Signorini Cylindrical Waves and Shannon Wavelets

Hyperelastic materials based on Signorini’s strain energy density are studied by using Shannon wavelets. Cylindrical waves propagating in a nonlinear elastic material from the circular cylindrical cavity along the radius are analyzed in the following by focusing both on the main nonlinear effects and on the method of solution for the corresponding nonlinear differential equation. Cylindrical waves’ solution of the resulting equations can be easily represented in terms of this family of wavelets. It will be shown that Hankel functions can be linked with Shannon wavelets, so that wavelets can have some physical meaning being a good approximation of cylindrical waves. The nonlinearity is introduced by Signorini elastic energy density and corresponds to the quadratic nonlinearity relative to displacements. The configuration state of elastic medium is defined through cylindrical coordinates but the deformation is considered as functionally depending only on the radial coordinate. The physical and geometrical nonlinearities arising from the wave propagation are discussed from the point of view of wavelet analysis. 1. Introduction In this paper, cylindrical waves arising from the nonlinear equation of hyperelastic Signorini materials [1–6] are studied. In particular, it will be shown that cylindrical waves can be easily given in terms of Shannon wavelets. Hyperelastic materials based on Signorini’s strain energy density [7, 8] were recently investigated [1–6, 9, 10], because of the simple form of the Signorini potential, which has the main advantage to be dependent only on three constants, including the two classical Lamé constants . Hyperelastic materials and composites are interesting for the many recent advances both in theoretical approaches and in practical discoveries of new composites, having extreme behaviors under deformation [9]. However, Signorini hyperelastic materials, as a drawback, lead to some nonlinear equations, to be studied in cylindrical coordinates [11–13]. The starting point, for searching the solution of these equations, is the Weber equation, which is classically solved by the special functions of Bessel type. Thus the main advantage of three parameters’ potential is counterbalanced by the Bessel function approximation. It has been recently shown [14] that Bessel functions locally coincide with Shannon wavelets, thus enabling us to represent cylindrical waves by the multiscale approach [9, 15, 16] of Shannon wavelets [14, 17–20]. In this way, Shannon wavelets might have some physical meaning through the cylindrical waves propagation.