AG Every

B Castagnede

BA Auld

E Schreiber

FI Fedorov

GFD Duff

JJ Hall

MJP Musgrave

RFS Hearmon

S Sengupta

W Sachse

Shen, G

Hagg, EUO

Rabie, ABM

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European Journal of Orthodontics

Condylar response to forward mandibular positioning

A. G. Every

Wolfgang Sachse

K. Y. Kim

Lin Niu

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Determination of Elastic Constants of Anisotropic Solids from Group Velocity Data

Most determinations of elastic constants of anisotropic solids are based on acoustic wave phase velocity measurements [1,2]. The methods for recovering elastic constants from phase velocity data are fairly straightforward and well-established. For high symmetry directions certain elastic constants, or combinations thereóf, can be obtained directly from individual velocity measurements [3]. When data pertaining to off-symmetry directions is used, usually an optimized fitting procedure is invoked to obtain the full set of elastic constants [4]. There is growing use of techniques that measure the group velocities of acoustic waves or phonons in solids. In an anisotropic solid the phase and group velocities do not in general coincide, even in the absence of dispersion and attenuation. Existing methods for recovering elastic constants from phase velocity data are therefore not directly applicable. The aim of this paper is to present a novel method that allows the elastic constants of an anisotropic solid to be recovered from group velocities measured in arbitrary directions in a sample. We demonstrate the method by applying it to computer generated velocity data on cubic and transversely isotropic solids and to experimental data obtained on single crystals of silicon.</p

Every, A.

Sachse, Wolfgang

Kim, K. Y.

Niu, Lin

Digital Repository @ Iowa State University (ISU)

