A comparative study of reflection and refraction of SH waves across a tri-layered thin-walled micro-structure

The current paper uses mathematical analysis to investigate the behaviour of reflected and transmitted SH-waves through a three-level structure with interface comprises of thin-walled micro-structure. The geometry of the structure consists of a slab with finite thickness sandwiched between two infinite spaces. Using the couple stress theory (Mindlin and Tiersten in Arch. Ration. Mech. Anal. 11:415-448, 1962), the mechanical behaviour of the successive interfaces has been characterized by means of two cases namely surface elasticity model and strain-gradient thin membrane model. The closed form expression of Transmission Coefficient (TC), Reflection Coefficient (RC), and the phase angle have been derived by means of suitable analytical treatment in each of the two models. Furthermore, they are reduced to classical elasticity model to validate the finding. Several special cases have been also derived and discussed to support the current investigation. Numerical results have been simulated by categorizing incident SH-wave through rarer to denser and denser to rarer medium in all possible orders. The core objective of the paper is to illustrate the influence of interfacial elasticity parameters, phase angles, characteristic length, and wavenumbers on TC and RC for both the models. This has been supported by the help of contour plots. A region of Total Internal Reflection (TIR) has been obtained which critically refracts the transmitted waves along the interface. It should be pointed out that the present study involves multi-field knowledge of surface waves, couple stress theory, and surface elasticity theory.