Statistical models of fracture

Abstract

Disorder and long-range interactions are two of the key components that make material failure an interesting playfield for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture in which a network of elastic beams, bonds, or electrical fuses with random failure thresholds are subject to an increasing external load. These models describe on a qualitative level the failure processes of real, brittle, or quasi-brittle materials. This has been particularly important in solving the classical engineering problems of material strength: the size dependence of maximum stress and its sample-to-sample statistical fluctuations. At the same time, lattice models pose many new fundamental questions in statistical physics, such as the relation between fracture and phase transitions. Experimental results point out to the existence of an intriguing crackling noise in the acoustic emission and of self-affine fractals in the crack surface morphology. Recent advances in computer power have enabled considerable progress in the understanding of such models. Among these partly still controversial issues, are the scaling and size-effects in material strength and accumulated damage, the statistics of avalanches or bursts of microfailures, and the morphology of the crack surface. Here we present an overview of the results obtained with lattice models for fracture, highlighting the relations with statistical physics theories and more conventional fracture mechanics approaches.

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Acknowledgments

MJA would like to thank the Centre of Excellence program of the Academy of Finland and Tekes – The National Technology Agency of Finland – for financial support. A number of students, colleagues, and partners in research (K. Niskanen, V. Räisänen, L. Salminen, E. Seppälä, J. Rosti, J. Lohi) have contributed to the related research. PKVVN is sponsored by the Mathematical, Information and Computational Sciences Division, Office of Advanced Scientific Computing Research, US Department of Energy under contract number DE-AC05-00OR22725 with UT-Battelle, LLC. In addition, PKVVN acknowledges the discussions with and, the support received from Dr. Srdjan Simunovic. SZ wishes to thank A. Baldassarri, D. Bonamy, G. Caldarelli, A. Hansen, H. J. Herrmann, F. Kun, M. Minozzi, A. Petri, P. Ray, S. Roux, H.E. Stanley, A. Vespignani, and M. Zaiser for collaborations, discussions, and remarks on the topics discussed in this review.

Notes

¹The standard KPZ term is , a non-local KPZ term is written in Fourier space as .