VALIDITY OF THE GAUSS-HERMITE ßEAM MODEL IN AN ANISOTROPIC

Transcription

1 VALIDITY OF THE GAUSS-HERMITE ßEAM MODEL IN AN ANISOTROPIC LAYERED MEDIUM: COMPARISON TO THE FINITE ELEMENT METHOD A. Minachi, Z. You, R.ß. Thompson, and W. Lord Center for Nondestructive Evaluation and College of Engineering Iowa State University, Ames, Iowa INTRODUCTION Under certain conditions, cast stainless steels develop highly aligned grain structures. When viewed macroscopically, such polycrystalline aggregates exhibit considerable elastic anisotropy and variation of wave speeds with direction. Stainless steels with aligned microstructures are often found in nuclear reactor components and require special attention in the design and interpretation of ultrasonic testing. Such phenomena as beam skewing and excess beam divergence, which are caused by elastic anisotropy, can severely confuse or constrain the detection and evaluation of flaws. Furthermore, when these components are integrated into structures by welding, the direction and degree of alignment in adjacent regions may be different and hence inhomogeneities are presented. Proper inspection of these layered, anisotropie materials is made more effective by the availability of a numerical model to predict the propagation of the ultrasonic fields through them. There are numerous existing software codes which use finite difference, finite element and ray tracing methods to describe ultasonic wave propagation in anisotropie materials. The finite difference [1] and finite element [2] techniques produce exact solutions and can treat very complex geometries and material inhomogeneities. However, they are very computationally intensive. Ray tracing [3] is much faster, but does not provide a full description of the elastic fields or treat beam spread properly. As an intermediate tool having features somewhat between the above, an approximate Gauss-Hermite model [4] has been recently developed which predicts the propagation of ultrasonic waves through homogeneous, isotropie and anisotropie materials [5]. This model is computationally fast and simple and explicitly incorporates beam spreading. Moreover, simple procedures exist which allow the propagation of beams through interfaces to be treated. There are, however, some limitations to this model due to the use of the (paraxial) Fresnel approximation which was responsible for the enhanced computational speed. Last year a comparison of predictions of the Gauss-Hermite model and finite difference methods was reported [6]. A fully quantitative comparison of these results was impossible because of certain differences in the initial assumptions, but the results were sufficiently encouraging to warrant further study. In this paper a more intense comparison between Review 0/ Progress in Quamitalive Nondestructive Evaluation, Vol. 11 Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New York,

2 the predictions of the Gauss-Hermite and finite element methods is presented, with care being taken to be certain that all input parameters are identical. THEORETICAL BACKGROUND Finite Element Analysis Theoretical aspects of the numerical modeling code used in this study have been described in other publications [7-9], together with details of experimental and analytical validation studies. From a users viewpoint, the code involves discretizing the region of interest and providing transducer input data and specimen boundary conditions. The code automatically forms the global matrix equation which, when solved, yields the nodal displacement values associated with every mesh node. Typical output plots are included in this paper. Gauss-Hermite Beam Model Basic Theory The Gauss-Hermite beam model [5] is an approximate solution of the elastic wave equation, which predicts the ultrasonic fields radiated into isotropic and anisotropic materials through planar or simply curved interfaces by focused or unfocused transducers. The model is derived by first representing the initial profile of the beam as a summation of Gauss-Hermite eigenfunctions, weighted properly. The radiation of each of these basis functions is determined by expressing the fields as an angular spectrum of plane waves and then using the Fresnel approximation to allow some of the integrals to be evaluated analytically. This approximation assumes that the energy of ultrasonic waves propagate in a sufficiently narrow range of angles that the slowness surface can be represented by a Taylor series expansion near the propagation direction of the central ray. In the case of an anisotropic material, this expansion will have the form ( k) (kx) (k y) (k x )2 (kx)(k Y) (k y )2 ~ "'So+A Ul +B Ul +C Ul +D Ul Ul +E Ul (1) where (~) is the slowness or inverse velocity, So is the slowness along the central ray, and kx and ky are the components of the wave vector in a coordinate system that has z-axis along the central ray direction. A,B,C,D and E are constants that are related to derivatives of the slowness with respect to each component kx and ky. These parameters control the beam divergence and skewing. In the case of isotropic materials, all of these constants, with the exception of SO, are zero. The use of these approximations leads to a numerical solution with considerably enhanced computational speeds but los ses in accuracy with respect to exact methods of solving the wave equation. Propagation Through Interfaces Procedures exist to use the Gauss-Hermite model to trace the evolution of a beam as it passes through a planar interface or an interface with bicylindrical curvature. At each interface several parameters have to be determined. First, the transmitted angle of the central ray is computed. Then the transmission and reflection coefficients of a plane wave passing through a planar interface at this angle is computed by applying the conservation of displacement and stresses at the boundary. Finally, the anisotropy parameters (A-E) are found to define the approximate slowness surface. The beam is then propagated to the next interface using the Gauss-Hermite model and the process is repeated until the layer at which the fields are to be predicted is reached. Procedures to deal with curved interfaces are also described in Ref. [5]. 1022

