基于渐近均匀化的力-电-湿耦合光滑有限元法

doi:10.13229/j.cnki.jdxbgxb.20230414

Abstract

Abstract:

Aiming at the mechanical property analysis problem of piezoelectric composite materials with microstructure-based moisture-electro-mechanical multi-physical-field coupling， the moisture-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization was proposed. The theoretical basis of this method includes the basic equation of piezoelectric composite materials， the effective performance parameters predicted by asymptotic homogenization， and the moisture-electro-mechanical coupling effect of piezoelectric composite materials. The dynamic control equation of this method was deduced， and the structural dynamic problems of piezoelectric composite materials were solved by applying the Wilson-θ method. The effects of moisture variation on the structure natural frequency and dynamic response were studied. The results were compared with those of the finite element method to verify the correctness and validity of the method. Therefore， this method has a broad application prospect for analyzing the multi-field coupling mechanical properties of piezoelectric composite components.

Key words: solid mechanics, finite element methods, moisture-electro-mechanical coupling, asymptotic method

CLC Number:

TB115

Jian-xiao ZHENG,Wen-bo WANG,Jin-song LIU,Li-ming ZHOU,Yu LI. Moisture-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization[J].Journal of Jilin University(Engineering and Technology Edition), 2024, 54(7): 1876-1886.

Figures/Tables 18

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Table 1

Material parameters of fibers and matrix in piezoelectric composites"

材料参数	BaTiO₃（基体）	PZT-7（纤维）
C₁₁/GPa	150	133.4
C₁₂/GPa	65.63	83.2
C₁₃/GPa	65.94	78.8
C₃₃/GPa	145.5	101.2
C₄₄/GPa	43.86	16.7
e₃₁/（C·m^-2）	-5.6	-4.924
e₃₃/（C·m^-2）	17.36	13.96
e₁₅/（C·m^-2）	11.404	13.31
λ₁₁/（10^-9C²·Nm^-2）	12.84	13.04
λ₃₃/（10^-9C²·Nm^-2）	15.05	9.76
p₁=p₂/（10^-4C·m^-2K）	—	-2.5
β₁=β₃/（mC·kg^-1）	—	0
$α 1 M$ /（m³·kg^-1）	—	0
$α 3 M$ /（10^-4 m³·kg^-1）	—	1.1
$α 1 θ$ = $α 3 θ$ /（10^-6·K^-1）	—	2.0
ρ/（kg·m^-3）	7 600	7 600

Table 1

Table 2

Equivalent performance parameters of piezoelectric composite cantilever beam predicted based onasymptotic homogenization method"

等效性能参数	数值	等效性能参数	数值	等效性能参数	数值
$C ˉ 11$ /GPa	139.0	$e ˉ 15$ /（C·m^-2）	12.72	$λ ˉ 11$ /（nF·m^-1）	13.05
$C ˉ 12$ /GPa	77.5	$e ˉ 33$ /（C·m^-2）	15.06	$λ ˉ 33$ /（nF·m^-1）	11.51
$C ˉ 13$ /GPa	74.1	$e ˉ 13$ /（C·m^-2）	?5.2	$p ˉ 1$ /（10^-4C·m^-2K）	?2.5
$C ˉ 33$ /GPa	115.2	$β ˉ 1$ = $β ˉ 3$ /（mC·kg^-1）	0	$p ˉ 3$ /（10^-4C·m^-2K）	?2.5
$C ˉ 44$ /GPa	25.6	$α ˉ 1 M$ /（m³·kg^-1）	0	$α ˉ 1 θ$ /（10^-6·K^-1）	2.0
$ρ ˉ$ /（kg·m^-3）	7600	$α ˉ 3 M$ /（10^-4m³·kg^-1）	1.1	$α ˉ 3 θ$ /（10^-6·K^-1）	2.0

Table 2

Fig.6

Fig.7

Fig.8

Table 3

Fig.9

Fig.10

Fig.11

Fig.12

Table 4

Material parameters of PZT-5 and polymer"

材料参数	聚合物（基体）	PZT-5（纤维）
C₁₁/GPa	3.86	121
C₁₂/GPa	2.57	75.4
C₁₃/GPa	2.57	75.2
C₃₃/GPa	3.86	111
C₄₄/GPa	0.64	21.1
e₃₁/（C·m^-2）	—	?5.4
e₃₃/（C·m^-2）	—	15.8
e₁₅/（C·m^-2）	—	12.3
λ₁₁/（10^-9C²·Nm^-2）	—	8.11
λ₃₃/（10^-9C²·Nm^-2）	—	7.35
$α 1 M$ /（10^-4m³·kg^-1）	—	0
$α 3 M$ /（10^-4m³·kg^-1）	—	0.44
β₁=β₃/（mC·kg^-1）	—	0
$α 1 θ$ /（10^-6·K^-1）	—	8.53
$α 3 θ$ /（10^-6·K^-1）	—	1.99
p₁=p₂/（10^-5C·m^-2·K）	—	?2.5
ρ/（kg·m^-3）	7 600	7 600

Table 4

Table 5

Equivalent performance parameters of piezoelectric composite sensors predicted based on asymptotic homogenization method"

等效性能参数	数值	等效性能参数	数值	等效性能参数	数值
$C ˉ 11$ /GPa	13.278 01	$e ˉ 15$ /（C·m^-2）	0.052 676	$λ ˉ 11$ /（nF·m^-1）	0.420 31
$C ˉ 12$ /GPa	7.019 85	$e ˉ 33$ /（C·m^-2）	12.984 637	$λ ˉ 33$ /（nF·m^-1）	5.105 80
$C ˉ 13$ /GPa	7.872 01	$e ˉ 13$ /（C·m^-2）	?0.394 201	$p ˉ 1$ /（10^-5C·m^-2·K）	?2.5
$C ˉ 33$ /GPa	42.316 62	$α ˉ 1 M$ /（10^-4·m³·kg^-1）	0	$p ˉ 3$ /（10^-5C·m^-2·K）	?2.5
$C ˉ 44$ /GPa	3.188 67	$α ˉ 3 M$ /（10^-4·m³·kg^-1）	0.44	$α ˉ 1 θ$ /（10^-6K^-1）	8.53
$ρ ˉ$ /（kg·m^-3）	7 600	$β ˉ 1$ = $β ˉ 3$ /（m·C·kg^-1）	0	$α ˉ 3 θ$ /（10^-6K^-1）	1.99

Table 5

Fig.13

References 19

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计算方法	单元总数/计算时间
FEM	60/0.107 s	240/2.184 s	480/3.694 s	2 160/196.524 s
CS-FEM	60/0.144 s	240/2.307 s	480/3.689 s	2 160/193.869 s

Moisture-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization

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