Thickness effects in polycrystalline thin films: Surface constraint versus interior constraint

The uniaxial tension behavior of polycrystalline thin films, in which all grain boundaries (GBs) are penetrable by dislocations, is investigated by two-dimensional discrete dislocation dynamics (DDD) method with a penetrable dislocation-GB interaction model. In order to study thickness effect on the tensile strength of thin films with and without surface treatment, three types of thin films are comparatively considered, including the thin films without surface treatment, with surface passivation layers (SPLs) of nanometer thickness and with surface grain refinement zones (SGRZs) consisting of nano-sized grains. Our results show that thickness effects and their underlying dislocation mechanisms are quite distinct among different types of thin films. The thicker thin films without surface treatment are stronger than the thinner ones; however, opposite thickness effects are captured in the thin films with SPLs or SGRZs. Moreover, the underlying dislocation mechanisms of the same thickness effects of thin films with SPLs and SGRZs are different. In the thin films with SPLs, the thickness effect is caused by the sharp increase of dislocation density near the film-passivation interface, while it is mainly due to the sharp decrease of dislocation density within the refined surface grains of the thin films with SGRZs. No matter in what type of thin films, thickness effect gradually disappears when the number of grains in the thickness direction is large enough. Our analysis reveals that general mechanism of those thickness effects lies in the competition between the exterior surface-constraint and interior GB-constraint on gliding dislocations.     

Tensile response of passivated films with climb-assisted dislocation glide

The tensile response of single crystal films passivated on two sides is analysed using climb enabled discrete dislocation plasticity. Plastic deformation is modelled through the motion of edge dislocations in an elastic solid with a lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation incorporated through a set of constitutive rules. The dislocation motion in the films is by glide-only or by climb-assisted glide whereas in the surface passivation layers dislocation motion occurs by glide-only and penalized by a friction stress. For realistic values of the friction stress, the size dependence of the flow strength of the oxidised films was mainly a geometrical effect resulting from the fact that the ratio of the oxide layer thickness to film thickness increases with decreasing film thickness. However, if the passivation layer was modelled as impenetrable, i.e. an infinite friction stress, the plastic hardening rate of the films increases with decreasing film thickness even for geometrically self-similar specimens. This size dependence is an intrinsic material size effect that occurs because the dislocation pile-up lengths become on the order of the film thickness. Counter-intuitively, the films have a higher flow strength when dislocation motion is driven by climb-assisted glide compared to the case when dislocation motion is glide-only. This occurs because dislocation climb breaks up the dislocation pile-ups that aid dislocations to penetrate the passivation layers. The results also show that the Bauschinger effect in passivated thin films is stronger when dislocation motion is climb-assisted compared to films wherein dislocation motion is by glide-only.     

On edge dislocation in a multilayered laminate

We investigate the elastic field induced by an edge dislocation in a multilayered laminated composite composed of (N ? 2) thin bonded elastic layers sandwiched between two semi-infinite elastic media. A simple closed-form solution is obtained when all the phases have equal shear modulus but different Poisson's ratios, and when the dislocation is located in the upper semi-infinite phase. The image force acting on the dislocation due to its interaction with the multilayered structure is also derived. Several specific examples are discussed in detail to illustrate the mobility of the edge dislocation. Some interesting behaviors of the dislocation are observed. Our results indicate that it is possible to find at most (N ? 2) equilibrium positions for the edge dislocation in an N-phase composite structure.     

