Explicit Backscattering Coefficient for Ultrasonic Wave Propagating in Hexagonal Polycrystals with Fiber Texture

  • Gaofeng ShaEmail author


Hexagonal polycrystalline materials such as titanium, zirconium and magnesium are common in engineering structures like car bodies and airplane engines. These components are usually specially designed in microstructure level to guarantee desired mechanical properties, so microstructure characterization is of great importance to the optimization of manufacturing processes. Compared with traditional destructive characterization methods like scanning electron microscopy (SEM), the ultrasonic approach is nondestructive and cost-effective. Although some progress has been made in ultrasonic microstructure characterization of polycrystal aggregates of hexagonal grains, some factors like macro texture and grain size distribution associated with realistic microstructures were not accounted for yet. Targeting one common texture component for hexagonal polycrystalline materials, {0001} basal fiber texture, this paper derived the explicit backscattering coefficient for aggregates of ellipsoidal grains through one texture parameter and obtained an analytical backscattering coefficient for microstructures with various grain size distributions. In this study, the basal fiber texture was quantified by the 1D Gaussian orientation distribution function (ODF), which merely includes one texture parameter, and its relationship with generalized spherical harmonics ODF was also addressed. Moreover, explicit expressions for effective elastic moduli and elastic constant covariances were derived. Furthermore, some computational examples were given to demonstrate the impacts of texture, frequency, grain geometry and grain size distribution on backscattering behavior. The theoretical results in this study will greatly benefit the later ultrasonic microstructure characterization of hexagonal polycrystalline materials.


Ultrasonic backscattering Polycrystalline materials Hexagonal grains Fiber texture 


