2005
DOI: 10.1007/s10778-005-0115-3
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Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

Abstract: A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinat… Show more

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Cited by 27 publications
(35 citation statements)
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“…The nonlinear wave equations describing the propagation of hyperelastic waves have been derived in [6]. This paper demonstrated a rigorous approach, passed over even in the textbooks [7,9,10,16], to deriving nonlinear wave equations in cylindrical (orthogonal) coordinates.…”
mentioning
confidence: 98%
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“…The nonlinear wave equations describing the propagation of hyperelastic waves have been derived in [6]. This paper demonstrated a rigorous approach, passed over even in the textbooks [7,9,10,16], to deriving nonlinear wave equations in cylindrical (orthogonal) coordinates.…”
mentioning
confidence: 98%
“…Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. Six different systems of wave equations are written Keywords: nonlinear continuum mechanics, rigorous approach, nonlinear hyperelastic cylindrical waves, quadratically nonlinear wave equations, geometrical and physical nonlinearities, axisymmetric stateThe nonlinear wave equations describing the propagation of hyperelastic waves have been derived in [6]. This paper demonstrated a rigorous approach, passed over even in the textbooks [7,9,10,16], to deriving nonlinear wave equations in cylindrical (orthogonal) coordinates.…”
mentioning
confidence: 99%
“…In the present paper, this approach is extended to another hyperelastic model-classical Signorini's model. The first part of the algorithm described in [5,6,[9][10][11][12][13][15][16][17] remains the same. It includes the following steps:…”
mentioning
confidence: 99%
“…Waves with curvilinear fronts have been studied much less than waves with plane fronts. Therefore, analysis of nonlinear cylindrical waves appears to be appropriate as expanding our knowledge of waves in materials.In the previous publications [17,18], an attempt was made to write, based on the general principles of the nonlinear mechanics of hyperelastic continua, the nonlinear equations of motion in cylindrical coordinates, with nonlinearity described by the Murnaghan potential and only quadratic nonlinearity retained in all the analytical representations of mechanical (displacement, strain, and stress) fields. Finally, all the equations of motion written in terms of displacements turned out to be quadratically nonlinear.…”
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confidence: 99%
“…Let us preserve inaccuracy adopted in(17) and neglect in (19) the terms that include the factors ( This is admissible for travel distances k r L = − 30 100.It is convenient to substitute(15) and(19)into the nonlinear right-hand side to produce Expression (20) (the coefficient ( / )( the approximation of the Hankel function(16). In a sense, the expression (20) may be considered a second harmonic, provided that the first one has the form(16).…”
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confidence: 99%