Size-dependent interaction of an edge dislocation with an elliptical nano-inhomogeneity incorporating interface effects

The elastic behavior of an edge dislocation, which is positioned outside of a nanoscale elliptical inhomogeneity, is studied within the interface elasticity approach incorporating the elastic moduli and surface tension of the interface. The complex potential function method is used. The dislocation stress field and the image force acting on the dislocation are found and analyzed in detail. The difference between the solutions obtained within the classical-elasticity and interface-elasticity approaches is discussed. It is shown that for the stress field, this difference can be significant in those points of the inhomogeneity-matrix interface, where the radius of curvature is smaller and which are closer to the dislocation. For the image force, this difference can be considerable or dispensable in dependence on the dislocation position, its Burgers vector orientation, and relations between the elastic moduli of the matrix, inhomogeneity and their interface. Under some special conditions, the dislocation can occupy a stable equilibrium position in atomically close vicinity of the interface. The size effect is demonstrated that the normalized image force strongly depends on the inhomogeneity size when it is in the range of several tens of nanometers, in contrast with the classical solution where this force is always constant. The general issue is that the interface elasticity effects become more evident when the characteristic sizes of the problem (inhomogeneity size, interface curvature radius and dislocation-interface spacing) reduce to the nanoscale.     

Surface/interface effects on elastic behavior of a screw dislocation in an eccentric core–shell nanowire

The elastic behavior of a screw dislocation which is positioned inside the shell domain of an eccentric core–shell nanowire is addressed with taking into account the surface/interface stress effect. The complex potential function method in combination with the conformal mapping function is applied to solve the governing non-classical equations. The dislocation stress field and the image force acting on the dislocation are studied in detail and compared with those obtained within the classical theory of elasticity. It is shown that near the free outer surface and the inner core–shell interface, the non-classical solution for the stress field considerably differs from the classical one, while this difference practically vanishes in the bulk regions of the nanowire. It is also demonstrated that the surface with positive (negative) shear modulus applies an extra non-classical repelling (attracting) image force to the dislocation, which can change the nature of the equilibrium positions depending on the system parameters. At the same time, the non-classical solution fails when the dislocation approaches very close to the surface/interface with negative shear modulus. The effects of the core–shell eccentricity and nanowire diameter on dislocation behavior are discussed. It is shown that the non-classical surface/interface effect has a short-range character and becomes more pronounced when the nanowire diameter is smaller than 20 nm.     

Interface effects on elastic behavior of an edge dislocation in a core–shell nanowire embedded to an infinite matrix

The elastic behavior of an edge dislocation located inside the core of a core–shell nanowire which is embedded in an infinite matrix is studied within the surface/interface elasticity theory. The corresponding boundary value problem is solved exactly by using complex potential functions. An important parameter so-called interface characteristic parameter which has the dimension of length and is a combination of the interface moduli enters the formulations. The stress field of the dislocation, image force acting on the dislocation, and the dislocation strain energy is calculated by considering the interface effect. The introduced characteristic parameter allows the examination of the core–shell size on the image forces acting on the dislocation. The repelling and attracting effects of the interface parameter on the image force are discussed. The equilibrium position of the dislocation is also studied. The dislocation strain energy in the interface elasticity framework is only slightly different from that of traditional elasticity when the dislocation is placed in the central region of the core and reaches its maximum value when it is located near the core–shell interface.     

Elastic field due to an edge dislocation in a multilayered composite

A numerical procedure is presented for the analysis of the elastic field due to an edge dislocation in a multilayered composite. The multilayered composite consists of n perfectly bonded layers having different material properties and thickness, and two half-planes adhere to the top and bottom layers. The stiffness matrices for each layer and the half-planes are first derived in the Fourier transform domain, then a set of global stiffness equations is assembled to solve for the transformed components of the elastic field. Since the singular part of the elastic field corresponding to the dislocation in the full-plane has been extracted from the transformed components, regular numerical integration is needed only to evaluate the inverse Fourier transform. Numerical results for the elastic field due to an edge dislocation in a bimaterial medium are shown in fairly good agreement with analytical solutions. The elastic field and the Peach–Kohler image force are also presented for an edge dislocation in a single layered half-plane, a two-layered half-plane and a multilayered composite made of alternating layers of two different materials.     

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.     

Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation

Nanoindentation experiments have shown that microstructural inhomogeneities across the surface of gold thin films lead to position-dependent nanoindentation behavior [Phys. Rev. B (2002), to be submitted]. The rationale for such behavior was based on the availability of dislocation sources at the grain boundary for initiating plasticity. In order to verify or refute this theory, a computational approach has been pursued. Here, a simulation study of the initial stages of indentation using the embedded atom method (EAM) is presented. First, the principles of the EAM are given, and a comparison is made between atomistic simulations and continuum models for elastic deformation. Then, the mechanism of dislocation nucleation in single crystalline gold is analyzed, and the effects of elastic anisotropy are considered. Finally, a systematic study of the indentation response in the proximity of a high angle, high sigma (low symmetry) grain boundary is presented; indentation behavior is simulated for varying indenter positions relative to the boundary. The results indicate that high angle grain boundaries are a ready source of dislocations in indentation-induced deformation.     

Study of size effects in thin films by means of a crystal plasticity theory based on DiFT

In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide given by Yefimov et al. [2004b. A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity simulations. J. Mech. Phys. Solids 52, 279-300]. In this dislocation field theory (DiFT) the transport equations for both the total dislocation density and geometrically necessary dislocation (GND) density on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested itself in the form of a back stress in addition to the external shear and the self-consistent internal stress. We here present the study of size effects in single crystalline thin films with symmetric double slip using the novel continuum theory. Two boundary value problems are analyzed: (1) stress relaxation in thin films on substrates subject to thermal loading, and (2) simple shear in constrained films. In these problems, earlier discrete dislocation simulations had shown that size effects are born out of layers of dislocations developing near constrained interfaces. These boundary layers depend on slip orientations and applied loading but are insensitive to the film thickness. We investigate the stress response to changes in controlled parameters in both problems. Comparisons with previous discrete dislocation simulations are discussed.     

Spinodal surface instability of soft elastic thin films

When the thicknesses of thin films reduce to microns or even nanometers, surface energy and surface interaction often play a significant role in their deformation behavior and surface morphology. The spinodal surface instability induced by the van der Waals force in a soft elastic thin film perfectly bonded to a rigid substrate is investigated theoretically using the bifurcation theory of elastic structures. The analytical solution is derived for the critical condition of spinodal surface morphology instability by accounting for the competition of the van der Waals interaction energy, elastic strain energy and surface energy. Detailed examinations on the effect of surface energy, thickness and elastic properties of the film show that the characteristic wavelength of the deformation bifurcation mode depends on the film thickness via an exponential relation, with the power index in the range from 0.749 to 1.0. The theoretical solution has a good agreement with relevant experiment results.     

Transmission of a screw dislocation across a coherent, non-slipping interface

Current research on nanocrystalline metals and nanoscale multilayer thin films suggests extraordinary plastic strength is due to confinement of slip to individual grains or layers. To assess the magnitude of confinement, a Peierls model of slip transmission of a screw dislocation across a coherent, non-slipping interface is presented. The results reflect that large interfacial barriers to transmission are generated by rapid fluctuations in dislocation line energy near the interface due to elastic modulus mismatch, stacking fault energy mismatch, and antiphase boundary energy for transmission into an ordered phase. Coherency stress is predicted to dramatically alter the dislocation core configuration and impart additional strength regardless of the sign. Contributions to strength are not additive due to nonlinear coupling via the dislocation core configuration. The predicted barrier strength for a coherent (0 0 1) Cu/Ni interface is comparable to atomistic (EAM) results but larger than estimates from hardness data.     

