# Magneto-Electric Effect on Guided Waves in Functionally Graded Piezoelectric–Piezomagnetic Fan-Shaped Cylindrical Structures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematics and Formulation

_{ph}= ω/k.

## 3. Numerical Results

_{2}O

_{4}, and the outer layer material is BaTiO

_{3}. Their material constants are listed in the Table 1 in the reference [34].

#### 3.1. Comparison with Rectangular Bar

^{−4}− 1) mm, b = 10

^{4}mm, and β = 10

^{−4}rad, respectively, to compare with a square steel bar using the two dimensional Rayleigh–Ritz method [35]. Its cross-section area is 1 mm

^{2}. Their phase velocity curves are shown in the Figure 2, where the lines are the authors’ results, and the dotted lines are results from reference [35]. c

_{p}represents the phase velocity, and c

_{s}represents the shear velocity. These results of two methods are overlapped completely. Accordingly, the correctness of the present method is confirmed.

_{p}*d)/c

_{s1}, where c

_{p}is the phase velocity value, and c

_{s1}is the shear velocity value of CoFe

_{2}O

_{4}). Here, Figure 3b is the partial enlarged drawing of Figure 3a. We can note from these figures that the phase velocity for the fan-shaped cylindrical structures is getting closer to that of rectangular bar as the radius-thickness ratio increases. Moreover, the dotted line and the line are overlapped completely as the radius-thickness ratio η = 1000.

#### 3.2. Convergence Confirmation

#### 3.3. The Magneto-Electric Effect

^{3}times) than that of the magnetic potential, which has a relationship with the electric and magnetic parameters. The phenomenon that the piezoelectric effect is stronger than the piezomagnetic effect is confirmed again. Furthermore, regardless of the amplitude, there are obvious differences between the distribution shapes at small wavenumbers, but they are similar at bigger wavenumbers.

^{6}times the dielectric coefficients. Hence, the influence of piezomagnetic is very weak, and the amplitude of electric potential is about 10

^{3}times of magnetic potential, due to the much higher magnetic permeability coefficients.

#### 3.4. The Influence of the Graded Functions

_{1}(r) = (r − a)/d and V

_{1}(r) = [(r − a)/d]

^{3}. They have the same geometry: η = 2, β = π/6. Figure 8 shows variation curves of material properties with different graded functions. Figure 9 illustrates their dispersion curves of the fourth to sixth modes. For a given mode, phase velocities with cubic function are higher than those with a linear function, since it is composed of more CoFe

_{2}O

_{4}, and the wave velocity of BaTiO

_{3}is lower than that of CoFe

_{2}O

_{4}.

_{3}decreases as n increases, and the piezoelectric effect mainly results from BaTiO

_{3}.

#### 3.5. The Influence of the Geometric Size

#### 3.6. Waves at High Frequencies

_{s}. However, the case for 2-D fan-shaped cylindrical structures is quite different. A homogeneous (V

_{1}(r) = 1) fan-shaped cylindrical structure with η = 2 and α = π/6 is taken into account. The phase velocity curves for the fifth and sixth mode are shown in Figure 16. The velocity values at high frequencies approximatively approach a value which is below c

_{s}. This is because wave motions at high frequencies mainly concentrate near corners of the fan-shaped cylindrical structures, while the other places remain almost motionless (see in the Section 3.8). The velocity for waves propagating near the edges at high frequencies is lower than that for waves propagating at a surface.

#### 3.7. The Stress, Electric, and Magnetic Displacement Distribution

#### 3.8. The Poynting Vector

_{3}. This is because elasticity modulus of BaTiO

_{3}is smaller than that of CoFe

_{2}O

_{4}. Accordingly, the stiffness of BaTiO

_{3}is smaller, the displacement is bigger, and the energy mainly transmits in this region. Moreover, the Poynting vector distributions are concentrated near the boundaries, especially near the corner.

## 4. Conclusions

- (1)
- If the radius-thickness ratio is bigger than 1000, fan-shaped cylindrical structures could be treated as a rectangular bar.
- (2)
- The variation in geometric size of the cross-section has remarkable influence on wave characteristics. The cut-off frequencies have a negative relationship with the cross-section area of the fan-shaped cylindrical structures.
- (3)
- For the FGPP fan-shaped cylindrical structures, the magneto-electric effect could be adjusted via altering the geometric size.
- (4)
- The phase velocities for higher modes at high frequencies approximatively approach a value which is below the shear velocity, and they also increase with the increase of β.
- (5)
- For the big wavenumber case, the Poynting vector distributions are concentrated in the region with more material of smaller elasticity modulus, and the Poynting vector distributions are also concentrated near the boundaries, especially near the corner.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 1.**Schematic drawing of a functionally graded piezoelectric–piezomagnetic (FGPP) fan-shaped cylindrical structure.

