Two-dimensional periodic structures modeling based on second strain gradient elasticity for a beam grid

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.

Automated summary

This paper explores the wave propagation characteristics of micro-sized structures using higher order gradient elasticity theories, specifically the Second Strain Gradient (SSG) theory. The study derives the governing equations and boundary conditions of the continuum model and provides a valuable Lattice Spring Model (LSM) to explain the SSG-based model. The dynamic properties of a 2D beam grid are analyzed within the Wave Finite Element Method (WFEM) framework, and the results demonstrate the significant potential of this approach in investigating the wave propagation characteristics of complex micro-sized periodic structures.