Interaction between periodic elastic waves and two contact nonlinearities

Stéphane Junca (1) , Bruno Lombard (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]

Propagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations. A typically nonlinear phenomenon is found to occur: under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. Perturbation analysis is performed to quantify the effect of this process, which offers non-destructive means to evaluate the cracks. The rest of the paper focuses on the existence, uniqueness and attractivity of periodic solutions. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition.

Domaines: Acoustique [physics.class-ph]; Acoustique [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Stéphane Junca, Bruno Lombard. Interaction between periodic elastic waves and two contact nonlinearities. 2010. ⟨hal-00549051v1⟩