ACOUSTIC WAVES GENERATED BY A TRANSIENT POINT SOURCE IN AN ANISOTROPIC MEDIUM: COMPARISON BETWEEN THEORY AND EXPERIMENT

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1 ACOUTIC WAVE GENERATED BY A TRANIENT POINT OURCE IN AN ANIOTROPIC MEDIUM: COMPARION BETWEEN THEORY AND EXPERIMENT B. Audoin, A. Mourad and M. Deschamps Universite Bordeaux I Laboratoire de Mecanique Physique, URA C.N.R Cours de la Liberation 3345 Talence cedex, France INTRODUCTION Many authors did study Lamb's problem for an anisotropic half space belonging to the hexagonal system of symmetry. The first application of the Cagniard - de Hoop method to study the line source configuration is due to Kraut [1]. Following this pioneering contribution, uh et al. [2] have extended such method to the point like source case. More recently, Payton [3] further developed to analytical calculations of the epicentre displacement fields. Other workers who considered that the Cagniard - de Hoop method is too cumbersome to be numerically handled have proposed alternative methods, e.g. see Eatwell et al. [4], Tewary and Fortunko [5]. Lately, Wang and Achenbach [6] described another approach derived from the Herglotz-Petrowsky [7] method. Other contributions, e.g. see Dubois et al. [8], are based on entirely numerical methods in order to include the heat diffusion during the laser impact generation of the elastic waves. However, the Cagniard - de Hoop method is more appropriate to cope with Lamb's problem. It is the only method which allows to include in a simple way the diffraction effects, and to respect the causality principle. The Lamb's problem for a point-like source, which is a generalization of the plane problem, can be decomposed as a bundle of line Review of Progress in Quantitative Nondestructive Evaluation. Vol. 15 Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New York,

2 sources. This feature permits to judiciously utilize the results obtained for the much simpler case of the line source [9]. The components of the Green's tensor are calculated for a solid having orthorhombic symmetry, and are further specialized ilicon (cubic symmetry). These results are then compared to experiment. THEORETICAL REULT The original Lamb's problem for a point - like source applied to on homogeneous elastic half-space having orthorhombic symmetry is considered. The surface of the halfspace is assumed to be a plane of symmetry. The purpose of this section is to express the displacement field in any location of the half-space, with exception to the free surface, by the Cagniard - de Hoop method. The displacement field is a function of the position vector X = (x}, X2, X3) and time, verify a system of partial differential equations, which is constitued by the wave equation, the boundary and initial conditions, the ommerfeld condition, and the causality principle. For the line source case [9, 1] the Laplace transform relative to time of the Green's tensor is written down: ioo G~ij(X,s)=-.- L ja ijk Q ik exp{-sr [p sino + T\k(P,~) coso]} dp 2m k;l-ioo 1 3 ioo = -.- L ja ijk Q ik exp{-s R [p sino cos(cj>-~) + T\k(P,~) coso]} dp (2) 2m k;l-ioo F(t): Force anisotropic half space Xl Figure 1. Geometry of the problem. 14

3 where the line source is oriented along direction ~ + rrl2 and where r is the projection distance of the X point on the free surface. The problem is translation invariant in the line source direction. The quantities appearing in equation (2) are the following: the projections of the phase slowness components on to the free surface p and along the axis l, 11k, the parameter ~ which belongs to [-Tt/2, rrl2], the parameter p which describes the imaginary axis of the complex plane, the parameter s of the Laplace transform, the spherical coordinates R, e and <p associated to the X position vector. The quantities Aijk, ilik are functions of the stiffness tensor Cijkl, the density, the angle and the parameter p. The quantities Aijk and ilik also functions of 11k. Their dependence on 11k can be formally written: ilik = <Xi (11k), the (Xi are the polynomial in 11k. Aijk are the transmission coefficients, while ilik are the polarisations. It is important to note that the values of Aijk depend on the source, i.e., ablation or thermoelastic regimes. The case of a punctual source is obtained by considering a large number of line sources scanning a limited area of the free surface, as following: _ 1t/2 G"(X s)=- J 1J, 2' m -1t/2 af\(x,s) p d~ = -s 1t/2 3 ioo -2 J L JAijk 4lt -1t12 k=l-ioo ilik exp{ -sr[p sin e cos( <p - ~) + 11k (p, ~) cos e]}pdpd~ (3) It should be pointed out that the Green's function for a line source corresponds to a temporal Dirac's function. On the other hand, the parameter s in equation (3) indicates that the Green's function for a point-like source corresponds to an heaviside step function. The Cagniard change of variable is written down: (4) A sixth order polynomial in p is obtained by isolating 11k in equation (4) and injecting its expression in the Christoffed equation. This is the so-called Cagniard - de Hoop polynomial whose coefficients are functions of <p, e, p, Cijkl and time t. The traces of the roots in the complex plane, whose parameter is time, define the Cagniard - de Hoop contours, from which the Pk (k = 1 to 3) values are calculated. Without giving any further detail of the calculations, the solution can be finally written: 15

