The development of several approximations appears to permit accurate and practical calculations of the scattering of elastic waves from volumetric and crack-like defects of simple shapes if the wavelength of the incident wave is larger than the characteristic length of the shape. These approximations, which I call the quasi-static and extended quasi-static, use static solutions of defects in uniform strains to predict scattered (dynamic) fields. Since static solutions for several simple defect shapes (oblate and prolate spheroid, ellipsoid, and circular and elliptical cracks) are available, scattering predictions are possible, and the results of such calculations are presented

Gubernatis, J. E.

English

The development of several approximations appears to permit accurate and practical calculations of the scattering of elastic waves from volumetric and crack-like defects of simple shapes if the wavelength of the incident wave is larger than the characteristic length of the shape. These approximations, which I call the quasi-static and extended quasi-static, use static solutions of defects in uniform strains to predict scattered (dynamic) fields. Since static solutions for several simple defect shapes (oblate and prolate spheroid, ellipsoid, and circular and elliptical cracks) are available, scattering predictions are possible, and the results of such calculations are presented.</p

Gubernatis, J

Digital Repository @ Iowa State University (ISU)

LONG WAVE SCATTERING OF ELASTIC WAVES FROM VOLUMETRIC AND CRACK-LIKE DEFECTS OF SIMPLE SHAPES J. E. Gubernatis Los Alamos Scientific Laboratory Los Alamos, New Mexico 87545 ABSTRACT The development of several approximations appears to permit accurate and practical calculations of the scattering of elastic waves from volumetric and crack-like defects of simple shapes if the wavelength of the incident wave is larger than the characteristic length of the shape. These approximations, which I call the quasi-static and extended quasi-static, use static solutions of defects in uniform strains to predict scattered (dynamic) fields. Si nce static solutions for several simple defect shapes (oblate and prolate spheroid, ellipsoid, and circular and elliptical cracks) are available, scattering predictions are possible, and the results of such calculations are presented. Introduction l was asked to preface my presentation with a few words about the current state of ultrasonic defect characterization studies in the ARPA/AFML program compared to their state at its start. Con-sequently, my presentation has two distinct parts: One part l call "Past to Present" in which, em-phasizing the role played by theoretical studies, l try to assess what I regard as benchmarks in ultrasonic flaw characterization studies. The other part I call "Long Wave Scattering from Simple Shapes" in which I discuss my recent work. Past to Present* A Chronology Zeroth Year (Krumhansl, Gubernatls) At the start of the ARPA/AFML program, the existing literature on the scattering of elastic waves from defects was not properly oriented to the problem of flaw characterization. No system-atic way existed to study the scattering from shapes more complicated than a sphere (or an in-finite cylinder). Since the sphere is the only shape of finite volume solvable in closed form, a need for the develo~nt of efficient numerical techniques or approxi~tions existed. first Year (Krumhansl, Gubernatis, Domany, and Huberman) As a first step, a formal theory of the scattering of elastic waves was developed.l,2 This theory has a strong analogy to scattering theories used in quantum mechanics and thus has the potential susceptibility to a variety of num-erical and approximate techniques. A decision was made to avoid costly numerical sol utions and concentrate on less costly and simple approxi-mate techniques, which would perhaps reveal some physical insight. An approximation, called the Born Approximation in quantum mechanics, was studied and was compared to exact scattering re-sults from a sphere. With some limitations, the approximation appeared quite useful.3 A special usefulness was the fact that the shape of a de-fect enters through easily calculable factors *This research was sponsored by the Center for Advanced NDE operated by the Science Center, Rockwell International, for the Advanced Research Projects Agency and the Air Force Materials Laboratory under contract F33615-74-C-5180. 