University of Oxford

Motivated by the tensile experiments on titanium alloys of Petrinic et al

(2006), which show the formation of cracks through the formation and

coalescence of voids in ductile fracture, we consider the problem of

formulating a variational model in nonlinear elasticity compatible both

with cavitation and with the appearance of discontinuities across

two-dimensional surfaces. As in the model for cavitation of Müller and

Spector (1995) we address this problem, which is connected to the

sequential weak continuity of the determinant of the deformation gradient

in spaces of functions having low regularity, by means of adding an

appropriate surface energy term to the elastic energy. Based upon

considerations of invertibility we are led to an expression for the

surface energy that admits a physical and a geometrical interpretation,

and that allows for the formulation of a model with better analytical

properties. We obtain, in particular, important regularity properites of

the inverses of deformations, as well as the weak continuity of the

determinants and the existence of minimizers. We show further that the

creation of surface can be modelled by carefully analyzing the jump set of

the inverses, and we point out some connections between the analysis of

cavitation and fracture, the theory of SBV functions, and the theory of

cartesian currents of Giaquinta, Modica and Soucek. (Joint work with

Carlos Mora-Corral, Basque Center for Applied Mathematics).

In this talk, I shall endeavour to explain to the uneducated and uninitiated the joys and pleasures one can have studying automata.

Modelling the behaviour of light in photonic crystal fibres requires

solving 2nd-order elliptic eigenvalue problems with discontinuous

coefficients. The eigenfunctions of these problems have limited

regularity. Therefore, the planewave expansion method would appear to

be an unusual choice of method for such problems. In this talk I

examine the convergence properties of the planewave expansion method as

well as demonstrate that smoothing the coefficients in the problem (to

get more regularity) introduces another error and this cancels any

benefit that smoothing may have.