  1. 1.
    Kocks, H.R.W.U.F., Tome, C.N.: Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  2. 2.
    Banerjee, D., Williams, J.C.: Perspectives on titanium science and technology. Acta Mater. 61(3), 844–879 (2013)CrossRefGoogle Scholar
  3. 3.
    Lemaignan, C., Motta, A.T.: Zirconium alloys in nuclear applications. In: Frost, B.R.T. (ed.) Materials Science and Technology. A Comprehensive Treatment, vol. 10B. VCH, New York (1994)Google Scholar
  4. 4.
    Kulekci, M.K.: Magnesium and its alloys applications in automotive industry. Int. J. Adv. Manuf. Technol. 39(9), 851–865 (2008)CrossRefGoogle Scholar
  5. 5.
    Zhu, Z.S., Liu, R.Y., Yan, M.G., Cao, C.X., Gu, J.L., Chen, N.P.: Texture control and the anisotropy of mechanical properties in titanium sheet. J. Mater. Sci. 32(19), 5163–5167 (1997)CrossRefGoogle Scholar
  6. 6.
    Cheadle, B.A., Ells, C.E., Evans, W.: The development of texture in zirconium alloy tubes. J. Nucl. Mater. 23(2), 199–208 (1967)CrossRefGoogle Scholar
  7. 7.
    Sagapuram, D., Efe, M., Moscoso, W., Chandrasekar, S., Trumble, K.P.: Controlling texture in magnesium alloy sheet by shear-based deformation processing. Acta Mater. 61(18), 6843–6856 (2013)CrossRefGoogle Scholar
  8. 8.
    Wang, Y.N., Huang, J.C.: Texture analysis in hexagonal materials. Mater. Chem. Phys. 81(1), 11–26 (2003)CrossRefGoogle Scholar
  9. 9.
    Pilchak, A.L., Szczepanski, C.J., Shaffer, J.A., Salem, A.A., Semiatin, S.L.: Characterization of microstructure, texture, and microtexture in near-alpha titanium mill products. Metall. Mater. Trans. A 44A(11), 4881–4890 (2013)CrossRefGoogle Scholar
  10. 10.
    Watanabe, H., Fukusumi, M.: Mechanical properties and texture of a superplastically deformed AZ31 magnesium alloy. Mater. Sci. Eng. A 477A(1), 153–161 (2008)CrossRefGoogle Scholar
  11. 11.
    Lutjering, G., Williams, J.C., Gysler, A.: Microstructure and mechanical properties of titanium alloys. In: Li, J.C.M. (ed.) Microstructure and Properties of Materials, vol. 2. World Scientific, Singapore (2000)Google Scholar
  12. 12.
    Dziubińska, A., Gontarz, A., Horzelska, K., Pieśko, P.: The microstructure and mechanical properties of AZ31 magnesium alloy aircraft brackets produced by a new forging technology. Procedia Manuf. 2, 337–341 (2015)CrossRefGoogle Scholar
  13. 13.
    Shon, J.H., et al.: Effect of particle size distribution on microstructure and mechanical properties of spark-plasma-sintered titanium from CP-Ti powders. Int. J. Precis. Eng. Manuf. 15(4), 643–647 (2014)CrossRefGoogle Scholar
  14. 14.
    Pilchak, A.L., Li, J., Sha, G., Groeber, M., Tucker, J., Rokhlin, S.: A quantitative assessment of microtexture in titanium alloys using destructive and nondestructive methods. Microsc. Microanal. (2014). CrossRefGoogle Scholar
  15. 15.
    Lobkis, O.I., Yang, L., Li, J., Rokhlin, S.I.: Ultrasonic backscattering in polycrystals with elongated single phase and duplex microstructures. Ultrasonics 52(6), 694–705 (2012)CrossRefGoogle Scholar
  16. 16.
    Pilchak, A.L., Li, J., Rokhlin, S.I.: Quantitative comparison of microtexture in near-alpha titanium measured by ultrasonic scattering and electron backscatter diffraction. Metall. Mater. Trans. A 45A(10), 4679–4697 (2014)CrossRefGoogle Scholar
  17. 17.
    Khorev, A.I., Krasnozhon, A.I., Babaréko, A.A.: Texture strengthening of titanium alloys. Met. Sci. Heat Treat. 19(2), 144–146 (1977)CrossRefGoogle Scholar
  18. 18.
    Mohr, D., Chevin, M.-A., Greve, L.: Deformation behavior of magnesium extrusions with strong basal texture: experiments and modeling. J. Appl. Mech. 80(6), 061002 (2013)CrossRefGoogle Scholar
  19. 19.
    Li, J., Yang, L., Rokhlin, S.I.: “Effect of texture and grain shape on ultrasonic backscattering in polycrystals. Ultrasonics 54(7), 1789–1803 (2014)CrossRefGoogle Scholar
  20. 20.
    Li, J., Rokhlin, S.I.: Elastic wave scattering in random anisotropic solids. Int. J. Solids Struct. 78–79, 110–124 (2016)CrossRefGoogle Scholar
  21. 21.
    Yalda-Mooshabad, I., Thompson, R.B.: Influence of texture and grain morphology on the two-point correlation of elastic constants: theory and implications on ultrasonic attenuation and backscattering. In: Thompson, D.O., Chimenti, D.E. (eds.) Review of Progress in Quantitative Nondestructive Evaluation, vol. 14, pp. 1939–1946. Springer, Boston (1995)CrossRefGoogle Scholar
  22. 22.
    Bai, X., Tie, B., Schmitt, J., Aubry, D.: Numerical modeling of grain size effect on the ultrasonic propagation in polycrystalline materials. Ultrasonics 87, 18–21 (2018)CrossRefGoogle Scholar
  23. 23.
    Yang, L., Rokhlin, S.I.: Ultrasonic backscattering in cubic polycrystals with ellipsoidal grains and texture. J. Nondestruct. Eval. 32, 142–155 (2013)CrossRefGoogle Scholar
  24. 24.
    Cho, J.H., Rollett, A.D., Oh, K.H.: “Determination of volume fractions of texture components with standard distributions in Euler space. Metall. Mater. Trans. A 35A(13), 1075–1086 (2004)CrossRefGoogle Scholar
  25. 25.
    Bunge, H.J.: Texture Analysis in Materials Science: Mathematical Methods. Butterworth-Heinemann, Oxford (1982)Google Scholar
  26. 26.
    Roe, R.J.: Description of crystallite orientation in polycrystalline materials. III. General solution to pole figure inversion. J. Appl. Phys. 36(6), 2024–2031 (1965)CrossRefGoogle Scholar
