Experimental investigation of the correspondence between quasistatic tension and longitudinal wave propagation in a polyethylene rod

The propagation of longitudinal waves created by the impact of a rigid mass against a polyethylene rod of finite length has been investigated. The experimental procedure is described and oscillograms of the process are presented. The relaxation spectrum is calculated from the experimental data on the mechanical behavior of the material over a broad range of strain rates. The results of the experiments are compared with a numberical solution of the problem of the impact of a rigid mass against a rod whose material is characterized by a relaxation spectrum.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 564–567, May–June, 1971.     

On the longitudinal wave propagation in a viscoelastic cylinder

Résumé L'auteur étudie le calcul des déplacements d'un matériau visco-élastique linéaire dans un cylindre d'étendue finie qui sont produits par des oscillations longitudinales. Il montre qu'avec une approximation pour le cas des basses fréquences, on arrive au même résultat que celui obtenu de façon rigoureuse par un autre auteur. De plus, il montre que cette méthode peut être utilisée dans la region de hautes fréquences.     

Experimental study of the propagation of perturbation waves in a long polyethylene rod

On the basis of the "kinematic method" of registration of the wave process [5] the author has investigated the propagation of longitudinal waves in a long polyethylene rod. The results of one of a series of experiments performed under identical conditions are presented in graph form. These results are used to justify a hypothesis concerning the form of the velocity influence function, and the applicability of the viscoelastic-viscoplastic [4] law of deformation to the case of dynamic extension of a polyethylene rod is investigated. The deformation characteristics are found to deviate from the usual conception of the influence of velocity on the physico-mechanical properties of materials.Mekhanika Polimerov, Vol. 2, No. 4, pp. 565–573, 1966     

Similarity representation of wave propagation in a nonlinear viscoelastic rod on a group theoretic basis

Similarity representation of wave propagation in a nonlinear viscoelastic rod, subjected to a velocity impact is constructed. By the use of a multiparameter dimensional group of transformations, under which the system of basic equations and auxiliary conditions are invariant, similarity transformations are obtained which are used to construct the similarity representation in the form of an ordinary nonlinear boundary value problem of the original system, consisting of a set of simultaneous nonlinear partial differential equations and the association boundary and initial conditions.     

1D wave propagation in a rod: analytic treatment for non-trivial boundary conditions

An analytic solution method for the one-dimensional wave propagation problem in a rod based on the Laplace transform is applied to a simple stamping model. While the rod is mounted at its lower end to a spring, it is hit at its upper end by an impacting mass, the separation of which can be correctly treated by the method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of wave propagation velocities in fluid mixtures

In order to determine the wave propagation velocities in fluid mixtures, the mixtures are approximated by block structures. These structures consist of identical cells containing eight blocks. The blocks may be filled with different fluids. In block structures, the passage to the limit is carried out under the conditions that the sizes of blocks tend to zero but the relative sizes of blocks remain constant. In the general case, the average wave field satisfies the equations of anisotropic fluids. Two special cases of mixtures of two fluids are considered. In the first case, both fluids are intermixed completely. In the second case, there are periodic inclusions of one fluid into the other. In both cases, the fluid mixtures are homogeneous and isotropic, and formulas for the velocities are obtained. These formulas determine the dependence of the velocities on the percent composition and the parameters of two mixed fluids. The velocity of propagation in the fluid mixture does not exceed the greatest velocity but may be less than the least velocity in mixed fluids. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 149–162.     

Wave propagation in a long polymer rod

The author examines the propagation of longitudinal waves in a semi-infinite rod whose material possesses elasticoviscous-plasticoviscous properties. Time conditions are introduced to take flow retardation into account, together with a hypothesis concerning the rate influence function. The Laplace transform method is used to obtain a solution close to the wave front; the solution for the plasticoviscous stage is obtained by a graphoanalytical method.Mekhanika Polimerov, Vol. 2, No. 2, pp. 253–262, 1966     

Investigation of wave propagation in models of cracked media

An effective two-phase model of wave propagation in a cracked medium in a half-space whose boundary is perpendicular to the cracks is investigated. The boundary conditions for this model are established in the case of the medium being in contact with other fluid and elastic media.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 116–127, 1989.     

