高阶剪切变形板理论下FG-GRC板的屈曲和弯曲分析

王壮壮; 马连生

doi:10.6052/j.issn.1000-4750.2021.11.0890

高阶剪切变形板理论下FG-GRC板的屈曲和弯曲分析

doi: 10.6052/j.issn.1000-4750.2021.11.0890

王壮壮,
马连生^,

潍坊科技学院建筑工程学院，山东，潍坊 262700

基金项目: 国家自然科学基金项目(11862012，12062010)；山东省自然科学基金项目(ZR2020KA006)

详细信息

作者简介:
王壮壮(1996−)，男，甘肃天水人，硕士生，主要从事功能梯度材料力学行为研究(E-mail: Wang_ZhuangZ@126.com)

通讯作者:
马连生(1963−)，男，山东临朐人，教授，博士，博导，主要从事新型材料结构力学行为研究(E-mail: lsma@lut.cn)

中图分类号: TB34
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出版历程
- 收稿日期: 2021-11-13
- 修回日期: 2022-04-03
- 录用日期: 2022-04-23
- 网络出版日期: 2022-04-23
- 刊出日期: 2023-06-25

BUCKLING AND BENDING ANALYSIS OF FG-GRC PLATES USING HIGH-ORDER SHEAR DEFORMATION PLATE THEORIES

School of Architecture and Engineering, Weifang University of Science and Technology, Weifang, Shandong 262700, China

摘要

摘要: 基于三阶剪切变形板理论(TSDPT)和正弦剪切变形板理论(SSDPT)，研究了功能梯度石墨烯增强复合材料(FG-GRC)板的屈曲和弯曲行为，并通过与一阶剪切变形板理论(FSDPT)计算结果的比较，分析了TSDPT、SSDPT与FSDPT在FG-GRC板屈曲和弯曲力学行为研究过程中的差异。材料的有效杨氏模量通过修正的Halpin-Tsai微观力学模型估算，有效泊松比通过混合律确定。利用最小势能原理推导出了包含五个未知量的控制方程，并获得了简支FG-GRC矩形板弯曲挠度和临界屈曲载荷Navier形式的解析解。数值结果表明：与TSDPT和SSDPT相比，FSDPT明显高估了FG-X型FG-GRC板的临界屈曲载荷而明显低估了其弯曲挠度，且略微低估了FG-O型FG-GRC板的临界屈曲载荷而略微高估了其弯曲挠度，而UD型和FG-A型FG-GRC板在三种理论下的计算结果几乎完全一致；TSDPT和SSDPT在计算FG-GRC板的弯曲挠度和临界屈曲载荷时结果十分相近；当板的总层数N_L小于10层~15层时，弯曲载荷比率和临界屈曲载荷比率的变化非常显著，当总层数N_L超过10层~15层时，弯曲载荷比率和临界屈曲载荷比率的变化趋于平缓；由于石墨烯纳米片(GPLs)极高的弹性模量，FG-GRC板中GPLs的重量分数f_G与板抵抗弯曲和屈曲的能力正相关。
- 石墨烯增强复合材料 /
- 功能梯度材料 /
- Navier法 /
- 高阶剪切变形板理论 /
- 弯曲 /
- 屈曲
Abstract: Based on the third-order shear deformation plate theory (TSDPT) and sinusoidal shear deformation plate theory (SSDPT), the buckling and bending behaviors of functionally graded graphene-reinforced composite (FG-GRC) plates were investigated. By comparing the calculated results with the first-order shear deformation plate theory (FSDPT), the differences among TSDPT, SSDPT and FSDPT in the buckling and bending mechanical behaviors of FG-GRC plates were analyzed. The effective Young's modulus is estimated using the modified Halpin-Tsai model, and the effective Poisson's ratio is determined using the rule of mixtures. The governing equations containing five unknown quantities are derived using the principle of minimum potential energy, and the analytical solutions in Navier form for the bending deflection and critical buckling load of simply supported FG-GRC rectangular plates are obtained. Numerical results show that: Compared with TSDPT and SSDPT, FSDPT significantly overestimates the critical buckling load and significantly underestimates the bending deflection of FG-X type FG-GRC plate, and slightly underestimates the critical buckling load and slightly overestimates the bending deflection of FG-O type FG-GRC plate, while the calculated results of UD type and FG-A type FG-GRC plate are almost identical using the three theories; The results of TSDPT and SSDPT are very similar in calculating the bending deflection and critical buckling load of FG-GRC plates; When the total number of layers N_L of the plate is less than 10~15, the variation of the ratio of bending load and the ratio of critical buckling load is significant. When N_L exceeds 10~15, The variation of these two ratios become small; Due to the extremely high modulus of elasticity of graphene nanoplatelets (GPLs), the weight fraction f_G of GPLs in FG-GRC plates is positively correlated with the capability of the plates to resist bending and buckling.
- graphene-reinforced composites /
- functionally graded materials /
- Navier solution technique /
- high-order shear deformation plate theories /
- bending /
- buckling

