Self-excitation of transverse waves in hyperelastic materials (third approximation)

Rushchitskii, Ya. Ya.; Коваленко, А. П.; Saveleva, Ekaterina

doi:10.1007/bf02091358

Int Appl Mech

1996

DOI: 10.1007/bf02091358

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Self-excitation of transverse waves in hyperelastic materials (third approximation)

Ya. Ya. Rushchitskii

А. П. Коваленко

Ekaterina Saveleva

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“…Thus, the effect of the third approximation is yet to be understood. The publications [11,25,26] analyze the second approximation for transverse waves in the special case where the second approximation coincides with the first one, while the third approximation is the first nonlinear approximation.The present paper sets out to derive an analytic expression for the third approximation and to analyze the contributions of the first three approximations to the overall wave pattern. …”

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Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

Rushchitsky

2009

Int Appl Mech

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The perturbation (small-parameter) method is used to analyze the propagation of a harmonic longitudinal plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The three first approximations are obtained, and the contribution of each of them into the wave pattern is analyzed. It is shown that the third approximation somewhat improves the prediction of the evolution of the initial waveprofile: the tendency to generate the second harmonic goes over into the tendency to generate the fourth harmonic Keywords: harmonic longitudinal plane wave, quadratic nonlinear wave equation, Murnaghan model, perturbation method, first three approximationsIntroduction. The subject of the present study is a harmonic longitudinal plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. Such a wave was studied chronologically first in [4] based on the nonlinear theory of elasticity. The results of this analysis are reported not only in journals, but also in some monographs [8,9,11,19]. Initially, longitudinal and transverse harmonic plane waves were studied assuming quadratic nonlinearity. Later, the new approach to the analysis of nonlinear elastic waves developed in the 1960s was extended to nonclassical structural models [7,11,19,21,22] and to cubic nonlinearity [13-15, 19, 25, 26]. Next, the class of waves was extended to rotational, cylindrical, torsional, and other waves [7,15,18,19,23,[28][29][30][31]. Also, the popular Murnaghan's classical nonlinear model was abandoned in favor of the less known, yet promising Signorini's model [15-17, 19, 24].Almost all the above-mentioned studies made use of two methods known in the nonlinear theory of waves: the method of successive approximations (small-parameter method) and the method of slowly varying amplitudes (Van der Pol method). The former was restricted to the first two approximations. The justification of this restriction was based on two arguments: (i) the two approximations coincide with the solution of the evolutionary equation found by the method of slowly varying amplitudes [1, 3, 7-9] and (ii) the nonlinear wave phenomena described by the two approximations and by the solution of the evolutionary equation are observed experimentally [2,6,20,32]. Thus, the effect of the third approximation is yet to be understood. The publications [11,25,26] analyze the second approximation for transverse waves in the special case where the second approximation coincides with the first one, while the third approximation is the first nonlinear approximation.The present paper sets out to derive an analytic expression for the third approximation and to analyze the contributions of the first three approximations to the overall wave pattern. Perturbation (Small-Parameter) Method Applied to the Nonlinear Wave Equation for a Hyperelastic Longitudinal PlaneWave. The motion of a quadratic nonlinear longitudinal plane wave is described by the following nonlinear wave equation [8,9,11,19]:

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Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

Rushchitsky

2009

Int Appl Mech

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Comparative Analysis of the Profile Evolutions of an Elastic Harmonic Wave Caused by the Second and Third Harmonics

Cattani

Rushchitsky

Sinchilo

2004

International Applied Mechanics

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Self-excitation of transverse waves in hyperelastic materials (third approximation)

Cited by 2 publications

References 2 publications

Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

Comparative Analysis of the Profile Evolutions of an Elastic Harmonic Wave Caused by the Second and Third Harmonics

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