Inverse problems of recovering external sources in the equation of longitudinal wave propagation

The inverse problems are considered for the equation of longitudinal wave propagation under the overdetermination conditions of the final and integral types. The main purpose of the study is to prove the existence of regular solutions of inverse problems of recovering the unknown external sources in addition to the solution. One of the suggested approaches bases on reducing the inverse problem to an integrodifferential equation.     

Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix

This paper reported the result of an investigation into the effect of magnetic field on wave propagation in carbon nanotubes (CNTs) embedded in elastic matrix. Dynamic equations of CNTs under a longitudinal magnetic field are derived by considering the Lorentz magnetic forces. The results obtained show that wave propagation in CNTs embedded in elastic matrix under longitudinal magnetic field appears in critical frequencies at which the velocity of wave propagation drops dramatically. The velocity of wave propagation in CNTs increases with the increase of longitudinal magnetic field exerted on the CNTs in some frequency regions. The critical/cut-off frequency increases with the increase of matrix stiffness, and the influence of matrix on wave velocity is little in some frequency regions. This investigation may give a useful help in applications of nano-oscillators, micro-wave absorbing and nano-electron technology.     

Ultrasonic wave propagation in a polymerizing medium

It is shown that by monitoring the propagation velocities of longitudinal and shear ultrasonic waves in a polymerizing system it is possible to keep a continuous track on the conversion of the monomer, estimate the effect of various technological factors — sensitizing additives, dose rate, absorbed dose — and trace the behavior of the dynamic moduli of elasticity and the dynamic Poisson's ratio of the polymeric material. Relations between the wavelength and the transverse dimension of the polystyrene specimens that satisfy the conditions for determining the speed of ultrasound in an infinite medium and in a rod are obtained.     

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

Waves induced by the longitudinal impact of a rod against a stepped rod in contact with a rigid barrier

The longitudinal elastic central impact of a rod system, consisting of a uniform rod having a pre-impact velocity against a stepped rod which is in a state of rest and interacts by means of unilateral constraints with a rigid barrier, is modelled.     

Two-dimensional wave propagation in a generalized elastic solid

The object of the present study is to investigate the propagation of two-dimensional waves in a weakly nonlinear and weakly dispersive elastic solid. The reductive perturbation method is directly applied to a Lagrangian whose Euler–Lagrange equations give the field equations for a quadratically nonlinear elastic medium with higher order gradients. In the long-wave approximation, it is shown that the long-time behavior of the two transverse waves is governed by the two coupled modified Kadomtsev–Petviashvili (CMKP) equations. Depending on the choice of the direction of perpendicular dynamics, various forms of the CMKP equations are obtained. Some special solutions are also presented for a simplified form of the CMKP equations.     

The longitudinal deformation of a rod in friction interaction with a compressing body

The deformation of a rod, confined in a fixed external housing, is considered. The friction forces in the contact surface are related to the deformation of the rod by a power relation. A wide range of variation of the friction parameter and the preliminary clearance parameter with which the rod is inserted into the housing is investigated and the characteristic features of the stress and strain distributions are revealed. The dissipation of energy due to friction and the formation of a hysteresis loop in the dependence of the stresses in the loaded end face on its displacement are considered. The problem is solved in a quasistatic formulation. Analytical relations are found for a number of important cases. Other results are obtained by numerical integration of the initial differential problem.     

Inverse problem of wave propagation in a random medium

The problem of finding the propagation velocity of waves is studied if this velocity is a random function of position.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 63–70, 1979.     

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.     

Determination of the physicomechanical characteristics of viscoelastic materials from the velocity of propagation and damping of longitudinal and transverse vibrations in a thin rod

The author examines the propagation of longitudinal and transverse waves in a thin viscoelastic rod. Relations are obtained for determining the components of the complex moduli and and Poisson's ratio from the propagation velocities and decrements of the longitudinal and transverse vibrations, together with the static and relaxational moduli of elasticity, the relaxation time, and the equivalent viscosity for materials whose law of deformation corresponds to a standard linear solid.Mekhanika Polimerov, Vol. 2, No. 3, pp. 453–460, 1966     

Optimal control of longitudinal vibrations of composite rods with the same wave propagation time in each part

We consider longitudinal elastic vibrations of a composite rod and find closedform expressions that describe optimal boundary controls bringing the rod from the quiescent state into a state with given displacement function φ(t) and velocity function ψ(t) in time T. We assume that the wave propagation time through each part of the rod is the same and T is a multiple of that time.     

Nonlinear wave propagation through a stratified atmosphere

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.