A novel AT1 phase-field framework is introduced for simulating quasi-static anti-plane shear fracture in geometrically linear elastic bodies. In this approach, \(\xi \) -based local mesh adaptivity is unified with an algebraically nonlinear strain energy density function to circumvent the physically inconsistent crack-tip singularities inherent to classical linear elastic fracture mechanics. A modified Francfort-Marigo energy functional with Ambrosio-Tortorelli-type regularization is proposed, in which the characteristic length of the damage zone, \(\xi \) , is dynamically optimized to enhance computational efficiency and approximation accuracy. The total energy functional–comprising nonlinear strain energy, evolving surface energy, and linear regularization–is minimized via variational principles to yield a coupled system of quasilinear partial differential equations, which are discretized using conforming bilinear finite elements. The formulation is governed by four parameters, including two asymptotically calibrated penalty terms. It is demonstrated that this spatially adaptive strategy is best suited for adaptive mesh simulation of quasi-static crack propagation and outperforms regular mesh refinement strategies, enabling accurate fracture propagation even while maintaining a significantly larger regularization length compared to non-adaptive counterparts.
