TRANSDUCER MODELS FOR THE FINITE ELEMENT SIMULATION OF ULTRASONIC NDT PHENOMENA. R. L. Ludwig, D. Moore and W. Lord

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1 TRANSDUCER MODELS FOR THE FINITE ELEMENT SIMULATION OF ULTRASONIC NDT PHENOMENA R. L. Ludwig, D. Moore and W. Lord Electrical Engineering Department Colorado State University Fort Collins, Colorado INTRODUCTION Numerical studies of realistic ultrasonic NDT situations in their most general form assume arbitrarily shaped defects excited by pulsed transducer waves all subject to the specimen's external boundaries. The combination of three features, transmitter model, ultrasound/defect interaction, and receiver model makes a thorough quantitative evaluation extremely difficult. Although a numerical finite element code capable of handling ultrasonic wave propagation and scattering has been developed [1,2] the overall system remains incomplete without apropriate transducer models. Reviewing some of the existing transducer models, such as the layered device structure [3,4] and the equivalent electronic circuits by Mason and others (S], one observes that all these models are one-dimensional approximations of three dimensional transducer behavior. Diffraction losses resulting in ultrasonic beam spread, nonuniform field variation across the aperture and, if the transducer is directly contacted to a solid, mode conversion are not included. In this paper a simple approach to incorporating the transducers within the finite element formulation is investigated. TRANSMITTER MODEL equation The basis for the transmitter model is the homogeneous elastic wave (1) To incorporate the transmitter excitation over a finite aperture a displacement vector is chosen such that (2) where 6(x) ensures application of the displacement vector on the surface of the specimen, w(y) specifies a window function, which for the present study is assumed to be rectangular, f(t-,(y)) accounts for the pulsed excitation 649

2 function with the phase delay ~(y) allowing for beam steering. It should be emphasized that f(t) can be any realistic transducer function, but for the present purpose the following expression is chosen (3) with w0=2nx1o6s-1 Incorporating (3) in a two dimensional, plane strain, finite element model allows the simulation of a line source, w(y)=&(y), acting on a half space with an otherwise stress-free boundary condition <~ n=o). The wave-front construction of the line source response can be ootained by displaying the displacement vector as a function of time. This is shown for the x component in Fig. 1. Adding several line sources over a finite aperture permits the simulation of a strip-like transducer based on the assumption that precise elastic, electric field coupling phenomena as well as couplant variation is neglected. The resulting x-components of the displacement vector are given in Fig. 2. Comparing Fig. 1 with Fig. 2 reveals the focusing effect of the finite aperture. RECEIVER MODEL In order to obtain signals that are directly comparable to practical A-scan measurements, a receiver model has been developed assuming the form (4) where a1=1: Couplant variation for longitudinal components a2=2: Couplant variation for shear components bi,ci=1: Amplitude factors for shear and longitudinal components N=11: Number of nodal points forming the aperture. Applying (4) to model a shear receiver, an aluminum block with a fixed L wave transmitter was used as shown in Figure 3. The receiver was scanned across the back wall in increments of 0.05 inches as indicated in Figure 3. The combined signals, as predicted by (4) and the finite element code are shown in Figure 4. For illustration purposes the incident L-wave, as produced by the fixed transmitter over the finite aperture, is also included. In this display one can observe the change in amplitude both for the incoming L-wave and S-wave. For three distinct receiver locations a comparison between numerical predictions and practical measurements is shown in Figure S. Comparing the predictions with the experiments (top line) shows good qualitative agreement. However, for more complex situations such as scattering or diffraction by defects this agreement can generally not be expected owing to the two dimensional nature of the numerical model. 650

3 1M: looo IICI!OS[CONOS 11W WICJIOS CONOS T1W[ J.OOO loicros[conds 11W -000 WICJIOS CONOS 11W ~. 000 loiciios[conos 11W WICI!OSI:CONOS Fig. 1. Time evolution for the x-component of the displacement vector excitated by a line source. 651

4 llw[ J.OOO WICROS CONDS nt.l( WICI!OS CONDS nt.l( ~.000 WICI!OS CONOS 1M: ~. 000 WICROS CONOS nt.l( WICI!OS CONOS nt.l( &.000 IIICIIOS CONOS Fig. 2. Combined effect of 9 line sources forming a finite aperture. 652

5 +1.0" r ~ +0.50" AI. specimen +0.25" y " -0.25" TRANSMITTER (fixed) -0.50" -1. 0"' ~ o. 0" 1. 5".Fig. 3 Test arrangement. 1- ~... oc;; >-' TIME IN MICROSEC Fig. 4. Combined receiver signals t

6 a b LO a.o a.o -.o... TIC N YCttO SlCONOS c nwt.. ~ $((0N05 Fig. S. Experimental A-scan (top line) versus numerical predictions. Receiver location at a) y=+.2s ", b) y=o.o, c) y=-0.25 ". 654

7 CONCLUSIONS The application of finite element modeling to ultrasonic nondestructive evaluation problems requires transducer models which both mimic reality and also lend themselves to incorporation into numerical code. This paper describes transmitter and receiver models which meet these requirements. Unfortunately exact quantitative confirmation of these models cannot occur until a full three dimensional code has been developed. ACKNOWLEDGMENTS Financial support for this work has been provided by the Electric Power Research Institute under Project RP CYBER 205 computer time has been provided by Colorado State University and the National Science Foundation. REFERENCES 1. R. Ludwig and W. Lord, A Finite Element Formulation for Ultrasonic NDT Modeling, Review of Progress in Quantitative NDE, Vol. 4A, D. 0. Thompson and D. E. Chimenti, Eds., (Plenum Press 1985). 2. R. Ludwig and W. Lord, Developments in the Finite Element Modeling of Ultrasonic NDT Phenomena, Review.of Progress in Quantitative NDE, Vol. SA, D. 0. Thompson and D. E. Chimenti, Eds., (Plenum Press, 1986). 3. J. Krautkraemer and H. Krautkraemer, Ultrasonic Testing of Materials, Springer Verlag (1977). 4. M. G. Silk, Ultrasonic Transducers for Nondestructive Testing, Adam Hilger (1984). 5. W. P. Mason, Electromechanical Transducer and Wave Filters, Van Nostrand (1948). 655