Formulas for the H/V ratio of Rayleigh waves in pre-stressed elastic half-spaces subject to an isotropic internal constraint

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Abstract
The present study addresses the propagation of Rayleigh waves in pre-stressed elastic half-spaces subjected to an isotropic internal constraint. The primary objective is to establish exact explicit expressions for the H/V ratio, which defines the amplitude relationship between horizontal and vertical displacements at the surface of half-spaces. By utilizing the surface impedance matrix methodology, a critical link is formed between the velocity and the H/V ratio of the Rayleigh wave. From this relationship and the secular equation, a cubic equation for the squared H/V ratio is derived. By applying the theory of cubic equations, we have formulated precise and explicit expressions for the H/V ratio, which are valid for an arbitrary strain energy function. Subsequently, these general formulas are specified for various strain energy functions and internal constraints. As the obtained formulas for the H/V ratio are totally explicit, they are a convenient tool to evaluate pre-stresses appearing in structures from measured values of the H/V ratio. To prove this point, two examples are considered and they show that for determining the pre-stresses we have only to solve nonlinear (algebraic) equations.
Funding Information
  • National Foundation for Science and Technology Development (107.02-2023.51)