Modeling the scattering processes on a crack in an elastic plate with the help of a zero-range potential

The problem of flexural wave scattering on a finite crack in elastic plate is considered. The zero-range potential is suggested as a model of a short crack. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 11–16, 1990. Translated by I. V. Andronov.     

Wave propagation in an isolated porous Biot layer with closed pores on the boundaries

On the boundaries of such an isolated porous Biot layer, the total stresses and normal relative displacement are equal to zero. For this layer, the symmetric and antisymmetric dispersion equations are established and investigated. The wave field consists of normal waves. In this layer, one bending wave, two plate waves, and infinitely many normal waves propagate. For all these waves, we determine dispersion curves by analytical methods. The velocities of the bending wave and the second plate wave for the infinite frequency are equal to the Rayleigh velocity. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 173–189.     

Nonstationary interference wave fields of SV type in a free elastic layer and the problem of ultrasonic simulation of plane seismic fields by using plate models

The problem of quantitatively studying wave fields in a free elastic layer excited by a nonstationary point source is discussed. Much attention is given to the low-frequency part of the Fourier spectrum. The roots of the dispersion equation of the layer are studied comprehensively, and on this basis efficient methods of quantitatively estimating interference wave modes are developed. Particular attention is given to the ultralow-frequency band of the spectrum. This allows us to make progress toward the mathematical justification of methods of seismic modeling of plane wave fields on plate models. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 121–239. Translated by N. S. Zabavnikova.     

On the propagation of Lamb waves in a sandwich plate made of compressible materials with finite initial strains

The propagation of flexural Lamb waves in a prestrained sandwich plate made from compressible highly elastic materials is investigated within the scope of a piecewise homogeneous body model by utilizing TLTEWISB. The mechanical relations of layer materials are described by a harmonic-type potential, and numerical results are obtained for the first and second vibration modes. According to the results, the influence of problem parameters and of the initial stretching strain along the layers on the wave propagation speed is examined. The asymptotic values of the speed are considered in the cases of short and long wavelengths, and the influence of the initial strains on these asymptotic (limit) values are also analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 231–244, March–April, 2008.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.      

Global existence of null-form wave equations in exterior domains

We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave equation. We instead rely upon a class of Keel–Smith–Sogge estimates for the perturbed wave equation. Using this, a notable simplification is made as compared to previous works concerning wave equations in exterior domains: one no longer needs to distinguish the scaling vector field from the other admissible vector fields.     

Normal waves in orthotropic plates and prismatic bodies with thin face coatings

We construct and study the dispersion equations that describe the spectra of normal waves with various kinds of wave motion symmetry in a cross section of the waveguides. We compute the dispersion curve diagrams of symmetric waves for several directions of propagation in the plane of a plate and for the axial direction of a prismatic body of monocrystal Rochelle salt. Four figures. Bibliography: 4 titles Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 90–97.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.     

Scattering of guided SH-wave by a partly debonded circular cylinder in a traction free plate

A solution of the scattering problem of guided SH-wave by a partly debonded circular cylinder centered in a traction free plate has been set up. The plate is divided up into three regions with two imaginary planes perpendicular to the plate walls. In the central region where the partly debonded cylindrical obstacle is posted, the wave field is expanded into the cylindrical wave modes and Chebyshev polynomials. In the other two exterior regions the fields are expanded into the plate wave modes. A system of fundamental equations to solve the problem is obtained according to the traction free boundary condition on the plate walls and the continuity condition of the traction and the displacement across the imaginary planes. The approximate numerical method termed mode-matching technique is used to construct a matrix equation to obtain curves showing the coefficient of reflection and transmission versus the ratio of the cylinder’s radius to the plate’s half-thickness and the angular width of the debonded region. A comparison of the numerical results between the welded interface condition and the debonded interface condition is made, and the results are discussed.     

Deformation of a composite plate on an elastic foundation by local loads

Bending of an elastic annular composite plate with a light filler lying on an elastic foundation is considered. The plate is subjected to local loads. To describe the kinematics of the package, asymmetric across its thickness, the hypotheses of broken normal is accepted. The reaction of foundation is described based on the Winkler model. A system of equilibrium equations is constructed, and its exact solution in displacements is found. Numerical solutions for a metal-polymer sandwich plate are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 109–120, January–February, 2007.     

Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

This paper is concerned with a diffusive and cooperative Lotka–Volterra model with distributed delays and nonlocal spatial effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185–232), sufficient conditions are established for the existence of traveling wave front solutions connecting the zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka–Volterra system with discrete delays.     

Steady-state vibrations of a thin two-layer plate with delamination

We consider the problem of harmonic vibrations of a thin two-layer plate with horizontal crack. The problem is solved with the help of the null-field approach. The influence of the shape of the crack contour on the amplitude-frequency characteristics of plate vibrations is investigated. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 83–89, April–June, 1998.     

Wave Breaking of the Camassa–Holm Equation

This paper gives a new and direct proof for McKean’s theorem (McKean in Asian J. Math. 2:867–874, 1998) on wave breaking of the Camassa–Holm equation. The blow-up profile is also analyzed.     

The passage of a bending wave through an inhomogeneity in an absolutely rigid rib backing an elastic plate

The wave properties of a system consisting of an elastic plate and an absolutely rigid infinite rib with a defect on a segment are examined. An elastic inclusion and a gap are two kinds of defects under study. The Green's function method is applied to the diffraction problem and transforms it to singular integro-differential equations on an interval. For the case of short defects, the nonresonance and resonance asymptotics of the scattering pattern are obtained. These results show that the coefficient of penetration for a gap is much larger than that for an elastic inclusion if the frequency is nonresonant. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210. 1994, pp. 22–29. Translated by I. V. Andronov.     

Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

This paper is concerned with a diffusive and cooperative Lotka–Volterra model with distributed delays and nonlocal spatial effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185–232), sufficient conditions are established for the existence of traveling wave front solutions connecting the zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka–Volterra system with discrete delays. Supported by the NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.     

Transfinite Thin Plate Spline Interpolation

Duchon’s method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e., interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo–Levi type to construct a semicardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo–Levi boundary conditions is in fact a thin plate spline, i.e., it minimizes a Duchon type functional. The construction and variational characterization of the Beppo–Levi polysplines are based on the analysis of a new class of univariate exponential ℒ-splines satisfying adjoint natural end conditions.     

Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of nonlinear viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We will prove that the energy associated to the system is unbounded. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. The key ingredient in the proof is a method used in Vitillaro (Arch Ration Mech Anal 149:155–182, 1999) and developed in Said-Houari (Diff Integr Equ 23(1–2):79–92, 2010) for a system of wave equations, with necessary modification imposed by the nature of our problem.