DETERMINATION OF ELASTIC CONSTANTS OF ANISOTROPIC SOLIDS FROM GROUP VELOCITY DATA A. G. Everyt, Wolfgang Sachse, K. Y. Kim and Lin Niu Department of Theoretical and Applied Mechanics Cornell University, Ithaca, NY 14853 t On leave from the University of the Witwatersrand Johannesburg, South Africa INTRODUCTION Most determinations of elastic constants of anisotropic solids are based on acoustic wave phase velocity measurements r1,2]. The methods for recovering elastic constants from phase velocity data are fairly straightforward and well-established. For high symmetry directions certain elastic constants, or combinations there6f, can be obtained directly from individual velocity measurements [3]. When data pertaining to off-symmetry directions is used, usuall;y an optimized fitting procedure is invoked to obtain the full set of elastic constants [4J. There is growing use of techniques that measure the group velocities of acoustic waves or phonons in solids. In an anisotropic solid the phase and group velocities do not in general coincide, even in the absence of dispersion and attenuation. Existing methods for recovering elastic constants from phase velocity data are therefore not directly applicable. The aim of this paper is to present a novel method that allows the elastic constants of an anisotropic solid to be recovered from group velocities measured in arbitrary directions in a sample. We demonstrate the method by applying it to computer generated velocity" data on cubic and transversely isotropic solids and to experimental data obtained on single crystals of silicon. The motivation for developing this method has been to provide a suitable framework for interpreting data acquired via the ultrasonic point-source/point-receiver (PS/PR) technique [5]. This technique affords flexibility as regards sample geometry and direction of measurement, it provides simultaneous measurement of longitudinal and transverse velocities, and its broadband feature proves particularly advantageous in the study of highly attenuating materials. The Rrinciple of the PS/PR technique, which has been described in greater detail previously [5], is as follows: some form of localized excitation, which may for example be a pulsed laser beam or a breaking capillary, launches acoustic waves into a sample. These waves are then detected by a small-aperture sensor located elsewhere on the surface of the sample. The source position can easily be varied to obtain data for a variety of directions in the sample. In the detected signal one discerns sharp features, such as spikes or discontinuities, that correspond to arrivals of waves that have propagated through the medium at their group velocity [6]. In the past, the treatment of these wave arrivals has ignored the difference between phase and group velocity [4,7]. The arrival times have been expressed in terms of phase velocities, and traditional methods have been used to recover the elastic constants. This approach is valid when applied to materials having a low degree of anisotropy, or where only data pertaining to the neighborhood of certain high symmetry directions is used. Elastic constants obtained on this basis, however, become progressively less reliable with increasing anisotropy and broader spread of measuring directions. Review oj Progress in Quantitative Nondestructive Evaluation, Vol. lOB Edited by D.O. Thompson and D.E. Chimenti, Plenum Press. New York, 1991 1663 The main problem to be confronted is that there is no self-contained mathematical expression available that relates only the elastic constants and the group velocity, which might conceivably be manipulated to obtain the elastic constants. Our method is therefore an indirect one, and takes the form of a two-step optimization process. The method requires the input of suitable approximate starting values of the elastic constants. These might be obtained, for example, from measurements made in certain high symmetry directions where phase and group velocities are equal, or even by means of the previously mentioned optimization process that ignores the difference between the phase and group velocities. METHOD Wave propagation in elastically anisotropic solids is governed by the Christoffel equations [3,8,9J (1) where Crism are the second-order elastic constants, n=(nt) is the wave normal, v=vn is the phase velocity and U=(Us) is the polarization vector of the wave, p is the density of the medium and hrs is the Kronecker delta. The secular equation (2) determines the directional dependence of the three phase velocities. Alternatively, given a sufficient number of phase velocities, Eq. 2 can be used to obtain the elastic constants. The group velocity V is given in terms of the phase velocity by [9J (3) There is no explicit equation presently available, analagous to Eq. 2, that relates only the elastic constants and the group velocity, which might conceivably be manipulated to obtain the elastic constants. Duff [101 has shown that this equation could be of degree as high as 150 in the components of V in the most general case, and there thus does not seem any realistic prospect of establishing it. The only presently available methods for calculating V, such as Eq. 3, all involve expressing it parametrically in terms of quantities such as v, n, the slowness s=v In and U. The inverse problem of obtaining the elastic constants from measured group velocities therefore needs to be approached indirectly. Our method for recovering elastic constants from measured group velocities consists, in summary, of the following steps: 1. Input of (a) suitable approximate values of the elastic constants, (b) a set of d 1 . f VI Vr h . d· f N1 Nr measure group ve OCI les exp ..... exp avmg lrec Ions .... . 