# Bifurcation and vibration of a surface-coated elastic block under flexure

Dryburgh, Gillian (1999) Bifurcation and vibration of a surface-coated elastic block under flexure. PhD thesis, University of Glasgow.

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Printed Thesis Information: https://eleanor.lib.gla.ac.uk/record=b1818179

## Abstract

The behaviour of a surface-coated rectangular, non-linearly elastic block subject to (plane-strain) flexure is investigated in this thesis. We consider a rectangular block of incompressible, isotropic elastic material coated with a thin elastic film on part of its boundary. Initially, the bulk material undergoes a non-homogeneous deformation and the equilibrium of the coated body is examined on the basis of the elastic surface coating theory derived by Steigmann and Ogden (1997a). Incremental displacements are then superimposed on the finitely deformed configuration in order to study possible bifurcation of the deformed block. Numerical bifurcation results pertaining to two particular strain-energy functions (for the bulk material) and a general energy function (for the coating material) are subsequently obtained. These results allow the influence of the surface coating on the bifurcation behaviour of the block to be determined and assessed with reference to corresponding results for an uncoated block.

Next, use is made of the dynamic equivalent of the static surface coating theory, developed by Ogden and Steigmann (1999), to establish incremental equations of motion for the coated block. Corresponding incremental governing equations for an uncoated, pre-flexed block then emerge as a special case. The resulting frequency equations are solved numerically, again on specialization of the form of strain-energy function. The numerical vibration results then provide evidence of the effect of surface coating on the dynamic behaviour of the considered coated block relative to the uncoated case.

Finally, we turn our attention to the (non-linear) shear responses of bonded elastic bodies. We examine the plane-strain problem of a rectangular compressible isotropic elastic block bonded to two rigid parallel plates. The deformation behaviour of the block is described by applying minimum energy and maximum complementary energy principles to obtain upper and lower bounds on the shear stress-strain relationship. Although maximum and minimum principles are not generally justifiable in non-linear elasticity we show that under certain conditions they are applicable and, for a particular form of strain-energy function, derive explicit energy bounds which we illustrate graphically.

Item Type: Thesis (PhD) Doctoral Q Science > QA Mathematics College of Science and Engineering > School of Mathematics and Statistics > Mathematics Ogden, Prof. Ray 1999 Elaine Ballantyne glathesis:1999-2023 Copyright of this thesis is held by the author. 04 Aug 2010 10 Dec 2012 13:50 http://theses.gla.ac.uk/id/eprint/2023