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

Source 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 [Formula: see text] functionals showcases a well-defined elastic limit and narrow transition regions before crack onset, as opposed to [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] functional, we first prove a [Formula: see text]-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 [Formula: see text]-convergence result. In the former, for technical reasons the [Formula: see text]-limsup estimate holds only for regular enough cracks, i.e. made by a finite union of [Formula: see text] 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 [Formula: see text] model is more accurate than the considered lower-order [Formula: see text] and [Formula: see text] models; this allows to employ larger mesh sizes, entailing a lower computational cost.