Amplitude modulation of nonlinear waves in a thick elastic tube filled with an inviscid fluid

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thick tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the reductive perturbation technique the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a nonlinear Schrödinger(NLS) equation. The range of modulational instability of the monochromatic wave solution with the initial deformation, material and geometrical characteristics is discussed for some elastic materials.     

Nonlinear wave modulation in a prestressed thin elastic tube filled with an inviscid fluid

In the present work, employing the nonlinear equations of anincompressible, isotropic and elastic thin tube and the approxiamteequations of an incompressible inviscid fluid, and then utilizingthe reductive perturbation technique the amplitude modulationof weakly nonlinear waves is examined. It is shown that equation.The result is compared with some previous works on the samesubject. The modulation instablity of the monochromatic wavesolution is discussed for some elastic materials and initialdeformation.     

Propagation of high-frequency harmonic elastic waves from an axially symmetric cavity

An asymptotic ray method is developed for studying propagation of high-frequency harmonic elastic waves from a smooth, convex, axially symmetric cavity. The dynamic stresses near a prolate spheroidal cavity, to whose surface is applied a normal, angle-independent load, are investigated as a specific numerical example. It is shown that account of three terms of the ray series makes it possible to find quite accurate solutions even when the wave length is comparable with the cavity sizes.Translated from Teorr eticheskaya i Prikladnaya Mekhanika, No. 18, pp. 87–95, 1987.     

Propagation of elastic waves in certain composites

Conclusions 1. Relationships have been obtained for determining nine elastic characteristics of orthotropic composite materials from the properties of the starting components and the assigned reinforcement scheme.2. Formulas are given for calculating the propagation velocity of three types of elastic flat waves for an arbitrary direction in one of the planes of elastic symmetry of a uniform orthotropic material.3. It has been shown that the velocity of the first arrival of a packet of ultrasonic vibrations which is recorded in an experiment is equal to the velocity of motion of the wave front in a limitless medium even for rather thin (5–10 mm) fiberglass-plastic specimens with unidirectional or cross-reinforced schemes.4. The dependences of elastic properties and rates of propagation of elastic vibrations on direction which are calculated theoretically from the properties of the starting components and the reinforcement scheme agree satisfactorily with experimental results.Translated from Mekhanika Polimerov, No. 3, pp. 531–536, May–June, 1978.     

Propagation of elastic waves in an infinite plate of cylindrically aelotropic material

Zusammenfassung Die Abhängigkeit der Wellengeschwindigkeit von der Wellenlänge elastischer Wellen, die sich in einer unendlichen Platte mit zylindrischer Aelotropie ausbreiten, ist auf numerischem Weg aus der vorgängig hergeleiteten «Frequenzgleichung» ermittelt worden. Die Ergebnisse sind denen des isotropen Falles ähnlich.     

Propagation of waves in a micropolar elastic layer with stretch immersed in an infinite liquid

The effect of liquid on the propagation of waves in a micropolar elastic layer with stretch has been investigated. The frequency and wave velocity equations for symmetric and antisymmetric vibrations are derived. Propagation of monochromatic waves in a micropolar elastic layer with stretch is discussed. Results of this analysis reduce to those without stretch.     

Torsional waves in an infinite elastic solid containing a penny-shaped crack

Zusammenfassung In dieser Arbeit wird das axialsymmetrische Problem der Torsionswellen in einem unendlichen, isotropen und homogenen elastischen Körper mit einem kreisförmigen Schlitz betrachtet. Es werden zwei Fredholmsche Integralgleichungen zweiter Art für das gestreute Feld aufgestellt, welche sich bei kleinen Frequenzen zur iterativen Behandlung eignen. Die erste davon ist insofern vorteilhafter, als sie die Verschiebungs- und die Schubspannungsverteilung auf der Schlitzoberfläche direkt liefert und bei iterativer Behandlung viel weniger Arbeit kostet als die zweite. Es werden Ausdrücke für das Verschiebungs- und das Spannungsfeld an der Schlitzoberfläche in Form von Potenzreihen des Frequenzfaktors erhalten. Ebenso wird ein Ausdruck für den Spannungsintensitätsfaktor als Potenzreihe von gewonnen.     

