The multi-parameter inversion of elastic wave in half-space

The multi-parameter inverse scattering problem of elastic wave equation with single frequency is investigated within Born approximation. By use of a wideband measuring scheme in which both transmitters and receivers scan over the half-space surface, the formula of the scattering field of elastic wave is derived. Four types of mode conversion of elastic wave (P→P,P→S,S→P,S→S) are separated from the scattering field. These components contain sufficient information for us to reconstruct the configurations of the density and Lamé parameters of the medium. The inverse formulas have the form of filtered back-propagation as in the acoustic diffraction tomography. Computer simulations are also obtained. Supported by Foundation of Ph.D Program of the State Education Commission of China.     

Interaction of slowly moving punches with an elastic half-space

The problem of slow dynamic contact interaction of a system of punches remote from each other with an elastic half-space surface in the absence of friction is studied under the assumptions that the diameters of the contact areas are smaller than the minimum distance between the punches and the time required for the shear wave to travel the distance equal to the punch diameter is comparable to the time scale of the process. A first-order asymptotic model is constructed. As an example, the case of steady-state vibrations of a system of two punches is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 198–206, November–December, 2008.     

THE ELASTIC FIELD INDUCED BY A HEMISPHERICAL INCLUSION IN THE HALF—SPACE

The elastic field induced by a hemispherical inclusion with uniform eigeustralns in asemi-infinite elastic medium is solved by using the Green‘s function method and series expansion tech-nique. The exact solutions axe presented for the displacement and stress fields which can be expressedby complete elliptic integrals of the first, second, and third kinds and hypergeometric functions. Thepresent method can be used to determine the corresponding elastic fields when the shape of the inclusionis a spherical crown or a spherical segment. Finally, numerical results axe given for the displacementand stress fields along the axis of symmetry (x3-axis).     

Contact interaction of an elastic punch and an elastic half-space with initial (residual) stresses

The contact interaction of an elastic punch of arbitrary cross-section and an elastic semi-space with initial (residual) stresses is studied. A general method to solve the problem is proposed. It allows solving contact problems for bodies with initial (residual) stresses when the solution of the corresponding elastic problem is known __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 28–40, December 2007.     

Response of an elastic half-space with power-law nonhomogeneity to static loads

In this paper a series of problems for an isotropic elastic half-space with power-law nonhomogeneity are considered. The action of surface vertical and horizontal forces applied to the half-space is studied. A part of the paper deals with the case of zero-valued surface shear modulus (for positive values of the power determining the nonhomogeneity). This condition leads to simple solutions for two-dimensional (2D) case when radial distribution of stresses exists for surface loads concentrated along an infinite line. Corresponding results for the three-dimensional (3D) case are constructed on the basis of the relationships between 2D and 3D solutions developed in the paper. A more complicated case, in which the shear modulus at the surface of the half-space differs from zero, is treated using fundamental solutions of the differential equations for Fourier–Bessel transformations of displacements. In the paper the fundamental solutions are built in the following two forms: (a) a combination of functions expressing displacements of the half-space under the action of vertical and horizontal forces in the case of zero surface shear modulus, and (b) a representation of the fundamental solutions using confluent hypergeometric functions. The results of numerical calculation given in the paper relate to Green functions for the surface vertical and horizontal point forces.     

Rayleigh-type waves in a water-saturated porous half-space with an elastic plate on its surface

The paper deals with the plane problem of steady-state time harmonic vibrations of an infinite elastic plate resting on a water-saturated porous solid. The displacements of the plate are described by means of the linear theory of small elastic oscillations. The motion of the two-phase medium is studied within the framework of Biot's linear theory of consolidation. The main interest is focused on the investigation of properties of the Rayleigh-type waves propagating alongside of the contact surface between the plate and the porous half-space. In particular, the dependence of the phase velocity and attenuation of the waves on the plate stiffness, mass coupling coefficient, and degree of saturation of the medium is studied. Besides, for the limiting case of an infinitely thin plate, the comparison of the wave characteristics is carried out with those of the pure Rayleigh waves.     

On the possibility of local buckling of the surface of an elastic half-space under compression

The possibility of local buckling of the free surface of the lower half-plane under compression is studied in a static formulation within the framework of plane deformation. It is shown that in some media small subcritical strains can lead to local buckling of the half-plane surface. It is found that two forms of local surface buckling correspond to one critical compression load. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 149–152, May–June, 2005.     

Application of transition matrix to scattering of elastic waves in half-space by a surface scatterer

A transition matrix formulism for the scattering problems of elastic waves using a surface scatterer of three-dimensional half-space is derived by applying Betti's third identity and orthogonality conditions. The basis functions and their corresponding regular parts are derived from Lamb's singular solutions. The new orthogonal functions are constructed based on the linear transform. The intrinsic properties of the scattering and transition matrices derived in this paper are investigated. Numerical results for verification are presented and discussed.     

