Vibration characteristic analysis of a cracked piezoelectric semiconductor curved beam

doi:10.1007/s10483-025-3307-6

Abstract

Abstract:

The fracture mechanics theory posits that cracks induce strain energy concentration near their tips in structural components, generating localized flexibility that impedes crack propagation. Theoretically, cracks are represented as dimensionless, massless spring models, effectively capturing crack characteristics and cross-sectional properties at the crack location. Leveraging this spring-based representation, this study establishes an open-crack model for a one-dimensional (1D) piezoelectric semiconductor (PSC) curved beam under dynamic loading. This model enables the investigation of vibration characteristics in cracked structures. The analytical solutions for the electromechanical fields of the beam are derived using the differential operator method, and the natural frequencies together with the corresponding generalized mode shapes of the beam are determined analytically. Furthermore, the effects of the crack parameters on the natural vibration characteristics of the PSC curved beam are analyzed.

Key words: piezoelectric semiconductor (PSC), curved beam, vibration, crack, analytical solution

2010 MSC Number:

O347.1

Qiaoyun ZHANG, Xiaoyan ZHANG, Jiahao XU, Zhicai SONG, Minghao ZHAO. Vibration characteristic analysis of a cracked piezoelectric semiconductor curved beam. Applied Mathematics and Mechanics (English Edition), 2025, 46(10): 1967-1982.

TrendMD

Figures/Tables 10

Fig. 1

Fig. 2

Table 1

Material constants of ZnO[42]"

Property	Parameter	Value	Unit
Elastic constant	$c 11$	209.7	GPa
	$c 12$	121.1	GPa
	$c 13$	105.1	GPa
	$c 33$	210.9	GPa
	$c 44$	42.47	GPa
Piezoelectric constant	$e 15$	$− 0.48$	C·m^-2
	$e 31$	$− 0.573$	C·m^-2
	$e 33$	1.32	C·m^-2
Dielectric constant	$κ 11$	$7.570 × 10 − 11$	C·V^-1·m^-1
Dielectric constant	$κ 33$	$9.031 × 10 − 11$	C·V^-1·m^-1
Electron mobility	$μ 11$	0.02	m²·V^-1·s^-1
Electron mobility	$μ 33$	0.02	m²·V^-1·s^-1
Diffusion constant	$d 11$	$5.2 × 10 − 4$	m²·s^-1
Diffusion constant	$d 33$	$5.2 × 10 − 4$	m²·s^-1
Mass density	$ρ$	5 680	kg·m^-3

Table 1

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 2

Natural frequencies between the intact and cracked ZnO curved beams"

Natural frequency	$ω 1$ / $(rad ⋅ s − 1)$	$ω 2$ / $(rad ⋅ s − 1)$	$ω 3$ / $(rad ⋅ s − 1)$
Intact structure	$6.676 5 × 109$	$1.763 5 × 1010$	$2.463 4 × 1010$
Damaged structure	$3.227 9 × 109$	$1.255 4 × 1010$	$1.895 8 × 1010$

Table 2

Fig. 7

Fig. 8

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