Analysis of harmonics propagating in pipes of quadratic material nonlinearity using shell theory

Higher harmonics in pipes of quadratic nonlinear material behavior have been analyzed in this paper. Using shell theory, the mixing of axisymmetric longitudinal waves and torsional waves, and the self-interaction of axisymmetric longitudinal waves, have been investigated. The dispersion curves of longitudinal waves derived from the linear version of the governing equations show excellent agreement with the corresponding curves obtained from thick shell theory and three dimensional theory, presented elsewhere. For torsional waves, only the lowest mode is taken into consideration. Using the perturbation method, analytical expressions for the resonant torsional waves generated by the mixing of longitudinal and torsional waves have been obtained. The resonant waves with difference frequencies propagate in the opposite direction of the corresponding primary wave. The back-propagation effect has potential application for nondestructive evaluation. The nonlinear shell theory is further simplified for applicability to thin pipes, to obtain expressions for the cumulative second longitudinal harmonics generated by self-interaction of longitudinal waves. For this case, the phase-match conditions, which are used to determine phase-match points, are also presented in analytical form.