Anisotropic Elasticity Theory Research Articles

Convergence of calculated dislocation core structures in hexagonal close packed titanium

The core structure of -type screw dislocations in hexagonal close packed titanium is investigated computationally using periodic supercells with quadrupolar configurations in combination with density functional theory (DFT) and a modified embedded atom method (MEAM) classical potential. Two arrangements of the quadrupolar supercell configurations are examined, and within each arrangement two initial dislocation positions are compared. (Meta)stable pyramidal and prismatic dislocation core structures exist within both DFT and MEAM methods, and the relaxed structure from a given configuration resulting from our anisotropic elasticity theory solution depends only on the assumed initial dislocation positions. Within DFT we find the ground state core structure to be spread on the pyramidal plane. We find that it is necessary to include the semi-core 3p electrons as valence states in the DFT calculations in order to converge the ground state dislocation core configuration and difference in energy between structures. In terms of k-point sampling, it is found that at least a k-point mesh is necessary to converge the dislocation core structure for a supercell one Burgers vector deep. Use of higher k-point densities or inclusion of additional semi-core electronic states as valence electrons results in the same core structure. With the MEAM potential considered in this work, we find the ground state core configuration to be spread predominantly on the prismatic plane, in contrast with the DFT results.

Interaction between semicoherent interfaces and Volterra-type dislocations in dissimilar anisotropic materials

Abstract

Elastic strain relaxation in interfacial dislocation patterns: II. From long- and short-range interactions to local reactions

The long- and short-range interactions as well as planar reactions between two infinitely periodic sets of crossing dislocations are investigated using anisotropic elasticity theory in face- (fcc) and body- (bcc) centered cubic materials. Two preliminary cases are proposed to examine the substantial changes in the elastic stress states and the corresponding strain energies due to a slight rearrangement in the internal dislocation geometries and characters. In general, significant differences and discrepancies resulting from the considered cubic crystal structure and the approximation of isotropic elasticity are exhibited. In a third scenario, special attention is paid to connecting specific internal dislocation structures from the previous cases with non-equilibrium configurations predicted by the quantized Frank-Bilby equation for the (111) fcc and (110) bcc twist grain boundaries. The present solutions lead to the formation of energetically favorable dislocation junctions with non-randomly strain-relaxed configurations of lower energy. In particular, the local dislocation interactions and reactions form equilibrium hexagonal-shaped patterns with planar three-fold dislocation nodes without producing spurious far-field stresses.Numerical application results are presented from a selection of cubic metals including aluminum, copper, tantalum, and niobium. In contrast to the fcc materials, asymmetric dislocation nodes occur in the anisotropic bcc cases, within which the minimum-energy paths for predicting the fully strain-relaxed dislocation patterns depend on the Zener anisotropic factor with respect to unity. The associated changes in the dislocation structures as well as the removal of the elastic strain energy upon relaxations are quantified and also discussed.

Elastic strain relaxation in interfacial dislocation patterns: I. A parametric energy-based framework

A parametric energy-based framework is developed to describe the elastic strain relaxation of interface dislocations. By means of the Stroh sextic formalism with a Fourier series technique, the proposed approach couples the classical anisotropic elasticity theory with surface/interface stress and elasticity properties in heterogeneous interface-dominated materials. For any semicoherent interface of interest, the strain energy landscape is computed using the persistent elastic fields produced by infinitely periodic hexagonal-shaped dislocation configurations with planar three-fold nodes. A finite element based procedure combined with the conjugate gradient and nudged elastic band methods is applied to determine the minimum-energy paths for which the pre-computed energy landscapes yield to elastically favorable dislocation reactions.Several applications on the Au/Cu heterosystems are given. The simple and limiting case of a single set of infinitely periodic dislocations is introduced to determine exact closed-form expressions for stresses. The second limiting case of the pure (010) Au/Cu heterophase interfaces containing two crossing sets of straight dislocations investigates the effects due to the non-classical boundary conditions on the stress distributions, including separate and appropriate constitutive relations at semicoherent interfaces and free surfaces. Using the quantized Frank–Bilby equation, it is shown that the elastic strain landscape exhibits intrinsic dislocation configurations for which the junction formation is energetically unfavorable. On the other hand, the mismatched (111) Au/Cu system gives rise to the existence of a minimum-energy path where the fully strain-relaxed equilibrium and non-regular intrinsic hexagonal-shaped dislocation rearrangement is accompanied by a significant removal of the short-range elastic energy.

Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity

Buckling of centrosymmetric anisotropic beams is studied within strain gradient theory. First, the three dimensional anisotropic gradient elasticity theory is outlined. Then the dimension of the three dimensional theory is reduced, resulting in Timoshenko beam as well as Euler–Bernoulli beam theories. The governing differential equations together with the consistent (classical and non-classical) boundary conditions are derived for centrosymmetric anisotropic beams through a variational approach. By considering von Kármán nonlinear strains, the geometric nonlinearity is taken into account. The obtained nonlinear formulation can be used to study the postbuckling configuration. The analysis of size effect on anisotropic beam structures is missing in the literature so far, while the present model allows one to characterize the size effect on the buckling of the centrosymmetric anisotropic micro- and nano-scale beam structures such as micropillars. As a specific case, the governing buckling equation is obtained for the more practical case of orthotropic beams. Finally, the buckling loads for orthotropic simply supported Timoshenko and Euler–Bernoulli beams as well as a clamped Euler–Bernoulli beam are obtained analytically and the effect of the internal length scale parameters on the buckling load is depicted.

界面特性を考慮した異方性弾性論によるナノスケール異材接合体の特異応力場解析

In recent advanced semiconductor products, elements of thickness and width are frequently in nanoscale. As the size of elements reduces to nanometer, a ratio of surface to volume increases. Surface properties, such as surface stresses and surface elasticity, influence on the distributions of bulk stresses near the surface. In the present study, the singular stress at the corner in an anisotropic two-dimensional multi-wedge joint consisting of three kinds of material is analyzed using molecular statics (MS) method and the anisotropic elasticity theory with a boundary condition considering interface properties. The joints are composed of Ni, Au and Cu. Several joints with different wedge angles at the interface are analyzed. Distributions of atomic stress calculated by the MS method are compared with those by the theory. In the results of the MS analysis, stress jumps exist at the interface in the circumferential direction of the stress distribution around the stress singular point. It is supposed that these are attributed to the influence of interface stresses. Then, interface stresses and elastic properties at the interfaces are investigated using the MS method. The obtained interface properties are used in the stress analysis using the Stroh’s formalism. Finally, the theoretical analysis considering the interface properties can be expressed the jumps of stress at the interfaces.

Hydrogen diffusion in the elastic fields of dislocations in iron

The effect of dislocation stress fields on the sink efficiency thereof is studied for hydrogen interstitial atoms at temperatures of 293 and 600 K and at a dislocation density of 3 × 1014 m–2 in bcc iron crystal. Rectilinear full screw and edge dislocations in basic slip systems 〈111〉{110}, 〈111〉{112}, 〈100〉{100}, and 〈100〉{110} are considered. Diffusion of defects is simulated by means of the object kinetic Monte Carlo method. The energy of interaction between defects and dislocations is calculated using the anisotropic theory of elasticity. The elastic fields of dislocations result in a less than 25% change of the sink efficiency as compared to the noninteracting linear sink efficiency at a room temperature. The elastic fields of edge dislocations increase the dislocation sink efficiency, whereas the elastic fields of screw dislocations either decrease this parameter (in the case of dislocations with the Burgers vector being 1/2〈111〉) or do not affect it (in the case of dislocations with the Burgers vector being 〈100〉). At temperatures above 600 K, the dislocations affect the behavior of hydrogen in bcc iron mainly owing to a high binding energy between the hydrogen atom and dislocation cores.

Numerical analysis of twin thickening process in magnesium alloys

A finite element study of the stress field evolution around the {101¯2} tensile twin of different thickness in magnesium alloy is performed. The system is represented by the 2D models of one and three elastic elliptical inclusions with different aspect ratios surrounded by the plastic matrix. Anisotropic elasticity and crystal plasticity theory are used to describe the material behavior. Numerical results for single inclusion overestimate the initial stress for the stable twin aspect ratio by a factor of two compared to the experimental results. Multiple inclusions case obtains values closer to the experimental ones. Therefore it is concluded that the process of twin thickening cannot be solved within the analysis of the single inclusion and an interaction between twins in the same and neighboring grains plays a crucial role during the twin thickening process.

Strain mapping in a nanoscale-triangular SiGe pattern by dark-field electron holography with medium magnification mode.