4 t/2 d G ij (X, t) = 21t2 L f Re(T k...ek. Pk) d<l>, with T k = Aijk Qik (5) k=1-1t/2 dt By using an adequate writing of function Tk and with an appropriate numerical scheme, the Cagniard - de Hoop method is numerically well suited to solve the Lamb's problem. Further details are available in [1]. COMPARAION WITH EXPERIMENT. The LAER used for the generation is a Nd:Yag LAER generating pulses 1 ns in duration, with a medium power output typically ranging from 6 to 25 mj per pulse at a wavelength of 164 nm. The optical beam is focused down to a spot size of approximately,1 mm. In these conditions, ablation occurs at the sample surface, and slightly marks it. The LAER interferometric probe for the detection is a Mach-Zehndder, heterodyne type [16]. The bandwidth used is of 18 MHz and the sensitivity is 1 mvi A. The GIl component of the Green's tensor is solely calculated, because it is the must easy to experimentally determine, either dealing with laser impact generation of elastic waves [11] or by breaking glass capillaries and detecting the acoustic event with a capacitive transducer [12]. Experiments has been done on 5 mm silicon thick plate, for which the normal direction to the free surface is along axis [11]. To compare the theoretical results with those obtained from the experiments, the calculated responses are obtained with the correct source,i. e., the correct function Aijk. ample Moving lens Laser interferometer receiver Miror Generating laser Figure 2. Experimental set up. 16

5 To fix ideas, figures 3 and 4 are for ablation regime, while figure 5 is for mixed regime. This last regime results in the calculation of arbitrary 5% of ablation regime and 5% of thermoelastic regime. Figures 3 show the calculated and measured waveforms at e =4 in the principal plan <I> =, while figures 4 present the similar results at e =3 From a general point of view, the important point to remark about these figures is the good agreement between the theoretically calculated waveforms and the measurements. 4 3 E- o " 3 2 :a <C L T Time (J.!s) a) theory. 3 T 2,5 E-.g :a 2 3 L 1,5 ~ 3L,5 -,5-1 Time (J.!s) b) experiment. Figure 3. Ablation regime: e = 4, <I> = o. 17

6 1,2,8 L,6,4 c,2 temps (~s) -,2,2,4,6,8 a) theory.,6,5,4,3,2,1 -,1 temps (~s) -,2,5 1,5 b) experiment. Figure 4. Ablation regime: e = 3, <I> = o. As a general rule, the analysis of the discontinuities is greatly facilitated by using some time versus angle diagrams, which represent the arrival of the various propagating modes as a function of the orientation of the line source. Because a point-like source can be modeled, after applying a temporal derivation, as a sum of many line sources, the observed cusps are easily explained in terms of the focusing effect of the ballistic phonons [1,14]. This phenomena is visible at e =4 and <I> = O. It is characterized by a hyperbolic discontinuity, that clearly appears in figures 3. This kind of discontinuity is similar to that exists for Rayleigh wave generation on a free surface. Note equally that the multiple reflection are not taken into account by the theory. 18

7 Concerning the mixed regime, it is noticeable that the peak associated to the longitudinal arrival time is characteristic of the ablation regime. On the contrary, the shear arrival time is issued from the thermoelastic regime (see figures 5). CONCLUION The use of the Cagniard - de Hoop method to calculate the Green functions for the transient loading of a half - space is the only method which provides expressions of waveforms generated by laser in an anisotropic medium the farthest away in the treatment. 2 -I -2 -=- <l " a -3 1 Temps 1) -4,2,4,6,8 1,2 a) theory.,4,2 L -,2 -,4 -=- <l " ~ -,6 ~ -,8 Time (1l),2,4,6,8 1,2 b) experiment. Figure 5. Ablation and thermoelastic regime: e =,<\l =. 19

8 In the present analysis, numerical calculations have been done and compared to experiments. Physical pheneomena can be then easely explained. For example, the focusing effects are predicted by calculations. everal advances that could be achieved in the future should be outlined. Dealing with theoretical works, the problem of the laser impact generation of Rayleigh wave should adequately be solved by maching this method. REFERENCE 1. Kraut E. A., Review of Geophysics 1, (1963), p uh. L., Goldsmith W, ackman 1. L. Taylor R. L., Int. 1. Rock. Mech. Min. ci., (1974). p Payton R. G., IAM 1. Appl. Math. 4, (1981), p Eatwell G. P., immons 1. A. and Willis G. R., Wave motion 4 (1982), p Tewary V. K. and Fortunko, 1. Acoust. oc. Am. 91, (1992), p Wang C. Y. and Achenbach 1. D., Wave motion 18, (1994), p Herglotz G., Berlin Verhandl. achs. Acad. Wiss. Leipzig. math. phys. K1. 78 (1926), p Dubois M., Enguehard F. and Bertrand L., Appl. Phys. Lett. 64 (5), (1994), p Mourad A., Deschamps M. and B. Castagnecte, Acta Acoustica (in press), 1. Mourad A., Ph. D thesis, University of Bordeaux I, (1995), 11. Mourad A. and Castagnede B, AME 1. Appl. Mech. 61, (1994), p , 12. Kim K. Y. and achse W., U. F. F. C. ympium (1992). 13. Kim K. Y. and Every A.G., 1. Acoust. oc. Am. 95, (1994), p Mourad A. and Deschamps M., 1. Acoust. oc. Am. 97, (1995), (in press). 11