# The Computation of Complex Dispersion and Properties of Evanescent Lamb Wave in Functionally Graded Piezoelectric-Piezomagnetic Plates

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematics and Formulation of the Problem

_{i}(i = x, y, z) denotes the mechanical displacement component in the ith direction. φ and ψ are the electric potential and magnetic potential. Comma in subscripts and superposed dot denote spatial and time derivatives, respectively.

_{i}(i = x, z) represents the amplitude of the displacements in the ith directions, X and Y represent the amplitude of electric potential and magnetic potential, respectively. k is the wave number, ω is the angular frequency, and i is the imaginary number.

^{(l)}is the coefficient. For homogeneous material, $f(z)={C}^{(0)}$, and when l > 0, f(l) is zero.

_{m}is the Legendre polynomial of order m.

**A**,

**B**,

**C**and

**M**are matrices of order 4(M + 1)·(M + 1), $\mathbf{p}={\left[{p}_{m}^{1}\hspace{0.17em}{p}_{m}^{2}\hspace{0.17em}{p}_{m}^{3}\hspace{0.17em}{p}_{m}^{4}\right]}^{T}$, the elements of the matrices are as following,

**I**is the identity matrix and

**Z**is a zero matrix.

## 3. Numerical Results and Discussion

_{2}O

_{4}(top) and Ba

_{2}TiO

_{3}(bottom), h = 1 mm. The material parameters are from literature [26] and are listed in Table 1.

_{B}and F

_{C}respectively represent the material property of the Ba

_{2}TiO

_{3}and CoFe

_{2}O

_{4}materials, and V

_{B}and V

_{C}are volume fraction.

_{C}(z) can be expressed as a power expansion, Here we consider four different gradient fields, ${V}_{C}(z)={(z/h)}^{n}$, n = 1, 2 and 3, namely linear, quadratic and cubic graded fields, and sinusoidal graded field ${V}_{C}(z)=\mathrm{sin}(0.5\pi z/h)$.

#### 3.1. Approach Validation and Convergence of the Problem

^{3}, C

_{11}= 281.757 GPa, C

_{12}= 113.161 GPa, C

_{44}= 84.298 GPa, and h = 10 mm. The non-dimensional frequency and wave number are defined as $\Omega =(\omega h\sqrt{\rho /C44})/\pi $, $\mathsf{\Psi}=kh/\pi $, respectively. The resulting dispersion curves are given in Figure 2. It clearly shows that the numerical results obtained by the present polynomial approach agree well with those obtained by the spectral collocation method, which validates our approach and program.

#### 3.2. Full Dispersion Curves of Lamb Wave

#### 3.3. Influences of Graded Field on Dispersion Curves and the Electromechanical Coupling Factor

_{2}O

_{4}content for the sinusoidal graded field is the highest, and the wave velocity of Ba

_{2}TiO

_{3}is slower than that of CoFe

_{2}O

_{4}.

^{2}is an important parameter for designing acoustic wave devices. A high magneto-electromechanical coupling factor is expected in engineering applications. It is defined as [29]

^{2}, we calculate the K

^{2}for S0 modes of four different FGPPM plates, as shown in Figure 10. We can find that the K

^{2}reaches a maximum at a certain wave number and tend to the same little value with increasing wave number, which implies the influence of the graded field on the energy propagation of Lamb wave in high-frequency zone is insignificant. It reaches a maximum from 4.4% for the sinusoidal graded field to 9.5% for the cubic graded field. They are located near kh = 2 and kh = 1.5 respectively. The K

^{2}for the cubic graded field is always bigger than that of the other three graded cases. Also the maximum of K

^{2}shifts to the smaller wave number when the graded power exponent is increasing.

#### 3.4. Wave Structure Analysis

_{z}and electric potential and magnetic potential distributions change along the z direction in a nearly anti-symmetric manner. The displacement u

_{x}exhibits a nearly symmetric manner. The distribution of displacement u

_{z}of the complex branch is very similar to that of the real branch, implying the evanescent wave mode converts into the propagating wave mode.

