The degradation of quasi-brittle materials encompasses micro-cracks propagation, interaction and coalescence in order to form a macro-crack. These phenomena are located within the Fracture Process Zone (FPZ). This paper aims at providing a further insight in the description of the FPZ evolution with the help of statistical analysis of damage.

The statistical analysis relies on the implementation of Ripley's functions [1], which have been developed in order to exhibit patterns in image analyses. Particularly, it was used to characterize the randomness in the spatial spreading of point distributions. It is of high interest in spatial ecology and has been used to characterize the development and spreading of different patterns, such as cell migration, tree and plant dissemination or disease spreading. Ripley's functions allow exhibiting a correlation length between fracture events from which a parallel can be drawn with the internal length entering in non-local models [2]. In the present approach, the analyses are carried out with the help of a lattice approach, which has been shown [3-4] to be representative of fracture of concrete, including size effect tests on notched and unnotched specimens [5].

First, three point bending tests on notched concrete specimens are considered. Ripley's analysis is compared with experimental data from acoustic emission. Comparisons between experimental and numerical evolutions of extracted correlation lengths are performed and they reveal that the lattice model captures very well the correlation of damage events observed experimentally. Then, numerical analyses are performed on specimens of different shapes and with different loads. It is shown that the correlation length can be very different depending on the geometry and boundary conditions, which suggests that the internal length in non local model should not be constant, but depends on how damage initiates (notch effect/boundary effect) especially [6].

Acknowledgment

Financial support from TOTAL Exploration & Production under the project "Fracture and permeability of heterogeneous quasi-brittle materials" is gratefully acknowledged.

The authors wish also to acknowledge the University of Bordeaux for the use of the cluster AVAKAS.

D. Grégoire and G. Pijaudier-Cabot are fellows of the Institut Universitaire de France.

References

[1] B.D. Ripley, Modelling spatial patterns, J R Stat Soc Ser B (Meth), 39, 172-212, 1977.

[2] Vincent Lefort, Gilles Pijaudier-Cabot, David Grégoire, Analysis by Ripley's function of the correlations involved during failure in quasi-brittle materials: Experimental and numerical investigations at the mesoscale, Engineering Fracture Mechanics, 147, 449-467, 2015.

[3] Peter Grassl, David Grégoire, Laura Rojas-Solano, Gilles Pijaudier-Cabot, Meso-scale modelling of the size effect on the fracture process zone of concrete, International Journal of Solids and Structures, 49 (13), 1818-1827, 2012.

[4] David Grégoire, Peter Grassl, Laura Verdon, Vincent Lefort, Jacqueline Saliba, Jean-Pierre Regoin, Ahmed Loukili, Gilles Pijaudier Cabot, Mesoscale analysis of failure in quasi-brittle materials: comparison between lattice model and acoustic emission data, International Journal for Numerical and Analytical Methods in Geomechanics, 39, 1639-1664, 2015

[5] David Grégoire, Laura Rojas-Solano, Gilles Pijaudier-Cabot, Failure and size effect for notched and unnotched concrete beams, International Journal for Numerical and Analytical Methods in Geomechanics, 37, 1434-1452, 2013.

[6] Vincent Lefort, Gilles Pijaudier-Cabot, David Grégoire, Analysis by Ripley's function of the correlations involved during failure in quasi-brittle materials: Experimental and numerical investigations at the mesoscale, Engineering Fracture Mechanics, 147, 449–467, 2015.