# Mechanic-Electric-Thermal Directly Coupling Simulation Method of Lamb Wave under Temperature Effect

^{*}

## Abstract

**:**

_{0}and S

_{0}mode of simulation signals match well with the experimental measurements. Additionally, the effect of temperature on LW propagation is consistent between simulation and experiment; that is, the amplitude increases, and the phase velocity decreases with the increment of temperature.

## 1. Introduction

## 2. Simulation Mechanisms of Temperature Influence on LW

#### 2.1. Excitation–Propagation–Sensing Model

#### 2.1.1. Excitation Model

**e**and

**E**refer to strain vector and electric field vector, respectively,

**s**is the elastic compliance matrix,

**σ**is the stress vector,

**ε**is the dielectric constant matrix, and

**d**and

**D**refer to the piezoelectric coefficient matrix and electric displacement vector, respectively.

_{act}and t

_{act}are the diameter and thickness of the PZT, respectively, t

_{bond}is the thickness of the adhesive layer, t

_{plate}is the thickness of the plate structure, G

_{bond}is the shear strength of the adhesive layer, ${Y}_{\mathrm{act}}^{E}$ is Young’s modulus of the PZT, and E

_{plate}is elastic modulus of the structure.

_{31}= d

_{32}and excite the LW by applying a voltage excitation signal in the 3-direction. Only the piezoelectric constant d

_{31}needs to be considered in m/V. The driving strain at the bottom of the excitation sensor along the x-direction is expressed in the following Equation (2), where V

_{in}is the excitation voltage.

#### 2.1.2. Propagation Model

_{L}and c

_{T}are the velocities of the longitudinal wave and transverse wave, respectively, as shown in Equations (8) and (9).

#### 2.1.3. Sensing Model

_{act}is the Poisson’s ratio of the PZT, e

_{33}is the dielectric constant of the PZT, s

_{13}is the elastic coefficient of the PZT, and C(Γ) is a function of the Γ shear-lag parameter, as shown in Equations (11) and (12).

#### 2.2. Material Parameters under Temperature Effect

^{−6}/°C. Barakat et al. [37] gave the curve of storage modulus with temperature. The numerical model for the variation of adhesive shear modulus with temperature is shown in Equation (15).

#### 2.3. Piezoelectric Constants under Thermal Stress Effect

^{−6}/°C, 54 × 10

^{−6}/°C, and 23 × 10

^{−6}/°C, respectively. The thermal stress is not negligible for the PZT because the difference between the three thermal expansion coefficients is large, and the piezoelectric sensor is sensitive to changes in stress.

_{31}is summarized as shown in Equation (19), where σ is the actual stress caused by the load in MPa.

_{31}and the influence of thermal stress on the piezoelectric constant d

_{31}are superimposed as the numerical model of the temperature-influenced LW propagation simulation, including thermal stress, as shown in Equation (20).

_{T}is the thermal stress, in MPa.

## 3. Simulation Method of LW under Temperature Effect

#### 3.1. Architecture of the Multiphysics Simulation Method

#### 3.1.1. D Geometry and Definitions

#### 3.1.2. Material Parameters Numerical Model of Temperature Effect

#### 3.1.3. Multiphysics Coupling under Temperature Effect

#### 3.1.4. Finite Element Meshes

_{0}mode and A

_{0}mode on a 2 mm thick aluminum plate are 5382 m/s and 1731 m/s, respectively, and the corresponding wavelengths are 27 mm and 9 mm, respectively. Therefore, the maximum mesh size of the aluminum plate shall be less than 1.5 mm. Considering that the thickness of PZTs and adhesive layers are 0.48 mm and 0.08 mm, the mesh size of the PZTs and the adhesive layers are set to 1 mm and 0.5 mm, respectively. Due to the small mesh size, the model contains 1,890,000 domain elements, 553,000 boundary elements, and 3800 edge elements, with 5,200,000 degrees of freedom.

#### 3.1.5. Stationary and Time-Dependent Solver Settings

^{−7}s, and the time range in the time-dependent study is from 0 s to 1.5 × 10

^{−4}s.

