Surface effects of network materials based on strain gradient homogenized media

The asymptotic homogenization of periodic network materials modeled as beam networks is pursued in this contribution, accounting for surface effects arising from the presence of a thin coating on the surface of the structural beam elements of the network. Cauchy and second gradient effective continua are considered and enhanced by the consideration of surface effects. The asymptotic homogenization technique is here extended to account for the additional surface properties, which emerge in the asymptotic expansion of the effective stress and hyperstress tensors versus the small scale parameters and the additional small parameters related to surface effects. Based on the elaboration of small dimensionless parameters of geometrical or mechanical nature reflecting the different length scales, we construct different models in which the importance of surface effects is dictated by specific choice of the scaling relations between the introduced small parameters. The effective moduli reflect the introduced surface properties. We show in particular that surface effects may become dominant for specific choices of the scaling laws of the introduced small parameters. Examples of networks are given for each class of the considered effective constitutive models to illustrate the proposed general framework.