In the present textbook a computational method in deriving stress and strain functions related to Fracture Mechanics is considered. This textbook contains a compilation of work available in the literature in a scatter form and, to a certain extent, selected experimental data of many researchers to justify the theoretical and computational fracture mechanics models in solving crack problems. It is a self-contained and detailed textbook for the reader (senior and graduate students, and engineers) involved in the analysis of failure using a computational approach for designing against fracture. In the present textbook, the author has attempted to cover the basic concepts of fracture mechanics for the linear elastic regime, and finite element implementation of the linear elastic crack simulation under static and dynamic loading. The material included in the textbook is based upon the development of analytical and numerical procedures pertinent to particular fields of linear elastic fracture mechanics (LEFM), including mixed-mode-loading interaction.