Phase field modeling of fracture nucleation and propagation: A theoretical and computational study

This thesis presents both theoretical results - most notably, a rigorous proof that steady-state phase-field evolutions satisfy Griffith’s criterion and Gamma-convergence analyses for new higher-order and cohesive models - and practical advances in the numerical simulation of complex fracture processes. Chapter 1 provides an introduction and an historical and conceptual overview of fracture mechanics, detailing sharp-crack models, diffuse phase-field formulations and their relationship. It also discusses the inherent numerical challenges of both frameworks, emphasizing the need for robust algorithms that ensure convergence and physically meaningful evolutions. In Chapter 2, a class of phase-field energies incorporating both volumetric–deviatoric and spectral decompositions is considered. The time-continuous limits of the resulting evolutions are analyzed, proving that in the steady-state regime they satisfy a phase- field version of Griffith’s criterion in terms of toughness and a suitably defined energy release rate. This analysis - valid independently of the adopted incremental scheme - clarifies the deep connection between the phase-field and sharp-crack approaches and is complemented by numerical simulations that compare their respective evolutions. Within the phase-field framework, several choices exist in the literature for both the elastic and fracture energies. Along this line, Chapter 3 introduces a novel fourth-order AT1 phase-field model. For the proposed functional, a rigorous Gamma-convergence result is established, and numerical simulations confirm its acuracy and efficiency. AT1 models naturally account for crack nucleation, however their effective strength surface in stress space is elliptic, failing to reproduce the strong asymmetry between tensile and compressive responses observed experimentally. Several extensions based on energy decompositions have been proposed to address this limitation; nevertheless, the flexibility of such models remains limited, reflecting an intrinsic feature of Griffith’s theory, which lacks a notion of material strength. Historically, the conceptual gap between strength-based and energy-based approaches was bridged by cohesive zone models and in Chapter 4 a cohesive phase-field functional recently proposed by Vicentini et al. is studied. This model enables explicit control over the shape of the strength surface while decoupling strength from the regularization parameter. A Gamma-convergence analysis toward a sharp cohesive fracture model is presented in both one- and two-dimensional (antiplane) settings, using a finite element discrete formulation and exploiting the strong localization of the damage variable. Numerical simulations explore the sensitivity of the model to mesh anisotropy, offering insight into both its theoretical robustness and its practical implementation. Finally, Chapter 5 introduces an alternative route toward enhanced flexibility in defining strength surfaces, grounded in the theory of generalized standard materials. The dissipation potential is allowed to depend explicitly on strain. This formulation permits direct control over the shape of the domains while maintaining a fully variational structure.