#### 3.2. Simulation Results

_{0}mode and A

_{0}mode are excited normally. The S

_{0}mode propagates faster than the A

_{0}mode, and the amplitude of the S

_{0}mode is weaker than that of the A

_{0}mode. It can be seen that the amplitude increases with the increment of temperature by comparing the color bar of maximum stress at different temperatures. In order to better study the effect of temperature on the LW phase, the region in the red box is selected and enlarged. The wave fields at t = 3 × 10

^{−5}s are given in Figure 5. It can be seen that the phase is delayed with the increment of temperature. The third wave packet just reaches the white line at 20 °C, exceeds the white line at −20 °C, and does not reach the white line at 60 °C.

_{0}mode is given to observe the changes in phase and amplitude better. It can be seen that the amplitude increases, and the phase delays with the increment of temperature.

## 4. Experimental Verification of the Simulation Method

#### 4.1. Experimental Setup

#### 4.2. Experimental Results

_{0}mode is given to observe the changes in phase and amplitude better. The amplitude increases, and the phase delays with the increase of temperature. Figure 10 shows the relationship between signal amplitude and temperature. The amplitude increases linearly with the increase in temperature.

#### 4.3. Comparison between Simulation and Experiment

_{0}mode is normalized. It can be found that the waveform of S

_{0}mode matches well, while the amplitude and phase of A

_{0}mode have small errors. The reason for the error may be that the wavelength of A

_{0}mode is less than S

_{0}; therefore, the mesh size of A

_{0}mode needs to be smaller to ensure sufficient accuracy.

_{Tem}is the amplitude of the LW signals at different temperatures, Amp

_{T0}is the amplitude of the LW signals at −20 °C, c

_{p}is the phase velocity, l

_{p}is the distance of LW propagation, and Δt is the time shift of the constant phase of the LW signals.

_{0}mode between simulation and experiment. Experiment 1 and experiment 2 represent channel 5–6 and channel 5–8, respectively. It can be seen that the amplitude increases, and the phase velocity decreases with the increment of temperature. However, there may be differences in the influence of temperature on material parameters between simulation and experiment, resulting in the amplitude variations rate of the experiment being greater than that of the simulation. The phase velocity variations match well between the simulation and experiment.

## 5. Conclusions

_{0}mode matches well, while the amplitude and phase of A

_{0}mode have small errors. In addition, the influence of temperature on the LW between simulation and experiment is also consistent; that is, the amplitude increases, and the phase velocity decreases with the increase of temperature. However, there may be differences in the influence of temperature on material parameters between simulation and experiment, resulting in the amplitude change rate of the experiment being greater than that of the simulation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Implication |