21 involving the Fou~ier transform of the volu~e V occupied by the defect (the shape factor)2,J, i.e., where~ fs the difference between the scattered and incident wave vectors. Tittmann1 meanwhile, measured the scattering of elastic waves from a sphere and found a good comparison between measured values and results calculated by exact theory. I regard the agree-ment as significant because I personally find as unconvincing explanations why What experimental-ists measure is what theorists calculate. In short, a useful and simple approximation now existed. Second Year (Krumhansl, Do.any, Teitel, Muzikar, and Wood) The Born Approximation was now applied to a spheroids, and the scattering was measured by Tittmann6 and Adler and Lewis7. Although the theory worked better for an oblate than for a prolate spheroid, when theory and Tittmann's ex-periment were compared, a correlation between the aspect ratio of spheroidal shapes and measured results was found; a definite relationship was established between scattering data and an identifying geometrical feature of a defect (a non-spherical one). In part, a flaw was charac-terized. The theorists began to examine other approxi-mations, and the scattering frOR crack-like flaws was computed.s Third Year (Krumhansl, Oomany, Rose, Teitel) The purpose of this meeting is to report the results for the third year; thus, I will conclude my view of the past after making several remarks that are difficult to time-sequence. Comments and Opinions The program has benefited greatly from a strong interaction between theory and experiment. This interaction is not accidental, but is indebted to the foresight shown in establishing the program when diffusion bonded samples containing identi-fiable defects of simple shapes (spheres and spheroids) were constructed. If the theorist had to work with the "flat bottom tmle". I doubt as much progress woula have occurred. Theory needs experiment, but can (and in some cases it has) progressed to the point where it is considerably cheaper to do the calculations than it is to make the measurements. For example, the Born Approximation, and to some extent another ap-proximation I discuss below, produce a data point for a fraction of a penny and 1 data base (about 1000 points) for several dollars. A comparably sized data base with experimental uncertainty cur-rently takes several days to measure which is several hundred dollars in salaries. This cost-effectiveness is germane to the work of Tony Mucciardi8 and of Eytan Domany.9 Tony needs a large data base on which to trtin his computer. Eytan, by being able to compute cheaply and ex-amine a variety of scattering cases. is seeing systematics translatable to experimental pro-cedures for identifying flaw snapes. The cost-effectiveness of theoretical studies canno~ emphasized strongly enough. -------Longwave Approximations for Simple Shapes** Introduction Completed studies used integral equation methods to describe the scattering. Recently, the theory of the scattering of elAStic waves from flaws by use of iQtegral equations was developed systematically. l,Z Previously, nearly all theo-retical studies of the scattering of elastic waves have used partial differential equations. With this approach the exact equations for scattering from a spherical flaw were fouad; however, the methods of partial differential equations have not been successful for shapes of finite volume other than the sphere: Boundary conditions are very dif-ficult to apply, and a systematic development of a perturbation theory for general finite volumes is equally difficult. By contrast, the boundary conditions , at least at the stlrt, are automati-cally built into the integral equation for all volumes, and a variety of systematically developed approximations is possible. Some Technical Details The basic scattering equation is1•2 where repeated subscripts implJ summation. The vector field u (r) represents the displacement field, and the1vector uy is the incident displace-ment field. The flaw is hoste4 in an infinite **Supported by USEROA. 22 elastic medium assumed isotropic and described by the Lame parameters~ and~. The Green's function for this ~edium gij(~.~·) equals (2) where w is the frequency of the incident wave, p the density of the host material, a the wave number of the longitudinal mode (P wave), P the wave number of the transverse mode (S wave), and R = 1!:_-!:.'1. The tensor field operator v;j(!:.