Variational modelling of cavitation and fracture in nonlinear elasticity

Thesis

Motivated by experiments on titanium alloys of Petrinic et al. (2006), which show the formation of cracks through the growth and coalescence of voids in ductile fracture, we consider the problem of formulating a variational model in nonlinear elasticity compatible both with cavitation and 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 derive 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 results for 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 modeled 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. In addition to the above, we extend previous work of Sivaloganathan, Spector and Tilakraj (2006) on the approximation of minimizers for the problem of cavitation with a constraint in the number of flaw points, and present some numerical results for this problem.

Authors: Henao Manrique, D

• Publication date: 2009
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: created surface; cavitation; invertibility; regularity of the inverse; ductile fracture; Cartesian currents; surface energy; distributional determinant; SBV; singular minimizers; weak continuity of the Jacobian; nonlinear elasticity
• Subjects: Calculus of variations and optimal control; Materials modelling; Partial differential equations; General mathematics; Solid mechanics; Mechanics of deformable solids (mathematics)