e | Strain vector |

d | Piezoelectric coefficient matrix |

E | Electric field vector |

s | Elastic compliance matrix |

σ | Stress vector |

D | Electric displacement vector |

ε | Dielectric constant matrix |

l_{act} | Diameter of PZT |

t_{act} | Thickness of PZT |

t_{bond} | Thickness of the adhesive layer |

t_{plate} | Thickness of the structure |

G_{bond} | Shear strength of the adhesive layer |

${Y}_{\mathrm{act}}^{E}$ | Young’s modulus of the PZT |

E_{plate} | Elastic modulus of the structure |

d_{31} | Piezoelectric constant |

V_{in} | Excitation voltage |

Γ | Shear-lag coefficient |

k | Wavenumber |

ω | Angel frequency of the LW |

c_{L} | Velocity of the longitudinal wave |

c_{T} | Velocity of the transverse wave |

ρ_{plate} | Density of the structure |

ν_{plate} | Poisson’s ratio of the structure |

e_{33} | Dielectric constant of PZT |

s_{13} | Elastic coefficient of PZT |

V_{out} | Response voltage |

C(Γ) | Function of the Γ shear-lag parameter |

I(ΓR) | Bessel function |

ΔT | Relative to the reference temperature |

Δσ_{T} | Thermal stress |

Ex | Excitation signal |

A | Amplitude of excitation signal |

f | Central frequency of excitation signal |

t | Wave propagating duration |

N | Number of cycles within the signal window |

Amp_{Tem} | Amplitude of LW signal at corresponding temperature |

Amp_{T0} | Amplitude of LW signal at −20 °C |

c_{p} | Phase velocity of LW signal |

l_{p} | Distance of LW propagation |

Δt | Time shit of the constant phase of LW signal |

## References

- Boller, C.; Chang, F.K.; Fujino, Y. Encyclopedia of Structural Health Monitoring; John Wiley & Sons: New York, NY, USA, 2009. [Google Scholar]
- Bekas, D.G.; Sharif-Khodaei, Z.; Aliabadi, M.H.F. An innovative diagnostic film for structural health monitoring of metallic and composite structures. Sensors
**2018**, 18, 2084. [Google Scholar] [CrossRef] [PubMed] - Yuan, S.F.; Ren, Y.Q.; Qiu, L.; Mei, H.F. A multi-response-based wireless impact monitoring network for aircraft composite structures. IEEE Trans. Ind. Electron.
**2016**, 63, 7712–7722. [Google Scholar] [CrossRef] - Chen, J.; Yuan, S.F.; Jin, X. On-line prognosis of fatigue cracking via a regularized particle filter and guided wave monitoring. Mech. Syst. Signal Process.
**2019**, 131, 1–17. [Google Scholar] [CrossRef] - Wang, Y.; Qiu, L.; Luo, Y.J.; Ding, R. A stretchable and large-scale guided wave sensor network for aircraft smart skin of structural health monitoring. Struct. Health Monit.
**2021**, 20, 861–876. [Google Scholar] [CrossRef] - Wandowski, T.; Malinowski, P.; Ostachowicz, W. Elastic wave mode conversion phenomenon in glass fiber-reinforced polymers. Int. J. Struct. Integr.
**2019**, 10, 337–355. [Google Scholar] [CrossRef] - Yuan, S.F.; Chen, J.; Yang, W.B.; Qiu, L. On-line crack prognosis in attachment lug using Lamb wave-deterministic resampling particle filter-based method. Smart Mater. Struct.
**2017**, 26, 085016. [Google Scholar] [CrossRef] - Ren, Y.Q.; Tao, J.Y.; Xue, Z.P. Design of a large-scale piezoelectric transducer network layer and its reliability verification for space structures sensors. Sensors
**2020**, 20, 4344. [Google Scholar] [CrossRef] - Xu, Q.H.; Yuan, S.F.; Huang, T.X. Multi-dimensional uniform initialization gaussian mixture model for spar crack quantification under uncertainty. Sensors
**2021**, 21, 1283. [Google Scholar] [CrossRef] - Gorgin, R.; Luo, Y.; Wu, Z.J. Environmental and operational conditions effects on lamb wave based structural health monitoring systems: A review. Ultrasonics
**2020**, 105, 106114. [Google Scholar] [CrossRef] - Su, Z.Q.; Zhou, C.; Hong, M.; Cheng, L.; Wang, Q.; Qing, X.L. Acousto-ultrasonics-based fatigue damage characterization: Linear versus nonlinear signal features. Mech. Syst. Signal Process.
