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On the asymptotic derivation of Winkler-type energies from 3D elasticity

Abstract : We show how bilateral, linear, elastic foundations (i.e. Winkler foundations) often regarded as heuristic, phenomenological models, emerge asymptotically from standard, linear, three-dimensional elasticity. We study the parametric asymptotics of a non-homogeneous linearly elastic bi-layer at- tached to a rigid substrate as its thickness vanishes, for varying thickness and stiffness ratios. By using rigorous arguments based on energy estimates, we provide a first rational and constructive justification of reduced foundation models. We establish the variational weak convergence of the three-dimensional elasticity problem to a two-dimensional one, of either a "membrane over in-plane elastic foundation", or a "plate over transverse elastic foundation". These two regimes are function of the only two parameters of the system, and a phase diagram synthesizes their domains of validity. Moreover, we derive explicit formulæ relating the effective coefficients of the elastic foundation to the elastic and geometric parameters of the original three-dimensional system.
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https://hal-ensta-paris.archives-ouvertes.fr//hal-01064163
Contributor : Andrés Alessandro León Baldelli <>
Submitted on : Friday, December 11, 2020 - 9:36:25 AM
Last modification on : Saturday, January 23, 2021 - 3:19:44 AM
Long-term archiving on: : Friday, March 12, 2021 - 6:13:29 PM

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Andrés Alessandro León Baldelli, Blaise Bourdin. On the asymptotic derivation of Winkler-type energies from 3D elasticity. Journal of Elasticity, Springer Verlag, 2015, 121 (2), pp.275-301. ⟨10.1007/s10659-015-9528-3⟩. ⟨hal-01064163⟩

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