A Stationarity Principle Generating Effective Boundary Conditions for Second-Order Homogenization

Abstract

We derive an effective model for a periodic chain of linearly-elastic springs, achieving second-order accuracy in the scale separation parameter \(\varepsilon \ll 1\). The chain has finite length and is made up of springs connecting both nearest- and next-nearest-neighbors: it serves as a one-dimensional prototype for higher-order periodic homogenization problems with boundaries. This type of problem has been approached by inserting two-scale expansions into the equations of equilibrium in the bulk and by matching them with boundary-layer solutions. We explore an alternative method operating at the energy level, bypassing the cumbersome matching procedure. We start from an ansatz of the microscopic displacement accounting for both boundary layers and for small-scale fluctuations in the bulk, and insert it into the discrete energy. This yields a continuous energy functional depending on the macroscopic displacement \(u\), in the form of a series expansion in powers of \(\varepsilon \). We call it a pseudo-energy \(\Phi _{\varepsilon } [u]\) as it is not positive when truncated at order \(\varepsilon ^{2}\). The boundary terms in the pseudo-energy account for boundary layers in an effective way. By making the pseudo-energy stationary order by order in \(\varepsilon \), we derive the homogenized equations of equilibrium along with effective boundary conditions. We provide quantitative validation showing that the effective model is correct to second order. We point out the special form of the effective higher-order tractions, which has been overlooked in the strain-gradient theories proposed so far.