On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

Epifanio G. Virga,Olivier Ozenda

doi:10.1007/s10659-021-09819-7

Abstract

The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard varGamma -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard varGamma -convergence also appears to be removed in the cases where contact with that method and ours can be made.

Highlights

Perhaps, some believe that the theory of elasticity is a dead subject
The distance of every point of the plate from the middle surface remains unchanged by the deformation. These statements have become to be known as the Kirchhoff-Love hypothesis; we shall stick to this not fully justified tradition
In preparation for this, we refresh the preliminaries of differential geometry of surfaces in a way that avoids local charts of coordinates, but resorts instead to a number of vector fields, which describe the correspondence between local movable frames in the reference and current configurations of a material surface

Summary

Introduction

Some believe that the theory of elasticity is a dead subject. It may rather be that it is just deceptively simple: it makes one believe that everything is understood and only routine computations need to be done, for which it suffices to devise the most appropriate algorithm (the distinguished job of computational mechanics). This theory extends the ideas underlying the isotropic Gaussian distribution of polymeric chains to chains made anisotropic by the mutual interactions of the nematogenic molecules appended to them It is no surprise if the free energy of nematic elastomers (of a purely entropic nature) turns out to be an anisotropic extension of the classical neo-Hookean formula of isotropic rubber elasticity. We wish to revisit this classical hypothesis in its natural environment, as it were, and show how its revision could possibly be used to derive reduced theories for nonlinear elastic plates in which stretching and bending energies are naturally combined together. The distance of every point of the plate from the middle surface remains unchanged by the deformation For mysterious reasons, these statements have become to be known as the Kirchhoff-Love hypothesis; we shall stick to this not fully justified tradition. This paper is closed by the Appendix, where we give explicit formulae for the mean and Gaussian curvatures of the deformed mid surface in terms of the mapping that describes it

Kinematics of Plates

Kinematic Preliminaries

Invariants of Cf

Cartesian Connectors

Incompressible Elastomer Plates

Polynomial Approximation

Gent’s Material

Compressible Plates

The Ciarlet-Geymonat Material

A Variant of the Saint-Venant-Kirchhoff Material

Conclusion