Low-frequency asymptotics of complex dispersion curves for lamb waves in layered elastic plate

Resumo

An asymptotic and iterative method is proposed to calculate complex dispersion curves for isotropically layered plates. At the first stage, a dispersion equation is derived in explicit form and its limiting form is obtained for the static case. Passages to the limit of coinciding materials or vanishingly small layer thicknesses are investigated. Specific case of materials with coinciding shear moduli is analyzed in detail. Asymptotics of static roots is deduced for a large value of the root magnitude, the error of asymptotics is estimated, and an iterative method is proposed for calculating exact root values. Long-wave asymptotics of dispersion curves is derived, and it is shown that every complex dispersion curve has a long flat initial segment. Asymptotics is the more accurate, the lower the frequency is and the higher the number of the curve is. Exact values of wave numbers on the dispersion curve are also evaluated by another iterative procedure. Examples of calculating the dispersion curves are presented and the efficiency of the algorithm is shown.