Scientific Article

Fast analytical models for the non-linear scattering of elastic waves at a contact interface

Université Paris-Saclay, CEA, List, F-91120 Palaiseau, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 31 March 2026
Accepted: 25 June 2026

Abstract

Non-Destructive Testing (NDT) of industrial components aims to detect defects early in service. Cracks can be notably initiated by fatigue due to cyclic loading or by stress corrosion under residual stress. For both failure mechanisms, the crack develops progressively, with the crack root opening first while a portion of the crack tip remains closed. Conventional linear ultrasounds are only sensitive to the open portion of a crack, leading to a systematic underestimation of the crack length. Nonlinear ultrasonic methods are able to detect micro-cracks, imperfect interfaces (kissing bonds, closed cracks, etc…) since they give rise to a measurable acoustic non-linearity such as the appearance of higher or lower harmonics. To quantify the nonlinear effects generated by closed cracks, several models have been developed to simulate their interaction with ultrasound. An existing 1D analytical model (using a unilateral contact law and the Coulomb friction law at a planar interface) and whose algorithm has been optimised can be much faster than a finite differences model. We then propose a 2D preliminary analytical model that yields good agreement with Finite Elements at low incident angles. Future work is planned to refine the model’s hypotheses and extend its validity.