#### 3.5. Merits of the Presented Method

- (1)
- The complex mathematical issue is reduced to solve an eigenvalue problem, which is capable of accurately determining all the real, imaginary and complex solutions of a transcendental dispersion equation.
- (2)
- The conventional approaches (root-finding routines or finite element simulations) require an iterative search procedure or a far greater coding effort, to find complex roots. The present method can avoid tedious iterative two-variable search and is simple to program. It needs to take a larger polynomial order to obtain solutions of the higher modes, which will cause more computer memory and long time.
- (3)
- The method is easy to implement and can be extended to complex structures such as multilayered or curved structures.

## 4. Conclusions

- (1)
- Superior to the conventional methods that necessitate an iterative search procedure to solve the complex roots of a dispersion equation, the presented analytic method can transform the set of differential equations for the acoustic waves into an eigenvalue problem in the form AX = kX to find the complex solutions.
- (2)
- Complex branches of the Lamb wave usually collapse onto the extremum of the real branches. They exhibit both local vibration and local propagation, and some can propagate a quite long distance (more than ten times of the plate thickness). They will turn into the propagating modes with increasing frequency.
- (3)
- Some evanescent modes have a noticeably higher phase velocity than the propagating modes. The phase velocity of the low order evanescent modes is more than four times larger than that of the propagating modes. Also, the wave dispersion of the evanescent mode is quite weak in a certain frequency range.
- (4)
- The magneto-electromechanical coupling factor of the guided wave in a FGPPM plate may be improved by adjusting the graded field. The coupling factor reaches a maximum from 4.4% for the sinusoidal graded field to 9.5% for the cubic graded field. The maximum of the magneto-electromechanical coupling factor for the S0 mode shifts to lower frequencies with increasing the gradient index.

## Author Contributions

**Funding:**This research was funded by the National Natural Science Foundation of China (No. U1504106), the fundamental research funds for the national outstanding youth project of Henan Polytechnic University (No. NSFRF140301), and the Program for Innovative Research Team of Henan Polytechnic University (T2017-3).

## Conflicts of Interest

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**Figure 2.**Dispersion curves of Lamb wave in a steel plate; hollow dots-our results, solid lines-literature results from the spectral collocation method; real branch in blue, purely imaginary branch in black, complex branch in red.

**Figure 3.**Dispersion curves of Lamb wave in a PZT-4 plate; red dots—our results, black dotted lines—literature results from the reverberation-ray matrix method.

**Figure 4.**Dispersion curves of propagating Lamb-like wave in a linear functionally graded piezoelectric-piezomagnetic material (FGPPM) plate with various “M”.

