# Modeling and Experimental Characterization of Bonding Delaminations in Single-Element Ultrasonic Transducer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Studied Single-Element Transducer

_{r}) is around 1.71 MHz. Material properties of PZ27 are given in Table 1 [22,23], where ρ is the density, ${c}_{ij}^{D}$ are the elastic constants, ${h}_{kj}$ are piezoelectric constants, ${\beta}_{kk}^{S}$ are the dielectric constants, ${\delta}_{m}$ and ${\delta}_{e}$ are the mechanical and dielectric loss factors, and Z is the acoustic impedance.

_{p}, Z

_{f}, and Z

_{m}are the acoustic impedance of the piezoceramic disk, the front loading, and the matching layer, respectively.

#### 2.2. Electromechanical Admittance

_{L}), and capacitive reactance (X

_{C}) is depicted in Figure 2. It is operated under an alternating voltage source. The complex admittance can be described by a real part (conductance G) and an imaginary part (susceptance B). Relationships between these terms can be expressed as

_{0}(ω) is the reference state.

_{R}) and bandwidth (BW

_{R}) can also be used to quantify the changes. As we know, conductance reaches its maximum value G

_{max}, i.e., A

_{R}, at the resonance frequency f

_{r}. The frequencies corresponding to ${G}_{\mathrm{max}}/\sqrt{2}$ are marked as f

_{1}and f

_{2}, then the −3dB bandwidth (${BW}_{R}^{-3dB}$) is given by

## 3. Finite Element Modeling

^{E}is the elastic stiffness tensor, e is the piezoelectric coupling tensor, and ${\epsilon}^{S}$ is the permittivity tensor. The apostrophe denotes matrix transpose.

^{®}version 5.4, COMSOLAB, Stockholm, Sweden. Figure 3 shows the FE model of an intact single-element ultrasonic transducer. The backing is always much thicker than the piezoceramic disk to ensure that no reflected waves come back. To reduce the computational cost of the entire structure, a perfectly matched layer (PML) [35,36,37] is added at the rear of the backing. Chosen parameters give this polynomial stretching type PML a maximum of 109 dB attenuation for normal incidence. All simulations are developed in vacuum instead of air, because the loading effect of air on the transducer vibration is negligible [38].

#### 3.1. Delamination Types

#### 3.2. Case I: Delaminations Between Ceramic and Backing

_{R}and ${BW}_{R}^{-3dB}$, versus the delamination ratio η. Resonance amplitude A

_{R}increases but the bandwidth ${BW}_{R}^{-3dB}$ decreases with the increasing η. On the other hand, it can be observed that the three delamination types show different variation patterns. The pattern of DT-III type is between those of the other two. In terms of the damaging influence of delamination on the performance of the ultrasonic transducer, such as the influence on the bandwidth, DT-I delamination makes the most, DT-III less, and DT-II even less. For example, a half delamination of the adhesive layer gives a decrease of ${BW}_{R}^{-3dB}$ from 52.2 kHz to 25.8 kHz (50.6%) for DT-I, 29.9 kHz (42.7%) for DT-III, and 44.8 kHz (14.2%) for DT-II.

#### 3.3. Case II: Delaminations Between Ceramic and Matching Layer

_{r}= 1.7 MHz) gradually appears and rises up. It seems that the two split peaks merge gradually towards the original T1 mode in the middle. This is because the matching layer is losing its function as η increases.

_{R}and ${BW}_{R}^{-3dB}$ since they are only suitable for the characterization of a specific peak. Nevertheless, the statistical based metrics RMSD and DI are still useful: Figure 8 shows the results of the calculation of RMSD and DI in the frequency range from 1 MHz to 2.4 MHz. From an intact state to a complete delaminated state, RMSD and DI increase monotonically up to 150% and 1.25, respectively. The variation patterns of the three types still shows differences, in which the one of DT-III delamination is again in the middle of the other two.

## 4. Experimental Validation

#### 4.1. Experiment Setup

#### 4.2. Comparison Between Experiment and FE modeling

_{R}and ${BW}_{R}^{-3dB}$ with delamination ratio η, extracted from the first split thickness mode peak Ts1. A good agreement is observed between numerical and experimental results. The evolution of the delamination indicators A

_{R}and ${BW}_{R}^{-3dB}$ coincides well and the difference between DT-I to DT-III is also confirmed by experimental results. The maximum discrepancy between experimental and numerical predicated values of A

_{R}(Table 3) are 5.7% for DT-I, 5.5% for DT-II, and 14.9% for DT-III delamination. The maximum discrepancies of ${BW}_{R}^{-3dB}$ (Table 4) are 8.9%, 16.9% and 12.1%, respectively, for these three types of delaminations.

## 5. Conclusions

_{R}and bandwidth ${BW}_{R}^{-3dB}$ of the first split thickness mode are recommended as quantitative delamination indicators. The maximum discrepancies between experimental and numerical predicated values of A

_{R}and ${BW}_{R}^{-3dB}$ are 14.9% and 16.9%, respectively. For the delaminations between ceramic and matching layer, RMSD and DI are recommended. A maximum discrepancy in RMSD of 37.6% is observed. Results obtained by the FE method and experiments are in reasonable agreement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagram of a single-element transducer made of a matching layer, a piezoelectric disk (PZT) and a backing.

