Modal expansion analyses of second-harmonic generation of the Lamb waves in layered structures

Through second-order perturbation approximation and modal expansion approach, the second-harmonic generation of a primary Lamb wave propagating in layered structures has been analyzed, and the general solution to second-harmonic displacement fields has been obtained. Because of the bulk elastic nonlinearities of solids, there are double frequency bulk driving forces in the interior of each layer and double frequency traction stress tensors at the two surfaces of each layer when a primary Lamb wave travels along layered structures. These driving forces and traction stress tensors of double the primary frequency generate a series of double frequency Lamb wave normal modes (abbr. DFLMs), and the summation of a series of DFLMs constitutes the second-harmonic fields of a primary Lamb wave. It is found that the DFLM component of a primary Lamb wave can grow with propagation distance down layered structures when its phase velocity equals that of a primary Lamb wave. Numerical simulations of second-harmonic displacement fields of a primary Lamb wave have been carried out for a layered structure consisting of aluminum-epoxy-aluminum. The results show that the properties of soft adhesive joint play a dominant role in generation of the second harmonics.