Machine-Learning Potentials Predict Orientation- and Mode-Dependent Fracture in Refractory Diborides

Fracture toughness ($K_\mathrm{Ic}$) and fracture strength ($\sigma_\mathrm{f}$) are key criteria in the selection and design of reliable ceramics. However, their experimental characterization remains challenging -- especially for ceramic thin films, where size and interfacial effects hinder accurate and reproducible measurements. Here, machine-learning interatomic potentials (MLIPs) trained on \textit{ab initio} datasets of single crystal models deformed up to fracture are used to characterize transgranular cleavage in pre-cracked ceramic diboride TMB$_2$ (TM = Ti, Zr, Hf) lattices through stress intensity factor ($K$)-controlled loading. Mode-I simulations performed across distinct crack geometries show that fracture is primarily driven by straight crack extension along the original plane. The corresponding macroscale fracture-initiation properties ($K_\mathrm{Ic} \approx 1.7$-2.9 MPa$\cdot\sqrt{\text{m}}$, $\sigma_\mathrm{f} \approx 1.6$-2.4 GPa) are extrapolated using established scaling laws. Considering TiB$_2$ as a representative system, additional simulations explore loading conditions ranging from pure Mode-I (opening) to Mode-II (sliding). TiB$_2$ models containing prismatic cracks exhibit their lowest fracture resistance under mixed-mode conditions, where the crack deflects onto pyramidal planes--as confirmed by nanoindentation tests on TiB$_2$(0001) thin films. This study establishes $K$-controlled, MLIP-based simulations as predictive tools for orientation- and mode-dependent fracture in ceramics. The approach is readily extendable to finite temperatures for evaluating fracture behavior under conditions relevant to refractory applications.