#
Improved SH0 Guided Wave Transducers Based on Piezoelectric Fiber Patches^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. A Short Review of SH0 Wave Generation with the SHPFP

#### 2.1.1. Piezoelectric Fiber Patches (PFPs) for Guided Wave Generation

_{33}effect for actuation. The polarization in each piezoelectric fiber is along the fiber’s direction with an alternating orientation (Figure 2). The applied electric field also has an alternating orientation so that the product of the polarization and the field has a constant sign (either positive or negative) along the fiber’s entire length with only some variation in amplitude. Assuming, for a moment, clamped conditions (i.e., there is no strain on the PFP due, for example, to gluing it onto a stiff substrate), the generated mechanical stress oscillates about a mean value along the fiber. This means that, on average, the forces that act on the substrate cancel out along the fiber and remain significant only at the fiber’s end.

#### 2.1.2. The Working Principle and Configuration of the SHPFP

#### 2.1.3. Optimization Criteria

#### 2.2. Modified Designs

#### 2.2.1. Change in the Transducer’s Shape (Rounded Corner SHPFP)

#### 2.2.2. The Dual Transducer (Dual SHPFP)

#### 2.3. Finite Element Method (FEM) Simulation Setup

#### 2.3.1. Homogenized PFP Model Verification

^{3}). To make the displacement, the PFP1 was driven by a voltage, while a closed-circuit boundary condition was imposed on PFP2. The material properties that were used in the studies [30,31] and in our numerical simulation are shown in Table 1.

#### 2.3.2. Modeling Details

^{3}. Its Young’s modulus, density, and Poisson’s ratio were 200 GPa, 7850 kg/m

^{3}, and 0.30, respectively. The original SHPFP was made of two homogenized PFPs with different tilted fiber directions (+45° and −45°). The material property matrix was also rotated +45° and −45° for the bottom and top PFPs, respectively, compared to that of the standard PFP shown in Figure 6. A perfect bond between both PFP layers and between the bottom layer and the substrate was assumed. This means that all displacement components ${u}_{i}$ and normal stress components ${\sigma}_{zi}\left(i=1,2,3\right)$ are continuous across the interface.

^{3}. The rounded corner design was modeled with the same length (40 mm) and width (10 mm) value of the original SHPFP design. It should be noted that the rounded corner design has a smaller active area compared to the original design. The dual SHPFP design was made of two SHPFPs. Each homogenized PFP had its own fiber direction and a rotated material property matrix (Figure 10). Again, perfect bonding interface conditions between the layers and between the bottom layer and the substrate were assumed. Because each SHPFP has a size of 40 × 10 × 0.6 mm

^{3}, the dimensions of the dual design were 40 × 20 × 0.6 mm

^{3}.

_{c}). Thereby, f

_{c}is the center frequency of a three-cycle, Hanning-windowed sinusoidal tone burst, which was used as the exciting signal, and λ is the wavelength of a horizontal shear wave in the plate at f

_{c}. The voltage of the applied signal was chosen to have an electric field of 200 V/mm through each PFP and its center frequency (f

_{c}) varied from 50 kHz to 200 kHz.

^{3}and there was an 8-mm gap between the two SHPFPs that formed a dual SHPFP (Figure 12). Details are provided in Section 2.4.

#### 2.4. Experimental Setup

^{2}was attached to the plate. Each measured signal was averaged over 128 excitations to improve the signal-to-noise ratio.

^{3}, but 48 × (30 + 8) × 0.6 mm

^{3}, because of the 8-mm gap between the two SHPFPs (Figure 12b). This problem can be solved by putting the electrode ends at the short edges (see Figure 7 in Ref. [10]), however this will not be discussed in this study.

^{2}. The PFPs were excited by a three-cycle, Hanning-windowed sinusoidal tone burst, as was done in the simulations. We chose a 200 V peak-to-peak voltage value to maintain the PFP’s electric field at 200 V/mm. Considering a shear wave velocity in steel of 3250 m/s, a center frequency of 80 kHz was chosen to match the wavelength λ of the SH0 wave to the width of the dual SHPFP.

## 3. Results

#### 3.1. Results from the Simulation of the Modified Designs

#### 3.1.1. Rounded-Corner SHPFP

#### 3.1.2. Dual SHPFP

#### 3.1.3. Dual SHPFP with an 8-mm Gap

^{2}, rather than 40 × 10 × 0.3 mm

^{3}, and there was an 8-mm gap between the two SHPFPs in the dual SHPFP (Figure 9).

