Two-dimensional periodic structures modeling based on second strain gradient elasticity for a beam grid. (15th May 2022) - Yang, Bo; Zine, Abdelmalek; Droz, Christophe; Ichchou, Mohamed
Abstract: Higher order gradient elasticity theories are widely applied to determine the wave propagation characteristics of micro-sized structures. The novelty of this paper, firstly, is using the Second Strain Gradient (SSG) theory to explore the mechanism of a micro-sized 2D beam grid. The strong formulas of continuum model including governing equations and boundary conditions are derived by using the Hamilton principle. Then, a valuable long-range Lattice Spring Model (LSM) is elaborated, providing a reasonable explanation for the model based on SSG theory. The dynamic continuum equations from LSM are calculated through the Fourier series transform approach. Finally, the dynamic properties of 2D beam grid are analyzed within the Wave Finite Element Method (WFEM) framework. The band structure and slowness surfaces, confined to the irreducible first Brillouin zone, are studied in frequency spectrum. The energy flow vector fields and wave beaming effects are discussed through SSG theory and Classical Theory (CT) of elasticity. The results show that the proposed approach is of significant potential for investigating the 2D wave propagation characteristics of complex micro-sized periodic structures. Graphical abstract: Highlights: Second Strain Gradient theory is used for the modeling of a micro-sized 2D beam grid. Continuum model including governing equations and boundary conditions are derived. Long-range lattice model is elaborated, giving reasonable explanation for SSG theory. The Wave Finite Element Method is applied for the vibration analysis of 2D beam grid.