A finite difference method for elastic wave scattering by a planar crack with contacting faces

This paper presents a finite difference time-domain technique for 2D problems of elastic wave scattering by cracks with interacting faces. The proposed technique introduces cracks into the finite difference model using a set of split computational nodes. The split-node pair is bound together when the crack is closed while the nodes move freely when open, thereby a unilateral contact condition is considered. The development of the open/close status is determined by solving the equation of motion so as to yield a non-negative crack opening displacement. To check validity of the proposed scheme, 1D and 2D scattering problems for which exact solutions are known are solved numerically. The 1D problem demonstrates accuracy and stability of the scheme in the presence of the crack-face interaction. The 2D problem, in which the crack-face interaction is not considered, shows that the proposed scheme can properly reproduce the stress singularity at the tip of the crack. Finally, scattered fields from cracks with interacting faces are investigated assuming a stick and a frictionless contact conditions. In particular, the directivity and higher-harmonics are investigated in conjunction with the pre-stress since those are the basic information required for a successful ultrasonic testing of closed cracks.     

Scattering of elastic waves by a surface-breaking crack

Scattering of incident surface waves and incident body waves by a surface-breaking crack is investigated in a two-dimensional geometry. By decomposing the scattered fields into symmetric and antisymmetric fields with respect to the plane of the crack, two boundary value problems for a quarter-plane have been obtained. The formulation of each boundary-value problem has been reduced to a singular integral equation which has been solved numerically. For incident surface waves the back-scattered and forward-scattered surface waves have been plotted versus the dimensionless frequency. Curves are also presented for the scattered displacement fields in the interior of the body generated by incident body waves, both versus the angle of incidence and versus the dimensionless frequency.     

Harmonic vibrations of a half-space with a surface-breaking crack under conditions of out-of-plane deformation

The paper presents a solution of the problem of determining the stress state in an elastic isotropic half-space with a crack intersecting its boundary under harmonic longitudinal shear vibrations. The vibrations are excited by a regular action of a harmonic shear load on the crack shores. The solution method is based on the use of the discontinuous solution of the Helmholtz equation, which allows one to reduce the original problem to a singular integro-differential equation for the unknown jump of the displacement on the crack surface. The solution of this equation is complicated by the existence of a fixed singularity of its kernel. Therefore, one of the main results is the development of an efficient approximate method for solving such equations, which takes into account the true asymptotics of the unknown function. The latter allows one to obtain a high-precision approximate formula for calculating the stress intensity factor.     

Interaction of an ultrasonic wave with a bubbly mixture

The interaction of an ultrasonic wave with a bubbly two-phase flow is studied both experimentally and theoretically. Brief theoretical reviews of acoustic wave generation by a piston and of the interaction of a plane wave with a single bubble are given. A theory relating ultrasonic wave transmission through a bubbly flow with two-phase flow parameters, notably the bubble size and the volumetric interfacial area, is derived and compared with preliminary data. The theoretical and experimental limitations concerning the application of ultrasonic transmission measurements to the study of bubbly flow are discussed in detail, and recommendations for future work in this area are made.Nomenclature A projected cross-sectional area of a bubble - a bubble radius - a _T emitter radius - c speed of sound - d equivalent bubble diameter - I intensity - I ₀ incident intensity - J ₀ Bessel function of zero order - J ₀ Bessel function of first order - j ₁ spherical Bessel function of the first kind and of order l - k wave number - n _l spherical Bessel function of the second kind (Nishi's notation) - n number of bubbles per unit volume or per unit area - P ₀ pressure amplitude at the emitter; equilibrium pressure in the liquid - p pressure perturbation - r spherical coordinate, radial distance to the x-axis - S total scattering cross-sectional area, surface of the piston - S _i scattering cross-sectional area of the i-th bubble - T transmittance - t time - U complex source strength divided by the source area - W ratio of radial distance from the axis on the emitter surface to the radius of the emitter - x axis coinciding with the direction of propagation of the plane wave; distance between the transducers - void fraction - interfacial area per unit volume - spherical coordinate - ultrasonic wavelength - density - angular frequency     

Scattering of elastic waves by a small inclined surface-breaking crack

I we examine the scattering of Rayleigh waves by an inclined two-dimensional plane surface-breaking crack in an isotropic elastic half-plane. We follow the method already introduced by the authors (A and W , 1992a, J. Mech. Phys. Solids 40, 1683) to obtain an analytical solution when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The procedure employed is the method of matched asymptotic expansions, which involves determining the scattered wave field both away from and near the crack. The outer solution is specialized from the general expansion in the first part of this work (A and W , 1992a, J. Mech. Phys. Solids 40, 1683), and the inner problem is exactly solved by the Wiener-Hopf technique. The displacement field and scattered Rayleigh waves are found uniformly to third order in , and concluding remarks are made about the general method as well as the results presented here.     

