Fracture mechanics is essential for predicting the failure of materials and structures. The Phase-Field Fracture (PFF) method has emerged as a powerful computational tool, overcoming the limitations of classical discrete approaches by modeling cracks as diffuse damage bands. This formulation eliminates the need for explicit crack tracking and naturally handles complex topologies like branching and merging. However, despite its success, the method faces significant numerical and physical challenges, particularly regarding the robust enforcement of inequality constraints in some regularization models, the accurate measurement of geometric quantities distorted by strain localization, and the high computational cost of cycle-by-cycle fatigue simulations.

The computational challenge of fatiguewhere phase-field fracture approaches typically rely on cycle-by-cycle simulations to accumulate damage over thousands of cycles, leading to prohibitive computational costsis addressed in this thesis through a framework that enables fatigue analysis using only monotonic loading. This is made possible by a specialized energy-controlled solver designed to robustly track the complete crack equilibrium path, capturing instabilities such as snap-back. By performing a single quasi-static simulation, the method numerically extracts the compliance rate of the structure with respect to the crack area, which is then used to bridge the results with Linear Elastic Fracture Mechanics (LEFM) and integrate Paris’ law. This allows for the prediction of fatigue life with drastically reduced computational effort while maintaining the topological flexibility of the phase-field method.

To ensure the accuracy of the geometric quantities extracted from these simulations, the work addresses the issue of crack area overestimation caused by numerical strain localization, which also affects the force and displacement values returned by the simulations. The Double Gradient Correction Method (DGCM) is proposed, founded on the principle of energy equipartition, where the energy contributions from the phase-field and its gradient are equal as the length-scale parameter approaches zero. Since numerical artifacts primarily affect the phase-field term while leaving the gradient term largely unperturbed, the crack area can be accurately approximated as twice the gradient-dependent energy. This method is inherently mesh-independent and readily applicable to the entire domain, including 3D simulations.

Finally, to enhance the robustness of constrained phase-field models beyond standard formulations, the Latent Variable Proximal Point (LVPP) algorithm is presented. In phase-field fracture simulations, the phase-field variable must remain within an admissible range typically between 0 (undamaged) and 1 (fully damaged) and satisfy an irreversibility condition to ensure damage does not decrease. While some standard models naturally respect the phase-field interval constraint, other regularizations require these physical constraints to be explicitly and robustly imposed. To address this, the LVPP framework reformulates these inequality-constrained problems as a sequence of unconstrained saddle-point problems via an auxiliary latent variable. This allows for the use of efficient Proximal Galerkin discretizations, providing a rigorous and robust solution that avoids the parameter sensitivity and convergence issues associated with traditional penalty methods.

All methodologies developed in this thesis are implemented in PhaseFieldX, an open-source software package designed to ensure reproducibility and to facilitate future research in the field.