Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

Rushchitsky, J. J.

doi:10.1007/s10778-009-0169-8

Cited by 8 publications

(14 citation statements)

References 23 publications

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“…We have derived wave equations to find the second approximation that includes second harmonics of the harmonic wave itself and its attenuating amplitude and is quadratically dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave. It is not unlikely that the subsequent approximations will contain the fourth and eighth harmonics [26][27][28].…”

Section: Theoretical Description Of the Nonlinear Elastic Deformationmentioning

confidence: 99%

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

Rushchitsky

Khotenko

2011

Int Appl Mech

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The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave Keywords: nonlinear elastic two-dimensional plane problem, Murnaghan model, nonlinear Rayleigh wave, second-order wave equations, second harmonicIntroduction. The Rayleigh wave [21] is classed among surface waves and its linear approximation is well studied [8,9].Linear Rayleigh waves are addressed in many fundamental theoretical studies [1, 2, 10, 13-15] which date back to 1885 when the theoretical solution for surface waves was first obtained [21]. Various approaches and methods were used in these studies. Currently, the Stroh formalism [30] is a well-tested method for studying surface waves.Later, Rayleigh waves were studied theoretically for various models (different from the linear elastic model). For example, Rayleigh waves in materials with initial (residual) stresses were analyzed in [3]. The microstructural elastic mixture model was used for this purpose in [7,22].Rayleigh waves were experimentally studied not only by simulation in laboratory [19], but also in natural conditions during earthquakes because Rayleigh waves are used for seismic monitoring [12]. Experimental studies of acoustic surface waves attract interest due to the wide range of applications of their properties.A quite extensive bibliography on nonlinear Rayleigh waves can be found in the old (1974) work [31] by Canadian scientists, the first of the studies [16-18, 32, 33] (1992-2008) by Zabolotskaya with coauthors, and Sgoureva-Philippakos' dissertation [29] (1999).Since nonlinearity in elasticity problems can be described in many ways, it makes sense to note that the constitutive equations in the first of the above-cited publications are a quadratic relationship among stresses, displacements, and displacement gradient in the form of an infinite series of "harmonics" of a linear Rayleigh wave and are solved numerically. The second publication (and the subsequent series of publications) follows the Hamiltonian formalism which incorporates the same quadratic nonlinearity, whereas the constitutive equations used in the third publication include the nonlinear John potential.

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Section: Theoretical Description Of the Nonlinear Elastic Deformationmentioning

confidence: 99%

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

Rushchitsky

Khotenko

2011

Int Appl Mech

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“…The more detailed analysis of the solution of the nonlinear wave equation in [6] leads to the unexpected result that the fourth harmonic becomes predominant when the third approximation is taken into account.…”

Section: Solving the Nonlinear Wave Equation (1) By The Perturbation mentioning

confidence: 99%

“…It is this method that was use in [6] to analyze the wave equation (1) (i.e., a longitudinal harmonic plane wave in a hyperelastic material described by the classical Murnaghan model, which is quadratic nonlinear, was considered). As indicated in [6], 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 [2,3,5,10].…”

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Generation of the second, fourth, eighth, and subsequent harmonics by a quadratic nonlinear hyperelastic longitudinal plane wave

2010

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The perturbation (small-parameter) method is used to obtain the first three approximations for the problem of a harmonic longitudinal plane wave propagating in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The subsequent approximations are discussed. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that the third approximation corrects the prediction of the evolution of the initial wave profile. Which of the harmonics dominates depends on the distance traveled by the wave: the second harmonic is generated first, then it transforms into the fourth harmonic, and finally, as the distance increases, the eighth harmonic shows Keywords: harmonic longitudinal plane wave, quadratic nonlinear wave equation, perturbation method, generation, fourth harmonic, eighth harmonic, subsequent harmonicsIntroduction. This paper deals with the well-known physical phenomenon of second-or third-harmonic generation by a harmonic wave propagating in a nonlinear medium of arbitrary nature (water, gas, thermomechanical, optical, or acoustic medium, etc.). The essence of the phenomenon is often and very effectively demonstrated in a second-harmonic generation experiment where an optical wave propagating in an ammonium dihydrogen phosphate crystal changes from ruby light (red) to ultraviolet [26]. More abstractly, a harmonic wave enters a quadratic or cubic nonlinear medium, interacts with itself while propagating, and leaves the medium as a second or third harmonic, respectively. Thus, experiments confirm that a harmonic wave does generate a second harmonic in a quadratic nonlinear medium [1-3, 8, 26]. This phenomenon is theoretically described as a property of the solution of the corresponding nonlinear wave equation. The generation of the second harmonic by a longitudinal harmonic plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model was described in [5, 17] by a method traditional for nonlinear optics (Van der Pol's method of slowly varying amplitudes). Let us outline this approach.The following nonlinear wave equation is known [2, 3, 5, 10, 17] to describe the motion of a quadratic nonlinear longitudinal plane wave:

