On well-posed local-nonlocal mixed integral model of piezoelectricity for dynamic stability and vibration analysis of piezoelectric Timoshenko nanobeams with general boundary constraints

doi:10.1007/s10483-026-3370-8

Abstract

Abstract:

Existing research has shown that nonlocal piezoelectric differential models often yield inconsistent dynamic responses for nanostructures. To address this issue, the two-phase local-nonlocal integral formulation has been proposed and has garnered increasing scholarly attention as an effective alternative. This study presents the first implementation of this theoretically consistent and paradox-free framework to investigate the size-dependent dynamic stability and free vibration behavior in piezoelectric Timoshenko nanobeams. The generalized boundary conditions are simulated through elastic constraints incorporating both translational and rotational springs at both beam ends. Departing from conventional approaches, the present formulation simultaneously accounts for size effects in both bending deformation and axial deformation caused by external voltages via the derivation of an equivalent differential representation of the well-posed local-nonlocal integral piezoelectric model. This formulation is rigorously complemented by a complete set of constitutive constraint conditions, ensuring mathematical well-posedness throughout the analytical framework. The generalized differential quadrature method (GDQM) is used to discretize the governing differential equations, enabling numerical determination of dynamic instability regions (DIRs) for various boundary configurations. Following comprehensive validation through comparative analyses, we systematically examine the influence of nonlocal parameters, static force factors, and boundary stiffness characteristics on the DIRs of the beams. Furthermore, this investigation underscores the significance of incorporating nonlocal effects into voltage-induced axial loading, addressing a critical gap in the current understanding of electromechanical coupling at nanoscale dimensions.

Key words: two-phase nonlocal integral model, piezoelectric beam, dynamic stability, nonlocal axial force, generalized differential quadrature method (GDQM)

2010 MSC Number:

O342

Pei ZHANG, P. SCHIAVONE, Luke ZHAO, Dongbo LI, Yanming REN, Hai QING. On well-posed local-nonlocal mixed integral model of piezoelectricity for dynamic stability and vibration analysis of piezoelectric Timoshenko nanobeams with general boundary constraints. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 815-838.

TrendMD

Figures/Tables 16

Fig. 1

Table 1

Required spring stiffness values for different boundary conditions"

BC	Requirement	Spring stiffness
Simply supported (S)	$M = 0, w = 0$	$k 0 R = 0, k 0 T = 1010$
Clamped (C)	$ϕ = 0, w = 0$	$k 0 R = k 0 T = 1010$
Free (F)	$M = 0, Q − (F (t) − N E) w ′ = 0$	$k 0 R = k 0 T = 0$
Elastically constrained (E)	$ϕ, w ≠ 0$	$k 0 R, k 0 T ≠ 0$

Table 1

Table 2

Mechanical and electrical properties adopted in numerical simulations"

Property	$c 11 / GPa$	$c 44 / GPa$	$e 31 / (C ⋅ m − 2)$	$e 15 / (C ⋅ m − 2)$	$μ 11$ / $(C ⋅ V − 1 ⋅ m − 1)$	$μ 33$ / $(C ⋅ V − 1 ⋅ m − 1)$
For Table 3 only	81.3	25.6	$− 10$	40.324 8	$6.712 × 10 − 9$	$10.27 × 10 − 9$
For others	132	26	$− 4.1$	14.1	$5.841 × 10 − 9$	$7.124 × 10 − 9$

Table 2

Table 3

Table 4

Comparative results of the first dimensionless frequencies of the piezoelectric Bernoulli-Euler beam made of PZT-4 with L=20 nm and h=0.01L for V0=0 mV and μ11=0"