2. Using a trial set of elastic constants the set of wave normals n1 ..... nr is determined, whose associated group velocities (of the appropriate polarization) V~alc .... V~alc point in the directions N1 .... Nr. 3. The closer the trial elastic constants are to the "correct" values, the smaller, on average, the differences between the measured and calculated velocities will be. The elastic constants are now varied so as to minimize r 2 1 . . X =r.1: (V~xp_V~alC)2. J=l (4) 1664 4. At each step of this minimization the n's are recalculated using the current values of the elastic constants. We treat the determination of the n's from the N's as a minimization problem, varying the polar angular coordinates 0 and <p of n so as to minimize the angular deviation between N and the ray vector calculated from n using Eqs. 2 and 3. The quantity that is minimized can be taken to be either (5) or (6) where e and i1> are the polar angular coordinates of N. In the case of transverse isotropy the velocities are independent of <p and so there is only the one angular coordinate 0 to consider. For this case we have used a bisection technique to find O. We have implemented this method for media of cubic symmetry and transverse isotropy in FORTRAN coding and run the programs on a CONVEX C210 supercomputer. The typical CPU time to fit 15 velocities for cubic symmetry and obtain the elastic constants is less than 2 minutes. The corresponding calculation for transverse isotropy requires a few seconds. As regards the transverse modes, caution has to be exercised in the selection of data. The transverse sheets of the group velocity surface often possess folded regions where the velocities are multivalued. Erroneous assignment of measured velocities to particular modes in the folded regions can easily occur in a computerized optimization scheme, and we have therefore found it expedient to avoid data from the folded regions. This requires that estimated values of the elastic constants be on hand so that the approximate shape of the wave surface can be determined. A convenient way 0f proceeding is to generate a polar plot of the rar vectors corresponding to a large number of wave normals distributed isotropically [11], using the estimated elastic constants. Figure 1 shows, as an example, polar plots for the ST and FT branches of germanium. The folded regions, which are to be avoided, are easily identified through being darker than the rest. (a) (b) Figure 1. Polar plot of the (a) ST and (b) FT ray distributions of germanium corresponding to an isotropic distribution of wave normals. Elastic constants for the calculation are from Ref. (3). 1665 APPLICATION TO COMPUTER GENERATED DATA Table 1 shows the results of calculations carried out on two cubic crystals, germanium and CsBr. The label a) indicates published values of the elastic constants that have been used in calculating various sets of group velocities that have been supplied to the fitting program. Label b) indicates values of the elastic constants that have been recovered for Ge from a set of 7 L, 3 ST and 4 FT velocities. The calculation of the velocities and the fitting has been done to 8-figure accuracy to test the program. As can be seen, the original elastic constants have been recovered to 6- or 7-figure accuracy. Label c), for CsBr, is for 6 Land 3 ST velocities and similar remarks apply. Label d) indicates values obtained from a combination of of Land T velocities, all of which have been randomly varied within an interval of ±0.5% to simulate statistical errors. All the elastic constants are still obtained to within ::::±1 % accuracy. Because of the restrictions imposed by the folding of the T sheets of the wave surface, it would be desirable if L mode velocity measurements sufficed to accurately determine all the elastic constants. This would be advantageous on experimental grounds also, since in PS/PR waveforms the L wave arrival is generally more easily and accurately identifiable than the T wave arrivals. We find, however, that given only L mode data, certain elastic constants and combinations can be accurately determined, but others cannot be recovered with the same high accuracy. Some constants are not obtainable at all. For cubic symmetry, in the principal directions the longitudinal velocity depends only on Cll and the combination Ce=C12+2C44 ' while the transverse velocities depend only on C44 and (Cll-C12)/2. Away from the principal directions, all three elastic constants are inextricably involved in determining each of the velocities, but the dependence of the longitudinal velocity on C12 and C44 individually is still in general too weak to allow the recovery of these two constants with much accuracy. Label e) indicates values of the elastic constants that have been recovered from a grid of 15 L velocities with 8-figure accuracy. We see that Cll and C l have been obtained with 4- or 5-fi'gure accuracy, but the optimization process has not been able to drive C12 ' C44 and Ct =(Cll-C12)/2 to their "correct" values, in spite of the very high accuracy of the calculations. Label f) indicates values that have been obtained from a grid of 15 L velocities scattered by ±0.5%. Again we see that C11 and C l have been determined with reasonable accuracy, but the other three constants have been