Symplectic electrothermomechanical buckling solutions for two-dimensional decagonal piezoelectric quasicrystal cylindrical shells

doi:10.1007/s10483-026-3410-8

Abstract

Abstract:

Piezoelectric quasicrystals (PQCs), characterized by unique phonon-phason coupling and piezoelectric effects, exhibit significant potential for use in next-generation smart structural devices. However, their complex electrothermomechanical buckling behavior remains a challenging analytical problem. This paper presents a symplectic electrothermomechanical buckling model for two-dimensional (2D) decagonal PQC cylindrical shells. By using the symplectic mathematics and Donnell’s thin shell theory, the governing buckling equations for axially compressed PQC cylindrical shells are reformulated into a Hamiltonian system. Consequently, the original buckling problem is transformed into a symplectic eigenproblem that can be solved directly, obviating the necessity of trial functions. By use of the symplectic eigenfunction expansion, analytical symplectic buckling equations are obtained, allowing the critical buckling loads and buckling mode shapes to be solved simultaneously. The results indicate that, in addition to the geometry, voltage, and temperature, the phonon-phason-electric coupling inherent in PQC materials significantly influences the critical buckling loads. These analytical results provide a reliable reference for validating other computational approaches.

Key words: piezoelectric quasicrystal (PQC) cylindrical shell, buckling, symplectic method, analytical solution

2010 MSC Number:

O343

Xin SU, Yuhang LI, Jufang JIA, Xinsheng XU, Andi LAI, Zhenhuan ZHOU. Symplectic electrothermomechanical buckling solutions for two-dimensional decagonal piezoelectric quasicrystal cylindrical shells. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1549-1568.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3410-8

https://www.amm.shu.edu.cn/EN/Y2026/V47/I7/1549

Figures/Tables 15

L,	longitudinal dimensions of piezoelectric quasicrystal (PQC) cylindrical shell;
R,	mid-plane radius;
h,	thickness;
$u i, ε i j,$	displacements and strains for the phonon field, respectively;
$w i, w i j,$	displacements and strains for the phason field, respectively;
$σ i j, H i j,$	stresses for the phonon and phason fields, respectively;
$C k l,$	elastic constants for the phonon field;
K_i,	elastic constants for the phason field;
$N i j, M i j,$	internal forces and bending moments for the phonon field, respectively;
$Y i j,$	internal forces for the phason field;
$Φ ˜,$	electric potential;
Φ,	electric potential induced by elastic deformation;
$Λ k,$	electric displacement;
$φ 0,$	initial external electric voltage;
$E x, E θ, E z,$	electric field intensities;
$L C, H,$	Lagrangian density function and Hamiltonian density function, respectively;
q, p,	original variable vector and dual variable vector, respectively;
H,	Hamiltonian operator matrix;
μ, η(x),	symplectic eigenvalue and symplectic eigenfunction vector, respectively.

Fig. 1

Table 1

Fig. 2

Table 2

Table 3

Critical buckling stresses and buckling modes for elastic cylindrical shells (L/R=1)"

Method	Result	$h / R = 1 / 200$	$h / R = 1 / 160$	$h / R = 1 / 140$	$h / R = 1 / 120$	$h / R = 1 / 100$
FEM	Mode
	Stress/Pa	$64.585 × 107$	$80.494 × 107$	$92.038 × 107$	$107.587 × 107$	$129.170 × 107$
Present	Mode
	Stress/Pa	$63.980 × 107$	$80.061 × 107$	$91.770 × 107$	$107.453 × 107$	$129.632 × 107$
	Error/%	0.94	0.54	0.29	0.12	0.36

Table 3

Table 4

Material properties"

Property	Value
Elastic constant for the phonon field/GPa	$c 11 = c 22 = 234.33,$ $c 12 = 57.41,$ $c 13 = c 23 = 66.63,$ $c 33 = 232.22,$ $c 66 = 88.46$
Elastic constant for the phason field/GPa	$K 1 = 122,$ $K 2 = 24$
Coupling constant/GPa	$R 1 = 8.846$
Piezoelectric constant for the phason field/(C·m^-2)	$d 112 = 0$
Piezoelectric constant/(C·m^-2)	$e 31 = e 32 = − 4.4,$ $e 33 = 18.6$
Dielectric constant/(C·V^-1·m^-1)	$s 11 = s 22 = 11.2 × 10 − 9,$ $s 33 = 12.6 × 10 − 9$
Thermal modulus/(N·K^-1·m^-2)	$β 1 = β 2 = 4.314 × 106,$ $β 3 = 3.486 × 106$

Table 4

Table 5

Critical buckling stresses of PQC cylindrical shells for various L/R and h/R"

h/R	L/R	Critical buckling stress/Pa
h/R	L/R	C-C	S-S	S-C
0.005	1	$62.372 × 107$	$61.938 × 107$	$61.979 × 107$
	2	$61.979 × 107$	$61.937 × 107$	$61.943 × 107$
	3	$61.949 × 107$	$61.936 × 107$	$61.945 × 107$
0.010	1	$126.248 × 107$	$123.932 × 107$	$124.046 × 107$
	2	$124.045 × 107$	$123.872 × 107$	$123.909 × 107$
	3	$123.922 × 107$	$123.869 × 107$	$123.889 × 107$
0.015	1	$191.977 × 107$	$185.979 × 107$	$186.719 × 107$
	2	$186.719 × 107$	$185.849 × 107$	$186.071 × 107$
	3	$186.258 × 107$	$185.799 × 107$	$185.924 × 107$
0.020	1	$260.267 × 107$	$247.738 × 107$	$249.508 × 107$
	2	$249.537 × 107$	$247.738 × 107$	$248.264 × 107$
	3	$248.643 × 107$	$247.733 × 107$	$247.970 × 107$

