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Article Dans Une Revue Journal of the Mechanics and Physics of Solids Année : 2024

Effective boundary conditions for second-order homogenization

Résumé

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.
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Dates et versions

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

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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⟩
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