A Variational Approach to Brittle Fracture

in collaboration with A. Chambolle (École Polytechnique, France), G. A. Francfort (Université Paris-Nord France), C. Larsen (Worcester Polytechnic Insitute, MA), and J-J. Marigo (Université Pierre et Marie Curie, France).

Fracture mechanics is a very active area of research, with vital applications. In recent years, the unexpected collapse of terminal 2E at Charles de Gaulle airport in France or the Columbia space shuttle disintegration upon re-entry illustrate the importance of a better understanding of the mechanism of fracture, as well as its numerical simulation.
In the area of brittle fracture, there has seen tremendous progress over the recent years, but some issues remain problematic. The most widely accepted theories, based on Griffith’s criterion, are limited to the propagation of an isolated, pre-existing crack along a given path.
   Extending Griffith’s theory into a global minimization principle, while preserving its essence, the concept of energy restitution in between surface and bulk terms, G. Francfort and J.-J. Marigo proposed in [FM98] a new formulation for the brittle fracture problem, capable of predicting the creation of new cracks, their path, and their interactions, in two and three space dimensions.

The essence of this model is the global minimization over all crack sets K and all admissible displacement fields u of an energy similar to

{\cal E}(u,K) = \frac{1}{2} \int_{\Omega\setminus K} W(e(u))\, dx +{\cal H}^{N-1}(K\cap \bar{\Omega})<br />{\cal E}(u,K) = \frac{1}{2} \int_{\Omega\setminus K} W(e(u))\, dx +{\cal H}^{N-1}(K\cap \bar{\Omega})<br />{\cal E}(u,K) = \frac{1}{2} \int_{\Omega\setminus K} W(e(u))\, dx +{\cal H}^{N-1}(K\cap \bar{\Omega})

where W denotes an elastic potential, and H is the Hausdorff measure.

The numerical minimization of such a functional is of course challenging.   The numerical experiments on this page are based on the concept of variational approximations of functionals by Γ-convergence, and are described in more detail in [BFM00].

Traction experiment

A cylinder is reinforced in its center by a ductile shaft. It is clamped on its lower end, and a force is applied to its upper end, along the direction of the symmetry axis. The brittle part is not displayed and the red helicoidal surface represents the crack surface, in the reference configuration.


CylCrackForce

See the movie:
[coarse mesh, 640x480, 2.6Mb] [coarse mesh, 320x240, 640kb]
[fine mesh, incomplete, 640x480, 1.7Mb] [fine mesh, incomplete, 320x240, 532kb]



Support for this work was provided in part by the National Science Foundation under grant DMS-0605320. The computations presented in these pages have been performed on LSU's supermike and on the teragrid supercomputers, under NSF Cyber-Infrastructure Partnership Development Allocation TG-DMS060007T.

References

[BFM00] B. Bourdin, G.A. Francfort and J.-J. Marigo, Numerical Experiments in Revisited Brittle Fracture, J. Mech. Phys. Solids, 48 (2000), no 4, 797-826. [Preprint] [DOI: 10.1016/S0022-5096(99)00028-9]
[FM98] G. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids. 46 (1998), no. 8, 1319-1342. [DOI: 10.1016/S0022-5096(98)00034-9]