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

Abstract

Functionally graded piezoelectric⁻piezomagnetic (FGPP) material simultaneously consists of piezomagnetic and piezoelectric phases, which are able to convert energy among mechanical, electric, and magnetic fields. The magneto-electric effect on waves in FGPP fan-shaped cylindrical structures is studied by exploiting the double Legendre orthogonal polynomial method. By means of the Heaviside function, the initial conditions are brought into wave motion equations. Dispersion properties, electric and magnetic potential, and the Poynting vector are calculated. Subsequently, the effect of the graded variation and geometric size on wave characteristics is analyzed. The FGPP fan-shaped cylindrical structures are of complex geometrical shape and material inhomogeneity, so their influences on the magneto-electric effect are the focus of discussion. Results reveal that the cut-off frequencies have a negative relationship with the cross-section area of the structure. The magneto-electric effect could be adjusted via altering the geometric size of the cross-section. These results can be utilized to design and optimize piezoelectric⁻piezomagnetic fan-shaped transducers.

Keywords: dispersion curves; fan-shaped cross-section; functionally graded piezoelectric–piezomagnetic material; magneto-electric effect; the Poynting vectors.

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 BaTiO₃; (b) e₁₅.

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 16

Phase velocity curves for the fifth and sixth mode with β = π/6 and η = 2.

Figure 17

Phase velocity curves for the fifth and six mode at η = 2.

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.

Figure 21

The partial enlarged drawings of the Figure 20. (a): mode 1; (b): mode 2.