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

TL;DR

This paper rigorously derives Winkler-type elastic foundation models from three-dimensional elasticity by analyzing a bi-layer system's asymptotic behavior as its thickness approaches zero, providing explicit formulas and phase diagrams.

Contribution

It offers the first rigorous, constructive derivation of reduced Winkler foundation models from 3D elasticity, including explicit coefficient formulas and regime classification.

Findings

Established weak convergence from 3D to 2D models

Derived explicit formulas for effective foundation coefficients

Mapped parameter regimes with a phase diagram

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 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 formulae relating the effective coefficients of the elastic foundation to the elastic and geometric parameters of the original three-dimensional system.