The characteristics of the condition to initiate ductile crack of steels has been quantitatively estimated using the “two-parameters, that is, equivalent plastic strain and stress triaxiality, criterion”. It has been demonstrated by authors using round-bar specimens with circumferential notch in single tension that the critical strain to initiate ductile crack from specimen center depends considerably on stress triaxiality, but surface cracking of notch root is in accordance with constant strain condition. This study provides the fundamental clarification of the effect of strength mis-matching, which can elevate plastic constraint due to heterogeneous plastic straining, loading mode and loading rate on critical condition to initiate ductile crack from notch root using two-parameters. The crack initiation testing using charpy type bend specimens with strength mis-matching near the notch were conducted under static and dynamic loading. To analyze the stress/strain state in the specimens especially under dynamic loading, thermal elastic-plastic dynamic FE-analysis considering the temperature rise was used. It has been shown that the strength mis-matching would not enhance the stress triaxiality at the notch root of bending specimen. The critical condition to initiate ductile crack from notch root for strength mis-matched bend specimens under both static and dynamic loading would be almost the same as that for homogeneous tensile specimens with circumferential sharp notch under static loading. It follows from these results that the ductile cracking from notch root surface under static and dynamic loading would be estimated by “critical plastic strain criteria” even if the strength mis-matching exists near the notch root