Laser-Ultrasonic Characterization of Strongly Anisotropic Materials by Transient Grating Spectroscopy



Transient grating spectroscopy (TGS) is a laser-ultrasonic method allowing measurement of the surface acoustic wave (SAW) velocity in an examined material for a given direction of the wave vector.


We explore the capability of TGS for determination of shear elastic coefficients (\(c^\prime\) and \(c_{44}\)) of strongly anisotropic cubic materials.


TGS is tested on a set of single crystals with an anisotropy factor up to \(A=25\). Using a numerical simulation based on a Ritz-Rayleigh approach, we show that strong anisotropy may lead to significant coupling of SAWs with bulk shear waves, which complicates TGS measurements in specific directions. Based on the obtained TGS data, we discuss the possibility of also using the TGS technique for assessing the longitudinal elastic coefficient (\(c_L\)).


Despite the energy focusing and other effects originating from the strong anisotropy, the TGS method can be used to reliably determine the directional dependence of the SAW velocity in these materials, and the resulting experimental datasets are sufficient for inverse determination of both the soft shear elastic constant (\(c^\prime\)) and the hard shear elastic constant (\(c_{44}\)). The longitudinal coefficient can be determined with lower accuracy.


TGS is a suitable experimental tool for contactless characterization of strongly anisotropic materials.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Lian J, Wang J (2014) Microstructure and Mechanical Anisotropy of Crab Cancer Magister Exoskeletons. Exp Mech 54(2):229–239

    Article  Google Scholar 

  2. 2.

    Dahl KB, Malo KA (2009) Planar strain measurements on wood specimens. Exp Mech 49(4):575–586

    Article  Google Scholar 

  3. 3.

    Wu TT, Ho ZH (1990) Anisotropic wave propagation and its applications to NDE of composite materials. Exp Mech 30(4):313–318

    Article  Google Scholar 

  4. 4.

    Sakaue K, Isawa T, Ogawa T, Yoshimoto T (2012) Evaluation of Viscoelastic Characteristics of Short-fiber Reinforced Composite by Indentation Method. Exp Mech 52(8):1003–1008

    Article  Google Scholar 

  5. 5.

    Lai Y, Wu Y, Sheng P, Zhang ZQ (2011) Hybrid elastic solids. Nat Mater 10(8):620–624

    Article  Google Scholar 

  6. 6.

    Kruisová A, Seiner H, Sedlák P, Landa M, Román-Manso B, Miranzo P, Belmonte M (2014) Acoustic metamaterial behavior of three-dimensional periodic architectures assembled by robocasting. Appl Phys Lett 105(21):211904

    Article  Google Scholar 

  7. 7.

    Nakanishi N (1980) Elastic constants as they relate to lattice properties and martensite formation. Prog Mater Sci 24(C):143-265

  8. 8.

    Lüthi B (2005) Physical Acoustics in the Solid State. Springer, Heidelberg

    Google Scholar 

  9. 9.

    Seiner H, Kopeček J, Sedlák P, Bodnárová L, Landa M, Sedmák P, Heczko O (2013) Microstructure, martensitic transformation and anomalies in c-softening in Co-Ni-Al ferromagnetic shape memory alloys. Acta Mater 61(15):5869–5876

    Article  Google Scholar 

  10. 10.

    Seiner H, Sedlák P, Bodnárová L, Drahokoupil J, Kopecký V, Kopeček J, Landa M, Heczko O (2013) The effect of antiphase boundaries on the elastic properties of Ni-Mn-Ga austenite and premartensite. J Phys Condens Matter 25(42):425402

    Article  Google Scholar 

  11. 11.

    Seiner H, Heczko O, Sedlák P, Bodnárová L, Novotný M, Kopeček J, Landa M (2013) Combined effect of structural softening and magneto-elastic coupling on elastic coefficients of NiMnGa austenite. J Alloy Compd 577:S131–S135

    Article  Google Scholar 

  12. 12.

    Seiner H, Stoklasová P, Sedlák P, Ševčík M, Janovská M, Landa M, Fukuda T, Yamaguchi T, Kakeshita T (2016) Evolution of soft-phonon modes in Fe-Pd shape memory alloy under large elastic-like strains. Acta Mater 105:182–188

    Article  Google Scholar 

  13. 13.

    Leisure RG, Willis FA (1997) Resonant ultrasound spectroscopy. J Phys Condens Matter 9(28):6001–6029

    Article  Google Scholar 

  14. 14.

    Maynard J (1996) Resonant ultrasound spectroscopy. Phys Today 49(1):26–31

    MathSciNet  Article  Google Scholar 

  15. 15.

    Zadler BJ (2005) Properties of elastic materials using contacting and non-contacting acoustic spectroscopy. Dissertation, Colorado School of Mines

  16. 16.

