Vol. 5, No. 3, 2010

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Chaotic vibrations in a damage oscillator with crack closure effect

Noël Challamel and Gilles Pijaudier-Cabot

Vol. 5 (2010), No. 3, 369–389

This paper deals with the dynamics of a single-degree-of-freedom unilateral damage oscillator. Using appropriate internal variables, the hysteretic dynamic system can be written as a nonsmooth autonomous system. The free dynamics of such a nonlinear system are simply reduced to periodic motion, eventually attractive trajectories, and divergent motion. The direct Lyapunov method is used to investigate the stability of the free damage system. A critical energy is highlighted that the oscillator can support while remaining stable. The natural frequency of the periodic motion depends on the stationary value of the damage internal variable. The inelastic forced oscillator, however, can exhibit very complex phenomena. When the damage parameter remains stationary, the dynamics are similar to those of an elastic oscillator with nonsymmetric stiffness. The dynamics appear to be controlled by the initial perturbations. Chaotic motions may appear in such a system, specifically for severely damaged oscillators. It is numerically shown that chaos is observed in the vicinity of the divergence zone (the collapse). This closeness of both behaviors—chaos and divergence—is probably related to the perturbation of the homoclinic orbit associated with the critical energy.

damage oscillator, nonlinear dynamics, seismic design, concrete structures, chaos, unilateral effect, crack closure, breathing crack, bimodular material
Received: 10 November 2008
Revised: 13 October 2009
Accepted: 19 October 2009
Published: 15 October 2010
Noël Challamel
Laboratoire de Génie Civil et de Génie Mécanique
INSA de Rennes
Université Européenne de Bretagne
20, Avenue des Buttes de Coësmes
35043 Rennes
Gilles Pijaudier-Cabot
Laboratoire des Fluides Complexes (UMR 5150)
Allée du Parc Montaury
64600 Anglet