This thesis aims to model the uniaxial deformation of a class of materials consisting of microscopic spherical shells embedded in a rubber matrix. These shells are assumed to buckle as the stress on the material increases. To motivate the analysis we consider the paradigm problem of the debonding of a distribution of cylindrical inclusions in an elastic material undergoing antiplane shear, with bonded and debonded inclusions playing the role of unbuckled and buckled shells respectively. We begin the modelling of the microsphere-containing material by considering the buckling of an isolated embedded shell inclusion with a uniaxial stress field at infinity, using Koiter's theory of shallow shells. The resulting energy functional is solved as an eigenvalue problem by the Rayleigh-Ritz method. Subsequently, we analyse the buckling criterion asymptotically in the limit as the thickness ratio tends to zero by analogy with the WKB analysis of a beam on a variable-stiffness substrate. To model the shell after buckling we consider the simplified case of an embedded shell with a crack around its equator. The system is solved by expressing the displacements in the shell and matrix as series of Love stress functions, with the resulting infinite system of equations solved numerically with the aid of a convergence acceleration method. Finally we consider a composite material consisting of a homogenised dilute distribution of buckled and unbuckled shells, with the proportion of each type of shell dependent on the stress applied to the material, according to an asymptotic formula relating the size of the inclusions and the critical buckling stress that was obtained previously.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:487266 |
| Date | January 2007 |
| Creators | Jones, G. W. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:2c91cdab-2e5e-4a74-af21-7ccb34128721 ; http://eprints.maths.ox.ac.uk/681/ |
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