Quasi-Static Evolutions in Brittle Fracture Generated by Gradient Flows: Sharp Crack and Phase-Field Approaches

Chapter
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 81)

Abstract

In this paper we will describe how gradient flows, in a suitable norm, are natural and helpful to generate quasi-static evolutions in brittle fracture. First, we will consider the case of a brittle crack running along a straight line according to Griffith’s law. Then, we will see how the same approach leads to quasi-static evolutions in the phase field setting, taking into account the alternate minimization scheme. In the latter, the norm associated to the gradient flow is not “user supplied”, however, the algorithm itself together with the separate quadratic structure of the energy defines a family of norms which, in the limit, characterizes the quasi-static evolution. Mathematically speaking, all of these evolutions are (parametrized) BV-evolutions.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PaviaPaviaItaly

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