An explicit finite difference procedure for contact-impact analysis of crack edges

Summary A finite difference procedure is proposed to simulate the frictional contact-impact response of crack edges. The procedure takes into account in an explicit way separation, stick contact and slip contact of the crack edges. Numerical examples are presented to show the influence of the contactimpacting of crack edges on the history of dynamic stress intensity factors.     

Singular perturbation analysis of dynamic fields in a thermoelastic solid with a small surface-breaking crack

An asymptotic analysis is presented for a dynamic problem of a semi-infinite isotropic thermoelastic solid with a small surface breaking crack. The exterior surface of the solid is subjected to a series of short thermal pulses. The crack surface is traction free and an ideal thermal contact is assumed across the crack. The stress intensity factor is asymptotically evaluated as a function of the crack depth and time. The effect of a boundary layer associated with the diffusive term is identified. The theoretical model is supplied with numerical simulations.     

多尺度有限差分方法求解波动方程

小波分析是多尺度分析方法，本文利用具有紧支集的正交小波变换对有限差分方程进行空间多尺度近似，提出适合于层状介质波传问题数值计算的多尺度有限差分方法，将波动方程的求解转换到小波域中进行。利用小波基的自适应性与消失矩特性，有效减少了计算量、提高了稳定性，扩大了可求解的速度范围。地球物理勘探中的数值实例显示了算法具有良好效率。     

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional(2D) tank are studied.The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid.A time-independent finite difference method,which is developed by Bang-fuh Chen,is used to solve the Euler equations for incompressible and inviscid fluids.The numer...     

Transient analysis of a propagating crack with finite length subjected to a horizontally polarized shear wave

In this study, the transient response of a finite crack subjected to an incident horizontally polarized shear wave and then propagated with a constant speed in an unbounded elastic solid is investigated. Initially, the finite crack with crack length l is stress-free and at rest. At time t = 0, an incident horizontally polarized shear wave strikes at one of the crack tips and will arrive at the other tip at a later time. Then, two crack tips propagate along the crack tip line with different velocities as the corresponding stress intensity factors reach their fracture toughness. The correspondent configuration is shown in Fig. 1

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Fig. 1. Configuration and coordinate systems of a finite crack in an unbounded medium.

. In analyzing this problem, diffracted waves generated by two propagating crack tips must be taken into account and it makes the analysis extremely difficult. In order to solve this problem, the transform formula in the Laplace transform domain between moving and stationary coordinates is first established. Complete solutions are determined by superposition of proposed fundamental solutions in the Laplace transform domain. The fundamental solutions to be used are from the problems of applying exponentially distributed traction and screw dislocation on crack faces and along the crack tip line, respectively. The exact transient solutions of dynamic stress intensity factor for the first few diffracted waves that arrive at two crack tips are obtained and expressed in compact formulations. Numerical calculations of dynamic stress intensity factors for both tips are evaluated and the results are discussed in detail.     

Lamb wave scattering by a symmetric pair of surface-breaking cracks in a plate

The scattering problem of a Lamb wave incident on a symmetric pair of surface-breaking transverse cracks in a plate is considered. The Lamb wave is assumed to be obliquely incident on the crack plane. Since the cracks are part-through, the scattered field will contain reflected as well as transmitted waves. The energy of the incoming wave is partitioned into reflected and transmitted wave modes. Energy coefficients of the reflected and transmitted waves are calculated as a function of incident frequency and crack depth. The incidence angle of the incoming wave is also treated as a parameter. Both the reflected and transmitted wave fields are considered as linear superpositions of all real and complex wave modes in the plate. Decomposition of modes is achieved with the help of an orthogonality condition based on the principle of reciprocal work. Continuity of displacement and stress fields is imposed at the crack plane. Energy coefficients for reflection and transmission are obtained from the mode amplitudes. Energy coefficients are shown to be a strong function of incident frequency and crack depth. Experiments are conducted with a PZT transducer network interacting with a symmetric pair of machined cracks in an aluminum plate. Trends predicted by the analysis are reflected in the experimental results.     

