Sequential weak continuity of the determinant and the modelling of cavitation and fracture in nonlinear elasticity
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Thu, 05/02/2009 12:30 |
Duvan Henao (University of Oxford) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
| 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). | |||