Cumulative second harmonics in weakly nonlinear plates and shells

The generation of cumulative second harmonic ultrasonic guided wave modes is analyzed with respect to their applications for nondestructive evaluation (NDE) and structural health monitoring (SHM). Due to the multimodal nature of guided waves, the selection of a primary wave mode that will generate a cumulative second harmonic is a critical first step for NDE and SHM applications. Thus, the nonlinear boundary value problem that must be solved by perturbation analysis is summarized and the results are tabulated for steel plates and circular cylindrical shells (pipes). The analysis includes shear-horizontal and Lamb waves in plates and axisymmetric torsional and longitudinal waves in pipes. Nonlinear finite element analyses that include kinematic and material nonlinearities are conducted for plate and pipe geometries. An excitation is applied to both plate and pipe samples by a simulated interdigitated transducer; SH1 for the plate and T(0,2) for the pipe. In each case a second harmonic mode having an orthogonal polarization is generated; S1 for the plate and L(0,4) for the pipe. In both cases the second harmonic grows linearly with propagation distance, and therefore is cumulative. A third example simulation is presented that demonstrates mode mixing in a pipe. T(0,3) and L(0,2) primary modes traveling in opposite directions intersect and generate significant harmonics at the sum and difference frequencies. Mode mixing provides a great opportunity to expand the potential of harmonic generation for NDE and SHM.