82 D. Kiracofe et al. Furthermore, this method could potentially be modified to be a reference-free method. Observe that the damage index corresponding to the intact structure is very periodic, with nearly every resonator having the same pattern. Therefore, if the DI δj of the intact structure was not available, it could be approximately reconstructed based on observing the repeating pattern for the non-damaged resonators in the DI corresponding to a damaged metastructure. 9.4 Damage Tolerance We call a metamaterial robust or damage-tolerant if a localized failure has a small effect on the overall performance (i.e., attenuation in the bandgap) and such localized failure does not propagate to a larger failure. Most well-designed metamaterials meet the first part of the definition. For example, the LR metamaterial illustrated in Fig. 9.1 and discussed in Sect. 9.3 has a sufficiently large number of local resonators such that the failure of any single resonator has virtually no effect on the ability of the overall structure to meaningfully attenuate vibration in the bandgaps (i.e., for an excitation at one end of the structure, the vibration response on the far end is minimally impacted by the failure of any single resonator in between). However, many metamaterials are susceptible to a cascading failure mode, illustrated in Fig. 9.10. If a resonator close to the excitation source fails (e.g., due to manufacturing defect, handling damage, corrosion, or high cycle fatigue), a localized mode is created. This mode may have very high stresses in resonators adjacent to the failure. Failure of these resonators can then create high stress in the next resonators, and so on. In this section, we present two examples on how to improve metamaterial robustness. Note: for computational efficiency, these results were generated using 1D beam elements (i.e., BEAM188 in ANSYS), instead of the plane stress elements in the prior section. The results are qualitatively similar, but with slight quantitative differences. 9.4.1 Damped Resonators For beam-like structures with beam-like resonators considered in this work so far, if the host and resonators are made out of the same material (i.e., cut out of the same piece of metal, or additively manufactured with the same material), then it would be reasonable to expect a similar damping ratio between the host and resonators. But consider what happens if the resonators (including the connection with the host beam) have a significantly higher damping ratio than the host structure (e.g., made out of a different material, or having an applied damping treatment such as constrained layer damping). The first observation is that this damping treatment has very little effect on the overall vibration attenuation of an intact structure, as shown in Fig. 9.11. However, for the failed structure, there is a considerable difference in the stresses in the local modes, as shown in Fig. 9.12. With damped resonators, the local modes are still present, but the peak stress is reduced by approximately half. 9.4.2 Geometry Optimization The damage tolerance can also be improved with modifications to the geometry. The local modes that are created with damage can vary depending on the construction of the metastructure, and some configurations will be more favorable than others. To give one example, consider the basic structure introduced in Fig. 9.1, but consider the thickness of the host beam to be variable, all other parameters kept constant.
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