Effect of mechanical nonlinearity on the electromagnetic response of a microwave tunable metamaterial

Tunable metamaterials functionalities change in response to external stimuli. Mechanical deformation is known to be an efficient approach to tune the electromagnetic response of a deformable metamaterial. However, in the case of large mechanical deformations, which are usually required to fully exploit the potential of the tunable metamaterials, the linear elastic mechanical analysis is no longer suitable. Nevertheless, nonlinear mechanical analysis is missing in the studies of mechanically tunable metamaterials. In this paper, we study the importance of considering nonlinearity in mechanical behavior when analyzing the response of a deformable metamaterial and its effects on electromagnetic behavior. We consider a microwave metamaterial formed by copper four-cut split ring resonators on a Polydimethylsiloxane (PDMS) substrate. Applying both displacement and force stimuli, we show that when the deformation is large, more than 10 percent strain, the use of nonlinear analysis considering the geometrical and material nonlinearities is imperative. We further show that the discrepancies between the linear and nonlinear analyses appear in overestimating the stress, underestimating the tunability of the metamaterial responses, and mispredicting the negative permeability regions.