#### 3.2. Experimental Results

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The scheme of a piezoelectric fiber patch (reprinted from Ref. [10] with permission from Elsevier). The finger-like electrodes are used for both the poling of the fibers and the generation of the electric fields in active operation.

**Figure 2.**A cross-section of the piezoelectric fiber patch (PFP) shown in Figure 1 along the line A–A (a corrected version of Figure 4 in [10]). The upper row shows the fiber with electrodes; the lower row is a magnification of the left fiber’s end. The application of electric voltage in the two orientations leads to electric fields ($E$) that are antiparallel (left) and parallel (right) to the polarization ($P$). Simply speaking, a resulting force (${f}_{x}~{\sigma}_{xx,x}$) is generated where the product ${\sigma}_{xx}~{P}_{x}{E}_{x}$ is changing from an average value to zero; that is, at the end of the polarized region of the fiber [22].

**Figure 3.**The two piezoelectric fiber patch (PFP) layers that overlie one another in a shear horizontal PFP (SHPFP) shown side-by-side. The red arrows indicate the resulting forces at the fiber ends of each individual layer (reprinted from [10] with permission from Elsevier).

**Figure 4.**The configuration of the SHPFP. The yellow boxes show the active area of each PFP. The green arrows show the forces that each PFP generates. The underlying PFP generates the dotted green arrows and the overlying PFP generates the solid green arrows. At each edge, the dotted and the solid green arrow pairs are perpendicular to each other in-plane. The purple arrows designate the forces that result from the summation of the green arrows at each edge. The blue and red arrows represent excited wave modes, the target fundamental order of the shear horizontal mode (SH0) wave, and the unwanted S0 Lamb wave, respectively.

**Figure 5.**Two modified SHPFP designs. The purple arrows visualize the generated surface traction forces that act as shear forces, and the blue arrows indicate the excited wave modes. The left part shows (

**a**) the rounded-corner design and the right part shows; (

**b**) the dual SHPFP design. The red crosses in (

**b**) indicate that the surface traction forces at the short edges act against each other and should (at least partially) cancel out. It should be noted that two PFPs overlie each other at each active area, even though the underlying and the overlying PFPs cannot be distinguished in the figure.

**Figure 6.**The schematic configuration of the homogenized PFP with its voltage boundary condition. The red arrow shows the direction of piezoelectric fibers in the PFP, and it also indicates the orientation of the material property matrix (piezoelectric, dielectric, and stiffness coefficients) of the homogenized PFP.

**Figure 7.**The homogenized PFP bending test’s configuration. Two PFPs (85 × 57 mm

^{2}in size) were attached to the front and the back of an aluminum beam. The PFP1 was driven by a voltage to be extended, while the PFP2 remained as a closed-circuit. In the experiment, 30 mm of the 330-mm-long aluminum beam was fixed by a clamp [31]. In the simulation, the aluminum beam was modelled with a length of 300 mm and with fixed boundary conditions at the left end.

**Figure 8.**Comparison of the results from the homogenized PFP bending test. (

**a**) The measurement point location was fixed, and the applied voltage was changed; (

**b**) The applied voltage was fixed and the measurement distance was changed. The blue dots show the simulation results and the orange triangles show the corresponding experimental results [31].

**Figure 9.**The schematic configuration of the homogenized original SHPFP with its voltage boundary condition. The red arrows indicate the direction of piezoelectric fibers and the rotated material property matrix of each homogenized PFP.

**Figure 10.**The schematic configuration of the homogenized dual SHPFP with its voltage boundary condition. The two voltage sources are exactly the same. The red arrows indicate the direction of the piezoelectric fibers and the rotated material property matrix of each homogenized PFP.

**Figure 11.**The modeling configuration. (

**a**) General view; (

**b**) top view. The size of the steel plate was 500 × 500 × 2 mm

^{3}, and the transducers were always located at the center. The red, green, and purple arrows indicate the radial (x), tangential (y), and out-of-plane (z) axes of the cylindrical coordinate system. The symbols (x,y,z) for the cylindrical coordinates were chosen to be in accordance with the output of ANSYS (compare Figure 14). The emitted waves were measured at a distance of 100 mm from the center.

**Figure 12.**Realization of the dual SHPFP design. (

**a**) An actual photo of the commercial PFP (M4815F1, Smart Materials GmbH) that was used to build the dual design. The size of the active area is 48 × 15 mm

^{2}and the PFP has electrode ends at both of its long edges; (

**b**) The configuration of the realized dual SHPFP design. Because of the electrode ends, there is an 8-mm gap between the SHPFP on the left side and the SHPFP on the right side.