Recent years have seen a great deal of progress in the development of transmission electron microscopy-based techniques for strain measurement. Dark-field electron holography (DFEH) is a new technique offering configuration of the off-axis principle. Using this technique with medium magnification (Holo-M), we carried out strain measurements in nanoscale-triangular SiGe/(001) Si with (004), (2-20) and (-111) diffraction spots. The reconstruction of holograms and interpretation of strain maps in term of strain precision were discussed and the strain distributions in the SiGe/(001) Si patterns were visualized. Based on linear anisotropic elastic theory for strain simulation, the simulated results obtained by the finite element method compared with the experimental results acquired byDFEH. The strain values were found to be 0.9-1.0%, 1.1-1.2% and 1.0-1.1%, for the (004), (2-20) and (-111) diffracted beams, respectively, and the strain precisions were determined to be ~2.1×10-3, 3.2×10-3 and 9.1×10-3 for the corresponding diffraction spots. As a result, DFEH is highlighted as a powerful technique for strain measurement, offering high-strain precision, high-spatial resolution and a large field of view.

Energetics of dislocation transformations in hcp metals

Dislocation core structures of hcp metals are highly complex and differ significantly among the hcp family. Some dislocations undergo unconventional transformations that have significant effects on the material plastic flow. Here, the energetics of dislocation dissociations are analyzed in a general anisotropic linear elastic theory framework for transformations in which changes in the partial Burgers vectors are small. Quantitative analyses on various transformations are made using DFT-computed stacking fault energies and partial Burgers vectors. Specifically, possible transformations of the mixed, edge, and screw 〈c+a〉 and screw 〈a〉 dislocations in 6 hcp metals (Mg, Ti, Zr, Re, Zn, Cd) are studied. Climb dissociation of mixed or edge 〈c+a〉 dislocations to the Basal plane is energetically favorable in all 6 metals and thus only limited by thermal activation. The 〈c+a〉 screw dislocation is energetically preferable on Pyramidal I for Ti, Zr, and Re, and on Pyramidal II for Zn and Cd. In Mg, the energy difference between screw 〈c+a〉 on Pyramidal I and II planes is small, suggesting relatively easy cross-slip. For the screw 〈a〉, Basal dissociation is energetically favorable in Mg, Re, Zn and Cd, while Prism dissociation is strongly favorable in Ti and Zr. Only Ti, Zr and Re show a metastable state for dissociation on the Prism plane, and the energy difference between screw 〈a〉 on the Prism and Pyramidal I planes is relatively small in all systems, suggesting relatively easy cross-slip of 〈a〉 in Ti and Zr. The elastic analysis thus provides a single framework able to capture the controlling energetics for different dissociations and slip systems in hcp metals. When the calculated energy differences are very small, the results point to the need for detailed modeling of the atomistic core structure. Moreover, the analyses rationalize broad experimental observations on dominant slip systems and dislocation behaviours, and provide predictions for possible transformations for the family of hcp metals.

Elastic interactions between interface dislocations and internal stresses in finite-thickness nanolayered materials

Anisotropic elasticity theory is used to analyze elastic strain interactions between heterophase interface dislocations and four types of internal stresses in finite-thickness nanolayered face-centered cubic materials. The first interaction is related to the mechanical forces acting on semicoherent interfaces, which arise from heterogeneous elastic fields between the neighboring materials with free surfaces. The second interaction is associated with the Peach-Koehler force exerted on lattice dislocations in free-standing bi- and tri-nanolayers, and is compared to the limiting case of infinite bicrystals. The third case yields to the interaction energy balance and the corresponding equilibrium distance between two Shockley partial dislocations. The fourth and last case is dedicated to the elastic interaction forces between vacancies and interfaces using the force dipole moment approximation.For such cases, the elastic interaction problems are treated by involving the effects of (i) the dislocation characters, (ii) the anisotropic (versus isotropic) elasticity calculations, and (iii) the additional third coherent MgO layer on bilayered Au/Al systems.

Uniqueness of weak solutions to dynamic problems in the elasticity theory with boundary conditions of Winkler and inertial types

A uniqueness theorem for the weak solution of an initial-boundary value problem in the anisotropic elasticity theory with the boundary conditions that “do not conserve” energy, namely, with the impedance and inertial type conditions is proved. The chosen method of proof does not require the positive definiteness of the elastic constant tensor (the case that may arise when solving the problems by the homogenization method for composite materials), but it requires to take the energy variation law as a postulate.

