Cohesive and Variational Methods for Fracture Mechanics in Statics and Fatigue

The widespread use of material over the past century has dramatically changed the world today. The reliability of any machine depends on the multiple com- plex interactions in the system leading to a failure. Hence, it is necessary to understand the failure mechanism of the structures with a multi-physics inter- action. The present thesis explores the role of complex multi-physics in failure to understand the overall structural performance, including an interface between different materials. Specifically, in Chapter-2, a phase field (PF) approximation of fracture for func- tionally graded materials (FGM) using a diffusive crack approach incorporat- ing the characteristic length scale as a material parameter is herein proposed. A rule of mixture is employed to estimate the material properties, according to the volume fractions of the constituent materials, which have been varied accord- ing to given grading profiles. Based on the ideas stemming from the study of size-scale effects, Γ-convergence for the proposed model is proved when the in- ternal length scale is either constant or a bounded function. Crack propagation events in conjunction with the differences with respect to their homogeneous surrogates are discussed through several representative applications, providing equivalence relationships for size-scale effects and demonstrating the applica- bility of the current model for structural analysis of FGMs. Failure processes in Laminated Fiber-Reinforced Composites (LFRCs) entail the development and progression of different physical mechanisms and, in partic- ular, the interaction between inter-laminar and intra-laminar cracking. Reliable modeling of such complex scenarios can be achieved by developing robust nu- merical predictive tools that allow for the interaction of both failure modes. In Chapter-3, a novel Multi Phase-Field (MPF) model relying on the Puck theory of failure for intra-laminar failure at ply level is coupled with a Cohesive Zone Model (CZM) for inter-laminar cracking. The computational tool is applied to qualitatively predict delamination migration in long laminated fiber-reinforced polymers composites comprising 44 cross-ply laminates. The reliability of the current approach is examined via the correlation with experimental results. Fi- nally, the present study is complemented with additional representative exam- ples with the aim of providing further insight into the potential role of different aspects of the system in the delamination migration, including (i) the variation of the ply angle in the migration zone, (ii) the load application point, and (iii) initial crack length. The analysis of fracture phenomena of thin-walled structures has been a matter of intensive research in the last decades. These phenomena notably restrict the applicability of slender structures, especially under the influence of tempera- ture. The research in Chapter-4 is concerned with the development of a thermo- dynamically consistent framework for the coupled thermo-mechanical phase- xx field model for thin-walled structures using a fully-integrated finite elements. This enables the use of three-dimensional constitutive thermo-mechanical mod- els for the materials. The proposed thermo-mechanical phase-field models are equipped with the EAS and ANS leading ti locking free element. Moreover, the same degradation function is used for both displacement field and thermal field. The coupled equations are numerically solved with ad hoc efficient solution schemes for nonlinear problems. Several numerical examples with and without phase-field (straight and curved shells) are provided to show the practicality and reliability of the proposed modeling framework. Moreover, the model is extended to incorporate FGM and the corresponding numerical examples are explored. Using the framework developed in the Chapter-4, the locking free solid shell element is extended to include the fatigue effects in Chapter-5. As a natural consequence of the developed model, SN curve and crack extension curves are recovered for straight and curved shells.