This work develops a locally adaptive isogeometric phase-field framework for two-dimensional quasi-static brittle fracture using cubic C1 polynomial splines over hierarchical T-meshes (PHT-splines). The aim is not to introduce a new crack-density functional or a new degradation law, but to provide a consistent variational-to-discrete setting in which second- and fourth-order phase-field regularizations can be treated within the same locally refined spline framework. Starting from the energy functional, the formulation is carried through admissible weak forms to the corresponding discrete residual equations. The second-order formulation is posed in an H1(Ω) setting, whereas the fourth-order model is treated directly through a Laplacian-based H2(Ω)-compatible approximation without auxiliary phase-field variables. The formulation combines history-field irreversibility, the tension–compression split of the elastic energy, and an adopted cubic degradation law with s=10−4, whose nonlinear tangent contribution is handled by a Taylor-stabilized staggered Newton scheme. Numerical tests on a single-edge notched tensile benchmark and a notched perforated beam under asymmetric bending show that local refinement captures the fracture zone while maintaining critical-load deviations of about 0.8% and 0.3%, respectively, relative to the reference critical loads used for the two benchmark problems. The contribution therefore lies in the coherent coupling of higher-order regularity, admissible weak forms, local PHT-spline adaptivity, and stabilized nonlinear degradation treatment within a spline-based phase-field fracture implementation.
Mahmi et al. (Fri,) studied this question.