ALGORITHM OF WINDING TENSION FOR CYLINDER WITH THICK-WALLED LINER

As a key factor in winding process, reasonable design of winding tension directly affects the performance of productions. An algorithm to design winding tension is proposed for the design of winding tension. Considering the deformations of mandrel, radial and circumferential stresses of winding layers under external pressure are obtained through analyzing on the basis of anisotropic composite elastic theory and isotropic thick-walled cylinder elastic theory. Within the scope of the elastic limit and based on the stress superposition principle, the analytic algorithm between residual tension distribution and winding tension is established. By analyzing three typical tension models of filament winding, residual tension formulas are given. Furthermore, according to these formulas, different analytical formulas of winding tension are estibilished through analyzing the uniform residual tension. The influence of inner to outer radius ratio of mandrel and taper coefficient on isotropic material residual tension distribution is investigated. The results indicate that changes of radius ration and taper coefficient have remarkable effects on residual tension distribution, while the algorithm to calculate the residual tension distribution is simple, reasonable and reliable. By analyzing uniform residual tension of the anisotropic composite, the result is found consistent with the existing netting theory, and the winding tensions of up and down layers change gently and are easy to be implemented.

Threading and Near-Surface Dislocations in InGaSb/AlSb Films with Blocking and Anti-Blocking Layers

In order to mitigate the formation of threading dislocations in an In0.1Ga0.9Sb heteroepitaxial buffer layer grown on a (001) GaSb substrate, an AlSb blocking layer was grown within the buffer layer. Transmission electron microscopy measurements revealed that the film with the AlSb blocking layer had a significantly lower threading dislocation density than a buffer layer of equivalent thickness in the absence of the blocking layer. Because the AlSb blocking layer is of greater mechanical stiffness than the In0.1Ga0.9Sb buffer layer, attractive image forces that act to draw dislocations to the film surface are countered by repulsive image forces traceable to the In0.1Ga0.9Sb/AlSb interface. These experimental results are consistent with calculations based on anisotropic elasticity theory for repulsive image forces acting on a dislocation beneath the blocking layer.

Robust atomistic calculation of dislocation line tension

The line tension of a dislocation is an important and fundamental property ubiquitous to continuum scale models of metal plasticity. However, the precise value of in a given material has proven difficult to assess, with literature values encompassing a wide range. Here results from a multiscale simulation and robust analysis of the dislocation line tension, for dislocation bow-out between pinning points, are presented for two widely-used interatomic potentials for Al. A central part of the analysis involves an effective Peierls stress applicable to curved dislocation structures that markedly differs from that of perfectly straight dislocations but is required to describe the bow-out both in loading and unloading. The line tensions for the two interatomic potentials are similar and provide robust numerical values for Al. Most importantly, the atomic results show notable differences with singular anisotropic elastic dislocation theory in that (i) the coefficient of the scaling with dislocation length L differs and (ii) the ratio of screw to edge line tension is smaller than predicted by anisotropic elasticity. These differences are attributed to local dislocation core interactions that remain beyond the scope of elasticity theory. The many differing literature values for are attributed to various approximations and inaccuracies in previous approaches. The results here indicate that continuum line dislocation models, based on elasticity theory and various core-cut-off assumptions, may be fundamentally unable to reproduce full atomistic results, thus hampering the detailed predictive ability of such continuum models.

Effect of external stresses on efficiency of dislocation sinks in BCC (Fe, V) and FCC (Cu) crystals

The efficiency of linear sinks for self-point defects (SPDs) elastically interacting (dislocations) and not interacting with sinks with the density of 3 × 1014 m–2 is calculated for BCC (Fe, V) and FCC (Cu) crystals at the temperature 293 K using the object kinetic Monte Carlo technique, depending on type and value of applied mechanical load (up to 200 MPa) and types of linear sinks. Full straight dislocations in slip systems [111](1*****0), [111](11*****), [100](001), and [100](011) for Fe and V and [100](001) for Cu are considered for dislocation sinks (DSs). Orientations of noninteracting linear sinks (NILSs) coincide with those of DSs. Interaction of SPDs with internal (dislocation) and external stress fields is calculated within the framework of anisotropic linear theory of elasticity. Relative changes in efficiency of different co-directional linear sinks (either interacting or not interacting with SPDs) under action of applied stress are approximately identical under low stress. Radiation creep rates are calculated for the considered crystals under uniaxial stress in the stationary regime of Frenkel pairs generation. The creep rate strongly depends on the loading direction and Burgers vector of dislocations in Fe and V, and it is almost independent of these parameters in Cu. At the same generation rate of Frenkel pairs, the radiation creep rate averaged over all loading directions is significantly higher in BCC (Fe, V) crystals containing dislocations with the Burgers vector a/2〈111〉 than in FCC (Cu) crystals.

