Fracture Mechanics

Fracture mechanics is a significant scientific field of great practical importance (Freund, 1990; Lawn, 1993). Recently the subject has been invigorated by a number of important accomplishments. From the viewpoint of fundamental science, there have been interesting new developments aimed at understanding fracture at the atomic scale (Abraham et al. 2002); simultaneously there has been active research focused on modelling, experimentation and computation at macroscopic scales (Willis, 2003).

Despite these important advances, existing mathematical models of fracture fall short of providing general theoretical formulations which place key existing calculations of fracture mechanics in a broader setting, while enabling new predictions of fracture initiation, growth and morphology to be made. In order to provide and analyse such formulations, one has to understand the physical processes involved in different types of fracture in different materials, and also to carefully relate them to an appropriate function-space setting which is not prejudiced regarding possible fracture geometries. For a survey of challenging open problems in the field, we refer to the recent review article by Cox et al. (2005).

Stimulated by the work of Mumford & Shah on image segmentation problems, a variational theory was initiated by Francfort and Marigo (1998) aimed at modelling brittle fracture via a static theory of free-discontinuity type. The conceptual appeal of this approach is that the crack set is a genuine unknown, and could in principle have very complex geometry. The model is close in spirit to Griffith’s model of brittle fracture (Griffith, 1921), while being free from the usual shortcomings of that theory, namely, the hypotheses of a preexisting crack and a well-defined crack path. Indeed, the Francfort-Marigo theory requires no a priori knowledge of the crack path or of its topology; moreover, it is capable of modelling crack initiation and propagation within the same theoretical framework. The model has also been shown to be a powerful tool whose predictive capabilities go beyond those of some more standard computational algorithms for the numerical simulation of fracture. A significant research effort has gone into analyzing the theory; see for example Francfort & Larsen (2003), Dal Maso, Francfort & Toader (2005). On the other hand, performing numerical computations in this framework is very difficult (Bourdin, Francfort & Marigo, 2000), and it is only a static model. There is no corresponding dynamic formulation, except in the quasistatic case.

The predictive computational modelling of fracture in solids subjected to loading is becoming extremely important as the streamlining in engineering design reaches toward the ultimate limits of material-carrying capacity. This is particularly so in situations of extreme loading caused by explosions or collisions in which safety aspects are imperative. Yet, the accurate prediction of fracture and failure processes remains a challenging task from the computational point of view.

The specific projects associated with this theme are

Mathematical and numerical analysis of free-discontinuity problems
Cavitation
Gradient flows and nonconvex energies
Continuum limits of atomistic energies and new computational models of fracture.

References

F.F. Abraham, R. Walkup, H. Gao, M. Duchaineau, T.D. De la Rubia and M. Seager. Simulating materials failure by using up to one billion atoms and the world’s fastest computer: brittle fracture. Proc. Natl. Acad. Sci. USA, 99(9):5777–5782, 2002.
B. Bourdin, G.A. Francfort and J.-J. Marigo. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, 48(4):797–826, 2000.
B.N. Cox, H. Gao, D. Gross and D. Rittel. Modern topics and challenges in dynamic fracture. J. Mech. Phys. Solids, 53(3):565–596, 2005.
G. Dal Maso, G.A. Francfort and R. Toader. Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal., 176(2):165–225, 2005.
G.A. Francfort and C.J. Larsen. Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56(10):1465–1500, 2003.
G.A. Francfort and J.-J. Marigo. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8):1319–1342, 1998.
L.B. Freund. Dynamic fracture mechanics, Cambridge monographs on mechanics and applied mathematics. Cambridge University Press, 1990.
A.A. Griffith. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. London, A221:163–198, 1921.
B.R. Lawn. Fracture of brittle solids. Second edition, Cambridge University Press, 1993.
J.R. Willis. Dynamic perturbation of a propagating crack: implications for crack stability. In: Asymptotics, Singularities and Homogenization in Problems of Mechanics, pages 201–210. Edited by A.B. Movchan. Kluwer, Dordrecht, 2003.