A New Approach to Inversion of Surface Wave Dispersion Relation for Determination of Depth Distribution of Non-Uniform Stresses in Elastic materials

摘要

The measurement of Stress Gradients with Depth using an analytical ultrasonic Raleigh wave dispersion relationship inversion procedure is discussed in this paper. Ultrasonic Surface (Raleigh) waves become dispersive when propagating on non-uniformly stressed media. In the light of this, the Acoustoelastic effect on their propagation in deformed but initially isotropic materials has been investigated in the past, using an energy perturbation approach that considerably reduces the complexity in the treatment of the Acoustoelastic effect. Inversion of the perturbation relation offers an advantageous route to obtaining the stress gradients. This paper presents a new mechanism for effecting this inversion. Crucial use is made of the fact that Raleigh waves diminish rapidly beyond a depth equaling the wavelength, and the governing integral equation, a Fredholm Integral equation of the first kind, is reduced to a Volterra integral equation of the first kind. The conditions of the problem allow a further conversion into the more stable and tractable Volterra Integral equation of the second kind, which is easily solvable by conventional analytical or numerical iterative techniques.