Effective boundary conditions for second-order homogenization - Archive ouverte HAL
Journal Articles Journal of the Mechanics and Physics of Solids Year : 2024

Effective boundary conditions for second-order homogenization

Abstract

Using matched asymptotic expansions, we derive an equivalent bar model for a periodic, one-dimensional lattice made up of linear elastic springs connecting both nearest and next-nearest neighbors. We obtain a strain-gradient model with effective boundary conditions accounting for the boundary layers forming at the endpoints. It is accurate to second order in the scale separation parameter $\varepsilon \ll 1$, as shown by a comparison with the solution to the discrete lattice problem. The homogenized modulus associated with the gradient effect (gradient stiffness) is found negative, as is often the case in second-order homogenization. Negative gradient stiffnesses are widely viewed as paradoxical as they can induce short-wavelength oscillations in the homogenized solution. In the one-dimensional lattice, the asymptotically correct boundary conditions are shown to suppress the oscillations, thereby restoring consistency. By contrast, most of the existing work on second-order homogenization makes use of postulated boundary conditions which, we argue, not only ruin the order of the approximation but are also the root cause of the undesirable oscillations.
Fichier principal
Vignette du fichier
standard-matching.pdf (570.78 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-04620932 , version 1 (22-06-2024)

Identifiers

Cite

Manon Thbaut, Basile Audoly, Claire Lestringant. Effective boundary conditions for second-order homogenization. Journal of the Mechanics and Physics of Solids, 2024, 190, pp.105707. ⟨10.1016/j.jmps.2024.105707⟩. ⟨hal-04620932⟩
108 View
43 Download

Altmetric

Share

More