} re-presents the flaw and is equal to (3) The quantity 8p is the difference between the den-sity of the flaw and its host; correspondingly, 6Cii~ is the difference between the elastic con-stants tensor of flaw and its host. Although the flaw is hosted in an isotropic medium, the flaw can have an arbitrary density and elastic proper-ties; ho.ever, the case of particular interest is the scattering from a void (p = c 1 j~ = 0). Eq~tion (1) describes the scattering of vector fields by a tensor "potential", and the scattered fields propagate from!:. to !:.' by a tensor Green's function (2). Additionally, be-cause an isotropic elastic medium has two modes of propagation, the far field displacement Is or the form iar e iPr u - u0 + A. ~ + 81. -r i I 1 r where the vector A; represents the amplitude of the longitudinal scattered wave and B; the transverse. As a consequence, for a given frequency, the dif-ferential cross-section is a(A + 2~) 1Ail 1 + 0~18il 2 a(A +Zit) Ia; I' + O!J ibil' (5) where a· and b. are the vector amplitudes of an in-cident longituAinal and a transverse displacement field. From Equation {5) it is apparent that even if lai1 2 or lbil' equals zero, the cross-section always involves contributions from two ~des of scattering associated with the same frequency (mode conversion). Because of the differential operators appear-ing in tile "potential", the scattered vector ampl i-tudes caa be regarded as functionals both of the displace-ent and strain fields internal to the flaw: A; = A;[ui; E ij] B; = Bi[U;; E ij) The Born Approximation corresponds to replacing the displacement field by the incident dis~lacement and the strain field by the strain field eyj associated with the incident displacement field: The results of this approximation have been com-puted for a variety of spherical flaws and compared to the exact solution.3 This approximation is found to describe backscattering well when the wavelengths are larger than the radius of a sphere. This observation was verified experimentally.6 for long wavelengths, the system is in a quasi-static condition. An alternative approxi-mation, called the ~uasi-static ap~roximation,lO is to replace the d splacement fie ds by the ampli-tude u~ of the incident mode and to replace the s~rain field by the associated static strain field c ij. A; 0 0 = Ai[ui; Eij] (6) Bi = Bi[u~; e~j] The results of this approximation are, for long wavelenyfhs, identical to the exact results for a sphere, i.e., it is not limited to back-scattering. By a systematic study of the iterative solution of the integral equation, it was recently shown that the approximation represented by Equation (6) 1 ~s exact for any finite shape at long wavelengths. This result pennits the exact cal-culation of the scattering of elastic waves from a flaw other than a sphere, albiet at long wave-lengths. The approximation determines exactly the ...,• contributions (the Rayleigh limit) to the scattering cross-section. Solutions of Elj are available for a number of geometries. The most famous a~g the most convenient is Eshelby's solution for a spheroid and ellipsoid. These shapes are extremely important, for by varying aspect ratios of the shapes one can distort them to resemble needle and disc crack-like geometries, types of flaws that one is most eager to detect. A more powerful approxjwation is the extended quasi-static approximation. In this approxl-mation A; 0 e11J.B.; o ei1,·B.] = Ai[ui ~ 1j (7) R1 0 e;~·R; ,o. eit,·g_] = 81 [ul lJ where 1, is the wave vector of the incident wave. An important point is that for ellipsoids and spheroids this approximation also uses Eshelby's solutions. This approximation, in contrast, to the Born and the quasi-static, is not well-defined in terms of a perturbation expansion. 23 A description of the full def~ils of these approximations is in preparation. What is more useful to present interests are calculated results for the variously shaped defects. Results Figure 1 illustrates the manner in which the scattering angles 8 and ~ are defined with respect to the direction of the incident wave !o· The Incident direction is always chosen along the z-axis, and the defects are always hosted in Ti-6Ai-4V. The differential cross-section is in decibels, and zero 1s equated to -100 db. X I I I I I SCATTERED ,.W POWER ~ z / / / I / y • INCIDENT POWER Figure 1. The basic scattering picture. Incident power scatters fran a flaw into a receiver in the di rection L· In Fig. 2 is a comparison of three approxi-mations, the Born, quasi-static, and extended quasi-static. with exact results for a sphere. The incident wave is a longitudinally polarized plane wave, the defect is a spherical void, k 1s the wavenumber of a longitudinal wave, and a is the radius of the sphere. The range of ka is to 2. For a void, the extended quasi-static ap-proximation appears to be quite useful up to ka ~ 1.5. Figure 2. Incident longitudinal plane wave scatter-ing from a spherical void. The differ-ential cross-section is calculated four ways: clockwise from lower left-hand corner, the Born Approximation, the extended quasi-static approximation, the quasi-static approximation, and the exact calculation. The remaining figures, because of the absence of exact results, were computed with the extended quasi-static approximation. The scattering of a longitudinally polarized plane wave from a stainless steel prolate spheroid is shown in Fig. 3. k is still the longitudinal wave number, but a refers to the semi-major axis. The axis of symmetry is perpendicular to the in-cident direction and along the y-direction. The major axes, along x andy, are 4 times the minor axis. Figure 3. Incident longitudinal plane wave scattering from a prolate spheroidal void calculated from the extended quasi-static approximation. The incident direction, along the z-axis, and the axis of symmetry are perpendicular, along the y-axis. The ratio of the z-axis to the y-axis is l/4. Clockwise from lower left, e equals 0, 30, 60, 90 degrees. 