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Hi Rong,
Thanks for sharing this. I still think a negative sign might be missing in the definition of stress triaxiality in Equation 4 in your screenshot. Stress triaxiality is commonly defined as the ratio of mean stress (hydrostatic stress) over equivalent stress. Pressure is the negative of hydrostatic stress.
In Johnson-Cook fracture criterion, fracture strain is inversely related to stress triaxiality. In other words, fracture strain should decrease by increasing the stress triaxiality. You can do a few checks using single-element simulation to ensure that the expected fracture trend is resolved in your model.