**Figure 2.**Phase velocity curves for the square bars: dotted lines—results from the reference [35], lines—the results of the present method.

**Figure 3.**Comparison of dispersion curves for the fan-shaped cylindrical structure cross-section with different η and square bar; (

**b**) is the enlarged drawing of (

**a**).

**Figure 4.**The phase velocity curves: the line—FGPP structure; dotted line—piezoelectric structure; dashed line—piezomagnetic structure; (

**b**) is the enlarged drawing of (

**a**).

**Figure 5.**The phase velocity curves, the line—elastic structure, dotted line—piezomagnetic structure; (

**b**) is the enlarged drawing of (

**a**).

**Figure 6.**The electric and magnetic potential of the first mode for the linearly cylindrical structure (η = 2 and β = π/6) at kd = 2.01.

**Figure 7.**The electric and magnetic potential of the first mode for the linearly cylindrical structure (η = 2 and β = π/6) at kd = 120.01.

**Figure 8.**The variation curves of material properties with different graded functions. (

**a**) Material volume content for BaTiO

_{3}; (

**b**) e

_{15}.

**Figure 9.**The phase velocity curves with power series functions: red lines—linear function, blue lines—cubic function.

**Figure 10.**The phase velocity curves with cubic function: the line—FGPP structure, dotted line—piezoelectric structure, dashed line—piezomagnetic structure; (

**b**) is the enlarged drawing of (

**a**).

**Figure 11.**The dispersion curves with different angular measure: red lines—β = π/4, blue lines—β = π/6.

**Figure 12.**The dispersion curves for fifth and sixth modes with different ratios of the radius-thickness.

**Figure 13.**The phase velocity curves for η = 3: the line—FGPP structure, dotted line—piezoelectric structure, dashed line—piezomagnetic structure; (

**b**) is the enlarged drawing of (

**a**).

**Figure 14.**The phase velocity curves for η = 2 and β = π/4: the line—FGPP structure, dotted line—piezoelectric structure, dashed line—piezomagnetic structure; (

**b**) is the enlarged drawing of (

**a**).

**Figure 15.**The phase velocity curves for η = 2 and β = π/8; the line—FGPP structure, dotted line—piezoelectric structure, dashed line—piezomagnetic structure; (

**b**) is the enlarged drawing of (

**a**).

**Figure 18.**The stress, electric, and magnetic displacement of the first mode for a linearly FGPP cylindrical structure (η = 2 and β = π/6) at kd = 2.01.

**Figure 19.**The Poynting vector of the first two modes for the linearly cylindrical structure (η = 2 and β = π/6) at kd = 2.01. (

**a**): mode 1; (

**b**): mode 2.

**Figure 20.**The Poynting vector of the first two modes for the linearly cylindrical structure (η = 2 and β = π/6) at kd = 120.01. (

**a**): mode 1; (

**b**): mode 2.

M,J | Mode1 | Mode2 | Mode3 |
---|---|---|---|

4,4 | 1054.86 | 1221.42 | 2564.77 |

4,5 | 1054.80 | 1221.34 | 2563.99 |

4,6 | 1054.79 | 1221.34 | 2563.12 |

5,4 | 1054.24 | 1220.87 | 2562.70 |

5,5 | 1054.19 | 1220.79 | 2561.84 |

5,6 | 1054.19 | 1220.79 | 2561.10 |

6,4 | 1054.27 | 1220.83 | 2560.59 |

6,5 | 1054.23 | 1220.75 | 2559.78 |

6,6 | 1054.22 | 1220.75 | 2559.17 |

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**MDPI and ACS Style**

Zhang, B.; Yu, J.; Elmaimouni, L.; Zhang, X.
Magneto-Electric Effect on Guided Waves in Functionally Graded Piezoelectric–Piezomagnetic Fan-Shaped Cylindrical Structures. *Materials* **2018**, *11*, 2174.
https://doi.org/10.3390/ma11112174

**AMA Style**

Zhang B, Yu J, Elmaimouni L, Zhang X.
Magneto-Electric Effect on Guided Waves in Functionally Graded Piezoelectric–Piezomagnetic Fan-Shaped Cylindrical Structures. *Materials*. 2018; 11(11):2174.
https://doi.org/10.3390/ma11112174

**Chicago/Turabian Style**

Zhang, Bo, Jiangong Yu, Lahoucine Elmaimouni, and Xiaoming Zhang.
2018. "Magneto-Electric Effect on Guided Waves in Functionally Graded Piezoelectric–Piezomagnetic Fan-Shaped Cylindrical Structures" *Materials* 11, no. 11: 2174.
https://doi.org/10.3390/ma11112174