Rectilinear Edge Research Articles

МОДЕЛИРОВАНИЕ РОСТА КРАЕВОЙ УСТАЛОСТНОЙ ТРЕЩИНЫ ПРИ ВЫСОКОЧАСТОТНОМ НАГРУЖЕНИИ

Multiple fracture surface investigations of real aviation industry products have proved that low-amplitude high-frequency oscillations can lead to an unexpected failure of aviation elements and other structures. High-frequency loading is the reason for this, as it produces a large number of loading cycles that often exceed an investigated area of a material fatigue behavior. This new fatigue failure mode (very-high-cycle fatigue) requires a special experimental study with the use of both experimental and mathematical modeling. This paper is focused on a numerical determination of stress intensity factors (SIF) in the specimen with the edge notch specimen loaded by harmonic high frequency displacements of small amplitudes. The numerical calculations were performed for the case of loading with frequency close to its natural frequency. A dimensionless adjusting function was determined for SIF that takes into account a change in modal characteristics of the resonance system (the specimen with rectilinear edge notch) due to crack propagation. The obtained equation was used to model the position of the curvilinear fatigue crack front. A general scheme of a piezoelectric fatigue testing machine is introduced with a technique of tensile-compression fatigue tests on the titanium crack growth specimen with an edge notch in the range of very high cycle fatigue. The analysis of the fracture surfaces with the identification of the front stop lines and mathematical modeling of the crack front evolution under high-frequency loading are carried out. The results of mathematical modeling are comparing with experimental data obtained during high-frequency fatigue tests on a piezoelectric fatigue testing machine. The numerical calculations have shown that this approach allows us to qualitatively and quantitatively simulate the evolution of the edge fatigue crack with a curvilinear front under very-high-cycle fatigue (high-frequency) loading mode.

The motion of pressure distribution over a free surface near the edge of ice sheet

The solution of a steady three-dimensional problem for the wave disturbances induced by a pressure distribution moving with uniform speed along the rectilinear edge of semi-infinite ice sheet is presented. This external load simulates the air-cushion vehicle. The problem is formulated within linear hydroelastic theory. The fluid is assumed to be inviscid and incompressible and its motion is potential. The ice sheet is treated as an elastic thin plate using the Kirchhoff-Love model. The solution of this problem is constructed using the Fourier transform and the Wiener-Hopf technique. The displacements of free surface and ice cover are determined, as well as power characteristics (wave resistance and side force) acting on the vehicle at various speeds of its movement: subcritical and supercritical relative to the minimum phase velocity of flexural-gravity waves in the ice cover. It is found that at speeds close to the critical velocity of flexural-gravity waves, the wave forces undergo sharp changes. It is shown that for some values of load speed, ice thickness and external pressure, the ice fracture near the edge is possible.

A solution to the problem of elastic half-plane with a cohesive edge crack

This paper considers the problem of extension of an elastic half-plane slackened by a rectilinear edge crack. The opposite edges of the crack are attracted to each other. The intensity of attracting forces – the forces of cohesion – depends on displacements of the edges; this dependence is nonlinear in the general case. External load and cohesive forces are related to each other by the condition of finite stresses at the crack tip. The authors apply Picard’s method of successive approximation. In each iteration, Irwin’s method is used to solve the problem of a half-plane with a crack, the edges of which are subjected to irregularly distributed load. The solution of the resulting integral equation is found by Galerkin’s method. The paper includes examples of calculations and their results. Some of them are compared with the data of previous studies.

Action of periodic surface pressure on an ice cover in the vicinity of a vertical wall

This paper presents the solution of the linear hydroelastic problem for steady forced vibrations of a semi-infinite ice cover under the effect of localized external load. The ice cover is simulated by a viscoelastic thin plate, the thickness of the fluid layer is assumed to be small, and the shallow water theory is used. The fluid is limited by a solid vertical wall, and the rectilinear edge of the elastic plate adjacent to the wall can be both free and clamped. The solution is obtained with the help of the Fourier integral transform. The behavior of the ice cover is studied depending on the frequency of the external load and boundary conditions on the edge of the plate. It is shown that, in the case of a free edge of the plate, there are considerable deflections on the edge, which could be comparable with deflections at the center of the pressure impact region. It is established that, due to the existence of wave movements of the type of edge waves, the external load energy is transferred to larger distances along the free edge, and there are significant bending moments on the edge of the clamped plate, which can lead to fracture of the ice cover with sufficiently great intensity of the external load.