Nonlinear time transformation method for strong nonlinear oscillation systems

In this paper, a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems. This method can be used to study the limit cycle behavior of the autonomous systems and to analyze the forced vibration of a strong nonlinear system. The project partly supported by the Foundation of Zhongshan University Advanced Research Center     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Abel inversion of axially-symmetric shock wave flows

Finite-fringe interferograms produced for axisymmetric shock wave flows are analyzed by Fourier transform fringe analysis and an Abel inversion method to produce density field data for the validation of numerical models. For the Abel inversion process, we use basis functions to model phase data from axially-symmetric shock wave structure. Steady and unsteady flow problems are studied, and compared with numerical simulations. Good agreement between theoretical and experimental results is obtained when one set of basis functions is used during the inversion process, but the shock front is smeared when another is used. This is because each function in the second set of basis functions is infinitely differentiable, making them poorly-suited to the modelling of a step function as is required in the representation of a shock wave.Received: 12 November 2003, Accepted: 21 October 2004, Published online: 31 March 2005[/PUBLISHED]PACS: 47.40.-x, 42.40.Kw, 02.30.Zz     

Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

In this paper we present a spectral technique for building asymptotic expansions which describe periodic processes in conservative and self-excited systems without assuming the oscillations to be weakly nonlinear. The small parameter of the expansion is connected with the ratio of the amplitudes of higher than the first harmonics in contrast to the traditional parameter connected with weak nonlinearity. In the case of an oscillator with power nonlinearity the frequency of the main harmonic and the complex amplitudes of higher harmonics are computed as the expansions of either integer (for weakly nonlinear oscillations) or algebraic (for strong nonlinearity) functions of the complex amplitude of the first harmonic depending on the character of the initial conditions and the maximum power of the nonlinear term in the equation. In the simplest case of weakly nonlinear oscillations the complete asymptotic expansion is shown to be valid in the whole domain of the periodic motions of definite type until the separatrix is reached. The expressions for the first terms of the expansion for concrete examples coincide with the expressions obtained both with the use of other methods and by expanding the exact solutions. For some special cases of the strongly nonlinear oscillations the comparison of the results with known exact solutions is carried out as well as the criteria of convergence of the expansions are determined.     

Primary wave transmission in systems of elastic rods with granular interfaces

We study analytically and numerically primary pulse transmission in one dimensional systems of identical linearly elastic non-dispersive rods separated by identical homogeneous granular layers composed of n beads. The beads interact elastically through a strongly (essentially) nonlinear Hertzian contact law. The main challenge in studying pulse transmission in such strongly nonlinear media is to analyze the ‘basic problem’, namely, the dynamical response of a single intermediate granular layer, confined from both ends by barely touching linear elastic rods subject to impulsive excitation of the left rod. The analysis of the basic problem is carried out under two basic assumptions; namely, of sufficiently small duration of the shock excitation applied to the first layer of the system, and of sufficiently small mass of each bead in the granular interface compared to the mass of each rod. In fact, the smallness of the mass of the bead defines the small parameter in the asymptotic analysis of this problem. Both assumptions are reasonable from the point of view of practical applications. In the analysis we focus only in primary pulse propagation, by neglecting secondary pulse reflections caused by wave scattering at each granular interface and considering only the transmission of the main (primary) pulse across the interface to the neighboring elastic rod. Two types of shock excitations are considered. The first corresponds to fixed time duration (but still much smaller compared to the characteristic time of pulse propagation through the length of each rod), whereas the second type corresponds to a pulse duration that depends on the small parameter of the problem. The influence of the number of beads of the granular interface on the primary wave transmission is studied, and it is shown that at granular interfaces with a relatively low number of beads fast time scale oscillations are excited with increasing amplitudes with increasing number of beads. For a larger number of beads, primary pulse transmission is by means of solitary wave trains resulting from the dispersion of the original shock pulse; in that case fast oscillations result due to interference phenomena caused by the scattering of the main pulse at the boundary of the interface. Considering a periodic system of rods we demonstrate significant reduction of the primary pulse when transmitted through a sequence of granular interfaces. This result highlights the efficacy of applying granular interfaces for passive shock mitigation in layered elastic media.     

Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load

Complex potentials in common form for compressible and incompressible elastic bodies are used to formulate and solve the problem of stationary motion of a prestressed two-layer elastic half-space under a moving surface load. The results presented are similar to those obtained earlier using the Fourier transform __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 3–15, May 2008.     

