# On the Directivity of Lamb Waves Generated by Wedge PZT Actuator in Thin CFRP Panel

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## Abstract

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## 1. Introduction

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- Geometric spreading;
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- Material damping;
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- Wave dispersion;
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- Dissipation into adjacent media.

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- Viscoelastic nature of matrix and/or fiber materials;
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- Damping due to interphase interaction;
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- Damping due to damage;
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- Viscoplastic damping due to the presence of high stress and strain concentration that exists in local regions between fibers;
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- Thermo-elastic damping due to the cyclic heat flow from the region of compressive stress to the region of the tensile stress.

_{x}, σ

_{y}, and τ

_{xy}. The proposed approach was used to obtain the variation of elastic moduli and the damping capacities of the beams with the different angle-ply orientations. The calculation results, which were confirmed experimentally, revealed very intensive angular dependencies of the damping parameters for the orthotropic composites. Moreover, the structural damping increased with the increase of the deviation angle from the main axis of anisotropy [27].

_{2s}plates are useful in design, but the stress wave inspection is difficult due to high attenuation of both Lamb modes. It is obvious that this conclusion relates to all CFRP and GFRP plates with complex lay-up and anisotropy. The authors of the paper [8] concluded that knowledge of the attenuation and directivity of the waves, depending on the anisotropy of the elastic and damping properties of the material for specific types of waves and their excitation frequencies, will benefit future acoustic based SHM and NDE applications in composite material design and technology.

## 2. Materials and Methods

#### 2.1. Dispersion Analysis of the Waves That Can Be Excited in CFRP Panel under Study

_{2s}to produce a symmetric balanced laminate. Its elastic properties were determined by two independent methods. The longitudinal E

_{1}, transversal E

_{2}, in-plane shear G

_{12}moduli, and Poisson ratio ν

_{12}were calculated, using prepreg’s manufacturer data of lamina elastic properties and equations of the classical lamination theory [26]. In order to obtain more realistic values of the elastic constants of the ready composite, the experimental technique described in our paper [19] was used. This technique included the determination of the effective moduli experimentally, according to standards ASTM D 3039-95 [42], D 5379-93 [43], D 2344-89 [44], and a refinement technique to reduce the effect of non-ideal experimental conditions [21]. The through thickness module E

_{3}, the interlaminar shear moduli G

_{23}, G

_{31}, and the Poisson ratios ν

_{23}and ν

_{31}, unavailable for reliable experimental measurement, were calculated by using FE simulation of the static tests for the multilayered composite [45]. The confidence intervals for all elastic constants were calculated, using the experimentally measured ones, obtained for every 5 tested specimens. The final results of the engineering constants of the studied CFRP panel, which are presented in Table 1, are typical for an orthotropic structural symmetry of composites.

#### 2.2. The Finite-Element Model of the Angle-Beam Wedge Actuator, Generating the Wave in a Thin CFRP Panel

^{®}[48], the outer dimensions of the used actuator (see Figure 3), the frequency response functions for the electric current through PZT active plate, and out-of-plane displacement amplitude on the actuator’s footprint (see Figure 4a,b), which have been determined experimentally. Only averaged normal displacement amplitudes of the actuator’s contact surface were measured experimentally by using the VibroGo single-point vibrometer (Polytec Co.

^{®}, Irvin, CA, USA), which was sensitive to the normal acceleration at the frequencies up to 100 kHz. The displacement amplitudes were calculated after the double integration of the sensor’s signal over time for each frequency. In order to compare the experimental and the simulation results, the displacements of the modeled actuator contact surface were averaged within the footprint. The tangent displacement could not be measured experimentally by the available means. They are presented in Figure 4c,d as the results of computer simulations of the actuator’s FE model, whose geometry is present in Figure 5. This model was implemented in the Comsol Multiphysics FE software (3.5a, COMSOL AB, Stockholm, Sweden), which contains built-in Structural Mechanics and Piezoelectric computations modules designed to solve classical equations of anisotropic three-dimensional elasticity and the piezoelectricity [26,45,46]. This ability allows it to easily tune elastic, damping properties of each actuator’s component, and to orient the polarization direction of the active PZT plate along the sound path made of Lucite.

#### 2.3. Investigation of the Wave Propagation Generated by the Oriented Angle-Beam Wedge Actuator in Orthotropic CFRP Panel: Experimental and Numerical Studies

^{0}and C

^{1}continuities (of the damping coefficients distribution and their spatial derivatives, respectively) at the boundary of the circle. By such a way, the perfectly matched layer was designed to eliminate the wave reflection. On average, for different frequencies, the model consisted of 250–300 thousand of the finite elements; the number of degrees of freedom of the problem was 1.2 × 10

^{6}–1.5 × 10

^{6}. The calculation time of one variant with 7–8 fluctuations in the control voltage varied from 4 to 8 h on a computer with an i7 processor and 32 GB of RAM.

## 3. Results and Discussion

#### 3.1. Comparison of the Experimentally Measured and Numerically Calculated A0 Wave Directivities

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- Wave propagation mainly in the direction of greatest structural stiffness and minimal attenuation;
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- Deviation of the maximum of the radiation directivity in the direction of the main axis of orthotropic anisotropy, which is confirmed by the fact that when the actuator rotates around an axis normal to the panel surface by a certain angle θ from the main axis, the directivity’s lobe rotates by a smaller angle (see Figure 11);
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- The reverse orientation of the wave relative to the orientation of the directional actuator when certain vibration modes are excited in it (see Figure 11c).