24 By letting the short axis of an ellipsoid or oblate spheroid go to zero, one pryguces ellipti-cal and circular disc-like cracks. Figure 4 shows the scattering of a longitudinal plane wave incident normal to the face of a crack lying in the xy-plane. k is the longitudinal wave number, and a is the radius of the crack. On the left is the longitudinal-to-longitudinal scattering; on the ri9ht, longitudinal-to-transverse (mode con-verted) scattering. Figure 4. Incident longitudinal plane wave scatter-ing from a circular crack. The incident direction, along the z-axis, is normal to the crack plane. On the left is longitudinal -to-longitudinal scattering; on the right, longitudinal-to-trans-verse scattering. In Fig. 5, a transversel.v polarized plane wave is scattered off the edge of an elliptical crack. The cracks lie in the xz-plane the semi-major axis, along x, is twice the semi-minor axis. The angle~ = 90°. On the left is the transverse-to-longitudinal scattering (mode converted}; on the right the transverse-to-transverse scattering. Figure 5. Incident transverse plane wave scatter-ing from an ellipti cal crack. The incident direction is along the z-axis, and the polarization is along the x-axis. The crack is in the xz-plane; its major axis, along x, is twice its minor axis. e • go·. On the left i s transverse-to-longitudinal scattering; on the right, transverse-to-transverse scattering.  Several conclusions are evident. The scattering is not isotropic. Different shapes produce different angular distributions. For acoustic and many quantum mechanical problems the scattering at long wavelengths is isotropic . This illustrates the fact that the elastic wave scattering problem has distinguishing features which can prevent the blind adoption of techniques and concepts successful in these other areas. Scattering signatures e~ist, but there is a need for systematic study to exploit them fully. The amount of data that can be easily generated is enormous, but it can be done cheaply. Time and space penmit the showing of only a small sample of nearly 100 calculations. The com-plete set of calcul~tions is being prepared as a Los Alamos Report. 15 References 1. J. E. Gubernatis, E. Domany, M. Huberman, and J. A. Krumhansl, in 1975 Ultrasonics Proceed-ings, (IEEE, New York,1975). p. 107.---2. J. E. Gubernatis, E. Domany, and J. A. Krumhans 1 , J. App 1. Phys. , June 1977. 3. J. E. Gubernatis, E. Domany, J. A. Krumhansl, and M. Huberman, J. Appl. Phys. June 1977. 4. B. R. Tittmann, AFML technical report, AFML-TR-75-212, p. 1g5. 5. J. A. Krumhansl, E. Domany, J. E. Gubernatis, P. Mulikar, S. Teitel, and D. Wood, Rockwell International (Science Center) technical re-port, "Interdisciplinary Program for Quanti-tative Flaw Definition; Special Report, Second Year Effort", p. 102 6. B. R. Tittmann, Rockwell International (Science Center) technical report, "Interdisciplinary Program for Quantitative Flaw Definition; Special Report, Second Year Effort", p. 102. 25 7. L. Adler and D. K. Lewis, Rockwell Inter-national (Science Center) technical report, "Interdisciplinary Program for Quantitative Flaw Definition; Special Report, Second Year Effort", p. 140 8. A. Mucciardi, this proceedings. 9. E. Domany, this proceedings. 10. In Ref. 5, this approxiaation was called Mal and Knopoff. 11. A. K. Mal and L. Knopoff, J. Inst. Math. Applic. 1. 376 (lg67). 12. J. E. Gubernatis, unpublished. S. K. Datta has shown this for ellipsoids ["Report on the Workshop on Application of Elastic Waves in Electrical Devices, Non-Destructive Testing, and Sei smo 1.ogy", Northwestern University, May 24-26, 1976, ed. by J. D. Achenbach, Y. H. Pao, and H. F. Tierstein, p. 355). 13. J. D. Eshelby, Proc. Roy. Soc. A24l, 376 (1957) . 14. In Ref. 5, thi s approxi•ation was called quasi-static. 15. J. E. Gubernatis, unpublished. 16. The work on disc-like cracks was done with E. Domany. 

Long Wave Scattering of Elastic Waves from Volumetric and Crack-Like Defects of Simple Shapes

Gubernatis, J.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=4456&amp;context=qnde

Long Wave Scattering of Elastic Waves from Volumetric and Crack-Like Defects of Simple Shapes

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Digital Repository @ Iowa State University (ISU)

Digital Repository @ Iowa State University (ISU)