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Acoustoelastic Coefficients in Thick Steel Plates under Normal and Shear Stresses


Ultrasonic monitoring of the integrity of structural materials utilizes the acoustoelasticity of materials (stress-dependence of ultrasonic wave velocity) obtained from the higher order strain terms in the constitutive equations. This method has been applied to measure residual and applied stresses in thin metals under principal stresses without shear. Here, acoustoelastic coefficients are determined in thick steel plates in normal, orthogonal, and angled directions by means of an array of ultrasonic sensors. A three-dimensional material model is developed which includes Murnaghan hyper-elasticity and can determine the effects of plate thickness and excitation frequency on the acoustoelastic coefficients. This model is experimentally validated by tensile loading of a thick steel plate by measuring the ultrasonic signals in three directions. Numerical and experimental results agree within the measurement uncertainties of each method. The 1.0 MHz ultrasonic frequency has the highest resolution for measuring normal and shear stresses in structural plates typically used in highway bridges.

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  1. 1.

    Liao M, Okazaki T, Ballarini R, Schultz A, Galambos T (2011) Nonlinear finite-element analysis of critical gusset plates in the I-35W bridge in Minnesota. J Struct Eng 137(1):59–68

    Article  Google Scholar 

  2. 2.

    Steinzig M, Ponslet E (2003) Residual stress measurement using the Hole-Drilling method and laser speckle interferometry: PART 1. Exp Tech 27(3):43–46

    Article  Google Scholar 

  3. 3.

    Withers PJ, Bhadeshia HKDH (2001) Residual stress, part 1 – measurement techniques. J Mater Sci Technol 17:355–365

    Article  Google Scholar 

  4. 4.

    Prevey PS (1996) Current applications of X-ray diffraction residual stress measurement. In: Vander Voort G, Friel J (eds) Developments in materials characterization technologies. ASM International, Materials Park, OH, pp 103–110

  5. 5.

    Ruud CO (1982) A review of selected non-destructive methods for residual stress measurement. NDT Int 15:15–23

    Article  Google Scholar 

  6. 6.

    Fitzpatrick ME, Fry AT, Holdway P, Kandil FA, Shakleton J, Souminen L (2005) Determination of residual stresses by x-ray diffraction, measurement good practice guide No 52. National Physics Laboratory, UK

    Google Scholar 

  7. 7.

    Wang Z, Gu Y, Wang Y (2012) A review of three magnetic NDT technologies. Magn Magn Mater 324:382–388

    Article  Google Scholar 

  8. 8.

    Rossini N, Dassisti M, Benyounis K, Olabi A (2012) Methods of measuring residual stresses in components. Mater Des 35:572–588

    Article  Google Scholar 

  9. 9.

    Murnaghan FD (1937) Finite deformations of an elastic solid. Am J Math 59(2):235–260

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Hughes DS, Kelly JL (1953) Second order elastic deformation of solids. Phys Rev 92(5):1145–1149

    Article  MATH  Google Scholar 

  11. 11.

    Lanza di Scalea F, Rizzo P, Seible F (2003) Stress measurement and defect detection in steel strands by guided stress waves. J Mater Civ Eng 15(3):219–227

    Article  Google Scholar 

  12. 12.

    Chaki S, Bourse G (2009) Stress level measurement in prestressed steel strands using acoustoelastic effect. Exp Mech 49:673–681

    Article  Google Scholar 

  13. 13.

    Salamone S, Bartoli I, Srivastava A, Philips R, Nucera C, Scalea FLD (2009) Nonlinear ultrasonic guided waves for stress monitoring in Prestressing tendons for post-tensioned concrete structures. ASNT NDE/NDT for Highways and Bridges, New York

    Google Scholar 

  14. 14.

    Gokhale S (2007) Determination of applied stresses in rails using the acoustoelastic effect of ultrasonic waves. M.SC Thesis. Texas A & M University

  15. 15.

    Szelazek J (2013) Sets of piezoelectric probeheads for stress evaluation with subsurface waves. J Nondestruct Eval 32:188–199

    Article  Google Scholar 

  16. 16.