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图 1 N_L=6时石墨烯增强功能梯度(FG-GRC)板的示意图

Figure 1. Schematic of a functionally graded graphene reinforced composite (FG-GRC) plate when N_L=6

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图 2 GPLs的四种分布模式

Figure 2. Four distribution modes of GPLs

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图 3 板理论采用的剪切函数ψ(z)

Figure 3. The shear shape functions employed in plate theories

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图 4 总层数N_L对FG-GRC板R_w的影响

Figure 4. Effect of total number of layers N_L on R_w of FG-GRC plates

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图 5 总层数N_L对FG-GRC板R_N的影响

Figure 5. Effect of total number of layers N_L on R_N of FG-GRC plates

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表 1 不同参数下Al/(ZrO₂)-1功能梯度方板无量纲中心挠度的计算结果(n=0.5)

Table 1. Calculation of dimensionless central deflection of Al/(ZrO₂)-1 functionally graded square plates with different parameters (n=0.5)

板理论	总层数	R=S=1	R=S=10	R=S=20
FSDPT^[9]	N_L=10	0.2408	0.2319	0.2318
	N_L=20	0.2413	0.2323	0.2323
	N_L=30	0.2414	0.2324	02324
TSDPT	N_L=10	0.2402	0.2313	0.2312
	N_L=20	0.2407	0.2318	0.2317
	N_L=30	0.2408	0.2319	0.2318
SSDPT	N_L=10	0.2401	0.2312	0.2312
	N_L=20	0.2406	0.2317	0.2317
	N_L=30	0.2407	0.2318	0.2318

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表 2 Al/(ZrO₂)-1功能梯度方板在均布载荷下的无量纲中心挠度 (a/h=5，N_L=10，R=S=10)

Table 2. Dimensionless central deflection of Al/(ZrO₂)-1 functionally graded square plate under uniform load (a/h=5，N_L=10，R=S=10)

板理论	n
板理论	0	0.5	1	2
文献[23]	0.1703	0.2232	0.2522	0.2827
文献[24]	0.1722	0.2403	0.2811	0.3221
FSDPT^[9]	0.1717	0.2319	0.2716	0.3121
TSDPT	0.1716	0.2313	0.2715	0.3140
SSDPT	0.1716	0.2312	0.2715	0.3140

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表 3 不同GPLs重量分数下FG-GRC板的无量纲中心挠度和R_w

Table 3. The dimensionless central deflection and R_w for different weight fractions of GPLs in FG-GRC plates

