Computational homogenization of nano-materials accounting for size effects via surface elasticity

Abstract

The objective of this contribution is to establish a first-order computational homogenization framework for micro-to-macro transitions of porous media that accounts for the size effects through the consideration of surface elasticity at the microscale. Although the classical (first-order) homogenization schemes are well established, they are not capable of capturing the well-known size effects in nano-porous materials. In this contribution we introduce surface elasticity as a remedy to account for size effects within a first-order homogenization scheme. This proposition is based on the fact that surfaces are no longer negligible at small scales. Following a standard first-order homogenization ansatz on the microscopic motion in terms of the macroscopic motion, a Hill-type averaging condition is used to link the two scales. The averaging theorems are revisited and generalized to account for surfaces. In the absence of surface energy this generalized framework reduces to classical homogenization. The influence of the length scale is elucidated via a series of numerical examples performed using the finite element method. The numerical results are compared against the analytical ones at small strains for tetragonal and hexagonal microstructures. Furthermore, numerical results at small strains are compared with those at finite strains for both microstructures. Finally, it is shown that there exists an upper bound for the material response of nano-porous media. This finding surprisingly restricts the notion of “smaller is stronger”.