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# Deformation and buckling of isolated and interacting thin shells in an elastic medium

[Thesis]. Manchester, UK: The University of Manchester; 2016.

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## Abstract

This thesis 'Deformation and buckling of isolated and interacting thin shells in an elastic medium', is submitted by Maria Thorpe to The University of Manchester for the degree of PhD in September 2015.This thesis aims to model the effects of interaction and buckling upon pairs of micro-shells embedded within an elastic medium under far field hydrostatic pressure.This analysis is motivated by the role shell buckling plays in the nonlinear nature of the pressure relative volume curve of elastomers containing micro-shells. Current models of the effective properties of these types of composites assume shells are in a dilute distribution within the host medium, and as such assume shells will buckle at the pressure of the associated isolated embedded shell model. For composites with a high volume fraction of micro-shells, or in poorly mixed composites, the dilute distribution model may provide a first approximation to the effective properties of the composite, however, interaction between shells must be considered to find a more accurate model.We begin the process of modelling the buckling of interacting embedded shells by considering the buckling of an isolated embedded thin spherical shell. For a host medium undergoing far field hydrostatic pressure we demonstrate the parameter ranges in which Jones et al. thin shell buckling theory agrees with the thin shell buckling theory of Fok and Allwright. We then use scalings to increase the range of validity of the thin shell approximation used in the Jones et al. theory to include composites with a high contrast between medium and shell materials. This enables more accurate predictions of buckling pressures of embedded shells under far field axially symmetric pressures to also be found, as is demonstrated for an embedded shell under far field axial compression.We model the linear elastic deformation of pairs of embedded micro-shells using the Boussinesq-Papkovich stress function method, before employing the thin shell linear analysis method developed in previous chapters to calculate the critical buckling pressure and buckling patterns of the pair of embedded shells.

## Layman's Abstract

This thesis 'Deformation and buckling of isolated and interacting thin shells in an elastic medium', is submitted by Maria Thorpe to The University of Manchester for the degree of PhD in September 2015.This thesis aims to model the effects of interaction and buckling upon pairs of microscopic thin spherical shells embedded within an elastic host material which experiences far field hydrostatic pressure.Current models of the macroscopic properties of these types of composites assume shells are dilutely distributed within the host medium and behave as if they were isolated from all other shells. For composites with a large concentration of shells the dilute distribution model may provide a first approximation to the macroscopic properties of the composite, however how the shells interact must be considered to find a more accurate model.We begin the process of modelling the buckling of interacting embedded shells by considering the buckling of an isolated embedded thin spherical shell. For a host medium undergoing far field hydrostatic pressure we demonstrate the parameter ranges in which Jones et al. thin shell buckling theory agrees with the thin shell buckling theory of Fok and Allwright. We then use scalings to increase the range of validity of the thin shell approximation used in the Jones et al. theory to include composites with a high contrast between medium and shell materials. This enables more accurate predictions of buckling pressures of embedded shells under far field axially symmetric pressures to also be found, as is demonstrated for an embedded shell under far field axial compression.We model the linear elastic deformation of pairs of embedded micro-shells using the Boussinesq-Papkovich stress function method, before employing the thin shell linear analysis method, developed in previous chapters, to calculate the critical buckling pressure and buckling patterns of the pair of embedded shells.