  27. 27.
    Yang, L., Turner, J.A., Li, Z.: Ultrasonic characterization of microstructure evolution during processing. J. Acoust. Soc. Am. 121(1), 50 (2007)CrossRefGoogle Scholar
  28. 28.
    Li, J.Y.: Effective electroelastic moduli of textured piezoelectric polycrystalline aggregates. J. Mech. Phys. Solids 48(3), 529–552 (2000)CrossRefGoogle Scholar
  29. 29.
    Chen, W., Boehlert, C.J.: Texture induced anisotropy in extruded Ti–6Al–4V–xB alloys. Mater. Charact. 62(3), 333–339 (2011)CrossRefGoogle Scholar
  30. 30.
    Baird, J.C., et al.: Localized twin bands in sheet bending of a magnesium alloy. Scr. Mater. 67(5), 471–474 (2012)CrossRefGoogle Scholar
  31. 31.
    Wang, F., Sandlöbes, S., Diehl, M., Sharma, L., Roters, F., Raabe, D.: In situ observation of collective grain-scale mechanics in Mg and Mg-rare earth alloys. Acta Mater. 80, 77–93 (2014)CrossRefGoogle Scholar
  32. 32.
    Paroni, R.: Optimal bounds on texture coefficients. J. Elast. 60(1), 19–34 (2000)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Man, C.S., Huang, M.: A simple explicit formula for the Voigt–Reuss–Hill average of elastic polycrystals with arbitrary crystal and texture symmetries. J. Elast. 105(1–2), 29–48 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Böhlke, T., Lobos, M.: Representation of Hashin–Shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with application in materials design. Acta Mater. 67, 324–334 (2014)CrossRefGoogle Scholar
  35. 35.
    Van Pamel, A., Sha, G., Rokhlin, S.I., Lowe, M.J.S.: “Finite-element modelling of elastic wave propagation and scattering within heterogeneous media. Proc. R. Soc. London A Math. Phys. Eng. Sci. (2017). CrossRefGoogle Scholar
  36. 36.
    Ryzy, M., Grabec, T., Sedlák, P., Veres, I.A.: Influence of grain morphology on ultrasonic wave attenuation in polycrystalline media with statistically equiaxed grains. J. Acoust. Soc. Am. 143(1), 219–229 (2018)CrossRefGoogle Scholar
  37. 37.
    Bate, P., Lundin, P., Lindh-Ulmgren, E., Hutchinson, B.: Application of laser-ultrasonics to texture measurements in metal processing. Acta Mater. 123, 329–336 (2017)CrossRefGoogle Scholar
  38. 38.
    Dixon, S., Edwards, C., Palmer, S.B.: Texture measurements of metal sheets using wideband electromagnetic acoustic transducers. J. Phys. D Appl. Phys. 35(8), 816–824 (2002)CrossRefGoogle Scholar
  39. 39.
    Thompson, B.R.: Elastic-wave propagation in random polycrystals: fundamentals and application to nondestructive evaluation. Mater. Sci. 257, 233–257 (2002)Google Scholar
  40. 40.
    Hirsekorn, S.: The scattering of ultrasonic waves in polycrystalline materials with texture. J. Acoust. Soc. Am. 77(3), 832–843 (1985)CrossRefGoogle Scholar
  41. 41.
    Turner, J.A.: Elastic wave propagation and scattering in heterogeneous, anisotropic media: textured polycrystalline materials. J. Acoust. Soc. Am. 106(2), 541 (1999)CrossRefGoogle Scholar
  42. 42.
    Yang, L., Lobkis, O.I., Rokhlin, S.I.: Shape effect of elongated grains on ultrasonic attenuation in polycrystalline materials. Ultrasonics 51(6), 697–708 (2011)CrossRefGoogle Scholar
  43. 43.
    Arguelles, A.P., Turner, J.A.: Ultrasonic attenuation of polycrystalline materials with a distribution of grain sizes. J. Acoust. Soc. Am. 141(6), 4347–4353 (2017)CrossRefGoogle Scholar
  44. 44.
    Ranganathan, S.I., Ostoja-Starzewski, M.: Universal elastic anisotropy index. Phys. Rev. Lett. 101(5), 3–6 (2008)CrossRefGoogle Scholar
  45. 45.
    Lan, B., Lowe, M., Dunne, F.P.E.: Experimental and computational studies of ultrasound wave propagation in hexagonal close-packed polycrystals for texture detection. Acta Mater. 63, 107–122 (2014)CrossRefGoogle Scholar
  46. 46.
    Hirsekorn, S.: The scattering of ultrasonic waves in polycrystalline materials with texture. J. Acoust. Soc. Am. 77(3), 832–843 (1985)CrossRefGoogle Scholar
  47. 47.
    Man, C.S., Paroni, R., Xiang, Y., Kenik, E.A.: On the geometric autocorrelation function of polycrystalline materials. J. Comput. Appl. Math. 190(1–2), 200–210 (2006)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Roebuck, B.: Measurement of grain size and size distribution in engineering materials. Mater. Sci. Technol. 16(10), 1167–1174 (2000)CrossRefGoogle Scholar
  49. 49.
    Núñez, C., Domingo, S.: Statistical considerations on uniform grain size. Metall. Mater. Trans. A 19A(12), 2937–2944 (1988)CrossRefGoogle Scholar
  50. 50.
    Rose, J.H.: Ultrasonic backscatter from microstructure. In: Rev. Prog. Quant. Nondestruct. Eval. Vol. 11B; Proceedings of the 18th Annu. Rev. Brunswick, ME, July 28–August 2, 1991 (A93-19582 06-38), vol. 11, pp. 1677–1684 (1992)Google Scholar
  51. 51.
    Ghoshal, G., Turner, J.A., Weaver, R.L.: Wigner distribution of a transducer beam pattern within a multiple scattering formalism for heterogeneous solids. J. Acoust. Soc. Am. 122(4), 2009–2021 (2007)CrossRefGoogle Scholar
  52. 52.
    Yang, L., Lobkis, O.I., Rokhlin, S.I.: An integrated model for ultrasonic wave propagation and scattering in a polycrystalline medium with elongated hexagonal grains. Wave Motion 49, 544–560 (2012)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringThe Ohio State UniversityColumbusUSA

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