BC	ξ	Present GDQM solution $(n = 21)$			Bernoulli-Euler beam^[49]
BC	ξ	$κ = 0.05$	$κ = 0.075$	$κ = 0.1$	$κ = 0.05$	$κ = 0.075$	$κ = 0.1$
SS	0.5	9.900 9	9.840 0	9.764 7	9.900 9	9.840 3	9.764 6
SS	0.1	9.582 8	9.734 2	9.582 1	9.852 8	9.734 2	9.582 1
CC	0.5	21.199 9	20.544 3	19.946 6	21.199 9	20.544 3	19.946 6
CC	0.1	19.593 6	18.265 0	17.072 0	19.593 6	18.265 0	17.072 0
CS	0.5	15.022 3	14.728 8	14.438 2	15.022 3	14.728 8	14.382 0
CS	0.1	14.390 6	13.780 8	13.183 1	14.390 6	13.780 8	13.183 1
CF	0.5	3.447 4	3.401 1	3.357 4	3.447 3	3.401 1	3.357 3
CF	0.1	3.319 7	3.217 0	3.120 9	3.319 7	3.217 0	3.120 9

Table 4

Table 5

Comparative results of the first dimensionless frequencies of the piezoelectric Bernoulli-Euler beam made of PZT-4 with L=20 nm and h=0.01L for ξ=0.01, κ=0.05, and μ11=0"

BC	Mode	Present GDQM solution $(n = 51)$			Bernoulli-Euler beam^[49]
BC	Mode	$V 0 = − 0.15 mV$	$V 0 = 0 mV$	$V 0 = 0.15 mV$	$V 0 = − 0.15 mV$	$V 0 = 0 mV$	$V 0 = 0.15 mV$
SS	1st	12.142 0	9.838 9	6.795 6	12.142 0	9.838 9	6.795 6
SS	2nd	40.617 7	38.043 3	35.281 5	40.617 7	38.043 3	35.281 5
CC	1st	20.243 3	18.808 2	17.243 2	20.243 3	18.808 2	17.243 2
CC	2nd	52.460 0	50.382 5	48.211 0	52.460 0	50.382 6	48.211 0
CS	1st	15.911 8	14.077 9	11.945 4	15.911 8	14.077 9	11.945 4
CS	2nd	46.526 1	44.231 0	41.809 2	46.526 1	44.231 0	41.809 2

Table 5

Table 6

Dimensionless frequencies of SS piezoelectric nanobeams with h=10 nm and V0=0 mV"

$L / h$	κ	Timoshenko beam			Bernoulli-Euler beam
$L / h$	κ	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$
6	$0 +$	9.306 4	9.306 4	9.306 4	9.954 2	9.954 2	9.954 2
	0.1	8.820 1	8.774 3	8.720 5	9.580 1	9.549 4	9.516 2
	0.2	8.013 9	7.878 6	7.718 7	8.801 1	8.677 9	8.529 6
	0.3	7.245 7	7.003 3	6.711 4	8.016 8	7.770 9	7.461 1
10	$0 +$	9.701 4	9.701 4	9.701 4	9.955 5	9.955 5	9.955 5
	0.1	9.279 7	9.242 7	9.200 2	9.581 2	9.550 5	9.517 4
	0.2	8.487 6	8.359 1	8.206 0	8.802 5	8.679 4	8.531 3
	0.3	7.708 1	7.463 4	7.161 3	8.018 3	7.772 7	7.463 3
16	$0 +$	9.853 5	9.853 5	9.853 5	9.955 9	9.955 9	9.955 9
	0.1	9.459 5	9.426 2	9.389 6	9.581 7	9.551 0	9.517 9
	0.2	8.675 1	8.549 8	8.399 7	8.803 1	8.680 1	8.532 2
	0.3	7.892 6	7.647 4	7.341 2	8.019 1	7.773 7	7.464 5

Table 6

Table 7

Dimensionless frequencies of CC piezoelectric nanobeams with h=10 nm and V0=0 mV"

$L / h$	κ	Timoshenko beam			Bernoulli-Euler beam
$L / h$	κ	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$
6	$0 +$	17.266 3	17.266 3	17.266 3	22.505 5	22.505 5	22.505 5
	0.1	13.688 2	13.305 6	12.831 8	17.028 4	16.457 4	15.756 4
	0.2	11.137 3	10.536 9	9.837 4	13.512 0	12.653 6	11.667 7
	0.3	9.503 5	8.743 4	7.898 1	11.418 7	10.352 6	9.187 8
10	$0 +$	20.094 6	20.094 6	20.094 6	22.528 3	22.528 3	22.528 3
	0.1	15.556 3	15.075 9	14.483 4	17.044 3	16.472 6	15.770 7
	0.2	12.490 2	11.751 9	10.897 6	13.524 2	12.664 8	11.677 7
	0.3	10.602 2	9.676 3	8.655 3	11.429 1	10.361 7	9.195 4
16	$0 +$	21.479 2	21.479 2	21.479 2	22.542 7	22.542 7	22.542 7
	0.1	16.417 5	15.885 9	15.231 8	17.054 2	16.482 0	15.779 5
	0.2	13.093 9	12.286 8	11.356 7	13.531 7	12.671 6	11.683 7
	0.3	11.086 7	10.079 8	8.975 2	11.435 4	10.367 1	9.199 9