**Figure 13.**A normalized polar diagram of the wave amplitude (particle velocity) of (

**a**) the rounded-corner SHPFP and (

**b**) the SHPFP. The blue lines show the SH0 wave amplitude, while the orange lines show the unwanted S0 Lamb wave amplitude.

**Figure 14.**The tangential component of the simulated wave field at a time of 50 µs for (

**a**) the original SHPFP and (

**b**) the dual SHPFP. The excitation has a center frequency of 80 kHz. For the complete videos of the wave propagation, see the section Supplementary Materials at the end of the paper.

**Figure 15.**A comparison of the SH0 wave directivity of the SHPFP (the red dotted line) versus the dual SHPFP (the green solid line) at a center frequency of 80 kHz. The values were normalized to the maximum value of the dual SHPFP transducer.

**Figure 16.**Polar plots of the maximum values of the surface velocity components. The plots of the original SHPFP and the dual SHPFP are compared for different center frequencies. The blue lines show the SH0 wave’s amplitude and the orange lines show the unwanted S0 Lamb wave’s amplitude. The values in each polar diagram were normalized to the maximum value of each graph. The scale is in dB and all values below −35 dB are cut off.

**Figure 17.**Result of the simulation of the dual SHPFP with an 8-mm gap. Snapshot of the tangential wave field component at 50 µs after excitation with ${f}_{c}=80\mathrm{kHz}$.

**Figure 18.**A comparison of measured wave field snapshots between (

**a**) the dual SHPFP and (

**b**) the original SHPFP at 50.24 µs after excitation. The wave packets in the solid green circles are the SH0 waves, and the ones in the dotted red circles are the unwanted S0 Lamb wave. For the complete videos of the wave propagation, see the section Supplementary Materials at the end of the paper.

**Figure 19.**Polar diagrams showing the SH0 amplitude (blue lines) together with the S0 Lamb wave amplitude (orange lines) of (

**a**) the dual SHPFP and (

**b**) the original SHPFP.

**Figure 20.**A polar diagram for the direct comparison of the original SHPFP and the dual SHPFP as a SH0 wave transducer. The solid green and the dotted red lines show the SH0 amplitude of the dual SHPFP and the original SHPFP, respectively. The values are scaled to the maximum of the dual SHPFP.

**Table 1.**The effective material properties of the homogenized M8557P1 PFP [31]: Young’s modulus (E), shear modulus (G), Poisson’s ratio ($\gamma $), piezoelectric coupling coefficients (${d}_{ij}$), and relative permittivity at constant strain (${\epsilon}_{ij}^{s}$).

Properties | Value |
---|---|

${E}_{33}$ (GPa) | 29.4 |

${E}_{11}$ (GPa) | 15.2 |

${G}_{31}$ (GPa) | 6.06 |

${\gamma}_{31}$ | 0.312 |

${\gamma}_{13}$ | 0.16 |

${d}_{33}$ (pm/V) | 467 |

${d}_{32}$ (pm/V) | −210 |

${d}_{31}$ (pm/V) | −210 |

${\epsilon}_{11}^{s}$ | 712 |

${\epsilon}_{22}^{s}$ | 1.7 |

${\epsilon}_{33}^{s}$ | 737 |

$\mathbf{Center}\text{}\mathbf{Frequency}\text{}{\mathit{f}}_{\mathit{c}}/\mathit{k}\mathit{H}\mathit{z}$ | SHPFP Purity Ratio | Dual SHPFP Purity Ratio | Increment |
---|---|---|---|

80 | 3.977 | 6.493 | 160% |

100 | 5.304 | 10.707 | 202% |

120 | 6.010 | 7.427 | 124% |

140 | 5.372 | 7.053 | 131% |

160 | 5.488 | 6.081 | 111% |

180 | 5.452 | 5.814 | 107% |

200 | 5.477 | 5.595 | 102% |

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**MDPI and ACS Style**

Kim, Y.; Gaul, T.; Köhler, B.
Improved SH0 Guided Wave Transducers Based on Piezoelectric Fiber Patches. *Sensors* **2019**, *19*, 2990.
https://doi.org/10.3390/s19132990

**AMA Style**

Kim Y, Gaul T, Köhler B.
Improved SH0 Guided Wave Transducers Based on Piezoelectric Fiber Patches. *Sensors*. 2019; 19(13):2990.
https://doi.org/10.3390/s19132990

**Chicago/Turabian Style**

Kim, Yongtak, Tobias Gaul, and Bernd Köhler.
2019. "Improved SH0 Guided Wave Transducers Based on Piezoelectric Fiber Patches" *Sensors* 19, no. 13: 2990.
https://doi.org/10.3390/s19132990