Equation of motion and subsonic-transonic transitions of rectilinear edge dislocations: A collective-variable approach

A theoretical framework is proposed to derive a dynamic equation motion for rectilinear dislocations within isotropic continuum elastodynamics. The theory relies on a recent dynamic extension of the Peierls-Nabarro equation, so as to account for core-width generalized stacking-fault energy effects. The degrees of freedom of the solution of the latter equation are reduced by means of the collective-variable method, well known in soliton theory, which we reformulate in a way suitable to the problem at hand. Through these means, two coupled governing equations for the dislocation position and core width are obtained, which are combined into one single complex-valued equation of motion, of compact form. The latter equation embodies the history dependence of dislocation inertia. It is employed to investigate the motion of an edge dislocation under uniform time-dependent loading, with focus on the subsonic/transonic transition. Except in the steady-state supersonic range of velocities---which the equation does not address---our results are in good agreement with atomistic simulations on tungsten. In particular, we provide an explanation for the transition, showing that it is governed by a loading-dependent dynamic critical stress. The transition has the character of a delayed bifurcation. Moreover, various quantitative predictions are made, that could be tested in atomistic simulations. Overall, this work demonstrates the crucial role played by core-width variations in dynamic dislocation motion.

Interpolation–free discrete modeling with gradient matrix: Case study of edge dislocation in linearly elastic crystal

A meshless and interpolation-free (MIF) method of numerical modeling in solid mechanics has been formulated with a gradient matrix extending the gradient operation to discrete data. Nodal strains and, consequently, stresses in this method are expressed immediately in terms of nodal displacements and the stress divergence in terms of nodal stresses that makes it possible to get the stress balance equation in a truly discrete form. A trial MIF model where nodal points correspond to atom positions is employed for a rectilinear edge dislocation in a linearly elastic crystal. Both the resulting stress level at the dislocation core, close to the theoretical crystal strength, and respective core dimensions prove to be realistic physically whereas calculated long-range stresses asymptotically approach the related continuous fields known in an analytical form for the dislocation in linearly elastic isotropic continuum.

Nonradial edge crack in a circular disk

We deduce a singular integral equation of the problem of stress-strain state of a circular disk weakened by a rectilinear nonradial edge crack whose lips are subjected to the action of an arbitrary self-balanced load. For the case of a pressure uniformly distributed over the crack lips, we compute the stress intensity factors as functions of the orientation and crack length.

A cylindrical plate with a weakly fixed curvilinear edge made of a transversally isotropic material

Free oscillations and stability under an axial compression of a thin cylindrical plate with a weakly fixed rectilinear edge made of a transversally isotropic material with low stiffness with respect to transverse displacements are considered. The curvilinear edges of the plate are assumed to be hingedly supported. The oscillation frequencies and the critical load for a plate with a free or weakly fixed edge are smaller than those for a shell closed in the circumferential direction. The shapes of oscillations and the forms of stability loss localized near the weakly fixed edge and damped at a distance from it are considered. The Timoshenko-Reissner model is used. Localized forms are analyzed by using a system of equations for Timoshenko-Reissner shallow shells, which is derived for this purpose. The main special feature of this system is that it contains a separate equation describing a solution with large variability. For the example of the stability problem under consideration, the error involved in the system of equations for Timoshenko-Reissner shallow shells is studied. The critical load values obtained with the use of the Kirchhoff-Love and Timoshenko-Reissner models are compared.

Relationship between J-integral and averaged strain-energy density for U-notches in the case of large control volume under Mode I loading

The main purpose of this technical note is to present a relationship between J-integral and averaged strain-energy density ( W ¯ ) in U-notches under Mode I loading for brittle or quasi-brittle materials. In this work, control volume includes the rectilinear edge of the notch in addition to semi-circular arc of the notch root. A dimensionless function ( f) between J and W ¯ has been presented in this paper. Finite element analysis has been used for verification. It is found that this relationship is identical for tension or bending loading.