    Kato M, Sato T, Kameyama K, Ninoyu H (1995) Estimation of the stress distribution in metals using nonlinear acoustoelasticity. J Acoust Soc Am 98:1496–1504

    Article  Google Scholar 

  17. 17.

    Kino GS, Hunter JB, Johnson GC, Selfridge AR, Barnett DM, Hermann G, Steele CR (1978) Acoustoelastic imaging of stress fields. J Appl Phys 50(4):2607–2613

    Article  Google Scholar 

  18. 18.

    Allen DR, Sayers CM (1984) The measurement of residual stresses in textured steel using an ultrasonic velocity combinations technique. J Ultrasonics 22:179–188

    Article  Google Scholar 

  19. 19.

    Kobayashi H, Arai Y, Ohsawa H, Nakamura H, Todoroki A (1992) Nondestructive measurement of welding residual stresses by acoustoelastic technique and prediction of fatigue crack growth. ASME J Press Vessel Technol 114:417–421

    Article  Google Scholar 

  20. 20.

    Leon-Salamanca T, Bray DE (1996) Residual stress measurement in steel plates and welds using critically refracted longitudinal (LCR) waves. Res Nondestruct Eval 7:169184

    Article  Google Scholar 

  21. 21.

    Lu H, Liu XS, Yang JG, Zhang SP, Fang HY (2008) Ultrasonic stress evaluation on welded plates with Lcr wave. Sci Technol Weld Join 13:7075

    Google Scholar 

  22. 22.

    Walaszek H, Hoblos J, Bourse G, Robin C (2002) Effect of microstructure on ultrasonic measurements of residual stresses in welded joints. In: Dias AM, Pina J, Batista AC, Diogo E (eds) ECRS 6 : proceedings of the 6th European Conference on Residual Stresses, Coimbra, Portugal, 10-12 July, 2002. Mater Sci Forum, vol 404. Trans Tech Publications, Switzerland, pp 875–880

  23. 23.

    Uzun F, Bilge AN (2011) Immersion ultrasonic technique for investigation of total welding residual stress. Procedia Eng 10:30983103

    Article  Google Scholar 

  24. 24.

    Fukuhara M, Kuwano Y, Saito K, Hirasawa T, Komura I (1998) Performance of nondestructive evaluation by diffracted SH ultrasonic waves in predicting degree of fatigue in cyclic bending of ferritic steel. J Nondestruct Eval Int 31:211–216

    Google Scholar 

  25. 25.

    Berruti T, Gola MM (1996) Acoustoelastic determination of stresses in steel using Rayleigh ultrasonic waves. In: Batros AL, Green RE, Ruud CO (eds) Nondestructive characterization of materials VII. Transtec Pub. Ltd, Lebanon, pp 171–178, Part 1

    Google Scholar 

  26. 26.

    Sanderson RM, Shen YC (2010) Measurement of residual stress using laser-generated ultrasound. J Press Vessels Pip 87:762–765

    Article  Google Scholar 

  27. 27.

    Kim N, Hong M (2009) Measurement of axial stress using mode-converted ultrasound. J NDT & E Int 42:164–169

    Article  Google Scholar 

  28. 28.

    Javadi Y, Akhlaghi M, Ahmadi Najafabadi M (2003) Using finite element and ultrasonic method to evaluate welding longitudinal residual stress through the thickness in austenitic stainless steel plates. J Mater Des 45:628–642

    Article  Google Scholar 

  29. 29.

    Todaro ME, Capsimalis GP (1986) Acoustoelastic effect for Rayleigh surface waves in the presence of a non-uniform stress field. Proc IEEE Ultrason Symp 1:229232

    Google Scholar 

  30. 30.

    Deputat J (1990) Application of acoustoelastic effect in measurements of residual stresses. Arch Acoust 15(1-2):69–92

    Google Scholar 

  31. 31.

    Sgalla M, Vangi D (2003) A device for measuring the velocity of ultrasonic waves: an application to stress analysis. J Exp Mech 44(1):85–90

    Article  Google Scholar 

  32. 32.

