Numerical and experimental analysis of the critically refracted longitudinal beam S. Chaki ⇑ , W. Ke, H. Demouveau Ecole des Mines de Douai, Département Technologie des Polymères et Composite et Ingénierie Mécanique (TPC & IM), 941 Rue Charles Bourseul, B.P. 10838, 59508 Douai Cedex, France article info Article history: Received 5 January 2012 Received in revised form 30 March 2012 Accepted 30 March 2012 Available online 7 April 2012 Keywords: Critically refracted longitudinal wave Creeping wave Ultrasonic beam profile Liquid–solid interface Finite element simulation abstract The critically refracted longitudinal (L CR ) wave can be used in numerous non-destructive testing (NDT) applications, such as characterization of surface geometric aspects, subsurface defect detection and mostly for residual stress measurement. However, very few works characterize the associated ultrasonic beam. This paper deals with characterization of the L CR beam profile both numerically and experimentally in order to optimize the incident angle choice in order to have sufficient energy in the experimental sig- nal. The simulations are performed in time and frequency domains concerning solid elastic, homogenous and isotropic materials taking into account the liquid–solid interaction of the excitation by a water- coupled transducer. In the obtained results all components of the refracted acoustical field are demon- strated, as well as energy distributions of L CR wave obtained with different incident angles. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction L CR wave, which is generated with longitudinal wave incident at the first critical angle, propagates just underneath the surface of specimens, thus demonstrates surface and subsurface characteris- tics by the wave properties linked to material elasticity or existing defects. Very often, Rayleigh waves are employed for that purpose because they are easily excited and only weakly attenuated along more or less ‘‘perfect’’ surfaces. However, they have the disadvan- tage of being very sensitive to surface roughness, as well as expo- nential decay within a few wavelengths normal to the surface, which prevents quantitative characterization of larger cracks or deep residual stresses. In contrast to that, L CR wave is a bulk wave which travels just below the surface with a longitudinal velocity, thus is more sensitive to a stress field in a finite thickness and not just at the surface [1]. Among the first theoretical studies about L CR waves, that of Basatskaya and Ermolov was based on a 2D analytical calculation in the case of harmonic waves [2]. While, the typical acoustic field of L CR wave was illustrated by Langenberg in 1990 it only numer- ically using an elastodynamic finite integration technique [3]. Since this last date, there is no new research works in this field, espe- cially combining numerical calculation and experimental valida- tion. Meanwhile, applications of residual stress measurement with L CR waves are mainly carried out by Bray [4–10], and more recently by Qozam et al. [11]. In this paper, a time domain finite element (FE) simulation is performed to interpret the experimental signals, whereas simula- tions in frequency domain are carried out to characterize the angu- lar ultrasonic beam in a bulk sample, which is especially designed for this purpose. Besides, this research topic is quite rarely and partially developed in the literature, this paper also contributes to clarify many necessary aspects of ultrasonic refraction phenom- enon in the case of longitudinal waves excited at critical angles. 2. Numerical and experimental identification of the refracted longitudinal beam components 2.1. Numerical identification The acoustical field can be described by equation of motion: C ijkl @ 2 u l @x j @x k ¼ q @ 2 u i @t 2 i; j; k; l ¼ 1; 2; 3 ð1Þ where u represents the displacement vector, C ijkl denotes the elastic modulus, and q is the density of the medium. The typical excitation configuration of the L CR wave is as shown in Fig. 1, with emitter in- clined at around the first critical angle for longitudinal wave. In this case, if the projection of the transducer radius R on the boundary is a, which may be slightly changed with different incident angles, then the excitation on the boundary can be defined as an applied stress r 22 . For calculation, this applied stress is taken as: r 22 ¼ Ae iðkx 1 xtÞ in excitation region a 6 x 1 6 a 0 jx 1 j < 1 ( ð2Þ r 21 ¼ 0 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.03.014 ⇑ Corresponding author. E-mail address: salim.chaki@mines-douai.fr (S. Chaki). Ultrasonics 53 (2013) 65–69 Contents lists available at SciVerse ScienceDirect Ultrasonics journal homepage: www.elsevier.com/locate/ultras