much less accurately determined. Table 1. Recovery of elastic constants of cubic crystals from com~uter generated velocity data. The input elastic constants for Ge are from Ref. t3), and those for CsBr are from Ref. (12). Elastic constants in GPa C11 C12 C44 Cl Ct Comments Germanium data ~ 128.9 48.3 67.1 182.5 40.3 Input values 128.9000 48.3000 67.1000 182.5000 40.3000 71, 3ST, 4FT, 8-figures 128.3 48.5 67.5 183.6 39.9 Land T, ±0.5% scatter 128.9 49.3 66.6 182.5 39.8 15L, 8-figures 128.8 46.9 68.1 183.1 40.9 15L, ±0.5% scatter CsBr data ~ 30.7 8.4 7.49 23.38 11.15 Input values 30.7000 8.4000 7.4900 23.3800 11.1500 6L, 3ST, 8-figures 30.66 8.46 7.48 23.41 11.10 Land T, ±0.5% scatter 30.700 8.48 7.45 23.380 11.11 15L, 8-figures 30.74 9.46 6.99 23.45 10.64 15L, ±0.5% scatter 1666 Table 2 shows corresponding results for a transversely isotropic material, and similar remarks apply. In this case, from L mode velocity data alone only ell ' C33 ' and , Ct"'CI3+2C44 can be accurately recovered, while C13 and C44 individually are less accurately determined. The constants C12 and C66=(Cll-C12)/2 cannot be obtained at all since the L mode velocity does not depend on them. Table 2. Recovery of the elastic constants of a unidirectional glass fibre/polyester composite from computer generated velocity data. Input elastic constants are from Ref. (13). Elastic constants in GPa Cll C12 C13 C33 C' l Comments 42.1 42.1000 42.37 42.1000 41.85 16.5 10.2 56.4 15.5000 10.2000 56.4000 16.54 10.22 56.18 10.1997 56.4000 10.80 56.32 18.6 47.4 18.6000 47.4000 18.56 47.34 18.6002 47.4000 18.38 47.56 Input values 10L, lOST, 10 FT, 8-figures Land T, ±0.5% scatter 10L, 8-figures lOL, ±0.5% scatter Table 3. Elastic constants of silicon determined from 16 group velocity measurements made on <100> and <110> oriented single crystals. The density has been taken to be 2.332 g~cm3. Measured (m; and calculated (c) data on the group velocities is given. - ,<P -polar angles 0 V; O,rp -polar angles ef n. Elastic constants in GPa Cll C12 C44 Cl Ct Comments 165.1 65.0 80.2 225.5 50.0 This investigation 165.7 63.9 79.6 223.1 50.9 Values from Ref. (13) Branch Angles in degrees em <Pm 0c ¢c Velocities in mm/jJ1l Vm Vc L 90.0 0.6 90.0 0.3 8.26 8.41 L 90.0 33.7 90.0 25.2 8.93 9.00 L 90.0 39.8 90.0 35.0 9.20 9.11 L 90.0 31.9 90.0 22.9 9.06 8.96 L 90.0 25.9 90.0 16.3 8.89 8.80 L 90.0 19.2 90.0 10.9 8.72 8.64 L 89.5 44.7 89.7 44.3 8.93 9.15 L 66.1 26.7 72.6 18.6 9.23 9.07 L 76.7 35.4 80.4 28.2 9.28 9.11 L 85.9 42.1 86.9 39.3 9.29 9.14 L 87.4 43.2 88.1 41.4 9.26 9.15 ST 90.0 22.5 90.0 36.8 4.95 4.87 ST 90.0 32.0 90.0 40.5 4.77 4.71 FT 77.2 35.8 61.8 40.0 5.40 5.48 FT 72.7 32.3 59.9 39.5 5.35 5.40 FT 67.7 28.1 58.4 39.0 5.36 5.36 1667 EXPERIMENTAL RESULTS ON SILICON Table 3 lists the results of an experimental investigation on two disc-shaped silicon single crystals using the PS/PR technique based on laser excitation and piezoelectric sensing. The elastic constants that have been obtained are in good agreement with published values [14J. Also listed are the angular coordinates 8 m and 4> m and magnitude V m of the measured group velocities, the deduced polar angles Dc and 4>c of the wave normals, and the magnitudes V c of the calculated group velocities. The agreement between measured and calculated group velocities is better than 1 % in all but one case. The importance of distinguishing between the phase and group velocities is highlighted by the angular separation between the V's and their calculated n's. For the L mode data this angle ranges up to 9.6°, while for the FT modes it reaches 14.4°. ACKNOWLEDGMENTS This work has been supported by the NSF-funded Materials Science Center at Cornell University. A.G.E. also acknowledges support from the University of the Witwatersrand; W.S. acknowledges the support of the ONR, Physical Acoustics Program. M.O. Thompson is thanked for making available the laser which was used in the experiments on silicon. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13 14. 1668 E. Schreiber, O.L. Anderson and M. Soga, Elastic Constants and their Measurements (McGraw Hill, New York, 1973). S. Sengupta, Lattice Theory of Elastic Constants (Trans Tech, Aedermannsdorf, 1988). B. A. Auld, Acoustic Fields and Waves in Solids (Wiley, New York, 1973), Vol.l. See e.~. B. Castagnede, J.T. Jenkins, W. Sachse and S. Baste, J. Appl. Phys. 67, 2753 l1990). W. Sachse and K. Y. Kim, in Review of Quantitative Nondestructive Evaluation, 6A, edited by D. O. Thompson and D. E. Chimenti (Plenum, New York, 1986), p.311j also in Ultrasonic Materials Characterization II, edited by J. Boussiere, J.P. Monchalin, C.O. Ruud and R.E. Green (Plenum, New York, 1987), p.707. A.G. Every, W. Sachse, K.Y. Kim and M.O. Thompson, to be published. B. Castagnede, K.Y. Kim, W. Sachse and M.O. Thompson, to be published. M.J.P. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970). F.1. Fedorov, Theory of Elastic Waves in Crystals (Plenum, New York, 1968). G. F. D. Duff, Philos. Trans. R. Soc. London 252,249 (1960). A. G. Every, Phys. Rev. B 24, 3456 (1981). R.F.S. Hearmon, in Numerical Data and Functional Relationships in Science and Technology, Vol. 11 of Landolt-Bornstein, New Series, edited by K.H. Hellwege (Springer, Berlin,1979), Group III, p.l. B. Castagnede and W. Sachse, in Review of Progress in Quantitative Nondestructive Evaluation, 8B, edited by D. O. Thompson and D. E. Chimenti (Plenum, New York, 1989). J.J. Hall, Phys. Rev. 161, 756 (1967). 

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