**2014**, 45, 225–239. [Google Scholar] [CrossRef] - Qiu, L.; Fang, F.; Yuan, S.F. Improved density peak clustering-based adaptive Gaussian mixture model for damage monitoring in aircraft structures under time-varying conditions. Mech. Syst. Signal Process.
**2019**, 126, 281–304. [Google Scholar] [CrossRef] - Ren, Y.Q.; Qiu, L.; Yuan, S.F.; Fang, F. Gaussian mixture model and delay-and-sum based 4D imaging of damage in aircraft composite structures under time-varying conditions. Mech. Syst. Signal Process.
**2020**, 135, 106390. [Google Scholar] [CrossRef] - Singh, P.; Keyvanlou, M.; Sadhu, A. An improved time-varying empirical mode decomposition for structural condition assessment using limited sensors. Eng. Struct.
**2021**, 232, 111882. [Google Scholar] [CrossRef] - Qiu, L.; Yuan, S.F.; Chang, F.K.; Bao, Q.; Mei, H.F. On-line updating Gaussian mixture model for aircraft wing spar damage evaluation under time-varying boundary condition. Smart Mater. Struct.
**2014**, 23, 125001. [Google Scholar] [CrossRef] - Shen, Y.F.; Giurgiutiu, V. Effective non-reflective boundary for Lamb waves: Theory, finite element implementation, and applications. Wave Motion
**2015**, 58, 22–41. [Google Scholar] [CrossRef] - Ge, L.Y.; Wang, X.W.; Wang, F. Accurate modeling of PZT-induced Lamb wave propagation in structures by using a novel spectral finite element method. Smart Mater. Struct.
**2014**, 23, 95018. [Google Scholar] [CrossRef] - Hafezi, M.H.; Alebrahim, R.; Kundu, T. Peri-ultrasound for modeling linear and nonlinear ultrasonic response. Ultrasonics
**2017**, 80, 47–57. [Google Scholar] [CrossRef] - Radecki, R.; Su, Z.Q.; Cheng, L.; Packo, P.; Staszewski, W.J. Modelling nonlinearity of guided ultrasonic waves in fatigued materials using a nonlinear local interaction simulation approach and a spring model. Ultrasonics
**2018**, 84, 272–289. [Google Scholar] [CrossRef] - Shen, Y.F.; Giurgiutiu, V. Combined analytical FEM approach for efficient simulation of Lamb wave damage detection. Ultrasonics
**2016**, 69, 116–228. [Google Scholar] [CrossRef] - Gravenkamp, H.; Prager, J.; Saputra, A.A.; Song, C. The simulation of Lamb waves in a cracked plate using the scaled boundary finite element method. J. Acoust. Soc. Am.
**2012**, 132, 1358–1367. [Google Scholar] [CrossRef] - Kumar, A.; Kapuria, S. Finite element simulation of axisymmetric elastic and electroelastic wave propagation using local-domain wave packet enrichment. J. Vib. Acoust.
**2022**, 144, 021011. [Google Scholar] [CrossRef] - Ajay, R.; Carlos, E.S.C. Effects of elevated temperature on guided-wave structural health monitoring. J. Intell. Mater. Syst. Struct.
**2008**, 19, 1383–1398. [Google Scholar] - Radecki, R.; Staszewski, W.J.; Uhl, T. Impact of changing temperature on Lamb wave propagation for damage detection. Key Eng. Mater.
**2014**, 588, 140–148. [Google Scholar] [CrossRef] - Dodson, J.C.; Inman, D.J. Thermal sensitivity of Lamb waves for structural health monitoring applications. Ultrasonics
**2013**, 53, 677–685. [Google Scholar] [CrossRef] [PubMed] - Roy, S.; Lonkar, K.; Janapati, V.; Chang, F.K. A novel physics-based temperature compensation model for structural health monitoring using ultrasonic guided waves. Struct. Health Monit.
