# Elastic Properties Measurement Using Guided Acoustic Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurement Setup

#### 2.2. Numerical Model

#### 2.2.1. Derivation of Differential Equations

#### 2.2.2. Spectral Collocation Method

#### 2.3. Optimization

## 3. Results

#### 3.1. Simulated Data

#### 3.2. Experimental Data

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Rose, J.L. Ultrasonic Guided Waves in Solid Media, 2nd ed.; Cambridge University Press: New York, NY, USA, 2014. [Google Scholar]
- Su, Z.; Ye, L.; Lu, Y. Guided Lamb Waves for Identification of Damage in Composite Structures: A Review. J. Sound Vib.
**2006**, 295, 753–780. [Google Scholar] [CrossRef] - Workman, G.L.; Kishoni, D.; Moore, P.O. Ultrasonic Testing, 3rd ed.; American Society for Nondestructive Testing: Columbus, OH, USA, 2007. [Google Scholar]
- Raghavan, A.; Cesnik, C.E.S. Review of Guided-Wave Structural Health Monitoring. Shock Vib. Digest
**2007**, 39, 91–114. [Google Scholar] [CrossRef] - Croxford, A.J.; Wilcox, P.D.; Drinkwater, B.W.; Konstantinidis, G. Strategies for Guided-Wave Structural Health Monitoring. Proc. R. Soc.
**2007**, 463, 2961–2981. [Google Scholar] [CrossRef] - Lindner, G. Sensors and Actuators Based on Surface Acoustic Waves Propagating Along Solid–Liquid Interfaces. J. Phys. D: Appl. Phys.
**2008**, 41, 123002. [Google Scholar] [CrossRef] - Tietze, S.; Singer, F.; Lasota, S.; Ebert, S.; Landskron, J.; Schwuchow, K.; Drese, K.S.; Lindner, G. Monitoring of Soft Deposition Layers in Liquid-Filled Tubes with Guided Acoustic Waves Excited by Clamp-on Transducers. Sensors
**2018**, 18, 526. [Google Scholar] [CrossRef] [Green Version] - Tietze, S.; Schlemmer, J.; Lindner, G. Influence of surface acoustic waves induced acoustic streaming on the kinetics of electrochemical. Micro/Nano Mater. Devices Syst.
**2013**, 8923, 89231B. [Google Scholar] - Redissi, A.; Miller, S. Experimental Characterization of the Propagation of Guided Acoustic Waves in Pipe Strings. J. Acoust. Soc. Am.
**2018**, 143, 1385–1391. [Google Scholar] [CrossRef] - Groth, E.B.; Iturrioz, I.; Clarke, T.G.R. The Dispersion Curve Applied in Guided Wave Propagation in Prismatic Rods. Lat. Am. J. Solids Struct.
**2018**, 15. [Google Scholar] [CrossRef] - Groth, E.B.; Clarke, T.G.R.; Schumacher da Silva, G.; Iturrioz, I.; Lacidogna, G. The Elastic Wave Propagation in Rectangular Waveguide Structure: Determination of Dispersion Curves and Their Application in Nondestructive Techniques. Appl. Sci.
**2020**, 10, 4401. [Google Scholar] [CrossRef] - Barshinger, J.N.; Rose, J.L. Guided Wave Propagation in an Elastic Hollow Cylinder Coated with a Viscoelastic Material. IEEE Trans. Ultrasonics Ferroelectr. Frequency Control
**2004**, 51, 1547–1556. [Google Scholar] [CrossRef] - Liu, Y.-C.; Hwang, Y.-F.; Huang, J.-H. Dispersion Relations and Modal Patterns of Wave in a Cylindrical Shell. In Wave Processes in Classical and New Solids; Giovine, P., Ed.; InTech: Rijeka, Croatia, 2012. [Google Scholar]
- Sarkar, A.; Sonti, V.R. Simplified Dispersion Curves for Circular Cylindrical Shells Using Shallow Shell Theory. J. Sound Vib.
**2009**, 322, 1–7. [Google Scholar] [CrossRef] - Lowe, M.J.S. Matrix Techniques for Modeling Ultrasonic Waves in Multilayered Media. IEEE Trans. Ultrasonics Ferroelectr. Frequency Control
**1995**, 42, 525–542. [Google Scholar] [CrossRef] - Maghsoodi, A.; Ohadi, A.; Sadighi, M. Calculation of Wave Dispersion Curves in Multilayered Composite-Metal Plates. Shock Vib.
**2014**, 2014, 1–6. [Google Scholar] [CrossRef] - Vaziri Astaneh, A.; Guddati, M.N. Efficient Computation of Dispersion Curves for Multilayered Waveguides and Half-Spaces. Comput. Methods Appl. Mechan. Eng.
**2016**, 300, 27–46. [Google Scholar] [CrossRef] [Green Version] - Shao, G.-Z.; Li, Q.-C.; Liang, Z.-Q. A Study on Dispersion Curves of Guided Wave in Layered Media with Overlying Liquid Surface. Chin. J. Geophys.
