Two Elastodynamic Incremental Models: The Incremental Theory of Diffraction and a Huygens Method.

The elastodynamic geometrical theory of diffraction (GTD) has proved to be useful in ultrasonic nondestructive testing (NDT) and utilizes the so-called diffraction coefficients obtained by solving canonical problems, such as diffraction from a half-plane or an infinite wedge. Consequently, applying GTD as a ray method leads to several limitations notably when the scatterer contour cannot be locally approximated by a straight infinite line: when the contour has a singularity (for instance, at a corner of a rectangular scatterer), the GTD field is, therefore, spatially nonuniform. In particular, defects encountered in ultrasonic NDT have contours of complex shape and finite length. Incremental models represent an alternative to standard GTD in the view of overcoming its limitations. Two elastodynamic incremental models have been developed to better take into consideration the finite length and shape of the defect contour and provide a more physical representation of the edge diffracted field: the first one is an extension to elastodynamics of the incremental theory of diffraction (ITD) previously developed in electromagnetism, while the second one relies on the Huygens principle. These two methods have been tested numerically, showing that they predict a spatially continuous scattered field and their experimental validation is presented in a 3-D configuration.

A semi-numerical model for near-critical angle scattering.

Numerous phenomena in the fields of physics and mathematics as seemingly different as seismology, ultrasonics, crystallography, photonics, relativistic quantum mechanics, and analytical number theory are described by integrals with oscillating integrands that contain three coalescing criticalities, a branch point, stationary phase point, and pole as well as accumulation points at which the speed of integrand oscillation is infinite. Evaluating such integrals is a challenge addressed in this paper. A fast and efficient numerical scheme based on the regularized composite Simpson's rule is proposed, and its efficacy is demonstrated by revisiting the scattering of an elastic plane wave by a stress-free half-plane crack embedded in an isotropic and homogeneous solid. In this canonical problem, the head wave, edge diffracted wave, and reflected (or compensating) wave each can be viewed as a respective contribution of a branch point, stationary phase point, and pole. The proposed scheme allows for a description of the non-classical diffraction effects near the "critical" rays (rays that separate regions irradiated by the head waves from their respective shadow zones). The effects include the spikes present in diffraction coefficients at the critical angles in the far field as well as related interference ripples in the near field.

The Uniform geometrical Theory of Diffraction for elastodynamics: Plane wave scattering from a half-plane.

Diffraction phenomena studied in electromagnetism, acoustics, and elastodynamics are often modeled using integrals, such as the well-known Sommerfeld integral. The far field asymptotic evaluation of such integrals obtained using the method of steepest descent leads to the classical Geometrical Theory of Diffraction (GTD). It is well known that the method of steepest descent is inapplicable when the integrand's stationary phase point coalesces with its pole, explaining why GTD fails in zones where edge diffracted waves interfere with incident or reflected waves. To overcome this drawback, the Uniform geometrical Theory of Diffraction (UTD) has been developed previously in electromagnetism, based on a ray theory, which is particularly easy to implement. In this paper, UTD is developed for the canonical elastodynamic problem of the scattering of a plane wave by a half-plane. UTD is then compared to another uniform extension of GTD, the Uniform Asymptotic Theory (UAT) of diffraction, based on a more cumbersome ray theory. A good agreement between the two methods is obtained in the far field.

A refinement of the Kirchhoff approximation to the scattered elastic fields.

We study two canonical problems, diffraction of a plane elastic wave by a thin crack and diffraction of a plane elastic wave by a wedge, both in the high-frequency regime. In applications this regime is usually treated using the so-called Kirchhoff approximation. It is very easy to implement but there are situations when it is known to give distorted results. We discuss an easy correction procedure, which is applicable not only in geometrical regions but inside penumbras too. The procedure involves a version of the Physical Theory of Diffraction that relies on the Geometrical Theory of Diffraction rather than the full solution of the corresponding canonical problem.