    Buenos AA, Pereira P Jr, Mei PR, dos Santos AA (2014) Influence of grain size on the propagation of LCR waves in low carbon steel. J Nondestruct Eval 33:562–570

    Article  Google Scholar 

  33. 33.

    Jassby K, Saltoun D (1982) Use of ultrasonic Rayleigh waves for the measurement of applied biaxial surface stresses in aluminum 2024-T351 alloy. J Mater Eval 40(2):198–205

    Google Scholar 

  34. 34.

    Husson D, Bennett SD, Kino GS (1982)Measurement of surface stresses using Rayleigh waves. IEEE, San Diego, pp 889–892. doi:10.1109/ULTSYM.1982.197962

  35. 35.

    Vangi D (2001) Stress evaluation by pulse-echo ultrasonic longitudinal wave. J Exp Mech 41:277–281

    Article  Google Scholar 

  36. 36.

    Duquennoy M, Ouaftouh M, Ourak M (1999) Ultrasonic evaluation of stresses in orthotropic materials using Rayleigh waves. J NDT & E Int 32:189–199

    Article  Google Scholar 

  37. 37.

    Akhshik S, Moharrami R (2009) Improvement in accuracy of the measurement of residual stress Due to circumferential welds in thin-walled pipe using Rayleigh wave method. J Nucl Eng Des 239(10):2201–2208

    Article  Google Scholar 

  38. 38.

    Hu E, He Y, Chen Y (2009) Experimental study on the surface stress measurement with Rayleigh wave detection technique. J Appl Acoust 70:356–360

    Article  Google Scholar 

  39. 39.

    Gandhi N (2010) Determination of dispersion curves for acoustoelastic lamb wave propagation, M.SC. Thesis. Georgia Institute of Technology University

  40. 40.

    Rogerson GA, Sandiford KJ (1999) Harmonic wave propagation along a non-principal direction in a pre-stressed elastic plate. J Eng Sci 37:1663–1691

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Destrade M, Ogden RW (2005) Surface waves in a stretched and sheared incompressible elastic material. J Non-Linear Mech 40:241–253

    Article  MATH  Google Scholar 

  42. 42.

    Destrade M, Otténio M, Pichugin AV, Rogerson GA (2005) Non-principal surface waves in deformed incompressible materials. J Eng Sci 43:1092–1106

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Connor P, Ogden RW (1995) The effect of shear on the propagation of elastic surface waves. J Eng Sci 33(7):973–982

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Connor P, Ogden RW (1996) The influence of shear strain and hydrostatic stress on stability and elastic waves in a layer. J Eng Sci 34(4):375–397

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Shi F, Michaels JE, Lee SJ (2012) In situ estimation of applied biaxial loads with lamb waves. J Acoust Soc Am 133(2):677–687

    Article  Google Scholar 

  46. 46.

    Egle DM, Bray DE (1976) Measurement of acoustoelastic and third-order elastic constants for rail steel. J Acoust Soc Am 60(3):741–744

    Article  Google Scholar 

  47. 47.

    Piersol AG (1981) Time delay estimation using phase data. J IEEE Trans Acoust 29(3):471–477

    Article  Google Scholar 

  48. 48.

    Makhort FG, Gushcha OI, Chernoochenko AA (1990) Theory of acoustoelasticity of Rayleigh surface waves. J Int Appl Mech 26(4):346–350

    MATH  Google Scholar 

  49. 49.

    Abbasi Z, Cunningham M, Ozevin O (2015) The quantification of errors in the measurement of nonlinear ultrasonics. SAE Conference, Michigan

    Book  Google Scholar 

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This investigation was supported by National Science Foundation Award No. 133552. The support from the sponsor is gratefully acknowledged. We acknowledge Dr. Daniel P. Bailey for detailed copy editing and discussion of the manuscript. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the organization acknowledged above.

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Correspondence to D. Ozevin.

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Abbasi, Z., Ozevin, D. Acoustoelastic Coefficients in Thick Steel Plates under Normal and Shear Stresses. Exp Mech 56, 1559–1575 (2016).

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  • Shear stress
  • Acoustoelasticity
  • Ultrasonics
  • Frequency