Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties

It is known that an anti-plane shear surface wave does not exist in anisotropic elastic half-space x2 ⩾ 0 when the material is homogeneous. For a functionally graded material for which the elastic stiffness C44, C45, C55 and the mass density ρ depend on the depth x2 of the half-space, an anti-plane shear surface wave may exist. Exact solutions for four cases of special graded materials are presented here. In all four cases C44 and ρ have the same function form. C55 and C45 are related but they need not have the same function form as C44 and ρ. In fact C45 can depend on x2 arbitrarily. In Case 4, C44 and ρ are periodic function in x2. For a general graded material, an asymptotic solution for large wave number k is obtained and conditions for the existence of an anti-plane shear surface wave are presented. Regardless of exact or asymptotic solutions, as k → ∞, the surface wave speed v   approaches vˆ which is the speed of the exceptional body wave evaluated for the material property at x2 = 0. The surface wave solution has an interesting phenomenon when the elastic stiffness C45 is non-zero.