## Colloquium - Flaviana Iurlano - Phase-field approximation of cohesive fracture energies

- Calendar
- Mathematics & Statistics

- Date
- 02.11.2022 10:30 am - 11:30 am

### Description

__Speaker: Flaviana Iurlano (Laboratoire Jacques-Louis Lions (LJLL) Sorbonne Université)__

https://sites.google.com/site/flavianaiurlano

Title: Phase-field approximation of cohesive fracture energies

Abstract: Variational models in Fracture Mechanics are effectively described through functional spaces with discontinuities. The most renowned example is Griffith's energy for brittle fracture, describing a situation in which already for the smallest opening there is no interaction between the two sides of the crack. In ductile materials, crack proceeds rather through the opening of a series of voids separated by thin filaments, which produce a weak bond between the lips at moderate openings (cohesive fracture).

A large literature has been devoted to the derivation of models including interfaces from more regular models, like damage or phase-field models, mainly within the framework of Gamma-convergence. These approximations can be interpreted both as microscopic physical models, so that the Gamma-convergence justifies the macroscopic model, and as regularization, therefore they can be used for example in numerical simulations. The first work of this sort is Ambrosio--Tortorelli '90, which provides a phase-field approximation of the Mumford-Shah functional for brittle fracture in the case of scalar-valued displacements. The corresponding result for cohesive fracture has been only recently obtained in Conti--Focardi--Iurlano '16. Numerical tests and experiments have been provided in Freddi--Iurlano '17.

The general case of vector-valued displacements, in a geometrically nonlinear cohesive framework, is the object of a work in preparation. In our phase-field models the elastic coefficient is computed from the damage variable $v$ through the function $f_\varepsilon(v):=\min\{1,\varepsilon^{\frac{1}{2}} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting absolute continuous density is quasiconvex with 1-growth at infinity, while the fracture density, depending on the opening of the crack and on the normal vector to the crack set, is given in terms of an asymptotic n-dimensional formula, is bounded, and is one-homogeneous for small openings.

We will conclude this colloquium by discussing the possibility of including time in our models and constructing cohesive quasi-static evolutions. We will present a rigorous analysis in 1D (Bonacini--Conti--Iurlano '21). In higher dimensions, even in the case of scalar-valued displacements, this is an open problem. The available results in literature require knowing the crack path in advance.

Location: Virtual

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