**2014**, 13, 321–342. [Google Scholar] [CrossRef] - Francesco, L.; Salamone, S. Temperature effects in ultrasonic Lamb wave structural health monitoring systems. J. Acoust. Soc. Am.
**2008**, 124, 161–174. [Google Scholar] - Marzani, A.; Salamone, S. Numerical prediction and experimental verification of temperature effect on plate waves generated and received by piezoceramic sensors. Mech. Syst. Signal Process.
**2012**, 30, 204–217. [Google Scholar] [CrossRef] - Esfarjani, S.M. Evaluation of effect changing temperature on lamb-wave based structural health monitoring. J. Mech. Energy Eng.
**2020**, 3, 329–336. [Google Scholar] [CrossRef] - Attarian, V.A.; Cegla, F.B.; Cawley, P. Long-term stability of guided wave structural health monitoring using distributed adhesively bonded piezoelectric transducers. Struct. Health Monit.
**2014**, 13, 265–280. [Google Scholar] [CrossRef] - Lonkar, K.P. Modeling of Piezo-Induced Ultrasonic Wave Propagation for Structural Health Monitoring. Ph.D. Thesis, Stanford University, Palo Alto, CA, USA, 2013. [Google Scholar]
- Han, S.J.; Palazotto, A.N.; Leakeas, C.L. Finite-element analysis of Lamb wave propagation in a thin aluminum plate. J. Aerospace Eng.
**2018**, 22, 185–197. [Google Scholar] [CrossRef] - Yule, L.; Zaghari, B.; Harris, N.; Hill, M. Modelling and validation of a guided acoustic wave temperature monitoring system. Sensors
**2021**, 21, 7390. [Google Scholar] [CrossRef] [PubMed] - Liu, A.Q. Research on the Propagation and Simulation of Lamb Wave under Temperature Effect. Master’s Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, China, 2020. [Google Scholar]
- Giurgiutiu, V. Tuned Lamb wave excitation and detection with piezoelectric wafer active sensors for structural health monitoring. J. Intell. Mater. Syst. Struct.
**2005**, 16, 291–305. [Google Scholar] [CrossRef] - Brammer, J.A.; Percival, C.M. Elevated-temperature elastic moduli of 2024 aluminum obtained by a laser-pulse technique. Exp. Mech.
**1970**, 10, 245–250. [Google Scholar] [CrossRef] - Barakat, S. The Effects of Low Temperature and Vacuum on the Fracture Behavior of Organosilicate Thin Films. Masters’s Thesis, University of Waterloo, Waterloo, ON, Canada, 2011. [Google Scholar]
- Lee, H.J.; Saravanos, D.A. The Effect of Temperature Dependent Material Nonlinearities on the Response of Piezoelectric Composite Plates; NASA Technical Memorandum; NASA: Brook Park, OH, USA, 1997; pp. 1–20. Available online: https://ntrs.nasa.gov/citations/19980017194. (accessed on 2 May 2022).
- Qiu, L.; Yan, X.X.; Lin, X.D.; Yuan, S.F. Multiphysics simulation method of lamb wave propagation with piezoelectric transducers under load condition. Chin. J. Aeronaut.
**2019**, 32, 1071–1086. [Google Scholar] [CrossRef] - Yang, C.H.; Ye, L.; Su, Z.Q.; Bannister, M. Some aspects of numerical simulation for Lamb wave propagation in composite laminates. Compos. Struct.
**2006**, 75, 267–275. [Google Scholar] [CrossRef] - Qiu, L.; Yuan, S.F.; Wang, Q.; Sun, Y.J.; Yang, W.W. Design and experiment of PZT network-based structural health monitoring scanning system. Chin. J. Aeronaut.
**2009**, 22, 505–512. [Google Scholar] - Nader, G.; Carlos, E.; Silva, N.; Adamowski, J.C. Effective damping value of piezoelectric transducer determined by experimental techniques and numerical analysis. ABCM Symp. Ser. Mechatron.
**2004**, 1, 271–279. [Google Scholar]

