The problem of damage in heterogeneous materials has received particular attention in recent years. The numerical models currently used in the simulation of damage require an internal length that is not currently related to a characteristic length of the material components. However, understanding damage regarding the size of the heterogeneities of the material is of crucial importance, particularly in civil engineering. The Fiber Bundle Model has been widely used to qualitatively address the issue of damage in heterogeneous media by studying the statistics of failure events during damage. The so-called ZIP model derives from Fiber Bundle Model to mimic crack propagation. In this work, a spatial correlation of tensile strength of fibers is added to the ZIP model to highlight the role of heterogeneity size in statistics of failure events during crack propagation. The addition of spatial correlation into the ZIP model modifies the distribution of failure events. Indeed, for a simulated material without spatial correlation, failure events follow two regimes. By adding a spatial correlation to the material, a transitional regime appears. The influence of spatial correlation on fiber rupture avalanche strongly depends on the ratio between the sizes of the shapes of the stress field and of the heterogeneities.

This work deals with fracture mechanisms in quasi-brittle materials, focusing on the characterization of the Fracture Process Zone (FPZ) of specimens under tensile load. Particularly, paper was used as model material. Experiments were conducted on notched and unnotched specimens. Based on an image analysis of these observations, a stochastic finite element model was developed, using both a nonlocal stress-based approach and a discretized random field modelling of the Young's modulus. The proposed methodology allowed characterizing the damage zone and the size of the FPZ, analysing the influence of the mesostructure, composed of flocs (fiber aggregates where the basis weight is larger than the average one) and antiflocs (complement of flocs). The area of the active FPZ and the normalized stress drop were linked using a surface energy dissipated in the active FPZ. The stress drop, until limiting value, increased with the width of the active FPZ. Finally, a relationship between the surface energy and the nonlocal internal length was established.

Mechanical properties of fiber based materials, such as paper, are governed by the heterogeneous microstructure induced by the formation process. For example, in paper materials during tensile tests, lower strains are observed in the flocs (fiber aggregates where the basis weight is larger than the average one) than in the antiflocs (complement of flocs). To explain such behavior, we investigated the mechanical and structural properties of both flocs and antiflocs. Using tensile tests on flocs and antiflocs, we measured their elastic modulus and found that the elastic modulus of flocs was larger than the antiflocs' one. To understand these differences, a multiscale structural analysis was conducted on 3D X-ray images of flocs and antiflocs at two different spatial resolutions. We found that the floc was thicker than the antifloc, whereas the microscale analysis showed the studied floc and antifloc exhibit the same bulk porosity. Then, it was concluded that the difference in elastic modulus between flocs and antiflocs is mainly due to their thickness difference.