Scattering of elastic waves in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).     

Scattering of elastic waves by three-dimensional surface topographies

Diffraction of plane harmonic waves by three-dimensional surface irregularities is investigated through the use of an indirect boundary integral equation method. The irregularity of an arbitrary shape is embedded in an elastic half-space and subjected to incident P, SV, SH, and Rayleigh waves. The material of the half-space is assumed to be linear, weakly anelastic, homogeneous and isotropic.

The accuracy of the method is demonstrated through comparison of the results with existing axisymmetric solutions. Several numerical examples for non-axisymmetric canyons are presented. The resulting amplification patterns exhibit strong sensitivity on type and angle of the incident waves and on the location of the observation point. Systematic comparisons of three-dimensional and corresponding two-dimensional models demonstrate similarity of the amplification pattern. The amplification is larger in some three-dimensional models than in two-dimensional ones. Strong coupling between SH and P-SV modes is observed for off-azimuthal incident waves. This phenomenon is specially pronounced for incident SH waves and it is intrinsic to three-dimensional scattering.     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Scattering of plane, elastic waves by a plane, rigid strip

The diffraction of time-harmonic, vertically polarized, plane elastic waves by a rigid strip is investigated with the aid of the integral-equation method. Using the integral representation for the particle displacement of the scattered wave, it is shown that the resulting integral equations of the first kind uncouple for this kind of obstacle. In them, the amounts by which the shearing stress and the tensile stress jump across the strip occur as unknown quantities. The integral equations are solved numerically. Normalized power scattering characteristics and scattering cross-sections are computed.The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).     

Scattering of elastic waves by an inclusion with an interface crack

A method of perturbation is used to derive an integral representation of the displacement field for the scattering of a plane wave from an inclusion with an interface crack. In the long-wave approximation it is shown that the solution of only an associated static problem is required and formal expressions are derived for the scattered far field amplitudes and scattering cross section. In the case of a cylindrical inclusion the solution of the associated static problem is then used to find in a closed form the corresponding expressions for plane incident P- and S-waves.     

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

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.     

Scattering of implusive sound waves by a rigid cylinder

Summary Analytic time solutions for the scattering of impulsive waves (the time-space Green function) by a rigid circular cylinder in an annular domain are obtained through a finite integral transform of the spatial variable. By a similar generalized procedure the scattering of impulsive waves by a rigid cylinder in an infinite medium is described in order to obtain the Green function of the reduced wave equation in terms of a series of propagation modes.

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Sommario In questo articolo si presentano alcuni metodi di soluzione analitica di problemi di diffrazione di onde impulsive in un dominio anulare. Le soluzioni sono ottenute con l'impiego di una trasformata integrale finita della variabile spaziale. Attraverso un procedimento analogo generalizzato viene descritta la diffrazione di onde impulsive causate da un cilindro circolare rigido in un mezzo infinito.

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This work was supported by C.N.R., Committee for Mathematical Sciences.     

Interaction of plane harmonic waves with a thin elastic inclusion of zero flexural rigidity

We solve the problem on the interaction of plane elastic harmonic waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) that is under plane strain conditions. The normal forces applied by the medium to the inclusion side edges are taken into account. Because of the small thickness of the inclusion, we assume that its flexural rigidity is zero and that the shear displacements at any of its points coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself can be found from the corresponding equation of the theory of plates. The solution method consists in representing the displacements as discontinuous solutions of the Lamé equations and then determining the unknown jump from a singular integral equation. This equation is solved numerically by the collocation method, and formulas for the approximate calculation of the stress intensity factors near the inclusion ends are obtained.     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.     

Scattering of elastic waves by non-planar cracks

A modification of the null field approach is used to study the scattering of elastic waves by non-planar cracks. A fictitious surface is added to the crack so that a convenient closed surface is obtained and the surface fields on this closed surface are expanded in vector spherical harmonics. The edge conditions are introduced into these expansions and this is shown to be essential for the numerical convergence. Total cross sections and backscattering amplitudes as functions of frequency are computed numerically for rotationally symmetric cracks which are part of spherical or spheroidal surfaces. By integration in frequency backscattered pulses are also computed. Some cases with two cracks are also considered.     

Interaction between a rigid line inclusion and an elastic circular inclusion

I.IntroductionManypracticalproblemsinengineering,suchascompositematerial,weldedjointorribbedslab,needustostudytheinteractionproblemoflineinclusionandcircularinclusionasshowninFig.1.Sotheproblemwasdiscussedinthispaper.Proceedingfromthestressfieldofplanecon…     

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.     

Interaction of one-periodic disk-shaped cracks under an incident elastic harmonic wave

We study a symmetric problem of harmonic wave propagation in an elastic space with a one-periodic array of interacting disk-shaped cracks. Using the Green function obtained by the Fourier transform, we reduce the problem to a boundary integral equation (BIE) for the function characterizing the displacement discontinuity on one of the cracks and numerically determine the desired function by solving the BIE. We present graphs of the dynamic stress intensity factors near a circular crack versus the wave number for various distances between the defects.     

Vertical vibrations of a rigid edge inclusion under a harmonic load

The paper considers the problem of vibrations of a rigid edge inclusion, which lies in an elastic half-plane and emerges on the surface perpendicular to that half-plane. The vibrations are initiated by a harmonic force acting on the end of the inclusion, which emerges on the surface. The field of translations in the half-plane is shown to be represented by the superposition of two discontinuous solutions with discontinuities at the boundary between the half-plane and the line of the inclusion. The unknown discontinuities are determined from the boundary conditions and the conditions of the inclusion-medium interaction. The problem is thus reduced to one of solving a singular integral equation with an immobile singularity for the jump in shear stresses on the line of the inclusion. The equation obtained is solved numerically by the method of mechanical quadratures. The amplitudes of the inclusion vibrations and the stressed state of the medium near it are studied.Odessa State Marine Academy, Odessa, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 46–55, July, 1995.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Interaction between a compliant disk-shaped inclusion and a crack upon incidence of an elastic wave

The propagation of harmonic elastic wave in an infinite three-dimensional matrix containing an interacting low-rigidity disk-shaped inclusion and a crack. The problem is reduced to a system of boundary integral equations for functions that characterize jumps of displacements on the inclusion and crack. The unknown functions are determined by numerical solution of the system of boundary integral equations. For the symmetric problem, graphs are given of the dynamic stress intensity factors in the vicinity of the circular inclusion and the crack on the wavenumber for different distances between them and different compliance parameters of the inclusion.