**Figure 2.**Schematic diagram of LW excitation [35].

**Figure 12.**Comparison for variations of S

_{0}mode between simulation and experiment. (

**a**) Variations of amplitude; (

**b**) variations of phase velocity.

LW Propagation Characteristics | Expressions | Temperature Effects |
---|---|---|

Propagation velocity | ${c}_{\mathrm{L}}=\sqrt{\frac{{Y}_{\mathrm{plate}}^{E}(1-{\nu}_{\mathrm{plate}})}{{\rho}_{\mathrm{plate}}(1+{\nu}_{\mathrm{plate}})(1-2{\nu}_{\mathrm{plate}})}}$ ${c}_{\mathrm{T}}=\sqrt{\frac{{Y}_{\mathrm{plate}}^{E}}{2{\rho}_{\mathrm{plate}}(1+{\nu}_{\mathrm{plate}})}}$ | Temperature affects the propagation velocity of LW by influencing the elastic modulus ${Y}_{\mathrm{plate}}^{E}$, density ρ_{plate}, and Poisson’s ratio ν_{plate} of the structure. |

Response amplitude | ${V}_{\mathrm{out}}\left(t\right)={d}_{31}^{\mathrm{act}}{C}_{\mathrm{act}}\left(\mathsf{\Gamma}\right){C}_{\mathrm{sen}}\left(\mathsf{\Gamma}\right){\left[\frac{{d}_{31}}{{e}_{33}{s}_{13}\left(1-{\nu}_{\mathrm{act}}\right)}\right]}_{\mathrm{sen}}{V}_{\mathrm{in}}\left(t\right)$ ${\mathsf{\Gamma}}^{2}=\frac{{G}_{\mathrm{bond}}\text{}{l}_{\mathrm{act}}^{2}}{{t}_{\mathrm{bond}}}\left(\frac{1}{{Y}_{\mathrm{act}}^{E}{t}_{\mathrm{act}}}+\frac{\alpha}{{E}_{\mathrm{plate}}\text{}{t}_{\mathrm{plate}}}\right)$ | Temperature affects the LW amplitude by influencing piezoelectric coefficient d_{31} including the effects of thermal stress, the dielectric constant e_{33} and elastic flexibility coefficient s_{13} of the PZT, shear modulus G_{bond} and shear-lag constant of the adhesive layer. |

Structure | Geometry | Excitation Signal Frequency | Temperature |
---|---|---|---|

2024 Aluminum plate | 500 mm × 500 mm × 2 mm (length × width × thickness) | 150 kHz, 200 kHz | −20 °C to 60 °C |

Material | Parameter | Value |
---|---|---|

2024 Aluminum plate | Elastic modulus | ${E}_{\mathrm{plate}}\left(\Delta T\right)\left(\mathrm{GPa}\right)=73.5-0.06\Delta T$ |

Poisson’s ratio | $v\left(\Delta T\right)=0.344+5.13\times {10}^{-5}\Delta T$ | |

Density | 2700 (kg/m^{3}) | |

Coefficient of thermal expansion | 23.1 × 10^{−6}(/K) | |

Adhesive | Shear modulus | ${G}_{\mathrm{bond}}\left(\Delta T\right)\left(\mathrm{GPa}\right)=3.61-0.01\Delta T$ 0.3 |

Poisson’s ratio | ||

Density | 1110 (kg/m^{3}) | |

Coefficient of thermal expansion | 54 × 10^{−6} (/K) | |

PZT-5A | Piezoelectric constant | $\begin{array}{l}{d}_{31}(\Delta T,\Delta {\sigma}_{T})=-167.7-0.194\times \Delta T\\ -167.7\left(-1.1\times {10}^{-5}\times \Delta {\sigma}_{T}{}^{2}+4.2\times {10}^{-3}\times \Delta {\sigma}_{T}\right)\end{array}$ |

Relative permittivity | ${e}_{33}\left(\Delta T\right)=4.12\times \Delta T+2155$ | |

Coefficient of thermal expansion | 3 × 10^{−6} (/K) |

3 × 10^{−5} s | 4.5 × 10^{−5} s | 6 × 10^{−5} s | |
---|---|---|---|

−20 °C | |||

20 °C | |||

60 °C |

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**MDPI and ACS Style**

Yang, X.; Xue, Z.; Zheng, H.; Qiu, L.; Xiong, K.
Mechanic-Electric-Thermal Directly Coupling Simulation Method of Lamb Wave under Temperature Effect. *Sensors* **2022**, *22*, 6647.
https://doi.org/10.3390/s22176647

**AMA Style**

Yang X, Xue Z, Zheng H, Qiu L, Xiong K.
Mechanic-Electric-Thermal Directly Coupling Simulation Method of Lamb Wave under Temperature Effect. *Sensors*. 2022; 22(17):6647.
https://doi.org/10.3390/s22176647

**Chicago/Turabian Style**

Yang, Xiaofei, Zhaopeng Xue, Hui Zheng, Lei Qiu, and Ke Xiong.
2022. "Mechanic-Electric-Thermal Directly Coupling Simulation Method of Lamb Wave under Temperature Effect" *Sensors* 22, no. 17: 6647.
https://doi.org/10.3390/s22176647