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The classical theory of univalent functions and quasistatic crack propagation

Published online by Cambridge University Press:  19 June 2006

GERARDO E. OLEAGA
GERARDO E. OLEAGA
Affiliation:
Departamento de Matemática Aplicada Facultad de Matemáticas, Universidad Complutense de Madrid, Ciudad Universitaria s/n 28040, Madrid, Spain email: oleaga@mat.ucm.es
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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 Löwner 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.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 17 , Issue 2 , April 2006 , pp. 233 - 255
Copyright
2006 Cambridge University Press

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