13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- Another approach would be to base the weakest-link statistics on the microstructure information, in this case the size distribution of crack-initiation features, the pores. This is similar to the approach used in the modelling of cleavage fracture, where second-phase particles are considered to be cleavage initiators. In order to show whether this is a realistic approach, a comparison is made between the probability density of pore sizes distributed in the lattice and pore sizes contained in the maximal component at failure. The results are shown in Fig. 6 for the pores in the entire lattice (a) and three of the loading cases as depicted. The results shown correspond to one and the same random assignment of pore sizes. Evidently, the probability density of the pores belonging to the final fracture surface is different from the lattice distribution and depends on the loading mode. While the initial damage may start at one and the same location in the system, the nature of loading develops the main crack in different ways and the final failure cannot be described as a weakest-link event using the statistics of the sizes of the failure initiation sites. This makes it difficult to derive a load-independent, purely micro-structure based relation between the macroscopic damage and the probability of failure. The outcome supports further the suggestion that macroscopic failure analysis needs to be performed with an underlying lattice-based analysis of local micro-crack propagation. 4. Conclusions • A microstructure-informed strategy for analysis of damage evolution in quasi-brittle materials was presented, whereas damage results from the formation, growth and interaction of a population of micro-cracks. • It was demonstrated that in cases of non-uniaxial extension, such as plane stress or plane strain found ahead of a main crack, the micro-crack population development was responsible for elastic anisotropy with extreme variations of longitudinal shear moduli. • It was shown that the damage-induced anisotropy was a complex function of the crack population structure. A load-independent damage evolution law might not be achievable and explicit analysis of crack population development, e.g. using a lattice model, might be necessary to complement continuum finite element analysis of failure. • It was shown that the maximal connected component of the crack population, i.e. the largest crack, became dominant very early in the process of macroscopic damage and controlled the ultimate failure. The analysis if this component suggested that the global failure could not be treated as a weakest-link event. • The graph-theoretical approach to the analysis of micro-crack populations showed significant potential to reveal the underlying topological structure of the cracked surface. Further work is required to link the topological structure to a measure for global probability of failure. Acknowledgements The author is grateful to EPSRC for the support for this work via grant EP/J019763/1, as well as to BNFL for the endowment allowing his continuous research. References [1] G. Pijaudier-Cabot, Z.P. Bazant, Nonlocal damage theory. J Eng Mech 113 (1987) 1512-1533. [2] J.W. Ju, On energy-based coupled elastoplastic damage theories: constitutive modelling and computational aspects. Int J Solids Struct 25 (1989) 803-833.
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