On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid

For incompressible Navier–Stokes equations in primitive variables, a method of setting absorbing outflow boundary conditions on an artificial boundary is considered. The advection equations used on the outflow boundary are convenient for finite difference (FD) methods, where a weak formulation of a problem is inapplicable. An unsteady viscous incompressible Navier–Stokes flow in a channel with a moving damper is modeled. An accurate comparison and analysis of numerical and mechanical situations are carried out for a variety of boundary conditions and Reynolds numbers. The proposed outflow conditions provide that the problem with Dirichlet boundary conditions should be solved on each time step.     

Plane problems of a finite disc containing an internal crack

Using complex variable methods in elasticity.this paper deals with the plane problemsof a finite dise containing an internal linear crack at any position under general loads,obtains the general forms of complex stress functions and stress-intensity factors.expressed in terms of series,and to these problems discusses three special cases,i.e.thecases of the crack under a uniform pressure,a uniform shear stress and the case of the discrotating uniformly.In these cases the approximate formulas calculating the stress-intensityfactors are also presented.The calculated results show that for the middle and small crackssituated inside the disc and not near the external boundary,these approximate formulasgive good or better approximation.     

Nonlocal interaction of a dislocation with a crack

Summary Based on the classical stress analysis, the stress field of a screw dislocation near a semi-infinite crack is investigated by a conformal mapping method. The classical image force on the dislocation due to the crack is presented by the obtained classical image stress. To eliminate the singularity of the stress field at the crack tip, the nonlocal elasticity theory is adopted. A maximum of the image stress is found near the crack tip and the nonlocal image force is then acquired by the obtained nonlocal stress field. Both the stress and the image force show much difference from the classical ones and seem to be more reasonable physically.

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Nichtlokale Wechselwirkung zwischen einer Versetzung und einem Riß

Übersicht Basierend auf der klassischen Spannungsanalyse einer Schraubenversetzung neben einem halb-unendlichen Riß mittels einer konformen Abbildung wird die klassische Bildkraft auf die Schraubenversetzung als Folge des Risses vorgestellt. Um die Spannungssingularitäten neben der Rißspitze zu beseitigen, wird die nichtlokale Elastizitätstheorie herangezogen, mittels ihrer wird das Maximum der Spannung neben dem Riß gefunden und die nichtlokale Bildkraft ermittelt. Die Spannungen und die Bildkraft der nichtlokalen Elastizitätstheorie zeigen große Unterschiede gegenüber der klassischen Analyse und sind physikalisch vernünftiger.

     

Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

A class of finite difference schemes for interface problems with an HOC approach

In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing a recently developed higher‐order compact methodology with special interface treatment for the points just next to the points of discontinuity. The overall order of accuracy of the scheme is at least second. We first formulate the scheme for one‐dimensional (1D) problems, and then extend it directly to two‐dimensional (2D) problems in polar coordinates. In the process, we also perform convergence and related analysis for both the cases. Finally, we show a new direction of implementing the methodology to 2D problems in cartesian coordinates. We then conduct numerous numerical studies on a number of problems, both for 1D and 2D cases, including the flow past circular cylinder governed by the incompressible Navier–Stokes equations. We compare our results with existing numerical and experimental results. In all the cases, our formulation is found to produce better results on coarser grids. For the circular cylinder problem, the scheme used is seen to capture all the flow characteristics including the famous von Kármán vortex street. Copyright © 2016 John Wiley & Sons, Ltd.     

Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid

The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and for the transmission coefficient as functions of the dimensionless frequency.     

Investigation into the interaction of a shock wave with an acute cone

A numerical investigation has been made into the nonstationary axisymmetric flow which arises from the interaction of a shock wave with a fixed acute cone. A solution has been obtained in self-similar coordinates for regimes in which a shock wave attached to the point of the cone is formed in the gas flow. The region of existence of these regimes has been established and the method of stabilization used to calculate the gas-dynamic functions in the disturbed region. The main attention is devoted to studying the transition from regular reflection to Mach reflection, a study of the features of double Mach reflection, and comparison of the results for the axisynuaetric and two-dimensional cases. The pressure distributions along the surface of the cone are given in a form which makes it possible to obtain dependences valid in a wide range of variation of the intensities of the incident shock wave and for different values of the adiabatlc exponent of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 98–104, May–June, 1980.