Propagation of plane waves in microstretch fluid

The propagation of plane harmonic waves are studied in a microstretch fluid medium. It is found that five basic waves can propagate at distinct speeds in an infinite linear homogeneous isotropic microstretch fluid. Out of these five waves, one is a longitudinal micro-rotational wave, two are coupled longitudinal waves and remaining two are coupled transverse waves. The longitudinal micro-rotational wave travels independently and is not influenced by the microstretching of the medium, while the coupled longitudinal waves arise due to the presence of microstretching and coupled transverse waves arise due to the presence of micro-rotation in the medium. Speed of propagation of all the waves are found to be complex valued and dispersive at low frequency, but almost non-dispersive at high frequency. Due to complex valued speeds of propagation, all the waves are attenuating but differently. Coupled sets of longitudinal waves reduce to a longitudinal wave of micropolar fluid in the absence of microstretching. Reflection phenomena of a set of coupled longitudinal waves incident obliquely at the free surface of a microstretch fluid half-space has been investigated. Closed formulae for the reflection coefficients are presented and computed numerically for a particular medium. The real and imaginary parts of the complex speeds of all the waves and their corresponding attenuation coefficients have also been studied numerically and depicted graphically against frequency parameter.     

Mathematical modelling and study of the consolidation of an elastic saturated soil with an incompressible fluid by FEM

The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the stability of the method as well as the estimation of the error by using semi-discretization in time. Finally, we then solved this one by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data.     

Propagation of quasielastic Rayleigh waves in an inhomogeneous,anisotropic, elastic medium

The propagation of modulated oscillations along the smooth surface free from normal stresses of an inhomogeneous, anisotropic, elastic body with parameters which vary little over a characteristic wavelength is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 234–245, 1979.The author thanks the participants of the seminar on the propagation and diffraction of waves of the Leningrad Branch of the Mathematics Institute (LOMI), in particular, V. S. Buldyrev and V. M. Babich, for fruitful discussions and their attention to the work.     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

The contact of a solid with an elastic half-space through a thin coating

More accurate equations of the deformation of thin plates, which are more convenient for solving contact problems for bodies with coatings and containing, as a special case, the equations of all known applied theories, are derived by an asymptotic analysis of the first fundamental problem of the theory of elasticity. The equations of the deformation of thin-walled elastic bodies are classified, their qualitative correspondence to the equations of the theory of elasticity is clarified, and the forms of the features that arise along the shift lines of the boundary conditions in the corresponding contact problems are established. A criterion for selecting approximate models to describe the properties of the coatings depending on the geometrical and mechanical characteristics of the coating and the substrate and also on their degree of adhesion is given.     

Propagation of torsional waves in a circular elastic rod

Zusammenfassung Das Problem eines unendlichen kreiszylindrischen Halbstabes, dessen Endquerschnitt mit beliebigen Torsionsspannungen belastet ist, wird auf Grund der strengen dreidimensionalen Elastizitätstheorie gelöst. Die Lösungsmethode besteht aus einer Integraltransformationstechnik mit asymptotischer Inversion für eine frequenzmodulierte Stufenfunktion. Die Ergebnisse zeigen, dass die Gültigkeit eines «dynamischen Saint-Venant-Prinzips» von der betreffenden Frequenzkomponente abhängt.

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This work was initiated while the author was still employed by the IBM Corp. at Endicott, New York.     

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.     

Propagation of three-dimensional waves in elastic and elastic-fluid layered systems

With the aid of the matrix method one investigates three-dimensional processes of wave propagation in layered media. The characteristic matrices are reduced to quasidiagonal form, allowing to reduce the problem to already investigated problems. In addition, one solves two seismology problems which have a practical interest.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 116–129, 1983.