    Sedlák P, Seiner H, Zídek J, Janovská M, Landa M (2014) Determination of All 21 Independent Elastic Coefficients of Generally Anisotropic Solids by Resonant Ultrasound Spectroscopy: Benchmark Examples. Exp Mech 54(6):1073–1085

    Article  Google Scholar 

  17. 17.

    Landa M, Sedlák P, Seiner H, Heller L, Bicanová L, Šittner P, Novák V (2009) Modal resonant ultrasound spectroscopy for ferroelastics. Appl Phys A-Mater 96(3):557–567

    Article  Google Scholar 

  18. 18.

    Hatcher N, Kontsevoi OYu, Freeman AJ (2009a) Role of elastic and shear stabilities in the martensitic transformation path of NiTi. Phys Rev B 80(14):144203

    Article  Google Scholar 

  19. 19.

    Hatcher N, Kontsevoi OYu, Freeman AJ (2009b) Martensitic transformation path of NiTi. Phys Rev B 79(2):020202

    Article  Google Scholar 

  20. 20.

    Seiner H, Kopecký V, Landa M, Heczko O (2014) Elasticity and magnetism of Ni2 MnGa premartensitic tweed. Phys Status Solidi B 251(10):2097–2103

    Article  Google Scholar 

  21. 21.

    Stoklasová P, Sedlák P, Seiner H, Landa M (2015) Forward and inverse problems for surface acoustic waves in anisotropic media: A Ritz-Rayleigh method based approach. Ultrasonics 56:381–389

    Article  Google Scholar 

  22. 22.

    Wolfe JP (1998) Imaging Phonons. Cambridge University Press, Cambridge

    Google Scholar 

  23. 23.

    Maznev AA, Lomonosov AM, Hess P, Kolomenskii AA (2003) Anisotropic effects in surface acoustic wave propagation from a point source in a crystal. Eur Phys J B 35(3):429–439

    Article  Google Scholar 

  24. 24.

    Rogers JA, Fuchs M, Banet MJ, Hanselman JB, Logan R, Nelson KA (1997) Optical system for rapid materials characterization with the transient grating technique: Application to nondestructive evaluation of thin films used in microelectronics. Appl Phys Lett 71(2):225–227

    Article  Google Scholar 

  25. 25.

    Maznev AA, Nelson KA, Rogers JA (1998) Optical heterodyne detection of laser-induced gratings. Opt Lett 23(16):1319–1321

    Article  Google Scholar 

  26. 26.

    Dennett CA, Cao P, Ferry SE, Vega-Flick A, Maznev AA, Nelson KA, Every AG, Short MP (2016) Bridging the gap to mesoscale radiation materials science with transient grating spectroscopy. Phys Rev B 94:214106

    Article  Google Scholar 

  27. 27.

    Sermeus J, Sinha R, Vanstreels K, Vereecken PM, Glorieux C (2014) Determination of elastic properties of a MnO2 coating by surface acoustic wave velocity dispersion analysis. J Appl Phys 116:023503

    Article  Google Scholar 

  28. 28.

    Verstraeten B, Sermeus J, Salenbien R, Fivez J, Shkerdin G, Glorieux C (2015) Determination of thermoelastic material properties by differential heterodyne detection of impulsive stimulated thermal scattering. Photoacoustics 3(2):64–77

    Article  Google Scholar 

  29. 29.

    Grabec T, Sedlák P, Stoklasová P, Thomasová M, Shilo D, Kabla M, Seiner H, Landa M (2016) In-situ Characterization of Local Elastic Properties of Thin Shape Memory Films by Surface Acoustic Waves. Smart Mater Struct 25:127002

    Article  Google Scholar 

  30. 30.

    Grabec T, Sedlák P, Seiner H (2020) Application of the Ritz-Rayleigh method for Lamb waves in extremely anisotropic media. Wave Motion 96:102567

    MathSciNet  Article  Google Scholar 

  31. 31.

    Dennett CA, So KP, Kushima A, Buller DL, Hattar K, Short MP (2018) Detecting self-ion irradiation-induced void swelling in pure copper using transient grating spectroscopy. Acta Mater 145:496–503

    Article  Google Scholar 

  32. 32.

    Yamaguchi M, Zhao L, Chronister EL, Baer BJ (2002) Impulsive stimulated scattering study of surface acoustic waves in metal and semiconductor crystals under high pressure. P Soc Photo-Opt Ins 4812:77–81

    Google Scholar 

  33. 33.

    Heczko O, Seiner H, Stoklasová P, Sedlák P, Sermeus J, Glorieux C, Backen A, Fähler S, Landa M (2018) Temperature dependence of elastic properties in austenite and martensite of Ni-Mn-Ga epitaxial films. Acta Mater 145:298–305

    Article  Google Scholar 

  34. 34.

