AT1 fourth-order isogeometric phase-field modeling of brittle fracture

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 the one-dimensional setting (for both the continuum and discretized isogeometric formulations). This is 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. In the two-dimensional setting, we consider two energies: with and without energy split. In the latter case, we provide a complete Γ-convergence result. In the former, for technical reasons the Γ-limsup estimate holds only for regular enough cracks, i.e. made by a finite union of C2 arcs, possibly intersecting in their endpoints. In the evolution scheme, 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.