Analysis of edge threading dislocations b→=12〈110〉 in three dimensional Ge crystals grown on (001)-Si substrates

Threading dislocations (TDs) in germanium (Ge) crystals epitaxially grown on a patterned (001)-silicon (Si) substrate are investigated using transmission electron microscopy (TEM) techniques. Analysis of dislocations performed on the Ge crystals reveals 60° and edge TDs with Burgers vector b→=12〈110〉. High-angle annular dark-field scanning TEM (HAADF-STEM) is used to observe the core of the edge TDs at atomic scale. Pairs of TDs with b→=12〈110〉 are present in the material running parallel at small distances between them (0.5–1.5 nm). The observation of such parallel dislocation pairs in Ge has not been documented before. The interaction between the edge dislocation pairs is obtained experimentally from the high-resolution HAADF-STEM images by applying geometrical phase analysis. The experimental strain maps are compared to analytical calculations based on the anisotropic elastic theory demonstrating a good match between them.

Physical constraints on c13 and δ for transversely isotropic hydrocarbon source rocks

ABSTRACTBased on the theory of anisotropic elasticity and observation of static mechanic measurement of transversely isotropic hydrocarbon source rocks or rock‐like materials, we reasoned that one of the three principal Poisson's ratios of transversely isotropic hydrocarbon source rocks should always be greater than the other two and they should be generally positive. From these relations, we derived tight physical constraints on c13, Thomsen parameter δ, and anellipticity parameter η. Some of the published data from laboratory velocity anisotropy measurement are lying outside of the constraints. We analysed that they are primarily caused by substantial uncertainty associated with the oblique velocity measurement. These physical constraints will be useful for our understanding of Thomsen parameter δ, data quality checking, and predicting δ from measurements perpendicular and parallel to the symmetrical axis of transversely isotropic medium. The physical constraints should also have potential application in anisotropic seismic data processing.

Utilizing mechanical loads and flexoelectricity to induce and control complicated evolution of domain patterns in ferroelectric nanofilms

We have conducted a systematical investigation to reveal the stability and evolution path of various ferroelectric domain patterns in nanofilms subjected to mechanical loads and related flexoelectric field. Within a rigorous framework of flexoelectricity, a phase-field approach has been established for simulating the domain structure of ferroelectric nanofilms. The electromechanical fields of the nanofilms are numerically solved by a fast Fourier transform technique (FFT) based on the combination of Khachaturyan's microscopic elastic theory and Stroh's formalism of anisotropic elasticity. Using this approach, we simulate eight types of domain patterns that can be stabilized in the nanofilms. It is further demonstrated that these domain patterns can be significantly affected by the mechanical loads and related flexoelectric field and exhibit fruitful evolution paths. To adapt the applied mechanical strain and strain gradient, the domain pattern may remain stable, evolve into another polydomain pattern, or become a monodomain state (an effect of domain erasing). The domain fraction, detailed domain morphology, average stresses in the nanofilms, average polarization and temporal evolution characteristics of the domain patterns under various mechanical loads and sources of flexoelectric field have been analyzed. This investigation should provide instructive information for the practical application of ferroelectric nanofilms under complex and changeable mechanical conditions.

The non-singular Green tensor of Mindlin's anisotropic gradient elasticity with separable weak non-locality

In this paper, we derive the Green tensor of anisotropic gradient elasticity with separable weak non-locality, a special version of Mindlin's form II anisotropic gradient elasticity theory with up to six independent length scale parameters. The framework models materials where anisotropy is twofold, namely the bulk material anisotropy and a weak non-local anisotropy relevant at the nano-scale. In contrast with classical anisotropic elasticity, it is found that both the Green tensor and its gradient are non-singular at the origin, and that they rapidly converge to their classical counterparts away from the origin. Therefore, the Green tensor of Mindlin's anisotropic gradient elasticity with separable weak non-locality can be used as a physically-based regularization of the classical Green tensor for materials with strong anisotropy.