    Landa M, Verstraeten B, Sermeus J, Salenbien R, Sedlák P, Seiner H, Glorieux C (2011) Thermomechanical properties of single crystals evaluated by impulsive stimulated thermal scattering technique. J Phys Conf Ser 278(1):012023

    Article  Google Scholar 

  35. 35.

    Sedlák P, Seiner H, Landa M, Novák V, Šittner P, Ll Mañosa (2005) Elastic constants of bcc austenite and 2H orthorhombic martensite in CuAlNi shape memory alloy. Acta Mater 53(13):3643–3661

    Article  Google Scholar 

  36. 36.

    Ševčík M, Zídek J, Nejezchlebová J, Štefan J, Machová A, Seiner H, Uhnáková A, Čapek J, Lejček P (2019) Crack growth in Fe-Si (2 wt%) single crystals on macroscopic and atomistic level. Results Phys 14:102450

    Article  Google Scholar 

  37. 37.

    Landa M, Novák V, Sedlák P, Šittner P (2004) Ultrasonic characterization of Cu-Al-Ni single crystals lattice stability in the vicinity of the phase transition. Ultrasonics 42(1–9):519–526

    Article  Google Scholar 

  38. 38.

    Zoubková K (2019) Elastic response of ferromagnetic shape memory alloys in the vicinity of the critical point. Master’s Thesis, Czech Technical University in Prague

  39. 39.

    Johnson JA, Maznev AA, Bulsara MT, Fitzgerald EA, Harman TC, Calawa S, Vineis CJ, Turner G, Nelson KA (2012) Phase-controlled, heterodyne laser-induced transient grating measurements of thermal transport properties in opaque material. J Appl Phys 111:023503

    Article  Google Scholar 

  40. 40.

    Dennett CA, Short MP (2018) Thermal diffusivity determination using heterodyne phase insensitive transient grating spectroscopy. J Appl Phys 123:215109

    Article  Google Scholar 

  41. 41.

    Dennett CA, Short MP (2017) Time-resolved, dual heterodyne phase collection transient grating spectroscopy. Appl Phys Lett 110:211106

    Article  Google Scholar 

  42. 42.

    Every AG (2002) Measurement of the near-surface elastic properties of solids and thin supported. Meas Sci Technol 13(5):R21

    Article  Google Scholar 

  43. 43.

    Barnett DM, Lothe J (1985) Free surface (Rayleigh) waves in anisotropic elastic half space: The surface impedance method. Proc R Soc Lond A 402:135–152

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Every AG, Maznev AA, Grill W, Pluta M, Comins JD, Wright OB, Matsuda O, Sachse W, Wolfe JP (2013) Bulk and surface acoustic wave phenomena in crystals: Observation and interpretation. Wave Motion 50(8):1197–1217

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Sun B, Winey JM, Hemmi N, Dreger ZA, Zimmerman KA, Gupta YM, Torchinsky DH, Nelson KA (2008) Second-order elastic constants of pentaerythritol tetranitrate and cyclotrimethylene trinitramine using impulsive stimulated thermal scattering. J Appl Phys 104(7):073517

    Article  Google Scholar 

  46. 46.

    Sun B, Winey JM, Gupta YM, Hooks DE (2009) Determination of second-order elastic constants of cyclotetramethylene tetranitramine (\(\beta\)-HMX) using impulsive stimulated thermal scattering. J Appl Phys 106(5):053505

    Article  Google Scholar 

Download references


We would like to acknowledge Prof. V.A. Chernenko (BCMaterials & UPV/EHU Bilbao) and Prof. H. Hosoda (Tokyo Institute of Technology) for providing the Ni-Fe-Ga-Co sample for measurements. This work was financially supported by the Czech Science Foundation (project No. 20-12624S), by ERDF in the frame of the project ’Centre of Advanced Applied Sciences’ (project No. CZ.02.1.01/0.0/0.0/16_019/0000778), and by OP RDE, MEYS (ESS-Scandinavia-CZ-OP, CZ.02.1.01/0.0/0.0/16_013/0001794). KZ acknowledges support from the Grant Agency of the Czech Technical University in Prague, grant No. SGS19/190/OHK4/3T/14.

Author information



Corresponding author

Correspondence to H. Seiner.

Ethics declarations

Conflicts of Interest

The authors have no relevant financial or nonfinancial interests to disclose.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stoklasová, P., Grabec, T., Zoubková, K. et al. Laser-Ultrasonic Characterization of Strongly Anisotropic Materials by Transient Grating Spectroscopy. Exp Mech (2021).

Download citation


  • Laser-ultrasonics
  • Transient grating spectroscopy
  • Elastic anisotropy
  • Surface acoustic waves