Coupled DDD–FEM modeling on the mechanical behavior of microlayered metallic multilayer film at elevated temperature

To investigate the mechanical behavior of the microlayered metallic thin films (MMMFs) at elevated temperature, an enhanced discrete-continuous model (DCM), which couples rather than superposes the two-dimensional climb/glide-enabled discrete dislocation dynamics (2D-DDD) with the linearly elastic finite element method (FEM), is developed in this study. In the present coupling scheme, two especial treatments are made. One is to solve how the plastic strain captured by the DDD module is transferred properly to the FEM module as an eigen-strain; the other is to answer how the stress field computationally obtained by the FEM module is transferred accurately to the DDD module to drive those discrete dislocations moving correctly. With these two especial treatments, the interactions between adjacent dislocations and between dislocation pile-ups and inter-phase boundaries (IBs), which are crucial to the strengthening effect in MMMFs, are carefully taken into account. After verified by comparing the computationally predicted results with the theoretical solutions for a dislocation residing in a homogeneous material and nearby a bi-material interface, this 2D-DDD/FEM coupling scheme is used to model the tensile mechanical behaviors of MMMFs at elevated temperature. The strengthening mechanism of MMMFs and the layer thickness effect are studied in detail, with special attentions to the influence of dislocation climb on them.     

A computational method for dislocation-precipitate interaction

A new computational method for the elastic interaction between dislocations and precipitates is developed and applied to the solution of problems involving dislocation cutting and looping around precipitates. Based on the superposition principle, the solution to the dislocation-precipitate interaction problem is obtained as the sum of two solutions: (1) a dislocation problem with image stresses from interfaces between the dislocation and the precipitate, and (2) a correction solution for the elastic problem of a precipitate with an initial strain distribution. The current development is based on a combination of the parametric dislocation dynamics (PDD) and the boundary element method (BEM) with volume integrals.The method allows us to calculate the stress field both inside and outside precipitates of elastic moduli different from the matrix, and that may have initial coherency strain fields. The numerical results of the present method show good convergence and high accuracy when compared to a known analytical solution, and they are also in good agreement with molecular dynamics (MD) simulations. Sheared copper precipitates (2.5 nm in diameter) are shown to lose some of their resistance to dislocation motion after they are cut by leading dislocations in a pileup. Successive cutting of precipitates by the passage of a dislocation pileup reduces the resistance to about half its original value, when the number of dislocations in the pileup exceeds about 10. The transition from the shearable precipitate regime to the Orowan looping regime occurs for precipitate-to-matrix elastic modulus ratios above approximately 3-4, with some dependence on the precipitate size. The effects of precipitate size, spacing, and elastic modulus mismatch with the host matrix on the critical shear stress (CSS) to dislocation motion are presented.     

Size effects in indentation response of thin films at the nanoscale: A molecular dynamics study

The indentation response of Ni thin films of thicknesses in the nanoscale was studied using molecular dynamics simulations with embedded atom method (EAM) interatomic potentials. A series of simulations were performed in films in the [1 1 1] orientation with thicknesses varying from 4 to 12.8 nm. The study included both single crystal films and films containing low angle grain boundaries perpendicular to the film surface. The simulation results for single crystal films show that as film thickness decreases larger forces are required for similar indentation depths but the contact stress necessary to emit the first dislocation under the indenter is nearly independent of film thickness. The low angle grain boundaries can act as dislocation sources under indentation. The mechanism of preferred dislocation emission from these boundaries operates at stresses that are lower as the film thickness increases and is not active for the thinnest films tested. These results are interpreted in terms of a simple model.     

Dislocation core spreading at interfaces between metal films and amorphous substrates

Experiments with transmission electron microscopy have shown that in a strong electron beam the contrast of dislocations may gradually disappear at an incoherent interface between a metal thin film and an amorphous substrate. There are reasons to believe that this phenomenon is caused by radiation-induced dislocation core spreading at the interface. A quantitative model accounting for this effect will be necessary for a better understanding of dislocation structures and plastic deformation in metal thin films. As a first step toward this objective, we develop a number of mathematical solutions for dislocation core spreading at an incoherent interface. For simplicity, we consider screw dislocations, and consider the interface to be characterized by a shear adhesive strength, τ₀, below which no core spreading occurs, and above which spreading takes place in a viscous manner. We determine the final equilibrium core width and the rate of core spreading for single or planar arrays of dislocations in a homogeneous bulk material or at the interface between a thin film and a semi-infinite substrate where the film and substrate may have the same, or different, elastic constants. Some of our solutions are analytic and others are based on an implicit finite difference method with a Gauss-Chebyshev quadrature scheme. The phenomenon of dislocation core spreading is expected to have a dramatic effect on the strength of crystalline films deposited on amorphous substrates.     

