Nonlinear plane waves in Signorini's hyperelastic materials

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 plane hyperelastic waves. The nonlinearity is introduced by the Signorini potential and represents the quadratic nonlinearity of all governing relations with respect to displacements.Aconfiguration (state) of the elastic medium dependent on the abscissa 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. Nonlinear wave equations describing the propagation of plane longitudinal and transverse waves in Signorini’s materials are derived, and the strain and true-stress tensors are analytically expressed in terms of the deformation gradient. These wave equations are compared with those based on the Murnaghan model. Their similarities and differences are indicated. It is shown that the new Signorini constant can be identified from the Lamé and Murnaghan constants