In this paper, some numerical tools for dispersion analysis of periodic structures are presented, with a focus on the ability of the methods to deal with dissipative behaviour of the systems. An adaptive phononic crystal based on the combination of metallic parts and highly dissipative polymeric interface is designed. The system consists in an infinite periodic bidirectional waveguide. The periodic cylindrical pillars include a layer of shape memory polymer and Aluminum. The mechanical properties of the polymer depend on both temperature and frequency and can radically change from glassy to rubbery state, with various combination of high/low stiffness and high/low dissipation. A fractional derivative Zener model is used for the description of the frequency-dependent behaviour of the polymer. A 3D finite element model of the cell is developed for the design of the metamaterial. The ”Shifted-Cell Operator” technique consists in a reformulation of the PDE problem by ”shifting” in terms of wave number the space derivatives appearing in the mechanical behaviour operator inside the cell, while imposing continuity boundary conditions on the borders of the domain. Damping effects can easily be introduced in the system and a quadratic eigenvalue problem yields to the dispersion properties of the periodic structure. In order to validate the design and the adaptive character of the metamaterial, results issued from a full 3D model of a finite structure embedding an interface composed by a distributed set of the unit cells are presented. Various driving temperature are used to change the behaviour of the system. After this step, a comparison between the results obtained using the tunable structure simulation and the experimental results is presented. Two states are obtained by changing the temperature of the polymeric interface: at 25°C, the bandgap is visible around a selected frequency. Above the glass transition, the phononic crystal tends to behave as an homogeneous plate.
K. Billon, M. Ouisse, E. Sadoulet-Reboul, M. Collet, G. Chevallier, and A. Khelif, "Design and experimental validation of an adaptive phononic crystal using highly dissipative polymeric material interface," Proc. SPIE 10164, Active and Passive Smart Structures and Integrated Systems 2017, 101640O (Presented at SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring: March 27, 2017; Published: 11 April 2017); https://doi.org/10.1117/12.2259889.
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