Abstract The Variational Crack Element Method (VCEM) (Ghoniem 2026a) is a constrained energy-minimization framework that represents cracks through distributed Burgers-density fields, preserving the exact crack-tip singular structure while enabling scalable computation for interacting, branching, and kinked geometries. We establish the numerical fidelity of VCEM through systematic comparison with analytical solutions: straight cracks under mode-I, mode-II, and mixed-mode biaxial loading; circular-arc cracks; and kinked and branched configurations for which complex-variable solutions are available. Stress intensity factors, crack-tip fields, and displacement discontinuities are reproduced with high accuracy across all benchmarks. The method is then applied to the diametrically compressed (Brazilian) disk, a finite-domain problem with strongly non-uniform stress fields. Centerline cracks under force control reproduce the classical splitting–arrest sequence as the tips enter the compressive contact zones; inclined cracks reorient toward the maximum-opening direction with kink angles in quantitative agreement with experiment. Finally, simulations of evolving multi-crack networks demonstrate that fracture under force control can be interpreted as a connectivity phase transition, in which the applied load is the control parameter and the crack-network connectivity is the order parameter, driving the disk from an intact to a fragmented state through unstable growth, junction formation, and branching.
Nasr Ghoniem (Tue,) studied this question.
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