On the Asymptotic Derivation of Winkler-Type Energies from 3D Elasticity

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 attached 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.