A modified size-dependent core-shell model and its application in the wave propagation of square cellular networks

We propose a modified core-shell model to depict the size-dependent elastic properties of materials with several different cross-sections. By using the Young-Laplace equation, a modified Euler-Bernoulli equation, which has taken a power-law relation between the bulk and surface moduli into account, is derived. A finite element method of the modified Euler-Bernoulli equation is formulated, and assembled to investigate the dispersion relations of the infinite two-dimensional periodic square cellular networks. The effectiveness of the proposed core-shell model is verified by comparing with results of the experiments and the molecular dynamics simulations available in the literature. Numerical results show that surface effects play an important role on the cellular networks with small diameters, large aspect ratios and high wave frequencies. Meanwhile, the analytical expressions for the size-dependent elastic modulus may be useful for the study of the size-dependent elasticity of materials and structures at small length scales.