Nonlinear interactions of non-collinear guided wave beams in an isotropic elastic plate

Non-collinear interactions of time-harmonic guided wave beams in an isotropic elastic plate with quadratic material nonlinearity are theoretically investigated using a perturbation approach. Unlike previous studies limited to the interactions of straight-crested waves characterized by single wavevectors, the present approach represents the primary waves as Rayleigh-Lamb and shear-horizontal wave beams via a continuous superposition of time-harmonic Green functions with Gaussian weighting, thereby capturing the diffraction of the primary waves. The resulting driving body and surface forces at the sum and difference frequencies are then used to obtain the displacement fields of nonlinearly generated secondary waves by integration with the Green function at the corresponding frequency. The effects of initial beam width, propagation distance, and intersection angle of the primary waves on secondary wave generation are examined. The possibility of exciting two primary wave beams using separate complex point sources is also investigated, yielding secondary wave fields that closely match those obtained from continuous Gaussian distributions of real point sources. The displacement amplitudes of the secondary waves predicted by the present theory are demonstrated to be in close quantitative agreement with those from direct three-dimensional dynamic finite-element analysis.