#### 3.2. Interfacial Shear Stress and Radial Tangent Displacement Distributions: FE Simulation Results

#### 3.3. Attenuation of the Lamb Waves Generated by the Differently Oriented Wedge Actuator

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The dispersion curves for the wavespeeds (

**a**) and wavelengths (

**b**) of the anti-symmetric A0 Lamb waves propagating along (

^{1}) and across (

^{2}) the main axis of orthotropic symmetry.

**Figure 2.**The dependencies of the wavespeeds (

**a**) and the wavelengths (

**b**) on the frequency for the symmetric horizontally polarized shear waves SS0 propagating along and across the main axis ${x}_{1}$ of orthotropic symmetry.

**Figure 3.**The linear and angular outer dimensions of the used wedge actuator [41].

**Figure 4.**The frequency response functions for the Olympus actuator: (

**a**)—phase angle of PZT current; (

**b**–

**d**)—averaged displacement amplitudes on the contact footprint surface—normal (

**b**) and tangential (

**c**,

**d**).

**Figure 5.**The geometry of wedge actuator’s finite-element (FE) model [41].

**Figure 6.**The angular distribution of relative dimensionless attenuation coefficient for the anti-symmetric A0 waves propagating from the center of the orthotropic CFRP panel [20].

**Figure 7.**The experimentally studied CFRP panel with schematically shown coordinate systems of orthotropic symmetry $\left({x}_{1},{x}_{2}\right)$ and wedge actuator’s (X,Y) rotated relative to the first one by an angle, θ (

**a**); the arrangement of sensors for recording the A0 Lamb waves propagation (

**b**).

**Figure 8.**The driving voltage for the PZT actuator and the time interval for the sensor’s signal averaging.

**Figure 9.**The snapshots of the Lamb A0 (

**a**,

**c**) and SS0 (

**b**,

**d**) waves, which propagated in the studied CFRP panel at the frequencies 15 kHz (

**a**,

**b**) and 30 kHz (

**c**,

**d**). Color scales show the normal out-of-plane displacements (

**a**,

**c**) and the radial in-plane displacements (

**b**,

**d**) at the time instants when these displacements have the amplitude values.

**Figure 10.**The directivity diagrams of the A0 Lamb waves propagating at the different wedge actuator orientation for the excitation frequencies 15 kHz (

**a**,

**b**), 30 kHz (

**c**,

**d**), and 65 kHz (

**e**,

**f**).

**Figure 11.**Variation of the A0 Lamb waves directivity with the wedge actuator’s angular orientation relative to the main axis of orthotropic symmetry of the CFRP plate at the excitation frequencies 15 kHz (

**a**), 30 kHz (

**b**), and 65 kHz (

**c**).

**Figure 12.**In-plane contact radial stress distributions (amplitude values in MPa) for the waves excitation frequencies 15 kHz (

**a**,

**b**), 30 kHz (

**c**,

**d**), and 65 kHz (

**e**,

**f**) at the actuator orientation along (

**a**,

**c**,

**e**) and across (

**b**,

**d**,

**f**) the main axis of orthotropic panel, which is denoted by the arrows.

**Figure 13.**The distribution of the interfacial tangent radial displacements in μm at the excitation frequency 15 kHz. Actuator’s axis is oriented along (

**a**) and across (

**b**) the main orthotropy axis of the CFRP panel, which is denoted by the arrows.

**Figure 14.**The attenuation of the A0 Lamb waves propagating along the main lobes of the directivity diagrams at the different wedge actuator orientation: along the main orthotropic axes of the material (

**a**), and rotated 30, 60 and 90 degrees (

**b**,

**c**,

**d**, respectively) and the waves excitation frequencies.

Young’s Moduli, GPa | Shear Moduli, GPa | Poisson’s Ratios | ||||||
---|---|---|---|---|---|---|---|---|

E_{1} | E_{2} | E_{3} | G_{12} | G_{23} | G_{31} | ν_{12} | ν_{23} | ν_{13} |

63 ± 12 | 22.5 ± 2.5 | 5 ± 1.5 | 9.6 ± 1.2 | 7.3 ± 0.6 | 7.8 ± 0.6 | 0.46 ± 0.06 | 0.32 ± 0.08 | 0.6 ± 0.1 |

^{1}The bounds of the confidence intervals for the elastic moduli are determined by using measuring data.

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**MDPI and ACS Style**

Shevtsov, S.; Chebanenko, V.; Shevtsova, M.; Chang, S.-H.; Kirillova, E.; Rozhkov, E.
On the Directivity of Lamb Waves Generated by Wedge PZT Actuator in Thin CFRP Panel. *Materials* **2020**, *13*, 907.
https://doi.org/10.3390/ma13040907

**AMA Style**

Shevtsov S, Chebanenko V, Shevtsova M, Chang S-H, Kirillova E, Rozhkov E.
On the Directivity of Lamb Waves Generated by Wedge PZT Actuator in Thin CFRP Panel. *Materials*. 2020; 13(4):907.
https://doi.org/10.3390/ma13040907

**Chicago/Turabian Style**

Shevtsov, Sergey, Valery Chebanenko, Maria Shevtsova, Shun-Hsyung Chang, Evgenia Kirillova, and Evgeny Rozhkov.
2020. "On the Directivity of Lamb Waves Generated by Wedge PZT Actuator in Thin CFRP Panel" *Materials* 13, no. 4: 907.
https://doi.org/10.3390/ma13040907