3 THE MODEL PROBLEM Both the Gauss-Hermite and finite element models were applied to a model problem in whieh the wave propagated from an isotropie layer (ferritie steel) into an anisotropie layer (austenitie 316 steel with aligned [001] axes at orientations of 0, 15 and 40 degrees with respeet to the interface normal. The elastie eonstants of isotropie ferritie steel are [1] CII~C22~C33~274.9 GPa p ~ 7.9x10 3 kgm- 3 The elastie eonstants for transversely anisotropie astenitie steel are [11] C 33 ~ GPa C 12 ~ 98.2 GPa Ferritie steel z x 1.5 ern Austenitie 2.5 ern Observation Plane ~ I r ~ x-axis (ern) Figure 1. Geometrieal eonfiguration of the model problem. 1023

4 C.. =Css =129.0 GPa C 66 =82.3 GPa p = 8.12xI0 3 kgm- 3 The waves are generated by a two-dimensional (i.e., strip) 2 MHz, em diameter, longitudinal wave transdueer on the surfaee of the isotropie layer. The transdueer was assumed to aet as a piston, produeing a uniform displaeement over its surfaee, and the displaeement was taken to be zero over the rest of the surfaee at the isotropie layer. Figure 1 shows the geometrieal eonfiguration of the model problem. RESULTS AND DISCUSSION o Degree Crystalline Axis Orientatjon In this ease, the erystalline axis of the anisotropie medium eoineides with propagation direetion. As ean be seen from the slowness eurve, the energy direetion and propagation direetion are the same and there is no beam skewing. (See Figure 2) Also, the beam profile is symmetrical with respeet to the z-axis, and the beam spread is maximum due to the high curvature of the slowness curve in this direction. The comparison between the two models shows good agreement in the vicinity of the center of the beam (x~o), but as the observation point moves along the x-direction away from the central ray, the disagreement inereases. We believe the error to be in the G-H model due to the paraxial approximation. However, most of the energy of the beam is concentrated in eentral lobe, and there is good agreement between the two models in this region. 15 Degree Crystalline Axis Orientation In this ease, the angle of the crystalline axis to the interface normal is 15 degrees. The slowness eurve shows that the energy direction is not the same as the propagation direction, and there is substantial skewing of the beam. (See Figure 3) The beam is not symmetrical about the z-axis; however, the beam spread is not as great as for the zero degree case beeause of the smaller curvature of the slowness surface. The eomparison between the two models shows the best agreement in the direetion of maximum energy flow (x - -1 cm) rather than the propagation direetion (x ~ 0 em). Thus agreement is again best in the central lobe. The G-H model prediets a symmetrieal beam with respect to the energy direction beeause of the use of the Fresnel approximation. However, this symmetry is not exhibited in the more accurate finite element predietions. Nevertheless, the differences are not great within the central lobe. 40 Degree Crystalline Axis Orientation In this ease the angle of the crystalline axis to the vertical is 40 degrees. The slowness eurve shows some skewing effect, but not as much as in the 15 degree ease. (See Figure 4) Also, due to flatness of the slowness eurve near the propagation direction, the beam spread is the least in this ease. Again the G-H model produces a good agreement with finite element model in the vieinity of the central lobe. As the distance of the observation points away from the beam axis increases along the x-axis, the dis agreement also inereases. For all eases shown there is good agreement on the peak amplitude with the major differenees being in the pulse shape at large time. 1024

