On the asymptotic reduction of a bifurcation equation for edge-buckling instabilities

Ciprian D Coman

doi:10.1007/s00707-017-2036-8

Abstract

Weakly clamped uniformly stretched thin elastic plates can experience edge buckling when subjected to a transverse pressure. This situation is revisited here for a circular plate, under the assumption of finite rotations and negligible bending stiffness in the pre-buckling range. The eigenproblem describing this instability is formulated in terms of two singularly perturbed fourth-order differential equations involving the non-dimensional bending stiffness e>0. By using an extension of the asymptotic reduction technique proposed by Coman and Haughton (Acta Mech 55:179–200, 2006), these equations are formally reduced to a simple second-order ordinary differential equation in the limit e→0+. It is further shown that the predictions of this reduced problem are in excellent agreement with the direct numerical simulations of the original bifurcation equations.

Highlights

The presence of localised deformations in thin elastic bodies such as bars, plates and shells, tends to signal the existence of lower-dimensional mathematical structures that can capture the underlying physics of such phenomena
By re-visiting the recent study [8] regarding the edge-wrinkling of uniformly stretched thin elastic plates subjected to a transverse pressure, we have extended the asymptotic reduction strategy proposed by
The relative errors are illustrated on the right

Summary

Introduction

The presence of localised deformations in thin elastic bodies such as bars, plates and shells, tends to signal the existence of lower-dimensional mathematical structures that can capture the underlying physics of such phenomena. In the present investigation we are motivated to explore the applicability of the aforementioned asymptotic reduction strategy to a rather different edge-wrinkling situation, recently studied by Coman et al [9, 12] within the context of the Foppl-von Karman (FvK) nonlinear plate theory In this new case, a weakly clamped uniformly stretched circular elastic plate is subjected to a transverse pressure; due to the gradual development of compressive stresses near the edge of the plate, for a critical value of the loading the plate experiences a regular wrinkling pattern confined to that region. The paper concludes with a further discussion of the results reported and some remarks on their possible extension

The bifurcation equations

The asymptotic reduction

Concluding remarks

Findings

A Laplace’s Expansion Theorem