Abstract: The concepts are clarified and a classification is suggested with respect to thestrong-discontinuous, weak-discontinuous, micro-discontinuous andall-continuous interfaces. A mechanical model is established forimpact fracture problems of functionally gradient elastic bi-materialinterfaces with weak discontinuity. Integral transform is used to deriveCauchy singular integral equation for the crack, and the allocation methodis used to obtain numerical solutions. It is indicated that, the weakdiscontinuity and the micro-discontinuity have significant effects on stressintensity factors, and the micro-discontinuity is a kind of connection of interfacial mechanical property better than the weakdiscontinuity. By assuming the mechanical property of the FGM at one side of theinterface to be the lower-rank terms of Taylor series with respect to that at the otherside, the interface will become micro-discontinuous. The first rankmicro-discontinuity is enough to reduce the stress intensity factorsnotably, however, the higher-rank terms, which are equal to or higher thanthe second rank, have less effect on stress intensity factors. To reduce theweak discontinuity or to make the interface micro-discontinuous are allbeneficial to the enhancement of the capacity of the FGMs interface toresist impact fracture, i.e. to the enhancement of the interfacialtoughness.