Energy release caused by the kinking of a crack in a plane anisotropic solid

The problem of the deformation of a homogeneous, elastic, anisotropic body with a rectilinear edge cut having a small kink ▪ τ of fairly arbitrary shape is analysed. The asymptotic solution of this problem in the case of small values of the dimensionless parameter τ, characterizing the kink size is constructed using a modified method of matched asymptotic expansions. The amount of elastic energy released is expressed in terms of a set of stress intensity factors (SIFs) at the tip of the unperturbed crack, the integral characteristics of the kink ▪ τ (the components of the enlarged energy release matrix) and the integral characteristics of the initial solid (the SIFs of the weighting functions). The results are compared with existing results in the case of an isotropic solid.

Compliant effect in twist-bonded systems

We describe how a buried twist-bonded interface closely set below the substrate surface alters the plastic relaxation in heteroepitaxy. For this, first—considering from an atomistic point of view, two twist-bonded semi-infinite substrates—we calculate and compare the elastic energy for several twist angles. We find that for ‘low twist angles’ (up to around 20°), the twist-bonded interface can be reconstructed without involving any dangling bonds, but for large twist angles, the twist-bonded interface becomes more unstable and involves some dangling bonds. Furthermore, for ‘low twist angles’, the strain field induced by such a twisted interface does not extend on more than several atomic layers. Second—considering a mesa made from a thin twist-bonded layer and a thick 4% compressive strained overlayer—we study the motion of edge dislocations while entering from the edge and moving towards the centre of the sample. The energy barrier at edges contravenes the introduction of a rectilinear edge dislocation at the mesa bottom interface. As far as misfit edge dislocations are concerned, it is easier to introduce it at the heterojunction between the compressive strained overlayer and the thin twist-bonded layer than at the mesa bottom twisted interface. Finally, we define a kinked edge dislocation which is able both to relax more energy than the rectilinear one and to weaken the energy barrier at the edges of a mesa, so that compliancy may become possible.

Spin-flip scattering of conduction electrons by dislocations in a metal

The spin-flip scattering of conduction electrons by dislocations in metals with a strong spin-orbit coupling is considered. Calculations are performed in terms of the model spin-orbit potential describing the spin-flip scattering of conduction electrons. It is shown that deformation of the crystal lattice in a metal leads to a change in the structure factor. The core of a rectilinear edge dislocation is calculated by the molecular dynamics method. The results obtained are compared with the experimental data on conduction-electron spin resonance (CESR) in copper.

Efficient moment method analysis of a rectilinear edge-slot radiator in rectangular waveguide

A moment method (MoM) model of a rectilinear edge slot in rectangular waveguide is developed using entire domain trigonometric basis functions. This model has been found to provide an optimum trade-off between computational efficiency and accuracy. The intention behind the analysis is to provide an accurate, computationally fast model for the inclined edge slot radiator for incorporation into an array synthesis program. The basis functions for the ‘three-dimensional’ slot geometries involved in the analysis are generated using a simple transmission line representation of the slot field distribution to identify the resonant ‘slot modes’. The results obtained are compared with alternative modelling approaches to this problem, based on finite element method and finite difference time domain techniques, and also with measurement.

Edge dislocations near phase boundaries in the gradient theory of elasticity

A solution of the boundary-value problem in the gradient theory of elasticity concerning a rectilinear edge dislocation parallel to the interface between phases with different elastic moduli and gradient coefficients is obtained. The interaction between the dislocation and the interface is considered on a nanoscopic level. It is shown that the stress field has no singularities on the dislocation line and remains continuous at the interface, unlike the classical solution, which is singular at the dislocation line and allows a discontinuity of two stress components at the interface. The gradient solution also removes the classical singularity of the image force for the dislocation on the interface. An additional elastic image force associated with the difference in the gradient coefficients of contacting phases is also determined. It is found that this force, which has a short range and a maximum at the interface, expels the edge dislocation into the material with a smaller gradient coefficient.