Table 5

Fig. 3

Table 6

Critical buckling stresses of PQC cylindrical shells for various φ0"

$φ 0$ /V	Critical buckling stress/Pa
	$Δ T = 0 ∘ C$			$Δ T = 30 ∘ C$
	C-C	S-S	S-C	C-C	S-S	S-C
$− 20 000$	$132.377 × 107$	$127.977 × 107$	$129.586 × 107$	$114.345 × 107$	$87.942 × 107$	$106.125 × 107$
$− 10 000$	$129.557 × 107$	$126.029 × 107$	$127.390 × 107$	$109.934 × 107$	$76.327 × 107$	$95.862 × 107$
0	$126.248 × 107$	$123.932 × 107$	$124.046 × 107$	$105.149 × 107$	$64.711 × 107$	$84.365 × 107$
10 000	$122.811 × 107$	$120.209 × 107$	$119.921 × 107$	$96.026 × 107$	$50.759 × 107$	$72.083 × 107$
20 000	$118.956 × 107$	$100.778 × 107$	$114.866 × 107$	$85.395 × 107$	$36.184 × 107$	$59.314 × 107$

Table 6

Table 7

Critical buckling stresses of PQC cylindrical shells for various ΔT"

$Δ T$ /°C	Critical buckling stress/Pa
	$φ 0 = 0$ V			$φ 0 = − 20 000$ V
	C-C	S-S	S-C	C-C	S-S	S-C
0	$126.248 × 107$	$123.932 × 107$	$124.046 × 107$	$132.377 × 107$	$127.977 × 107$	$129.586 × 107$
10	$120.150 × 107$	$104.243 × 107$	$116.453 × 107$	$127.252 × 107$	$124.605 × 107$	$125.115 × 107$
20	$113.052 × 107$	$84.477 × 107$	$103.226 × 107$	$121.321 × 107$	$107.708 × 107$	$117.976 × 107$
30	$105.149 × 107$	$64.711 × 107$	$84.368 × 107$	$114.345 × 107$	$87.942 × 107$	$106.117 × 107$
40	$88.671 × 107$	$40.532 × 107$	$63.161 × 107$	$106.722 × 107$	$68.177 × 107$	$87.892 × 107$
50	$67.874 × 107$	$15.730 × 107$	$40.866 × 107$	$91.876 × 107$	$44.880 × 107$	$66.979 × 107$

Table 7

Fig. 4

Fig. 5

Table 8

Critical buckling stresses of PQC cylindrical shells for various λK and λR"

$λ K$	$λ R$	Critical buckling stress/Pa
$λ K$	$λ R$	C-C	S-S	S-C
1/4	1/4	114.555×10⁷	88.133×10⁷	106.316×10⁷
	1/2	114.344×10⁷	87.942×10⁷	106.125×10⁷
	1	113.489×10⁷	87.171×10⁷	105.355×10⁷
	2	109.826×10⁷	83.938×10⁷	102.122×10⁷
	4	90.257×10⁷	68.153×10⁷	82.017×10⁷
1/2	1/4	114.593×10⁷	88.165×10⁷	106.347×10⁷
	1/2	114.488×10⁷	88.070×10⁷	106.250×10⁷
	1	114.063×10⁷	87.687×10⁷	105.869×10⁷
	2	112.309×10⁷	86.119×10⁷	104.322×10⁷
	4	104.275×10⁷	79.231×10⁷	97.368×10⁷
1	1/4	114.608×10⁷	88.181×10⁷	106.362×10⁷
	1/2	114.555×10⁷	88.133×10⁷	106.316×10⁷
	1	114.346×10⁷	87.942×10⁷	106.125×10⁷
	2	113.488×10⁷	87.171×10⁷	105.355×10⁷
	4	109.823×10⁷	83.938×10⁷	102.121×10⁷
2	1/4	114.619×10⁷	88.189×10⁷	106.372×10⁷
	1/2	114.592×10⁷	88.165×10⁷	106.347×10⁷
	1	114.484×10⁷	88.070×10⁷	106.253×10⁷
	2	114.061×10⁷	87.687×10⁷	105.872×10⁷
	4	112.309×10⁷	86.119×10⁷	104.326×10⁷
4	1/4	114.624×10⁷	88.193×10⁷	106.379×10⁷
	1/2	114.608×10⁷	88.181×10⁷	106.363×10⁷
	1	114.553×10⁷	88.133×10⁷	106.316×10⁷
	2	114.344×10⁷	87.942×10⁷	106.124×10⁷
	4	113.488×10⁷	87.171×10⁷	105.357×10⁷

Table 8

Fig. 6

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R/h	10	20	30	40	50
Ref. [36]	1.054 8	0.968 9	0.951 8	0.945 6	0.942 7
Present	1.061 9	0.972 1	0.955 5	0.949 6	0.947 0
Error/%	0.67	0.33	0.38	0.42	0.45

h/R	Ceramic			Metal
h/R	Ref. [48]	Ref. [49]	Present	Ref. [48]	Ref. [49]	Present
0.001 0	57.29	57.26	57.26	36.23	36.21	36.34
0.005 0	286.48	268.51	286.28	181.19	181.21	181.15
0.007 5	429.73	431.41	429.81	271.79	272.86	271.85
0.010 0	572.90	573.56	572.60	362.39	362.76	363.55