Modeling of laser thermoelastic generation of ultrasound in an orthotropic medium
Résumé
We present a numerical model that calculates the surface displacements generated by the absorption of a laser pulse by an orthotropic medium. This model solves the heat and acoustic wave equations using temporal Laplace and spatial two-dimensional Fourier transformations. This model allows us to calculate the normal and in-plane displacements on the front or back surface of an orthotropic plate over a complete area and for virtually any time and beam profiles of the laser excitation. Numerical simulations are compared to experimental results obtained on an aluminum sample and on a graphite-eppxy plate. The experimental and numerical results are in good agreement. Laser generation of ultrasound, especially when coupled to optical detection, has been recognized to be a powerful tool for nondestructive materials evaluation.'-3 In the thermoelastic generation regime, a localized temperature elevation in the sample induced by the absorption of a laser pulse results in a localized thermal expansion which, in turn, generates ultrasonic waves. Several approaches were proposed for modeling this thermoelastic process, essentially using Green functions4-7 or Laplace and Hankel transformations.8'9 However, the use of Green functions or Hankel transformations for modeling this process cannot be easily extended to the case of anisotropic materials such as materials with an orthotropic symmetry. Other numerical techniques, like the finite element technique, " are not suitable for the resolution of the laser generation of ultrasound. In this letter, we present a model based on temporal Laplace and spatial two-dimensional (2D) Fourier transformations that allows us to describe the thermoelastic generation of ultrasound in the case of an orthotropic sample. In our model, we consider an infinite plate of finite thickness L made of an orthotropic material. The coordinate =es XI, x2, and x3 correspond to the principal axes of the medium, with x3 being the optical axis of the incident laser radiation. The stiffness tensor [C] of the orthotropic medium has then only nine different nonzero components when the abbreviated notation is used.'r To describe the laser generation of ultrasound in the ther-moelastic regime, one must solve simultaneously the heat equation and the acoustic wave equation. In our model, we neglect the mechanical heat sources in the heat equation. This assumption is valid for the time scales of our simulations.7~gP12 We use the hyperbolic heat equation' even if in the cases studied, the classical one would be adequate. The temperature elevation field 6 is then given by d6 Bs v([k]w)=pc, at +7x-%dxlkk-l " 3f(t>, i 1 (1)