Table 7

Table 8

Dimensionless frequencies of CS piezoelectric nanobeams with h=10 nm and V0=0 mV"

$L / h$	κ	Timoshenko beam			Bernoulli-Euler beam
$L / h$	κ	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$
6	$0 +$	13.174 6	13.174 6	13.174 6	15.528 8	15.528 8	15.528 8
	0.1	11.281 9	11.077 6	10.824 5	13.163 6	12.907 2	12.588 9
	0.2	9.587 9	9.225 6	8.798 4	11.102 3	10.647 6	10.109 0
	0.3	8.351 2	7.858 1	7.294 6	9.640 3	9.021 6	8.309 2
10	$0 +$	14.534 3	14.534 3	14.534 3	15.538 3	15.538 3	15.538 3
	0.1	12.377 1	12.143 7	11.854 2	13.171 1	12.914 5	12.595 9
	0.2	10.474 6	10.060 5	9.570 9	11.108 5	10.653 6	10.114 7
	0.3	9.108 0	8.544 0	7.896 6	9.645 8	12.914 5	8.313 9
16	$0 +$	15.123 2	15.123 2	15.123 2	15.543 8	15.543 8	15.543 8
	0.1	12.844 5	12.597 7	12.291 4	13.175 5	12.918 8	12.600 1
	0.2	10.848 8	10.411 1	9.892 9	11.112 2	10.657 1	10.118 0
	0.3	9.425 9	8.830 0	8.144 6	9.649 1	9.029 8	8.316 7

Table 8

Table 9

Dimensionless frequencies of CF piezoelectric nanobeams with h=10 nm and V0=0 mV"

$L / h$	κ	Timoshenko beam			Bernoulli-Euler beam
$L / h$	κ	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$	$ξ = 0.1$	$ξ = 0.05$	$ξ = 0.01$
6	$0 +$	3.408 9	3.408 9	3.408 9	3.541 4	3.541 4	3.541 4
	0.1	3.011 4	2.961 8	2.897 1	3.116 4	3.063 4	2.994 4
	0.2	2.703 7	2.619 7	2.512 8	2.791 4	2.702 3	2.589 0
	0.3	2.465 6	2.355 5	2.218 2	2.542 0	2.425 7	2.280 9
10	$0 +$	3.493 1	3.493 1	3.493 1	3.543 4	3.543 4	3.543 4
	0.1	3.078 4	3.026 6	2.959 2	3.118 1	3.065 1	2.996 1
	0.2	2.759 8	2.672 6	2.561 6	2.792 9	2.703 8	2.590 4
	0.3	2.514 5	2.400 5	2.258 4	2.543 3	2.426 9	2.282 0
16	$0 +$	3.524 4	3.524 4	3.524 4	3.544 7	3.544 7	3.544 7
	0.1	3.103 1	3.050 6	2.982 2	3.119 1	3.066 1	2.997 0
	0.2	2.780 4	2.692 0	2.579 6	2.793 8	2.704 6	2.591 2
	0.3	2.532 5	2.417 0	2.273 2	2.544 1	2.427 7	2.282 7

Table 9

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

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BC	Present			Local model^[63]
BC	1st mode	2nd mode	3rd mode	1st mode	2nd mode	3rd mode
CC	0.223 5	0.611 3	1.187 3	0.225 1	0.616 5	1.197 3
CS	0.155 4	0.500 8	1.036 9	0.155 8	0.502 3	1.038 8
CF	0.035 5	0.221 8	0.617 0	0.036 0	0.222 4	0.622 0