, 1-dimensional topological defects in the crystal lattice. It is well known that glide motion of dislocations is the dominant microscopic mechanism for plastic deformation in many crystalline materials, most notably in metals. We propose a novel geometric language, built on the concepts of space-time “slip trajectories” and the “crystal scaffold” to describe the movement of (discrete) dislocations and to couple this movement to plastic flow. The energetics and dissipation relationships in our model are derived from first principles drawing on the theories of crystal modeling, elasticity, and thermodynamics. The resulting force balances involve a new configurational stress tensor describing the forces acting against slip. In order to place our model into context, we further show that it recovers several laws that were known in special cases before, most notably the equation for the Peach–Koehler force (linearized configurational force) and the fact that the combination of all dislocations yields the curl of the plastic distortion field. Finally, we also include a brief discussion on how a number of other effects, such as hardening, softening, dislocation climb, and coarse-graining, could be incorporated into our model.

XFEM simulation of dislocation in SixGe1-x alloy under thermal loads

SixGe1-x is an alloy made of two semiconductor materials from group Ⅳ-A of the periodic table. The performance of semiconductors results in higher temperatures due to electric field formation and also they are exposed to elevated atmospheric temperatures. Here, the SixGe1-x domain with one free surface at the left is considered under thermal loads at different compositions when the edge dislocation is present in the domain. The Extended Finite Element Method (XFEM) is used for numerical modeling of the problem. The lattice constant, thermal conductivity, electrical conductivity and thermal expansion coefficient are functions of the amount of Si (xSi) in the alloy. The dislocation Burgers vector is also dependent on the Si concentration as its magnitude is defined in terms of the lattice constant. The XFEM formulation introduces the Volterra type edge dislocation through the enrichment functions. Firstly, the discrete heat equilibrium equation is solved and then considering the thermal strains, the discrete force equilibrium equation is solved. The two cases of the heat flux acting normal to the glide plane and along the glide plane of the dislocation are considered for different temperatures. The results indicate an increase in the Peach-Koehler force with the increase in ‘xSi’ greater than 0.2 for both the cases of the heat flux application at all temperatures. However, at the lower concentration of Si i.e. below 0.2, the Peach-Koehler force is affected differently.

Topological Defects in Solids with Odd Elasticity.

Crystallography typically studies collections of point particles whose interaction forces are the gradient of a potential. Lifting this assumption generically gives rise in the continuum limit to a form of elasticity with additional moduli known as odd elasticity. We show that such odd elastic moduli modify the strain induced by topological defects and their interactions, even reversing the stability of, otherwise, bound dislocation pairs. Beyond continuum theory, isolated dislocations can self propel via microscopic work cycles active at their cores that compete with conventional Peach-Koehler forces caused, for example, by an ambient torque density. We perform molecular dynamics simulations isolating active plastic processes and discuss their experimental relevance to solids composed of spinning particles, vortexlike objects, and robotic metamaterials.

A gradient Eshelby force on twinning partial dislocations and associated detwinning mechanism in gradient nanotwinned metals

It is well known that the driving force on dislocation glide is the Peach-Koehler force which is proportional to the resolved shear stress on the slip plane. Here, we report a type of configurational force, referred to as the gradient Eshelby force, that can drive the motion of twinning partial dislocations on twin boundaries (TBs) in the absence of any resolved shear stress and cause detwinning in gradient nanotwinned (GNT) metals, an emerging class of multiscale metallic materials with exceptional mechanical properties and novel deformation mechanisms. Specifically, we consider the Eshelby-force-driven motion of twinning partial dislocations and associated detwinning mechanism in GNT metals made of preferentially oriented TBs and columnar grain boundaries (GBs) with spatially varying twin spacing and grain size. Large-scale molecular dynamics simulations validate the proposed Eshelby-driven detwinning mechanism, where twinning partial dislocations are nucleated from GBs and glide on TBs near the region with the steepest local gradient, leading to extensive TB migration and twin annihilation. This study demonstrates the important role of Eshelby force in controlling twinning partial glide and TB migration in GNT metals, which may have broad implications on plastic deformation in gradient nanostructured metals.

Dislocation blocking in elastically anisotropic semiconductor thin films

One strategy for decreasing the density of threading dislocations penetrating the surface of a heteroepitaxial semiconductor film is that in which the greater mechanical stiffness of a dislocation blocking layer acts to reduce the Peach–Koehler image forces acting on the leading segment of the half loop generated by dislocation multiplication sources at the heteroepitaxial interface situated below the blocking layer. Reducing the Peach–Koehler force, drawing the half loop to the film surface, helps prevent the two threading arms of the half loop from becoming threading dislocations once the half loop penetrates the film surface. The calculation of the Peach–Koehler force employs an analytical continuation formalism using anisotropic elasticity theory for treating dislocation image forces generated by three heteroepitaxial interfaces corresponding to the top and bottom interfaces of the blocking layer and the film surface. The system used in this calculation is that of a Ge film grown on a (001) Si substrate, using a SiGe blocking layer just below the critical thickness for dislocation generation. It is found that the dislocation blocking is favored by thinner blocking layers of greater mechanical stiffness, rather than thicker blocking layers of moderate mechanical stiffness. Specifically, for the blocking layers of composition Si0.2Ge0.8, Si0.3Ge0.7, and Si0.4Ge0.6, of thickness 50, 18, and 10 nm, respectively, it is the thinnest (and mechanically stiffest) layer (Si0.4Ge0.6, 10 nm) that brings about the greatest reduction in the Peach–Koehler force, drawing the leading segment of the half loop to the surface of the film.

Influence of precipitation on tension and compression twinning in Mg-6.5Zn alloy

Depending on the stress state, magnesium alloys can display tensile and/or compression twinning. The present letter exploits bimodal textures in alloy Mg-6.5Zn to examine how age hardening impacts on the two different twin types. We found that tensile twinning is stimulated by precipitates (i.e. higher twin volume fractions are seen at an equivalent plastic strain) whereas compression twinning is suppressed by precipitates (i.e. lower twin volume fractions are seen at an equivalent plastic strain). The difference is rationalized in terms of a lower twin surface energy and a larger twinning dislocation Burgers vector in compression twins. The larger Burgers vector renders compression twinning dislocations more resistant to bowing. The surface energy effect is more subtle. All else constant, twins with lower surface energies can propagate with a lower total content of twinning dislocations. Thus, a higher applied stress is required to attain the Peach-Koehler force needed to bypass precipitates.