The effect of cubic material nonlinearity on the propagation of torsional wave modes in a pipe

The effect of cubic material nonlinearity on the propagation in a pipe of the lowest axially symmetric torsional wave mode has been investigated in this paper. Two cases, one that the material of the whole pipe is nonlinear, and the second that a small segment of the pipe is nonlinear, have been considered. For the first case, a first and a third harmonic have been obtained by the perturbation method. Analytical expressions for the two cumulative harmonics have been derived. The second case leads to a scattering problem. The segment produces nonlinear terms in the equation of motion, which can be regarded as a distribution of body forces. The problem is then reduced to a linear scattering problem. An analytical expression for the backscattered wave can be easily obtained by using the elastodynamic reciprocity theorem. Due to the low amplitude of the backscattered wave, the authors propose to add another higher frequency wave to the primary wave, to increase the total magnitude of the scattered wave. An example that the originally scattered wave is amplified 50 times by selecting proper frequencies is presented. Both cases considered here have a potential application to determine the material properties in a region of nonlinear material behavior.