ABSTRACT
This chapter analyses the structures containing multi-scale resonators with the emphasis on a class of standing waves of sufficiently low frequencies. It describes the governing equations and geometry of the periodic structure. The chapter gives mechanism of the thermal pre-stress and presents an evaluation of the critical temperature required to buckle a thin ligament within the structure. It gives asymptotic estimates of the frequencies of standing waves, of rotational and translational types, and existing at the boundaries of the stop bands of the dispersion diagrams. The chapter presents the results of numerical simulations and comparison of asymptotic estimates and numerical computations. It derives the analytical asymptotic approximation for the corresponding eigenvalue. The chapter provides an asymptotic analysis of thin-legged resonators with cracks. Analytical and numerical results are consolidated into a class of important engineering applications. The chapter discusses an infinite periodic structure as well as a finite-thickness structured interface.
