Because the Earth’s crust is quite heterogeneous and consists of various types of rocks, many faults are located near material interfaces. Recent observational studies have revealed that some faults cross the interfaces, but theoretical treatment of their dynamic behaviour is still undeveloped. In this paper, we develop a boundary integral equation method (BIEM) to analyse the dynamic behaviour of antiplane rupture propagation crossing a material interface. First, we obtain an exact solution of the stress response due to a rectangular slip rate function on one discretized crack element embedded in one of the elastic half-spaces welded along a planar interface. This solution is helpful not only in BIEM simulations of rupture propagation but also in bench mark testing orextension of other numerical schemes. Next, we simulate the dynamic propagation of an antiplane rupture crossing the material interface by using the exact solution for a stress response in a BIEM framework. We find that a shear wave reflected from the material interface generates a significant change in the slip rate on the crack depending on the angle between the interface and the crack, contrast in the elastic properties, and rupture velocity. Moreover, we infer from our numerical results and previous related works that a backward propagating healing front emerge sunderrate-weakening friction owing to interaction between the reflected wave and friction.