Antiplane shear wave dynamics in functionally graded materials: A green’s function framework

The present study investigates horizontally polarized shear wave propagation in an inhomogeneous, anisotropic medium with depth-dependent elastic properties. A Green’s function formulation is developed to characterize transient SH-waves in a transversely isotropic medium whose material parameters follow a power-law variation along the depth in a Cartesian coordinate system. In particular, the elastic modulus and mass density are prescribed through distinct power-function profiles, enabling independent control of inhomogeneity effects. Under these assumptions, Green’s functions corresponding to the governing inhomogeneous scalar wave equations are derived, and closed-form expressions for the displacement fields associated with the wavefronts are obtained. The singular points of the solution are identified in terms of the wave arrival times, and the onset of a secondary diffracted wave generated by the incoming front is highlighted. Numerical results illustrate the influence of the inhomogeneity parameters associated with the elastic moduli and density, demonstrating that certain parameter regimes lead to upward shifting of wavefronts, whereas others result in downward shifting, thereby highlighting the significant role of material grading on wave kinematics in anisotropic environments.