Localization of nonlinear waves in elastic bodies with inclusions

Indeitsev, D. A.; Osipova, E. V.

doi:10.1134/1.1776219

Cited by 9 publications

(8 citation statements)

References 5 publications

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“…At first, consider the simplest case p = v 0 δ(τ ) (p F = v 0 ) that corresponds to the initial conditions (9). To estimate asymptotically the integral in the righthand side of (27) for large times τ , we use the method of stationary phase [27,28].…”

Section: Inhomogeneous Non-stationary Problemmentioning

confidence: 99%

Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity

2019

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We consider non-stationary oscillations of an infinite string with time-varying tension. The string lies on the Winkler foundation with a point inhomogeneity (a concentrated spring of negative stiffness). In such a system with constant parameters (the string tension), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the string oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations of the string with slowly time-varying tension using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations based on the finite difference method. The applicability of the analytical formulas was demonstrated for various types of external excitation and laws governing the varying tension. In particular, we have shown that in the case when the S.N. Gavrilov, corresponding author Institute for Problems in Mechanical Engineering RAS, V.O., trapped mode frequency approaches zero, localized lowfrequency oscillations with increasing amplitude precede the localized string buckling. The dependence of the amplitude of such oscillations on its frequency is more complicated in comparison with the case of a one degree of freedom system with time-varying stiffness.

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Section: Inhomogeneous Non-stationary Problemmentioning

confidence: 99%

Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity

2019

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“…We express Ψ ξ from (12) and substitute the result into Equation (10), having previously differentiated thereof for ξ. Neglecting the term εΨ ττ , we finally obtain…”

Section: The First Asymptoticsmentioning

confidence: 99%

“…The nonlinear fourth order equation is a resolving one for the main order of system ( 8)- (10), with respect to the small parameter ε. This equation can be called the generalized Boussinesq-Ostrovsky equation, since it contains dispersion terms specific to the usual and "improved" Boussinesq equation [17], as well as a linear term proportional to W, typical of the Ostrovsky equation [18][19][20][21].…”

Section: The First Asymptoticsmentioning

confidence: 99%

“…It is the very case when computational simulations enable confirming, specifying, or even rejecting original hypotheses [8]. This article focuses on the discussions of some of the problems that arise when modeling a particular class of problems, regarding the propagation of nonlinear waves in the Timoshenko-type thin elastic cylindrical shells [9] interacting with a nonlinear elastic Winkler medium [4,10,11]. This article is arranged as follows.…”

Section: Introductionmentioning

confidence: 99%

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Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

et al. 2021

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The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.

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“…This obviously relates to some facets of our problem. Note also that the phenomenon of trapped modes of oscillations also partake of the same class of dynamical phenomena (see, e.g., Indeitsev and Osipova [6]). …”

Section: Introductionmentioning

confidence: 98%

Elements of study on dynamic materials

2010

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As a preliminary study to more complex situations of interest in small-scale technology, this paper envisages the elementary propagation properties of elastic waves in one-spatial dimension when some of the properties (mass density, elasticity) may vary suddenly in space or in time, the second case being of course more original. Combination of the two may be of even greater interest. Toward this goal, a critical examination of what happens to solutions at the crossing of pure space-like and time-like material discontinuities is given together with simple solutions for smooth transitions and numerical simulations in the discontinuous case. The effects on amplitude, speed of propagation, frequency changes and the appearance of a Doppler-like effect are demonstrated although the whole physical system remains linear.

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Localization of nonlinear waves in elastic bodies with inclusions

Cited by 9 publications

References 5 publications

Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity

Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity

Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

Elements of study on dynamic materials

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