**Figure 5.**3D dispersion curves of Lamb wave: (

**a**) four quadrants, (

**b**) one quadrant; blue—real solutions, green—imaginary solutions, red—complex solutions.

**Figure 6.**The phase velocity dispersion and attenuation curves; propagating wave in blue, evanescent wave in red.

**Figure 7.**3D dispersion curves of Lamb wave: (

**a**) cubic graded field, (

**b**) sinusoidal graded field. blue—real branches, green—purely imaginary branches, red—complex branches.

**Figure 8.**Dispersion curves of propagating Lamb wave for FGPPM plates with different graded fields; (

**a**) Frequency spectra; (

**b**) Phase velocity spectra.

**Figure 11.**Distributions of the physical quantities when Ω = 1.01115, Ψ = 0.23216 − 0.04612i. (

**a**) displacement distribution, (

**b**) electric potential and magnetic potential distribution.

**Figure 12.**Distributions of the physical quantities when Ω = 1.01911, Ψ = 0.17938. (

**a**) displacement distribution, (

**b**) electric potential and magnetic potential distribution.

Materials | Property | ||||||||

C_{11} | C_{12} | C_{13} | C_{22} | C_{23} | C_{33} | C_{44} | C_{55} | C_{66} | |

Ba_{2}TiO_{3} | 166 | 77 | 78 | 166 | 78 | 162 | 43 | 43 | 44.6 |

CoFe_{2}O_{4} | 286 | 173 | 170 | 286 | 170 | 269 | 45.3 | 45.3 | 46.5 |

Property | |||||||||

e_{15} | e_{24} | e_{31} | e_{32} | e_{33} | ∈_{11} | ∈_{22} | ∈_{33} | ρ | |

Ba_{2}TiO_{3} | 11.6 | 11.6 | −4.4 | −4.3 | 18.6 | 196 | 201 | 28 | 5.8 |

CoFe_{2}O_{4} | 0 | 0 | 0 | 0 | 0 | 0.8 | 0.8 | 0.93 | 5.3 |

Property | |||||||||

q_{15} | q_{24} | q_{31} | q_{32} | q_{33} | μ_{11} | μ_{22} | μ_{33} | ||

Ba_{2}TiO_{3} | 0 | 0 | 0 | 0 | 0 | 5 | 5 | 10 | |

CoFe_{2}O_{4} | 550 | 550 | 580.3 | 580.3 | 699.7 | −590 | −590 | 157 |

_{ij}(10

^{9}N/m

^{2}), ∈

_{ij}(10

^{−10}F/m), e

_{ij}(C/m

^{2}), q

_{ij}(N/Am), μ

_{ij}(10

^{−6}Ns

^{2}/C

^{2}), ρ (10

^{3}kg/m

^{3}).

Material | Property | ||||||||

C_{11} | C_{12} | C_{13} | C_{22} | C_{23} | C_{33} | C_{44} | C_{55} | C_{66} | |

PZT-4 | 139 | 78 | 74 | 139 | 74 | 115 | 25.6 | 25.6 | 30.5 |

Property | |||||||||

e_{15} | e_{24} | e_{31} | e_{32} | e_{33} | ∈_{11} | ∈_{22} | ∈_{33} | ρ | |

PZT-4 | 12.7 | 12.7 | −5.2 | −5.2 | 15.1 | 65 | 65 | 56 | 7.5 |

_{ij}(10

^{9}N/m

^{2}), ∈

_{ij}(10

^{−10}F/m), e

_{ij}(C/m

^{2}), ρ (10

^{3}kg/m

^{3}).

M | Mode | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

10 | 0.31308 +0.67653i | 0.32055 +1.23591i ™ | 0.16593 +1.87713i ™ | 0.33292 +2.58928i ™ | 0.34536 +3.14395i ™ | 0.24222 +3.74226i ™ |

11 | 0.31308 +0.67653i | 0.32055 +1.23591i ™ | 0.16593 +1.87713i ™ | 0.33184 +2.59041i | 0.34722 +3.14407i ™ | 0.24481 +3.73460i ™ |

12 | 0.31308 +0.67653i | 0.32055 +1.23591i ™ | 0.16593 +1.87713i ™ | 0.33063 +2.59061i | 0.35203 +3.14418i ™ | 0.25047 +3.72524i |

13 | 0.31308 +0.67653i | 0.32055 +1.23591i ™ | 0.16593 +1.87713i ™ | 0.33063 +2.59061i | 0.35211 +3.14436i ™ | 0.25504 +3.72496i ™ |

14 | 0.31308 +0.67653i | 0.32055 +1.23591i ™ | 0.16593 +1.87713i ™ | 0.33063 +2.59061i | 0.35211 +3.14436i ™ | 0.25561 +3.72471i ™ |

20 | 0.31308 +0.67653i | 0.32055 +1.23591i ™ | 0.16593 +1.87713i ™ | 0.33063 +2.59061i | 0.35211 +3.14436i ™ | 0.25561 +3.72471i ™ |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Li, Z.; Yu, J.
The Computation of Complex Dispersion and Properties of Evanescent Lamb Wave in Functionally Graded Piezoelectric-Piezomagnetic Plates. *Materials* **2018**, *11*, 1186.
https://doi.org/10.3390/ma11071186

**AMA Style**

Zhang X, Li Z, Yu J.
The Computation of Complex Dispersion and Properties of Evanescent Lamb Wave in Functionally Graded Piezoelectric-Piezomagnetic Plates. *Materials*. 2018; 11(7):1186.
https://doi.org/10.3390/ma11071186

**Chicago/Turabian Style**

Zhang, Xiaoming, Zhi Li, and Jiangong Yu.
2018. "The Computation of Complex Dispersion and Properties of Evanescent Lamb Wave in Functionally Graded Piezoelectric-Piezomagnetic Plates" *Materials* 11, no. 7: 1186.
https://doi.org/10.3390/ma11071186