Nonlinear interaction of torsional and longitudinal guided waves in   hyperelastic cylinders
  The nonlinear forcing terms for the wave equation in general curvilinear coordinates are derived based on a hyperelastic material. The expressions for the nonlinear part of the first Piola-Kirchhoff stress are specialized for axisymmetric torsional and longitudinal fundamental wave fields in a cylinder. The matrix characteristics of the nonlinear forcing terms and secondary mode wavestructures are manipulated to analyze the higher harmonic generation due to the guided wave mode self interactions and mutual interactions. It is proven that both torsional and longitudinal mode secondary fields can be cumulative by specific type of guided wave mode interactions. A method for the selection of preferred fundamental excitations that generate strongly cumulative higher harmonics is formulated, and described in detail for second harmonic generation. Hyperelastic finite element models are built to simulate second harmonic generation by T(0,3) and L(0,4) modes. A linear increase of the modal amplitude ratio A2/A1^2 over the propagation distance is observed for both cases, which indicates mode L(0,5) is effectively generated as cumulative second harmonics. 
Yang Liu, Ehsan Khajeh, Cliff J. Lissenden, Joseph L. Rose
physics.class-ph