分布模式	板理论	f_G=0	f_G=0.5%	f_G=1.0%
UD	FSDPT	0.4549	0.1708(37.55%)	0.1052(23.13%)
	TSDPT	0.4549	0.1708(37.55%)	0.1052(23.13%)
	SSDPT	0.4549	0.1708(37.55%)	0.1052(23.13%)
FG-O	FSDPT	0.4549	0.2247(49.40%)	0.1495(32.86%)
	TSDPT	0.4549	0.2237(49.18%)	0.1487(32.69%)
	SSDPT	0.4549	0.2236(49.15%)	0.1487(32.69%)
FG-X	FSDPT	0.4549	0.1385(30.45%)	0.0818(17.98%)
	TSDPT	0.4549	0.1402(30.82%)	0.0832(18.29%)
	SSDPT	0.4549	0.1404(30.86%)	0.0833(18.31%)
FG-A	FSDPT	0.4549	0.1899(41.75%)	0.1242(27.30%)
	TSDPT	0.4549	0.1899(41.75%)	0.1241(27.28%)
	SSDPT	0.4549	0.1899(41.75%)	0.1241(27.28%)
注：表中括号内的值为R_w。

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表 4 简支各向同性板在单轴压缩和双轴压缩下的无量纲临界屈曲载荷

Table 4. Dimensionless critical buckling loads of simply supported isotropic plates in uniaxial compression and biaxial compression

板理论	单轴压缩(γ₁=−1，γ₂=0)		双轴压缩(γ₁=−1，γ₂=−1)
板理论	a/h=5	a/h=10	a/h=5	a/h=10
文献[10]	2.9513	3.4224	1.4756	1.7112
FSDPT^[9]	2.9498	3.4222	1.4749	1.7111
TSDPT	2.9512	3.4224	1.4756	1.7112
SSDPT	2.9524	3.4226	1.4762	1.7113

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表 5 FG-GRC板单轴压缩时的无量纲临界屈曲载荷和R_N

Table 5. Dimensionless critical buckling load and R_N for FG-GRC plate in uniaxial compression

分布模式	板理论	f_G
分布模式	板理论	0	0.2%	0.4%	0.6%	0.8%	1.0%	1.2%
UD	FSDPT^[9]	0.0035	0.0058	0.0082	0.0105	0.0128	0.0152	0.0175
	FSDPT^[9]	0.0035	(165.7%)	(234.3%)	(300.0%)	(365.7%)	(434.3%)	(500.0%)
	TSDPT	0.0035	0.0058	0.0082	0.0105	0.0128	0.0152	0.0175
	TSDPT	0.0035	(165.7%)	(234.3%)	(300.0%)	(365.7%)	(434.3%)	(500.0%)
	SSDPT	0.0035	0.0058	0.0082	0.0105	0.0128	0.0152	0.0175
	SSDPT	0.0035	(165.7%)	(234.3%)	(300.0%)	(365.7%)	(434.3%)	(500.0%)
FG-O	FSDPT^[9]	0.0035	0.0050	0.0064	0.0078	0.0093	0.0107	0.0121
	FSDPT^[9]	0.0035	(142.9%)	(182.9%)	(222.9%)	(265.7%)	(305.7%)	(345.7%)
	TSDPT	0.0035	0.0050	0.0064	0.0079	0.0093	0.0108	0.0122
	TSDPT	0.0035	(142.9%)	(182.9%)	(225.7%)	(265.7%)	(308.6%)	(348.6%)
	SSDPT	0.0035	0.0050	0.0064	0.0079	0.0093	0.0108	0.0122
	SSDPT	0.0035	(142.9%)	(182.9%)	(225.7%)	(265.7%)	(308.6%)	(348.6%)
FG-X	FSDPT^[9]	0.0035	0.0067	0.0099	0.0131	0.0163	0.0195	0.0227
	FSDPT^[9]	0.0035	(191.4%)	(282.9%)	(374.3%)	(465.7%)	(557.1%)	(648.6%)
	TSDPT	0.0035	0.0067	0.0098	0.0129	0.0160	0.0191	0.0222
	TSDPT	0.0035	(191.4%)	(280.0%)	(368.6%)	(457.1%)	(545.7%)	(634.3%)
	SSDPT	0.0035	0.0067	0.0098	0.0129	0.0160	0.0191	0.0222
	SSDPT	0.0035	(191.4%)	(280.0%)	(368.6%)	(457.1%)	(545.7%)	(634.3%)
注：表中括号内的值为R_N。

