Nonlinear cylindrical waves in Signorini’s hyperelastic material

This paper presents a rigorous procedure, based on the concepts of nonlinear continuum mechanics, to derive nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. Nonlinearity is introduced by the Signorini potential and represents the quadratic and cubic nonlinearities of all governing relations with respect to displacements. A configuration (state) of the elastic medium dependent on the coordinate r and independent of the coordinates ϑ and r is analyzed. Analytic transformations are used to go over from the Eulerian to the Lagrangian description of nonlinear deformation and from the invariants of the Almansi finite-strain tensor to the invariants of the Cauchy-Green finite-strain tensor. Possibly for the first time in the 60 years the Signorini model has been in existence, a nonlinear wave equation is derived for it and the strain and true-stress tensors are analytically expressed in terms of the deformation gradient. The quasilinear Signorini model and four cases of incorporating material and geometrical nonlinearities into the wave equation are discussed