**Invitation**

The Symposium is hosted by the International Research Center of Mathematics & Mechanics of Complex System **M&MoCS.**

The Symposium on “**MECHANICS OF FRACTURES AND SECOND GRADIENT THEORY**” will take place at a historical ”Palazzo Caetani”, in Cisterna di Latina, Italy.

**Brief Syllabuses**

Continuum mechanics is usually understood as a homogenized description of materials which are heterogeneous at the microscopic level. It is natural to expect from any general theory of continuum mechanics to be stable by homogenization procedures. The class of Cauchy continua does not enjoy this stability property. Indeed, it will be shown that the effective properties of some periodic elastic materials have to be described by a Second Gradient theory. Materials for which energy depends on the Second Gradient of the displacement cannot be considered as Cauchy continua otherwise one would be led to a thermodynamic paradox. This paradox can be removed by extending the thermodynamical framework but the fundamental point is that the Cauchy stress tensor is not sufﬁcient to describe internal forces. External forces concentrate along any edge of the boundary and the Cauchy theorem deﬁning the Cauchy stress tensor cannot be applied. Moreover, a supplementary boundary condition is needed to write well-posed problems, which is unusual and not intuitive. The simplest way to describe these media is to use the Second Gradient theory.

The objective of the symposium is to apply the Second Gradient Theory to the propagation of various defects like cracks, damage in Continua Media.

The Symposium is aimed at PhD students, postdocs and researchers in applied mathematics and mechanics.

**Schedule**

Monday

4 July 2011

10:00

Registration

10:30 – 11:30

OPENING

The Speakers will open the Symposium presenting briefly the topics of the event and the basic idea from which this event was born; in this section they establish, in collaboration with the participants, weekly program of the Symposium.

**Lecturers**

**Gilles Francfort** is Professor of Mathematics at the Université Paris-Nord. His research topics of interest lie primarily in the modelling and mathematical formulation of problems in Solid Mechanics. In recent years the main focus has been the development of mathematically consistent models for the propagation of various defects (cracks, damage, …) in an otherwise elastic material.

**Pierre Seppecher** is professor of the Institut de Mathématiques, Université du Sud-Toulon-Var. His main research areas are the Second Grandient Theory, theory of capillarity, differential geometry, relaxation methods and problems of singular perturbations, topology and measure theory. In particular his work in solid mechanics includes the properties of continua for which energy depends on the second gradient of the displacement.