[ \mathbfu^h(\mathbfx) = \sum_i=1^N_n \mathbfN_i(\mathbfx) , \mathbfu i, \qquad \phi^h(\mathbfx) = \sum i=1^N_n N_i(\mathbfx) , \phi_i, \tag5 ]

[ \Delta W = \int_\Gamma_N \mathbft\cdot \Delta\mathbfu,\mathrmdS . \tag7 ]

The load‑displacement curve obtained with the phase‑field model matches the analytical LEFM prediction for the critical stress intensity factor (K_IC= \sqrtE G_c). The computed (F_c= 4.58) kN is within 2 % of the analytical value. The crack path follows the straight line of the notch, confirming the absence of mesh bias.

: Phase‑field fracture, 2‑D crack propagation, brittle fracture, finite‑element method, variational formulation, adaptive mesh refinement. 1. Introduction Fracture in brittle materials is traditionally modelled by linear‑elastic fracture mechanics (LEFM) , which relies on singular stress fields and explicit tracking of crack fronts. While LEFM provides elegant analytical solutions for simple geometries, it becomes cumbersome for complex crack nucleation, branching, or interaction. Over the past two decades, phase‑field models of fracture have emerged as a powerful alternative because they regularise the sharp crack interface by a diffuse scalar field, thereby avoiding explicit geometry handling and naturally satisfying the Griffith criterion.