BriCoFra · New ideas for the Variational Approach to Brittle and Cohesive Fracture

New ideas for the Variational Approach to Brittle and Cohesive Fracture

During World War I the aeronautical engineer A.A. Griffith formulated a theory to explain failure of materials, based on the idea that crack growth is the result of the competition between surface energy spent to produce fracture and energy stored in the uncracked region. This found a rigorous mathematical setting in the 'variational formulation for quasistatic evolutions (QSE)' by Francfort-Marigo. They devised, in 1998, a general procedure trough an implicit time-sheme based on energy minimisation, which calls for new techniques in Calculus of Variations. Remarkably, it provided a general framework for also post-Griffith theories, in particular the one due to Barenblatt. Griffith and Barenblatt theories differ in the surface energy dissipated in the fracture: the first is proportional to the measure of the fracture set (surface in 3d, length in 2d), the latter depends also on the amplitude of the opening between the two sides of the crack. Microscopically, in the first case (brittle fracture) any material point is either broken or sound, in the latter (cohesive fracture) restorative forces, depending on the opening, are present between the lips of the crack set. In fact, in formation of cohesive cracks the role of plastic deformations is determinant: these deformations are dissipative (and irreversible), so that the material is not forced to dissipate energy only along cracks and the fracture process is slowed down; in contrast, in brittle fracture the solid is subject only to elastic deformations (reversible) in the uncracked zone, which causes sudden cracks and fast propagation. The application in Numerical Analysis relies on the approximation of fracture energies in terms of so-called damage energies. The minimisation of fracture energies is a classical 'free discontinuity problem', namely it contains a set variable (the crack) as unknown, which makes the numerical simulation hard. In damage models the role of fracture is played by a function variable, supported on a thin 3d-neighborhood of the crack surface, and the resulting energy is more regular in the damage variable. Brittle fracture is approximated by coupling of damage and elasticity, while coupling of damage and elasto-plasticity approximates cohesive fracture (cf. the mechanical interpretation above), in terms of Gamma-convergence, guaranteeing convergence of minima and minimisers of the associated (static) problem. Such approximations, called 'à la Ambrosio-Tortorelli', are nowadays employed in thousands of papers modelling fracture. (In figure, approximation of discontinuity through a smooth profile, restricted to 1d) The overall objective of the project is to progress the mathematical analysis of brittle/cohesive fracture, in synergy with that of the associated damage models.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз