Cubically Nonlinear Versus Quadratically Nonlinear Elastic Waves: Main Wave Effects

2005

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 coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly Keywords: nonlinear continuum mechanics, rigorous approach, nonlinear hyperelastic cylindrical waves, quadratically nonlinear wave equations, geometrical and physical nonlinearities, plane strain state Introduction. Preliminary Definitions.Nonlinear wave equations in cylindrical coordinates have long been of interest (see [1, 10, 14-17] for a bibliography on cylindrical waves). The approach discussed below is classical and the corresponding results are new.The formulation of the problem and derivation of the nonlinear wave equation necessarily involve a number of concepts of modern nonlinear continuum mechanics. Let us discuss the most important of them.A real substance is modeled by a material continuum, and then the deformation of this continuum rather than of the original substance (material) is studied.To study the motion (deformation) of a continuum, material particles are first identified or parametrized [2]. To this end, a coordinate system is chosen and the configuration of the body at some initial time is defined. The identification is performed in this configuration.The most common choice [2, 8] is a comoving Lagrangian coordinate system with the following coordinate

Section: Derivation Of Nonlinear Wave Equationsmentioning

confidence: 99%

Section: Derivation Of Nonlinear Wave Equationsmentioning

confidence: 99%

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

2005

“…The nonlinear theory of waves in materials is a developing division of mechanics. The recent advances in this theory are described in [2,6,[8][9][10]16]. Waves with curvilinear fronts have been studied much less than waves with plane fronts.…”

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confidence: 99%

“…Two types of nonlinearity of the five ones on the right-hand side of Eq. (10) have well been studied in nonlinear physics: the type ( ) u r 2 describes nonlinear polarization in wave optics and the type u u r rr r r , , describes the nonlinear interaction of plane displacement waves in acoustics [3,4,6,8,9]. This experience may be useful for the analysis of Eqs.…”

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confidence: 99%

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

2005

The propagation and interaction of hyperelastic cylindrical waves are studied. Nonlinearity is introduced by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. To analyze wave propagation, an asymptotic representation of the Hankel function of the first order and first kind is used. The second-order analytical solution of the nonlinear wave equation is similar to that for a plane longitudinal wave and is the sum of the first and second harmonics, with the difference that the amplitudes of cylindrical harmonics decrease with the distance traveled by the wave. A primary computer analysis of the distortion of the initial wave profile is carried out for six classical hyperelastic materials. The transformation of the first harmonic of a cylindrical wave into the second one is demonstrated numerically. Three ways of allowing for nonlinearities are compared Keywords: nonlinear continuum mechanics, rigorous approach, nonlinear hyperelastic cylindrical waves, distortion, initial wave profile, second harmonic generation Introduction. The nonlinear theory of waves in materials is a developing division of mechanics. The recent advances in this theory are described in [2,6,[8][9][10]16]. 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. Note that the assumption of quadratic nonlinearity is acceptable enough, and most problems on nonlinear waves in materials are solved under this assumption [3,4,6,13,14].Let us use the last three equations from [18], which represent an axisymmetric (depending only on the coordinate r and having Oz as a symmetry axis) state of the continuum. As indicated in [18], this state is typical of the classical cylindrical wave or Volterra distortions in a hollow cylinder. Therefore, we will further use these equations to analyze the evolution of a hyperelastic cylindrical wave.1. Wave Equations. Recall that we use the cylindrical coordinate system θ 1 = r, θ 2 = ϑ, θ 3 = z.As indicated earlier, we are considering an axisymmetric configuration (state) with Oz as a symmetry axis.

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confidence: 99%

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Axisymmetric and Other States

2005

A rigorous approach of nonlinear continuum mechanics is used to derive nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. Nonlinearity is introduced by means of metric coefficients, the Cauchy-Green strain tensor, and the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. Quadratically nonlinear wave equations are derived for three states (configurations): (i) axisymmetric configuration dependent on the radial and axial coordinates and independent of the angular coordinate, (ii) configuration dependent on the angular coordinate, and (iii) axisymmetric configuration dependent on the radial coordinate. 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. The approach is based on the concepts of modern nonlinear continuum mechanics. Nonlinearity was introduced by means of metric coefficients, the Cauchy-Green strain tensor, and the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. A configuration (state) of an elastic medium dependent on the coordinates r, ϑ and independent of the coordinate z was analyzed. This configuration is often called plane-strain state. For this case, the corresponding equations of motion and analytical expressions for the stress tensor in terms of the deformation gradient were derived. Four ways of introducing physical and geometrical nonlinearities to the wave equations were analyzed. For one of the ways, the nonlinear wave equations were written explicitly.The present paper extends the analysis performed in [6] to several partial configurations (states) omitted there. State I. It is an axisymmetric configuration dependent on the coordinates r and z and independent of the coordinate ϑ. The Oz-axis is the axis of symmetry. This state is typical of, for example, a longitudinal torsional wave propagating along a cylinder.State II. It is a configuration that depends only on the angular coordinate ϑ. Its axis of symmetry is Oz. This state is typical of, for example, a transverse torsional wave propagating along a cylinder.State III. It is an axisymmetric configuration that depends only on the coordinate r. Its axis of symmetry is Oz. This state is typical of, for example, a classical cylindrical wave or Volterra translational distortions in a hollow cylinder.As in [6], we introduce a cylindrical (orthogonal) coordinate system: θ θ θ 1 2 3 = = = r z , , ϑ . In this system, the length of a vec...