A homogenized model for free vibration analysis of finite phononic crystal rods using strain gradient theory

Abstract

While extensive research has been conducted on elastic wave propagation in infinite phononic crystal (PC) rods, practical applications often involve PC rods of finite length. Therefore, it is essential to consider not only their functional properties, such as wave filtering, but also their mechanical behavior, including free vibration. Motivated by this concern, this study aims to establish a homogenized model for a finite-length PC rod based on the strain gradient theory (SGT) and provide simple closed-form expressions for its natural frequencies. To achieve this objective, an innovative method is developed to determine the material length scale parameters within the SGT. The investigation begins by analyzing the propagation of longitudinal waves in an infinite PC rod. By using the dispersion obtained from Bloch's theorem as a benchmark, it is demonstrated that the material length scale parameters associated with strain energy and micro-inertial effects in the SGT can be effectively determined by fitting the dispersion obtained from the SGT to the benchmark dispersion obtained from Bloch's theorem. With the determined parameters, the study proceeds to investigate the free vibration of a finite PC rod. The natural frequencies obtained from both the classical continuum theory (CCT) and the SGT are compared with those calculated by finite element (FE) simulations, which are based on an actual periodic structure and serve as a reliable benchmark. This comparison clearly highlights the superiority of the developed mesoscopic model based on the SGT in accurately predicting the natural frequencies, particularly for higher-order modes, compared to the CCT.