The work proposed is a contribution to a better understanding the cracking mechanisms under mixed mode loading by analytical and numerical modelling in fracture mechanics in the three-dimensional medium. The fracture usually occurs as a result of a multitude of combined mechanical loads that are accompanied by multiple secondary disruptions, such as climatic variations over time. Predictive mathematical models are needed to quantify fracture parameters in the three-dimensional medium, and then predict the behaviour of cracking and assess the risk of failure. The three-dimensional modelling of the fracture is important because we do not have to perform destructive tests to conduct different investigations. The assumptions of plane strain or plane stress considered in two-dimensional case are not always valid. The third dimension in the mathematical model allows a better understanding of all the phenomena highlighted during the cracking, In particular in massive structures where cracking occurs inside the material.

In this work, a mathematical formulation based on an energy approach is developed for the study of mixed mode crack problem in three-dimensional medium. Thanks to Noether's theorem, it proposes a generalization of the surface and volume integrals in order to compute the energy release rate and its distribution along the crack front. This generalization will allow the fracture modes separation due to the introduction of a virtual displacement field.

A numerical validation, in terms of energy release rate, is carried out on a Mixed Mode crack Growth MMCG specimen under mixed-mode loading for different thickness. The finite element numerical computation are conducted to validate the invariance property of the proposed new integral, as well as comparisons with two-dimensional case. The invariance property of the integration path around the crack front is verified for the each fracture modes, as well as the influence of the fracture modes evaluated according to the thickness exploration value. The fracture mode separation is extended to evaluate out of plan complex loads; mode III, mixed mode I + II + III and mixed mode I + III.