Numerical modelling of crack propagation in quasi-brittle heterogeneous materials : a stochastic approach

Deformation and damage processes in brittle and quasi-brittle materials, such as rock and concrete, are strongly influenced by their heterogeneous nature, related to their formation processes. The presence of heterogeneities leads in fact to noticeable variation in material properties values: it is of extreme importance that a numerical model which aims to realistically, reliably reproduce with low computational effort deformation and damage processes is able to include the effect of laminations, micro-cracks, voids and other types of heterogeneities; this is even more important when a numerical models has to reproduce the propagation of fractures. This thesis presents the development of a numerical framework for the simulation of crack propagation in shale rocks and concrete which also looks at the optimisation problem in the sense of computational efficiency (defined as optimal computational time needed to obtain realistic and accurate results). The numerical framework for crack propagation developed in this thesis is a variational phase-field model based on a finite elements smeared approach, able to automatically and realistically capture crack initiation processes for a variety of loading conditions; this numerical framework is based on the relation between potential energy associated to body deformation and the energy released during fracture formation. Heterogeneity is considered in the model by means of a stochastic approach based on the assumption that some mechanical properties of heterogeneous brittle materials (such as fracture energy) follow a non-Gaussian Weibull distribution. To guarantee adequate convergence of the results, Monte Carlo Simulation (MCS) method has been used in combination with the developed stochastic methodology. A non-linear dimensionality reduction technique has been developed and incorporated in the algorithm to reduce the computational effort required for the generation of sample realisations. The methodology has been validated using experimental results from both laboratory tests on shale rocks and literature on fracture in concrete. Results show that the developed algorithm is capable of realistically reproducing the mechanical behaviour of the chosen case studies, showing an applicability to problems where cracks propagate in mode-I, mode-II and mixed-mode I and II, guaranteeing a fast generation of sampling realisations of realistic stochastic fields and convergence of results after a maximum of 130 MCS analyses. This methodology can be applied to materials with random spatially-distributed variations of mechanical properties and to those showing laminar natural formations