Reflection and transmission of plane waves at the interface between two different dipolar gradient elastic half-spaces

The reflection and transmission of plane waves at the interface between two different dipolar gradient elastic solids are studied. First, the auxiliary monopolar tractions and the auxiliary dipolar tractions are introduced and the explicit expressions of them are derived for the dipolar gradient elastic solid. Then, the nontraditional interface conditions requiring the auxiliary monopolar tractions and the auxiliary dipolar tractions continuous across the interface are used to determine the amplitude ratio of the reflection and transmission waves with respect to the incident wave. Our aim is to investigate the microstructure effects on the reflection and transmission waves at the interface of two microstructured solids, and to show the possible deviation from the classical elastic solids without microstructure effects. It is found that the microstructure effects make the propagating waves dispersive and create the evanescent waves that become the surface waves at interface. Some numerical examples are given for the amplitude ratio and the phase shift of reflection and transmission waves. The numerical results are validated by consideration of energy conservation. It is found from the numerical results that the reflection and transmission waves are significantly dependent upon the microstructure parameters and the microstructure effects become more pronounced as the incident wavelength is close to the characteristic length of microstructure.