Scattering of spin waves by a rectilinear edge dislocation

The propagation of volume spin waves in an unbounded easy-axis magnet containing a rectilinear edge dislocation is studied theoretically. The spin-wave scattering amplitudes are calculated in the Born approximation. It is shown that the spin-wave scattering amplitude vanishes for certain values of the scattering angle. The dependence of the scattering angle on the angle of incidence of the spin waves is found for this case. The transport scattering cross section of spin waves is found.

Growth of plasticity bands around the tip of an arbitrarily oriented crack in a semiinfinite thin plate

Elastoplastic equilibrium is considered for a thin semiinfinite plate containing an arbitrarily oriented rectilinear edge crack. It is assumed that the plastic strains are localized along a narrow band arising from the crack tip. The elastoplastic treatment is then reduced to a planar treatment in the theory of elasticity with an edge kinked crack whose edges are subject to defined forces. The solution is derived by means of singular integral equations. Numerical results have been obtained for a tensile load acting at infinity for a constant pressure acting on the crack edges. An estimate is made of how the free edge of the plate affects the magnitude and orientation of the plasticity band.

Detection and segmentation of man-made objects in outdoor scenes: concrete bridges

We present a new approach for the detection and segmentation of man-made objects in color images of natural scenes. The approach is based on detecting geometric structure in the image and combining the detected structure with color information to guide the segmentation. The central problem is the detection and semantic interpretation of large stationary man-made objects in color images of nonurban scenes. We describe the generation of color-based confidence functions for material selection in incremental segmentation. We focus on the segmentation of images of concrete bridges. These techniques are applicable to autonomous navigation, target acquisition, and several industrial computer vision problems. Large concrete objects often have rectilinear edge structures with many parallel relationships. We use these properties to guide our initial incremental segmentation toward concrete objects. The goal for our segmenter is to locate the representative faces of concrete material in the image as a starting point for the interpretation phase. These heuristics rely on the detection of straight line segments of the edge map of the gray-scale image. The straight line segments, once detected, are then grouped according to several perceptual grouping criteria. The straight line segments are then constrained further by region label restrictions. Finally, color cues are used to restrict the candidate artifacts further and to produce confidence measures of our initial belief in our estimation of the material of the identified rectilinear faces.

Equivalent currents for second-order diffraction by the edges of perfectly conducting polygonal surfaces

Equivalent, electric and magnetic, edge currents for arbitrary aspects of observation are derived for second-order diffraction by the edges of perfectly conducting, flat, polygonal surfaces. The physical model underlying the derivation is that each illuminated rectilinear edge segment excites a fringe current on the surface, acting as if it were part of an infinite straight edge. The surface wave associated with this fringe current traverses the surface along the grazing diffracted rays until it strikes an opposite edge segment, and its illumination area on the surface is delimited according to the finite length of the initiating edge segment. The "ray coordinate" measured along a grazing diffracted ray is chosen as the integration variable complementary to the edge coordinate in the fringe current radiation integral. The one-dimensional "radiation" integral over this coordinate is evaluated asymptotically in the high-frequency limit and reduced to the sum of two endpoint contributions. The upper integration limit contribution is east in the form of equivalent edge currents pertaining to the termination point of the grazing diffracted ray. These currents are responsible for the dominant part of the second-order edge diffraction. Their expressions incorporate the well-known Fresnel function (which in many eases may be replaced by its asymptotic approximation) and are finite for any combination of incidence and scattering directions, except when both of them coalesce with the grazing diffracted ray. The developed method applies both to flat plates and to plane faces of thick bodies. Examples of backscatter calculations for flat plates are given which exhibit certain improvements over previous calculations by Sikta, as compared with measured data.

Minimization of the drag of blunt edges of projecting elements of a hypersonic object

An approximate method is given for calculating the pressure on the critical line of a cylindrical edge of an element projecting above the surface of a hypersonic object. Correlations of the pressure distribution in the interference regions are proposed on the basis of the calculations and analysis of the experimental data. Variational problems of finding the form of a leading edge having minimum drag are set up and solved on the basis of the pressure calculation model obtained. The solutions are sought with the use of Pontryagin's maximum principle. The effect on the minimum drag profile of Mach numbers and blunting of the body is analyzed. The advantage of the optimal form of the edge in comparison with a corresponding rectilinear edge is shown.