**2007**, 50, 783–789. [Google Scholar] [CrossRef] - Bao, X.L.; Franklin, H.; Raju, P.K.; Überall, H. The Splitting of Dispersion Curves for Plates Fluid-Loaded on Both Sides. J. Acoust. Soc. Am.
**1997**, 102, 1246–1248. [Google Scholar] [CrossRef] - Maess, M.; Wagner, N.; Gaul, L. Dispersion Curves of Fluid Filled Elastic Pipes by Standard FE Models and Eigenpath Analysis. J. Sound Vib.
**2006**, 296, 264–276. [Google Scholar] [CrossRef] - Rogers, W.P. Elastic Property Measurement Using Rayleigh-Lamb Waves. Res. Nondestruct. Eval.
**1995**, 6, 185–208. [Google Scholar] [CrossRef] - Sale, M.; Rizzo, P.; Marzani, A. Guided Waves Based Approach for the Reconstruction of the Elastic Moduli of Plates. In Proceedings of the 2009 IEEE International Ultrasonics Symposium, Rome, Italy, 20–23 September 2009; pp. 1499–1502. [Google Scholar]
- Deán, J.L.; Trillo, C.; Doval, A.F.; Fernández, J.L. Determination of Thickness and Elastic Constants of Aluminum Plates from Full-Field Wavelength Measurements of Single-Mode Narrowband Lamb Waves. J. Acoust. Soc. Am.
**2008**, 124, 1477–1489. [Google Scholar] [CrossRef] [PubMed] - Ponschab, M.; Kiefer, D.A.; Rupitsch, S.J. Simulation-Based Characterization of Mechanical Parameters and Thickness of Homogeneous Plates Using Guided Waves. IEEE Trans. Ultrasonics Ferroelectr. Frequency Control
**2019**, 66, 1898–1905. [Google Scholar] [CrossRef] [PubMed] - Gao, W.; Glorieux, C.; Thoen, J. Laser Ultrasonic Study of Lamb Waves: Determination of the Thickness and Velocities of a Thin Plate. Int. J. Eng. Sci.
**2003**, 41, 219–228. [Google Scholar] [CrossRef] - Lašová, S.; Zemčík, R. Determination of Group Velocity of Propagation of Lamb Waves in Aluminium Plate Using Piezoelectric Transducers. Applied and Computational Mechanics
**2017**, 11. [Google Scholar] [CrossRef] - Pei, N.; Bond, L.J. Higher Order Acoustoelastic Lamb Wave Propagation in Stressed Plates. J. Acoust. Soc. Am.
**2016**, 140, 3834. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Karpfinger, F.; Valero, H.-P.; Gurevich, B.; Bakulin, A.; Sinha, B. Spectral-Method Algorithm for Modeling Dispersion of Acoustic Modes in Elastic Cylindrical Structures. Geophysics
**2010**, 75, 19–27. [Google Scholar] [CrossRef] - Quintanilla, F.H.; Lowe, M.J.S.; Craster, R.V. Modeling Guided Elastic Waves in Generally Anisotropic Media Using a Spectral Collocation Method. J. Acoust. Soc. Am.
**2015**, 137, 1180–1194. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Adamou, A.T.I.; Craster, R.V. Spectral Methods for Modelling Guided Waves in Elastic Media. J. Acoust. Soc. Am.
**2004**, 116, 1524–1535. [Google Scholar] [CrossRef] - Lan, B. Non-Iterative, Stable Analysis of Surface Acoustic Waves in Anisotropic Piezoelectric Multilayers Using Spectral Collocation Method. J. Sound Vib.
**2018**, 433, 16–28. [Google Scholar] [CrossRef] - Quintanilla, F.H.; Fan, Z.; Lowe, M.J.S.; Craster, R.V. Guided Waves’ Dispersion Curves in Anisotropic Viscoelastic Single- and Multi-Layered Media. Proc. R. Soc. A: Math. Phys. Eng. Sci.
**2015**, 471, 20150268. [Google Scholar] [CrossRef] - Zharnikov, T.; Syresin, D.; Hsu, C.-J. Application of the Spectral Method for Computation of the Spectrum of Anisotropic Waveguides. J. Acoust. Soc. Am.
**2013**, 133, 3456. [Google Scholar] [CrossRef] - Achenbach, J.D. Wave Propagation in Elastic Solids, 8th ed.; Elsevier: Amsterdam, The Netherlands, 1999. [Google Scholar]
- Boyd, J.P. Chebyshev & Fourier Spectral Methods; Springer: Berlin, Germany, 1989. [Google Scholar]
- Trefethen, L.N. Spectral Methods in MATLAB; Society for Industrial and Applied Mathematics: New York, NY, USA, 2000. [Google Scholar]
- Moler, C.B.; Stewart, G.W. An Algorithm for Generalized Matrix Eigenvalue Problems. SIAM J. Numer. Anal.
**1973**, 10, 241–256. [Google Scholar] [CrossRef] - Marquardt, D.W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Branch, M.A.; Cloeman, T.F.; Li, Y. A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems. SIAM J. Sci. Comput.
**1999**, 21, 1–23. [Google Scholar] [CrossRef]

