An embedded couple stress micro-/nano-obstacle with micro-inertia incident upon by SH-waves

An elliptic micro-/nano-obstacle bonded to an infinite body incident upon by SH-waves, where both domains are couple stress media with micro-inertia, is of major concern. The formulation of this problem in the mathematical framework of couple stress elasticity with micro-inertia leads to angular and radial Mathieu differential equations which are solved analytically. These equations carry two characteristic lengths which are peculiar to the discrete nature of each domain enabling the capture of size effect, dispersion phenomenon, as well as the enhancement of the accuracy of the results. For verification, the ratio of the semi-axis of the elliptic obstacle is set equal to 1, and the result associated with the circular fiber which was previously considered by Shodja et al. (Int J Solids Struct 58:73–90, 2015) is recovered. The distribution of the induced stresses along an elliptic obstacle boundary, depending on its major-to-minor semi-axis ratio (e.g., 2), can significantly differ from that pertinent to a circular one.