The Classical Theory of Univalent Functions and Quasistatic Crack Propagation. Gerardo E. Oleaga Departamento de Matem´ atica Aplicada Facultad de Matem´ aticas, Universidad Complutense de Madrid Ciudad Universitaria s/n 28040, Madrid (Spain) email: oleaga@mat.ucm.es November 29, 2005 Abstract We study the propagation of a crack in critical equilibrium for a brittle material in a Mode III field. The energy variations for small virtual extensions of the crack are handled in a novel way: the amount of energy released is written as a functional over a family of univalent functions on the upper half plane. Classical techniques developed in connection to the Bieberbach Conjecture are used to quantify the energy-shape relationship. By means of a special family of trial paths generated by the so-called L¨ owner equation we impose a stability condition on the field which derives in a local crack propagation criterion. We called this the “anti-symmetry” principle, being closely related to the well known symmetry principle for the in-plane fields. Keywords Crack propagation, Mode III, Univalent functions, Loewner equation, Schiffer’s method. 2000 Mathematics Subject Classification 74R10, 74B05, 74G70, 30C55, 30C70. Contents 1 Introduction 2 2 Basic facts 3 2.1 Out-of-plane fields......................................... 3 2.2 Complex representation and Slit Maps ............................. 4 3 Univalent functions on H. 7 3.1 Motivation for this Section ................................... 7 3.2 A set of conformal maps as trial paths.............................. 8 3.3 The Energy functional ...................................... 9 3.4 The best shape for optimal energy release rate. ........................ 10 3.5 Schiffer’s variational method ................................... 11 4 L¨ owner evolutions. 13 4.1 Constant forcing ......................................... 14 4.2 Linear forcing and Anti-Symmetry principle .......................... 15 5 Concluding remarks 17 1