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表 6 FG-GRC板双轴压缩时的无量纲临界屈曲载荷和R_N

Table 6. Dimensionless critical buckling load and R_N for FG-GRC plate in biaxial compression

分布模式	板理论	f_G
分布模式	板理论	0	0.2%	0.4%	0.6%	0.8%	1.0%	1.2%
UD	FSDPT^[9]	0.0018	0.0029	0.0041	0.0053	0.0064	0.0076	0.0088
	FSDPT^[9]	0.0018	(161.1%)	(227.8%)	(294.4%)	(355.6%)	(422.2%)	(488.9%)
	TSDPT	0.0018	0.0029	0.0041	0.0053	0.0064	0.0076	0.0088
	TSDPT	0.0018	(161.1%)	(227.8%)	(294.4%)	(355.6%)	(422.2%)	(488.9%)
	SSDPT	0.0018	0.0029	0.0041	0.0053	0.0064	0.0076	0.0088
	SSDPT	0.0018	(161.1%)	(227.8%)	(294.4%)	(355.6%)	(422.2%)	(488.9%)
FG-O	FSDPT^[9]	0.0018	0.0025	0.0032	0.0039	0.0046	0.0053	0.0061
	FSDPT^[9]	0.0018	(138.9%)	(177.8%)	(216.7%)	(255.6%)	(294.4%)	(338.9%)
	TSDPT	0.0018	0.0025	0.0032	0.0039	0.0047	0.0054	0.0061
	TSDPT	0.0018	(138.9%)	(177.8%)	(216.7%)	(261.1%)	(300.0%)	(338.9%)
	SSDPT	0.0018	0.0025	0.0032	0.0039	0.0047	0.0054	0.0061
	SSDPT	0.0018	(138.9%)	(177.8%)	(216.7%)	(261.1%)	(300.0%)	(338.9%)
FG-X	FSDPT^[9]	0.0018	0.0034	0.0050	0.0066	0.0082	0.0097	0.0113
	FSDPT^[9]	0.0018	(188.9%)	(277.8%)	(366.7%)	(455.6%)	(538.9%)	(627.8%)
	TSDPT	0.0018	0.0033	0.0049	0.0065	0.0080	0.0096	0.0111
	TSDPT	0.0018	(183.3%)	(272.2%)	(361.1%)	(444.4%)	(533.3%)	(616.7%)
	SSDPT	0.0018	0.0033	0.0049	0.0065	0.0080	0.0096	0.0111
	SSDPT	0.0018	(183.3%)	(272.2%)	(361.1%)	(444.4%)	(533.3%)	(616.7%)
注：表中括号内的值为R_N。

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参考文献(25)

