"The modelling of micromechanical mechanisms occurring during damage of solids is fundamental for the understanding of how materials fail and consequently structures collapse. For example, predominant mechanism of damage in brittle and quasi-brittle materials is the propagation of a pre-existing dominant crack-like defect, whereas micro-instabilities and localization are failure mechanisms in structured and multiphase materials. A challenge in the modern Solid and Structural Mechanics is the description of such mechanisms in advanced materials with complex heterogeneous microstructure. The theory of fracture for homogeneous bodies has been initiated with the pioneering work by Griffith on strength of glass fibers and, after the second world war, developed by Williams, Irwin, Rice, and others and now it is a well-developed discipline. However, for heterogeneous materials, the theory of fracture is still under development and many challenging problems are still unsolved. The motivation for the extension of the theory to heterogeneous materials comes from the observation that many natural as well as man-made composite materials may exhibit peculiar mechanical properties and high resistance to fracture propagation. The general goal of the research project is the advance in the fundamental understanding of crack propagation, localization and damage progression in complex heterogeneous materials. In this context, a special effort will be devoted to the analytical solution of still unsolved fundamental mathematical problems in view of the application to the description of material failure in composite solids."