Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

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 __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 73–82, July 2005.