# Variational Models of Fracture (16w5090)

Arriving in Banff, Alberta Sunday, May 8 and departing Friday May 13, 2016

## Organizers

(Louisiana State University)

(Université Paris Nord)

(Worcester Polytechnic Institute)

(Université Pierre et Maris Curie)

## Objectives

Since Francfort and Marigo initiated a revisiting of Griffith's classical theory of fracture of brittle fracture with modern mathematical tools, variational models in fracture mechanics have been the subject of intense activity. The interest of the mathematical community for this class of problem can be measured by a large body of work in areas including mathematical analysis, mathematical modeling, numerical analysis and computational science, several workshop and mini-symposia including weeklong meetings in Oberwolfach in 2007 and 2011 and at BIRS in 2011, or G.A. Francfort's plenary lecture at the 2011 ICIAM congress.

In recent years, the interest of the engineering community in this class of model has also rapidly increased, mainly motivated by the potential for rigorous, accurate and efficient numerical simulations. This interest can be measured for instance by the size of mini-symposia relating to phase--field'' approach in fracture mechanics congresses, or through M.~Ortiz and C.~Miehe's plenary lectures at the 2014 WCCM conference. The interest of various industries in this class of problems is also noticeable with ongoing projects at Corning, Chevron, Lafarge, and Airbus, amongst others.

Yet, as the mathematical understanding of this problem is reaching its maturity, we are reaching a tipping point where the mathematics and engineering or computational science communities are becoming increasingly divided, instead of mutually benefiting from each other's progress. The reasons of this growing schism are multiple: the mathematical literature can be very technical and deter engineering students. Conversely, mathematicians may have a hard time translating issues mentioned in the engineering and technical literature into well defined mathematical problems. This is especially true of recent extensions to dynamic or rate dependent problems or modeling of coupled problems in reservoir engineering, fracture of ferromagnetic materials or corrosion cracks in thin coatings, for which a rigorous mathematical understanding is lacking. Finally, graduate students and young researchers often lack a common culture with engineers being unfamiliar with modern mathematical tools such as geometric measure theory or $\Gamma$--convergence, and mathematicians lacking awareness of actual problems.

The goal of this workshop is to bring together a group of mathematicians, mechanicians, engineers and computational scientist sharing an interest in variational models of fracture mechanics in order to achieve a breakthrough in the mathematical understanding of current topics, tools and issues, and in the scope of the numerical applications of the current theories.

Our specific objectives are

1. To present the state of the art of the mathematical analysis of problems arising from variational models of fracture.

2. A better understanding of the mathematical issues arising in these problems is essential to reach a deep understanding of the numerical methods. Yet, there is a lack of concise and focussed literature at the graduate level. We will begin this workshop with a few introductory lectures on mathematical modeling and tools so as to give the more applied participants a (possibly critical) overview of the current state of the theory.

3. To gain a better understanding of the challenges facing this class of methods.

4. "Real life'' problems can be quite at odds with those favored by mathematicians and are often beyond the reach of rigorous analysis. Engineers and industry partners will be invited to present current or potential applications and algorithms related to the variational models of fracture. The rationale is that a better theoretical understanding of this problem can lead to more efficient numerical tools, while exposure to a broader range of problems will stimulate new theoretical developments.

5. To devise a set of reference problems that can be analyzed rigorously, then used in order to assess the accuracy and efficiency of algorithms.

6. The popularity of benchmark problems in fracture mechanics is highly skewed by the strength and weaknesses of classical methods. The resulting tests are often inappropriate or even non sensical from the standpoint of the variational approach to fracture. Devising proper numerical experiments highlighting specific properties of a model or implementation is difficult and time consuming. In addition to the lack of common reference tests, comparing methods is difficult. We propose to come up with a small set of problems which will be used in the years to come.