Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p2 with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x2 of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.

Research Highlights
► We studied exponentially graded materials for general anisotropic elastic materials.
► The coefficients of the sextic equation for p remain real.
► The different decay rates for the displacement and the stress hold for general anisotropic materials.
► A matrix differential equation is presented for surface waves in an arbitrarily graded anisotropic material with gravity.
► The existence of one-component surface waves in exponentially graded anisotropic elastic materials with or without gravity is discussed.