AT1 fourth-order isogeometric phase-field modeling of brittle fracture
Luigi Greco, Eleonora Maggiorelli, Matteo Negri, Alessia Patton, Alessandro Reali
TL;DR
This work introduces an AT1 fourth-order phase-field model for brittle fracture implemented within an isogeometric framework, merging a well-defined elastic limit with high-order accuracy to reduce computational cost. A rigorous Γ-convergence analysis provides an explicit toughness correction factor c_ρ and a compact, finite-width transition profile w_*, enabling accurate recovery of the sharp crack energy in both continuum and IgA contexts. Numerical experiments across pure traction, DCB, and SEN tests show the proposed model outperforms traditional AT1/AT2 formulations, achieving lower toughness errors and allowing larger mesh sizes, with clear guidelines linking mesh size to the optimal-profile width via R_*. A parametric study on the Laplacian weight ρ reveals a trade-off: larger ρ yields a broader, smoother transition and potential mesh savings, while excessively large values may affect crack-path fidelity in certain scenarios. Collectively, the results demonstrate that the AT1 fourth-order IgA formulation provides higher fidelity fracture predictions at reduced computational cost, with practical implications for complex structural analyses.
Abstract
A crucial aspect in phase-field modeling, based on the variational formulation of brittle fracture, is the accurate representation of how the fracture surface energy is dissipated during the fracture process in the energy competition within a minimization problem. In general, the family of AT1 functionals showcases a well-defined elastic limit and narrow transition regions before crack onset, as opposed to AT2 models. On the other hand, high-order functionals provide similar accuracy as low-order ones but allow for larger mesh sizes in their discretization, remarkably reducing the computational cost. In this work, we aim to combine both these advantages and propose a novel AT1 fourth-order phase-field model for brittle fracture within an isogeometric framework, which provides a straightforward discretization of the high-order term in the crack surface density functional. For the introduced AT1 functional, we first prove a Γ-convergence result (in both the continuum and discretized isogeometric setting) based on a careful study of the optimal transition profile, which ultimately provides the explicit correction factor for the toughness and the exact size of the transition region. Fracture irreversibility is modeled by monotonicity of the damage variable and is conveniently enforced using the Projected Successive Over-Relaxation algorithm. Our numerical results indicate that the proposed fourth-order AT1 model is more accurate than the considered lower-order AT1 and AT2 models; this allows to employ larger mesh sizes, entailing a lower computational cost.