5 z ::::::-----, 'x:.:...- =...::O~. O:...:c:.:.m:.:._..., I t-I~ x O.O V.. _ L...-~-'---'-----'~'---...I...-~---J ~ o G.H - F.E x:.:...- =~0:..:.;. 6"_c ::..:m.:..:..., 'x:.:...-=~i..:.:. 2:..;c :.:. m:..:.-..., " L--...o--'-.l..--'--~ '----J -1.2 L o..--'----'-_L..-...o----'- ----I 'x"--=-=i.:..:.8:...;c::..:.m:..: 'xc.:...- =..:::; 2.:...;.4-=c;.;.:m-=-- ---, o.o... --~s o.o fita -1.2 '--...o----'----''--...l.---'----''--..o..--' ' '----'_-'----'----'-_ Figure 2. Comparison of finite element and Gauss-Hermite model for zero degree orientation at different positions on the x-axis. The horizontal axis is time in microseconds. 1025

6 x 1.5 x =O.Ocrn G.H -F.E 1.5 x = 0.6crn 1.5 x = -0.6crn x-12crn - -, :x:.:...=_-..:.;1.;.::8-=c:..:.;m:.:......, ~... V ~ '--"""'"" _'--"""'""--' ' Figure 3. Comparison of finite element and Gauss-Hermite model for 15 degree orientation at different positions on the x-axis. The horizontal axis is time in microseconds. 1026

7 1.5 x =O.Ocm G.H 0 -F.E x = 0.6cm 1.5 x = 1.2cm x = -0.6 cm 1.5 x = -1.2 cm ~ V Figure 4. Comparison of finite element and Gauss-Hermite model for 40 degree orientation at different positions on the x-axis. The horizontal axis is time in microseconds. CONCLUSION The Gauss-Hermite model reproduces the major features of the finite element calculation, including beam spread and skewing. These results show that the Gauss-Hermite beam model conveniently treats multiple interface problems in anisotropie materials with high computational speed (the finite element model took more than 24 hours, the G-H less than a minute including beam spread and skewing). Some limitations of the Gauss-Hermite model are revealed in the comparison to the finite element method. Oue to the paraxial approximations in the Gauss-Hermite model, the accuracy decreases as the observation point moves away from the center of the beam. Moreover, the beam profile remains symmetrical in the simplest implementations. Nevertheless, the model is found to make quite useful predictions of radiation patterns, particular in the central lobe of the beam. 1027

8 ACKNOWLEDGEMENT This work was sponsored by the Eleetrie Power Research Institute under eontraet RP and was performed at the Ames Laboratory. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contraet No. W-7405-ENG-82. REFERENCES 1. A. H. Harker, J. A. Ogilvy and J. A. G. Temple, "Modeling Ultrasonie Inspeetion of Austenitie Weld," J. Nondestr. Eval. Vol. 9, , sept W. Lord, R. Ludwig and Z. You, "Development in Ultrasonie Modeling with Finite Element Analysis," J. Nondestr. Eyal. Vol. 9, , sept J. A. Ogilvy, "A Layered Media Model for Ray Propagation in Anisotropie, Inhomogeneous Material," UKAEA Report AERE-TP, 1306, Oetober B. P. Newberry and R. B. Thompson, "A Paraxial Theory for the Propagation of Ultrasonie Beam in Anisotropie Solid," J. beoust. Soe. ~ 85, (1989). 5. B. P. Newberry, A. Minaehi and R. B. Thompson, "Ultrasonie Beam Propagation in Cast stainless Steel," Review of Progress in ONDE 8, D. O. Thompson and D. E. Chimenti, (Plenum, New York, 1989), pp A. Minaehi and R. B. Thompson, "Ultrasonie Wave propagation in Inhomogeneous, Anisotropie Cast stainless Steel," ibid, Vol. 10 (1991) p R. Ludwig and W. Lord, "Finite Element Study of Ultrasonie Wave Propagation and Seattering in an Aluminum Block," Materials Eyaluation. ~, (1988). 8. R. Ludwig and W. Lord, "A Finite Element Formulation for the Study of Ultrasonie NDT Systems," IEEE Transactions on Ultrasonies. Ferroeleetries and Frequeney Control. 35, (1988). 9. R. Ludwig and W. Lord, "A Theoretieal and Numerieal Study of Transient Force Exeitations on an Elastie Half-Spaee," IEEE Transactions on Ultrasonies. Ferroeleetries and Frequeney Control. 36, (1989). 1028