**Figure 1.**A schematic diagram of the measurement setup, consisting of a signal generator (GEN), a digital storage oscilloscope (DSO) and two transducers, one for transmitting (T) and one for receiving (R) the guided acoustic wave signals, attached to the surface of the test specimen at a fixed distance (d).

**Figure 4.**Group velocities ratios (circles) derived from the acoustic signals at different frequencies and the corresponding fitted model (solid line) for (

**a**) simulated data and (

**b**) experimental data.

Parameter | Reference | Solution | Error (%) |
---|---|---|---|

${c}_{L}$ ($\mathrm{m}/\mathrm{s}$) | 6350.00 | 6418.44 | 1.08 |

${c}_{T}$ ($\mathrm{m}/\mathrm{s}$) | 3100.00 | 3121.98 | 0.71 |

h () | 1.00 | 1.01 | 0.90 |

Parameter | Reference | Solution | Error (%) |
---|---|---|---|

${c}_{L}$ ($\mathrm{m}/\mathrm{s}$) | 6718.75 | 6876.13 | 2.34 |

${c}_{T}$ ($\mathrm{m}/\mathrm{s}$) | - | 3151.11 | - |

h (mm) | 1.00 | 0.97 | 3.40 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fairuschin, V.; Brand, F.; Backer, A.; Drese, K.S.
Elastic Properties Measurement Using Guided Acoustic Waves. *Sensors* **2021**, *21*, 6675.
https://doi.org/10.3390/s21196675

**AMA Style**

Fairuschin V, Brand F, Backer A, Drese KS.
Elastic Properties Measurement Using Guided Acoustic Waves. *Sensors*. 2021; 21(19):6675.
https://doi.org/10.3390/s21196675

**Chicago/Turabian Style**

Fairuschin, Viktor, Felix Brand, Alexander Backer, and Klaus Stefan Drese.
2021. "Elastic Properties Measurement Using Guided Acoustic Waves" *Sensors* 21, no. 19: 6675.
https://doi.org/10.3390/s21196675