Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua

We study plane waves in second gradient solids and their reflection and transmission at plane displacement discontinuity surfaces. The needed extension of the treatment adopted to study plane wave propagation in Cauchy continua is not straightforward and is developed here. In particular, the balance of mechanical energy valid for second gradient continua is deduced. The presented results may be of interest as many boundary layer phenomena can be accounted for by second gradient models. We prove that second gradient elastic moduli may influence, in a measurable manner, how planar waves behave at discontinuity surfaces: the novel results presented here can be the basis of experimental procedures to estimate some among these moduli. We explicitly remark that reflection and transmission coefficients which we have estimated show a significant dependence on frequency, which indeed makes easier to conceive effective measurement methods.