<p>Tooth flank fracture ( defined as TFF ) is the contact fatigue failure behavior of high-cycle / ultra-high-cycle service damage of material damage initiation cracks in the deep position of the tooth. The purpose is to construct a parsing model for the risk threshold of TFF in the internal field of gear teeth according to the multi-axis stress method. Based on contact theory, gear meshing geometry theory and elastic mechanics theory, the multi-axial stress parsing model of equivalent contact internal field of gear pair was established, and the coupling mathematical model of contact fatigue risk prediction of gear gradient capacity was formed. By employing the explicit analytical solution of rectangular microelements and the discrete numerical efficient calculation method, the singular integral contact problem of the mathematical equations of elastic half-plane contact stress components was resolved. And the risk threshold of TFF in the internal field of gear teeth was studied by multiaxial stress evaluation method. The research shows that the risk threshold of the transition zone from the inner hardened layer to the core of the gear tooth is the highest under the multi-axial stress field, which is prone to crack initiation and fracture failure of the tooth surface. The model in the paper is basically consistent with the risk prediction and evaluation of ISO6336.4 technical specification, and is basically consistent with the risk area of the broken tooth position of the test gear. It is clarified that the model can be used for the initial design evaluation of the risk of anti-TFF, and can be used as a supplementary evaluation method for the safety design evaluation of TFF other than ISO6336. The proposed multi-axial stress TFF risk prediction model provides basic technical support for the design and manufacture of anti-TFF.</p>