This research develops two novel numerical methods for applications in fracture mechanics: (I) A new crack tip enrichment function in the extended finite element method (XFEM), and (II) a discrete damage zone model for quasi-static and fatigue delamination in composites. The first method improves XFEM when applied to general nonlinear materials when crack tip analytical solutions are not available. For linear elastic materials, Branch functions are commonly used as crack tip enrichments. Typically, these are four functions derived from linear elasticity theory and added as additional degrees of freedom. However, for general inelastic material behavior, where the analytical solution and the order of singularity are unknown, Branch functions are typically not used, and only the Heaviside function is employed. This however may introduce numerical error, such as inconsistency in the position of the crack tip. Hence, a special construction of Ramp function is proposed as tip enrichment, which may alleviate some of the problems associated with the Heaviside function when applied to general nonlinear materials, especially ones with no analytical solutions available. The idea is to linearly ramp down the displacement jump on the opposite sides of the crack to the actual crack tip, which may stop the crack at any point within an element, employing only one enrichment function. Moreover, a material length scale that controls the slope of the ramping is introduced to allow for better flexibility in modeling general nonlinear materials. Numerical examples for ideal and hardening elasto-plastic and elasto-viscoplastic materials are given, and the convergence studies show that a better performance is obtained by the proposed Ramp function in comparison with the Heaviside function. Nevertheless, when analytical functions, such as the Hutchinson-Rice-Rosengren (HRR) fields, do exist (for very limited material models), they indeed perform better than the proposed Ramp function. However, they also employ more degrees of freedom per node and hence are more expensive. The second method developed in this thesis is a discrete damage zone model (DDZM) to simulate delamination in composite laminates. The method is aimed at simulating fracture initiation and propagation within the framework of the finite element method. In this approach, rather than employing specific cohesive laws, we employ damage laws to prescribe both interface spring softening and bulk material stiffness degradation to study crack propagation. For a homogeneous isotropic material the same damage law is assumed to hold in both the continuum and the interface elements. The irreversibility of damage naturally accounts for the reduction in material strength and stiffness if the material was previously loaded beyond the elastic limit. The model parameters for interface element are calculated from the principles of linear elastic fracture mechanics. The model is implemented in Abaqus and numerical results for single-mode as well as mixed-mode delamination are presented. The results are in good agreement with those obtained from the virtual crack closure technique (VCCT) and available analytical solutions, thus, illustrating the validity of this approach. The suitability of the method for studying fracture in fiber-matrix composites involving fiber debonding and matrix cracking is demonstrated. Finally, the DDZM method is extended to account for temperature dependent fatigue delamination in composites. The interface element softening is described by a combination of static and fatigue damage growth laws so as to model delamination under high-cycle fatigue. The dependence of fatigue delamination on the ambient temperature is incorporated by introducing an Arrhenius type relation into the damage evolution law. Numerical results for mode I, mode II and mixed mode delamination growth under cyclic loading are presented and the model parameters are calibrated using previously published experimental data. Then, predictions are made under varying mode mix conditions and are compared with numerical results in the literature.
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