Interaction of a screw dislocation with a semi-infinite interfacial crack in a magneto-electro-elastic bi-material

The interaction between a screw dislocation and a semi-infinite interfacial crack in a transversely isotropic magneto-electro-elastic bi-material is investigated. The dislocation line is perpendicular to the isotropic basal plane of the bi-material. The elastic and electromagnetic fields induced by the dislocation are obtained through the use of the complex variable method together with the superposition scheme. The stress, electric displacement and magnetic intensity factors as well as the image exerted on the dislocation are given explicitly. We find that the intensity factors are expressed in terms of the so-called effective materials and the radial component of the image force is only dependent on the elastic modulus of the material with the dislocation. As an illustrative example, the bi-material that consists of piezoelectric and piezomagnetic phases is analyzed.     

一维六方准晶中螺型位错偶极子与直线纳米裂纹的干涉效应

关于准晶中螺型位错偶极子与直线纳米裂纹干涉效应的问题，提出了一种可获得精确解的新方法。首先，将Gurtin-Murdoch表/界面理论推广运用到准晶材料中的纳米裂纹问题，建立了裂纹表面上含声子场、相位子场的应力边界模型；然后，利用复变函数中的共形映射方法，将直线裂纹问题化为单位圆孔问题；再借助于柯西积分方法获得了问题的封闭解；最后，数值结果表明：当裂纹的尺寸减小到纳米量级时，表面效应将对声子场和相位子场中的应力场、像力及像力偶矩产生显著的影响，从而改变其力学性能，即裂纹尺寸越小，表面效应越强，声子场和相位子场中的应力场、像力及像力偶矩的数值变化越大；螺型位错偶极子力偶臂的方向对像力的极值和自身平衡亦有较大影响。     

A crystal plasticity analysis of length-scale dependent internal stresses with image effects

In this work, we present a stress functions approach to include image effects in continuum crystal plasticity arising from the long-range elastic interactions (LRI) between the GND density and free surfaces. The resulting length-scale dependent internal stresses augment those produced by the GND density variation. The formulation is applied to the case of a long, thin specimen subjected to uniform curvature. The analysis shows that under nominally uniform GND density distribution, internal stresses arise from two sources: (1) GND–GND LRI arising from the finite spatial extent of the uniform GND density field and (2) the LRI between the GND density and free surfaces appearing as image fields. A comparison with experimental results suggests that the length-scale for internal stresses, described as a correlation length-scale, should increase with decreasing specimen thickness. This observation is rationalized by associating the internal length-scale with the average slip-plane spacing, which may increase with decreasing specimen size due to paucity of dislocation sources. Finally, we also discuss the length-scale dependent image stress in terms of the Peach–Koehler force density proposed by Gurtin (2002).     

Interaction of a screw dislocation with an interfacial edge crack in a two-phase piezoelectric material

The interaction of a screw dislocation with an interfacial edge crack in a two-phase piezoelectric medium is investigated. Closed-form solutions of the elastic and electrical fields induced by the screw dislocation are derived using the conformal mapping method in conjunction with the image principle. Based on the electroelastic fields derived, the stress and electric displacement intensity factors, the image force acting on the dislocation are given explicitly. We find that the stress and electric displacement intensity factors depend on the effective electroelastic material constants. In the case where one of two phases is purely elastic, the stress intensity factor and image force are plotted to illustrate the influences of electromechanical coupling effect, the position of the dislocation and the material properties on the interaction mechanism. The project supported by the Doctoral Foundation of Hebei Province (B2003113)