Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second gradient materials

In this paper reflection and transmission of compression and shear waves at structured interfaces between second gradient continua is investigated. Two semi-infinite slabs of the same second gradient material are connected through an interface which is assumed to have its own material properties (mass density, elasticity and inertia). Using a variational principle, general balance equations are deduced for the bulk as well as jump duality conditions to be imposed at considered structured interfaces. The obtained equations include the effect of surface mass density, inertia and elasticity on the motion of the overall system. In principle, the surface elasticity considered here accounts for all deformation modes, including bending, even if simplified applications to one-dimensional wave propagation are treated in the second part of the paper. We clearly show that the effect on re ection and transmission of the surface material properties is tangibly di erent from the effect of the second gradient parameter. For three of the four used internal constraints, the effect of the second gradient parameter on reflection and transmission is seen to be definitely non-negligibleand clearly distinguishable from the e ect of the surface material properties. In particular, both the interface material properties and the second gradient parameter give rise to reflection and transmission coeffcients which are strongly frequency-dependent, in opposition to what happens for standard Cauchy continua and for surfaces without material properties. We show that, depending on the type of the considered constraint, the frequency-dependence related to second gradient effects is qualitatively and quantitatively different from the one related to the surface material structure.