**Figure 5.**Case I: Influence of DT-I delamination on the electromechanical admittance. (

**a**) frequency range 0–2.4 MHz; (

**b**) peak at the first split thickness mode Ts1.

**Figure 6.**Case I: Variations of resonance amplitude (

**a**) and −3 dB bandwidth (

**b**) with delamination ratio η for the first split thickness mode Ts1.

**Figure 7.**Case II: Influence of DT-I delamination on the electromechanical admittance. (

**a**) frequency range 0–2.4 MHz; (

**b**) peaks around the thickness mode.

**Figure 8.**Case II: Variations of RMSD (

**a**) and DI (

**b**) with delamination ratio η for the thickness mode.

**Figure 9.**Transducer components. (

**a**) a piezoceramic disk PZ27, a PLA backing and a copper-filled PLA matching layer; (

**b**) an intact mounted single-element transducer.

**Figure 10.**Samples and measurement setup. (

**a**) nine 3D printed PLA backings; (

**b**) a sample during the measurements; (

**c**) experiment setup.

**Figure 11.**Comparison between results from FE modeling (FE) and experiments (Exp) for case I. (

**a**) normalized A

_{R}versus η; (

**b**) normalized ${BW}_{R}^{-3dB}$ versus η.

**Figure 12.**Comparison between results from FE modeling (FE) and experiments (Exp) for case II. (

**a**) conductance versus frequency; (

**b**) RMSD versus η.

ϕ (mm) | t (mm) | ρ (kg/m^{3}) | ${\mathit{c}}_{\mathbf{11}}^{\mathit{D}}$ (GPa) | ${\mathit{c}}_{\mathbf{12}}^{\mathit{D}}$ (GPa) | ${\mathit{c}}_{\mathbf{13}}^{\mathit{D}}$ (GPa) | ${\mathit{c}}_{\mathbf{33}}^{\mathit{D}}$ (GPa) | ${\mathit{c}}_{\mathbf{44}}^{\mathit{D}}$ (GPa) |
---|---|---|---|---|---|---|---|

16 | 1.13 | 7879.5 | 148.6 | 106.1 | 87.5 | 144.4 | 36.6 |

${h}_{31}$ (GV/m) | ${h}_{33}$ (GV/m) | ${h}_{15}$ (GV/m) | ${\beta}_{11}^{S}$ (Gm/F) | ${\beta}_{33}^{S}$ (Gm/F) | ${\delta}_{m}$ (%) | ${\delta}_{e}$ (%) | Z (MRayl) |

−0.38 | 1.98 | 1.16 | 9.99 | 0.12 | 0.45 | 0.27 | 33.7 |

Components | ϕ (mm) | t (mm) | ρ (kg/m^{3}) | ${\mathit{c}}_{\mathit{L}}$ (m/s) | ${\mathit{c}}_{\mathit{T}}$ (m/s) | ${\mathit{\delta}}_{\mathit{m}}$ (%) | Z (MRayl) |
---|---|---|---|---|---|---|---|

Backing | 16 | 20 | 1135 | 1738 | 952.2 | 5 | 1.97 |

Matching | 16 | 0.18 | 3284 | 1246 | 740.2 | 5 | 4.09 |

Bonding | 16 | 0.002 | 1250 | 2369 | 994 | 1 | 2.96 |

**Table 3.**Discrepancy of resonance amplitude of the first split thickness mode depending on the delamination ratio (η).

η (%) | DT-I (%) | DT-II (%) | DT-III (%) |
---|---|---|---|

25 | 1.7 | 1.6 | 0.6 |

50 | 4.2 | 2.7 | 14.9 |

75 | 5.7 | 5.5 | 6.6 |

**Table 4.**Discrepancy of -3dB bandwidth of the first split thickness mode depending on the delamination ratio (η).

η (%) | DT-I (%) | DT-II (%) | DT-III (%) |
---|---|---|---|

25 | 8.9 | 1.3 | 9.2 |

50 | 7.7 | 10.4 | 0.1 |

75 | 1.8 | 16.9 | 12.1 |

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**MDPI and ACS Style**

Ding, W.; Bavencoffe, M.; Lethiecq, M.
Modeling and Experimental Characterization of Bonding Delaminations in Single-Element Ultrasonic Transducer. *Materials* **2021**, *14*, 2269.
https://doi.org/10.3390/ma14092269

**AMA Style**

Ding W, Bavencoffe M, Lethiecq M.
Modeling and Experimental Characterization of Bonding Delaminations in Single-Element Ultrasonic Transducer. *Materials*. 2021; 14(9):2269.
https://doi.org/10.3390/ma14092269

**Chicago/Turabian Style**

Ding, Wenxiang, Maxime Bavencoffe, and Marc Lethiecq.
2021. "Modeling and Experimental Characterization of Bonding Delaminations in Single-Element Ultrasonic Transducer" *Materials* 14, no. 9: 2269.
https://doi.org/10.3390/ma14092269