Secular Equations for Rayleigh and Stoneley Waves in Exponentially Graded Elastic Materials of General Anisotropy under the Influence of Gravity

Abstract

The Stroh formalism is employed to study Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity. The 6 × 6 fundamental matrix N is no longer real. Nevertheless the coefficients of the sextic equation for the Stroh eigenvalue p are real. The orthogonality and closure relations are derived. Also derived are three Barnett-Lothe tensors. They are not necessarily real. Secular equations for Rayleigh and Stoneley wave speeds are presented. Explicit secular equations are obtained when the materials are orthotropic. In the literature, the secular equations for Stoneley waves in orthotropic materials are obtained without using the Stroh formalism. As a result, it requires computation of a 4 × 4 determinant. The secular equation presented here requires computation of a 2 × 2 determinant, and hence is fully explicit. A Rayleigh or Stoneley wave exists in the exponentially graded material under the influence of gravity if the wave can propagate in the homogeneous material without the influence of gravity. As the wave number k → ∞, the Rayleigh or Stoneley wave speed approaches the speed for the homogeneous material.

Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects

“…[59,30]. For the statement of the problem one may consult the book [1] and the works [54,55,58,45,40,16,68,69,70,71,72].…”

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