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“…Various problem formulations can be used to consider this wave (plane strain state, axisymmetric state, etc.) [15][16][17].We choose the axisymmetric case. The first two approximations of such a problem were analyzed in [16,19,20].…”

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confidence: 99%

“…In the former case, the solution was restricted to the first two (zero-order (linear) and first-order) approximations because of the following two reasons [3,5]: (i) the first-order approximation coincides with the solution of the evolutionary equation found with the method of slowly varying amplitudes; (ii) the basic nonlinear wave phenomena described by the first-order solution and the solution of the evolutionary equation are in agreement with experimental observations. The analysis of the effect of the subsequent approximations is still an open issue if the publications [17,18] are regarded as just the beginning of such an analysis.The present paper suggests a general approach to the analysis of many approximations and obtains and comments the exact analytical representation of the solution in terms of Hankel functions found using the first two approximations. …”

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Analyzing the propagation of a quadratic nonlinear hyperelastic cylindrical wave

Rushchitsky

2011

Int Appl Mech

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The perturbation method (small-parameter method) is used to analyze the propagation of a cylindrical wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The exact representation of the first-order solution in terms of squared Hankel functions of zero and first orders and their product is derived. A simplification of the new solution is considered Keywords: cylindrical wave, quadratic nonlinear wave equation, Murnaghan model, perturbation method, first two approximations, exact representationIntroduction. The subject of the present study is a cylindrical wave [2, 5, 10] in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. Various problem formulations can be used to consider this wave (plane strain state, axisymmetric state, etc.) [15][16][17].We choose the axisymmetric case. The first two approximations of such a problem were analyzed in [16,19,20]. Since the zero-order (classical linear) approximation is expressed in terms of the Hankel function that can be represented as power series, such series expansions were used in [16,19,20] to represent the first nonlinear approximation in a form convenient for comparison with the first nonlinear approximation for a longitudinal plane wave.It should be noted that plane and cylindrical waves in Murnaghan materials were earlier analyzed using methods well 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) [3,9]. In the former case, the solution was restricted to the first two (zero-order (linear) and first-order) approximations because of the following two reasons [3,5]: (i) the first-order approximation coincides with the solution of the evolutionary equation found with the method of slowly varying amplitudes; (ii) the basic nonlinear wave phenomena described by the first-order solution and the solution of the evolutionary equation are in agreement with experimental observations. The analysis of the effect of the subsequent approximations is still an open issue if the publications [17,18] are regarded as just the beginning of such an analysis.The present paper suggests a general approach to the analysis of many approximations and obtains and comments the exact analytical representation of the solution in terms of Hankel functions found using the first two approximations. Perturbation (Small-Parameter) Method Applied to the Nonlinear Wave Equation for a Cylindrical Wave.We start with the nonlinear wave equation for an axisymmetric state (depending only on coordinate r and having the axis of symmetry Oz) of a continuum (this state is characteristic of a classical cylindrical wave) with physical and geometrical nonlinearities [15,16]:

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Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

Cited by 8 publications

References 23 publications

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

Generation of the second, fourth, eighth, and subsequent harmonics by a quadratic nonlinear hyperelastic longitudinal plane wave

Analyzing the propagation of a quadratic nonlinear hyperelastic cylindrical wave

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