A Critical Study of Baldelli and Bourdin's On the Asymptotic Derivation of Winkler-Type Energies From 3D Elasticity

TL;DR

This paper critically examines Baldelli and Bourdin's derivation of Winkler-type energies from 3D elasticity, revealing limitations related to Poisson's ratios, asymptotic scalings, and applicability to plates, while acknowledging its compliance with elasticity laws.

Contribution

The study identifies specific conditions and limitations of Baldelli and Bourdin's derivation, clarifying its valid regimes and highlighting its strengths in preserving elasticity laws.

Findings

Valid only for certain Poisson's ratios

Phase diagram is four-dimensional, not two-dimensional

Method cannot be applied to plates with planar loading unless displacement is zero

Abstract

In our analysis, we show that Baldelli and Bourdin's work is only valid when describing the behaviour of a film bonded to an elastic pseudo-foundation, where Poisson's ratios of both bodies are in between -1 and 0 or in between 0 and 0.5 (where both Poisson's ratios are sufficiently away from 0 and 0.5), and with an asymptotic condition that is different to what the authors present. We also show that, for all Poisson's ratios, the authors' phase diagram is four-dimensional and not two-dimensional. Also, due to the Poisson's ratio dependence, the asymptotic scalings that the authors present are insufficient to derive their proposed models. Furthermore, the authors' scaling of the displacement field implies that their method cannot be applicable to films (or strings) with planar loading, unless the normal displacement is zero. Finally, by deriving a Winkler foundation type solution for a plate supported by an elastic pseudo-foundation via the method implied by the authors, we show that the authors' method cannot be applied to plates due to the structure of the overlying body (i.e. the limits of integration of the plate) and the foundation (i.e. planar-stress free condition of the foundation), unless planar displacement field is identically zero. Despite the limitations of the authors' work, we highlight its strength by showing that unlike the classical derivation of the Winkler foundation equation, Baldelli and Bourdin's approach does not violate the volume conversation laws or violate the governing equations of mathematical elasticity.