Fracture analysis in NbZrTa–based high–entropy carbides using isogeometric adaptive phase–field method

Abstract This study investigates the fracture behavior of high-entropy carbides using an adaptive isogeometric phase-field framework that relies on an order of four model, incorporating a cubic stress-degradation function and PHT-spline-based adaptive mesh refinement. The fracture behavior of four experimentally characterized compositions - ( NbZrTa ) C , ( NbZrTaW ) C , ( NbZrTaHf ) C , and ( NbZrTaHfW ) C - is examined across five benchmark geometries: 2D single-edge notched tension, pure shear, three-point bending with varying crack offsets, 3D mode-I tension, and a cubic specimen with an inclined penny-shaped crack. ( NbZrTaW ) C exhibits the highest fracture resistance across most loading configurations, with peak critical loads of F c = 119.99 N under 2D tension and F c = 28.1 N under 3D mode-I loading, consistent with its highest critical energy release rate ( G c = 0.0379 N mm − 1 ). ( NbZrTa ) C , with the lowest G c ( 0.0217 N mm