Interaction between periodic elastic waves and two contact nonlinearities

Stéphane Junca 1, 2 Bruno Lombard 3
2 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
3 O&I - Ondes et Imagerie
LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
Abstract : 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, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. 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. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves.
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Stéphane Junca, Bruno Lombard. Interaction between periodic elastic waves and two contact nonlinearities. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2012, 22 (4), pp.1. ⟨10.1142/S0218202511500229⟩. ⟨hal-00549051v2⟩

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