Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

Abstract

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.

Physical constants for one type of nonlinearly elastic fibrous micro-and nanocomposites with hard and soft nonlinearities

“…(i) a traveling longitudinal plane wave (its evolution is due to the quadratic nonlinearity of the material [18]); (ii) a traveling transverse plane wave (its evolution is due to the cubic nonlinearity of the material [18]); and (iii) a cylindrical wave generated by harmonic oscillations at a cylindrical cavity (its evolution is due to the quadratic nonlinearity of the material [16]). Figures 1-3 show the dependence of the wave amplitude u x t x ( , ) * on the distance x traveled by each of the three waves by a certain time t * in the cases of soft (a) and hard (b) nonlinearities of the composite matrix, with all the other parameters (type of fiber, fiber volume fraction, initial wave amplitude, wave frequency, etc.)…”

“…However, only the first model was studied using a nonlinear theory with the Murnaghan potential [13][14][15][16][17] (model 1n). Therefore, only the effective moduli of the second and third orders were calculated on the assumption that the fiber volume fraction is small (from 1 to 10%).…”

Nonlinear cylindrical waves in Signorini’s hyperelastic material

“…One of the approaches is more logical and follows from Signorini's procedure [3,4,17,18]. He used the nonlinear Almansi strain tensor ε ik (1.6) to express a geometrically nonlinear elastic potential.…”

“…In the recent studies [5,6,[9][10][11][12][13][15][16][17], an approach was proposed to derive the nonlinear wave equations in terms of cylindrical coordinates, and theoretical and numerical analyses were made of nonlinear cylindrical waves in Murnaghan's hyperelastic material. In the present paper, this approach is extended to another hyperelastic model-classical Signorini's model.…”

“…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:…”