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

Andrés Alessandro León Baldelli (1) , Blaise Bourdin (1, 2)

1 CCT - Center for Computation and Technology [Baton Rouge]
2 DM - Department of Mathematics

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.

Mots clés: Winkler foundation; variational methods; thin films; dimension reduction; asymptotic analysis

Domaines: Mécanique des structures [physics.class-ph]; Mécanique des structures [physics.class-ph]; Physique mathématique [math-ph]

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