Simulation of multi-cracks in solids using nonlinear elastic wave spectroscopy with a time-reversal process

In non-classical nonlinear media, much characteristic information is contained in their dynamic elastic responses. A method combining nonlinear elastic wave spectroscopy (NEWS) with a time-reversal (TR) process is used in this numerical study, in which the presence of one defect and two defects acting with non-classical nonlinearity in an attenuating medium is simulated. Nonlinear defect behavior is introduced using a modified Preisach–Mayergoyz (PM) model. Two methods are used to determine retrofocal quality: harmonic filtering and modulated wave filtering. In the simulation, the nonlinear signal is filtered from the received continuous wave, then reversed and re-sent; a crack image can be obtained from the nonlinear signal in a lossy solid. By comparison with the actual defect, the image can reflect the distribution of one or two flaws, which show the feasibility and value of the NEWS–TR methodology for microdamage imaging of two defects. These results also show that images obtained with different harmonic and modulated frequencies can reflect the presence of defects. With increasing frequency, the crack positions obtained from the image change due to the influence of solid loss and interaction with sound waves.     

流体饱和弹性半空间裂隙对平面P_I波的衍射

采用间接边界元法,求解了饱和半空间裂隙对平面PI波的二维衍射问题。基于单层位势理论,将边界离散并直接在边界单元上施加虚拟荷载(水平作用力、竖向作用力和流量源的叠加)以构造散射波场,并由边界条件确定虚拟荷载密度,总波场由自由波场和散射波场共同组成。通过参数分析研究了入射波频率、入射倾角、埋深、孔隙率、边界渗透条件等因素对饱和半空间中裂隙对平面PI波衍射的影响规律。结果表明:裂隙随埋深增大,地表位移谱振荡加剧,峰值有所降低;随着入射频率增加,孔隙率影响逐渐增大;垂直入射时,水平位移的放大区域主要分布在裂隙两端,斜入射时,主要集中在裂隙正上方地表附近;透水和不透水两种情况下的地表位移幅值和相位差别较小,但干土情况与饱和情况下的位移幅值相差较大。     

Diffraction of a planar strong detonation wave by a moving three-dimensional thin body

An analytic solution is obtained for the diffraction of a planar strong detonation wave by a three-dimensional thin body moving in the opposite direction. The planform and the thickness distribution of the body can be arbitrary and the speed of the body can be either supersonic or subsonic relative to the undisturbed stream ahead of the wave or to that behind the wave. The solution is a generalization of the previous solution of Ting and Gunzburger for the shock diffraction.     

Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel

In this paper, a power series and Fourier series approach is used to solve the governing equations of motion in an elastic axisymmetric vessel with the assumption that the fluid is incompressible and Newtonian in a laminar flow. We obtain solutions for the wave speed and attenuation coefficient, analytically where possible, and show how these differ under a number of different conditions. Viscosity is found to reduce the wave speed from that predicted by linear wave theory and the nonlinear terms to increase the wave speed in comparison to the linear solution. For vessels with a wall stiffness in the arterial range, the reduction in the wave speed due to the viscous terms is approximately 10% and the increase due to the nonlinear terms is approximately 5%. This difference between the linear and nonlinear wave speeds was found to be largely constant irrespective of the number of terms considered in the power series for the velocity profile. The linear wave speed was found to vary weakly with stiffness, whilst the nonlinear wave speed was found to vary significantly with the stiffness, especially at low values of stiffness. The 10% variation in the wave speed due to the viscous terms was found to be constant with wall stiffness whilst the 5% variation due to the nonlinear terms was found to vary with wall stiffness. The importance of the number of terms considered in the power series is discussed showing that only a relatively small number is required in the viscous case to obtain accurate results.     

Investigation of wave fields in a fluid layer and elastic half-space excited by a near-bottom source of vibrations

Questions of a study of acoustic and seismic wave propagation in the ocean and the underlying medium, associated with problems of marine hydrolocation (sonar) and the investigation of tsunami wave predecessors caused by underwater volcanic eruptions are of great interest at this time. The simplest model, a point source of vibrations located in a fluid layer at a certain range from the bottom, is used to describe the mentioned wave processes. Wave fields in a fluid and elastic base are investigated in this paper, analytical formulas are obtained, and results of a numerical analysis are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 49–51, July–August, 1984.     

Use of reciprocity considerations for the two-dimensional BEM analysis of wave propagation in an elastic half-space with applications to acoustic emission

A simple numerical treatment of the infinite boundary in the BEM analysis of two-dimensional wave propagation problems in elastic half-spaces is proposed to avoid the spurious reflections of non-decaying Rayleigh waves introduced by the truncation of the boundary. The proposed method exploits the knowledge of the far-field asymptotic behavior of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution exclusively in terms of the boundary integrals of the original BEM system. The method is applied to the study of the acoustic emission from nucleating and propagating surface-breaking and buried cracks in a two-dimensional elastic half-space. It is shown to be particularly advantageous since it allows for an accurate calculation of the generated signal even when the observation point is located far from the acoustic emission source.