Abstract: Green’s function and complex function methods are used here to investigate the problem of the dynamic response for shallowly buried circular inclusions of arbitrary positions impacted by SH-wave in a bi-material half space. Firstly, the expression of the scattering wave field was constructed, satisfying the free boundary conditions in a right-angle plane by the method of ‘image’, then Green’s function was constructed. Secondly, the bi-material media was divided into two parts along the vertical interface using the idea of interface ‘conjunction’, then a series of Fredholm integral equations of first kind for determining the unknown forces could be set up through continuity conditions on the surface and Green’s function. Finally, some examples for the dynamic stress concentration factor of the cylindrical elastic inclusion are given. Numerical results show that the dynamic stress concentration factor is influenced by the interface, free boundary, circular inclusion, incident wave and so on.