[1]	SONG M, KITIPORNCHAI S, YANG J. Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets [J]. Composite Structures, 2017, 159: 579 − 588. doi: 10.1016/j.compstruct.2016.09.070
[2]	YANG B, YANG J, KITIPORNCHAI S. Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity [J]. Meccanica, 2017, 52(10): 2275 − 2292. doi: 10.1007/s11012-016-0579-8
[3]	YAGHOOBI H, TAHERI F. Characterization of the vibration, stability and static responses of graphene-reinforced sandwich plates under mechanical and thermal loadings using the refined shear deformation plate theory [J]. Journal of Sandwich Structures & Materials, 2022, 24(1): 35 − 65.
[4]	WANG Z, MA L. Effect of Thickness Stretching on Bending and Free Vibration Behaviors of Functionally Graded Graphene Reinforced Composite Plates [J]. Applied Sciences, 2021, 11(23): 11362. doi: 10.3390/app112311362
[5]	WANG M, XU Y G, QIAO P, et al. Buckling and free vibration analysis of shear deformable graphene-reinforced composite laminated plates [J]. Composite Structures, 2022, 280: 114854. doi: 10.1016/j.compstruct.2021.114854
[6]	LU L, WANG S, LI M, et al. Free vibration and dynamic stability of functionally graded composite microtubes reinforced with graphene platelets [J]. Composite Structures, 2021, 272: 114231. doi: 10.1016/j.compstruct.2021.114231
[7]	AL-FURJAN M S H, HABIBI M, GHABUSSI A, et al. Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory [J]. Engineering Structures, 2021, 228: 111496. doi: 10.1016/j.engstruct.2020.111496
[8]	PHUNG-VAN P, LIEU Q X, FERREIRA A J M, et al. A refined nonlocal isogeometric model for multilayer functionally graded graphene platelet-reinforced composite nanoplates [J]. Thin-Walled Structures, 2021, 164: 107862. doi: 10.1016/j.tws.2021.107862
[9]	SONG M, YANG J, KITIPORNCHAI S. Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets [J]. Composites Part B:Engineering, 2018, 134: 106 − 113. doi: 10.1016/j.compositesb.2017.09.043
[10]	THAI C H, FERREIRA A, TRAN T D, et al. Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation [J]. Composite Structures, 2019, 220: 749 − 759. doi: 10.1016/j.compstruct.2019.03.100
[11]	SONG M, YANG J, KITIPORNCHAI S, et al. Buckling and postbuckling of biaxially compressed functionally graded multilayer graphene nanoplatelet-reinforced polymer composite plates [J]. International Journal of Mechanical Sciences, 2017, 131: 345 − 355. doi: 10.1016/j.ijmecsci.2017.07.017
[12]	ZHAO Z, FENG C, WANG Y, et al. Bending and vibration analysis of functionally graded trapezoidal nanocomposite plates reinforced with graphene nanoplatelets (GPLs) [J]. Composite Structures, 2017, 180: 799 − 808. doi: 10.1016/j.compstruct.2017.08.044
[13]	YANG B, KITIPORNCHAI S, YANG Y, et al. 3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates [J]. Applied Mathematical Modelling, 2017, 49: 69 − 86. doi: 10.1016/j.apm.2017.04.044
[14]	YANG B, MEI J, CHEN D, et al. 3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates [J]. Composite Structures, 2018, 184: 1040 − 1048. doi: 10.1016/j.compstruct.2017.09.086
[15]	牛牧华, 马连生. 基于物理中面FGM梁的非线性力学行为[J]. 工程力学, 2011, 28(6): 219 − 225. NIU Muhua, MA Liansheng. Nonlinear mechanical behaviors of FGM beams based on the physical neutral surface [J]. Engineering Mechanics, 2011, 28(6): 219 − 225. (in Chinese)
[16]	马连生, 顾春龙. 剪切可变形梁热过屈曲解析解[J]. 工程力学, 2012, 29(2): 172 − 176. MA Liansheng, GU Chunlong. Exact solutions for thermal post-buckling of shear deformable beams [J]. Engineering Mechanics, 2012, 29(2): 172 − 176. (in Chinese)
[17]	邓先琪, 苏成, 马海涛. 功能梯度梁静力与动力分析的格林函数法[J]. 工程力学, 2020, 37(9): 248 − 256. doi: 10.6052/j.issn.1000-4750.2019.10.0615 DENG Xianqi, SU Cheng, MA Haitao. Green’s function method for static and dynamic analysis of functionally graded beams [J]. Engineering Mechanics, 2020, 37(9): 248 − 256. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.10.0615
[18]	赵伟东, 高士武, 黄永玉. 热环境中的功能梯度扁球壳非线性稳定性分析[J]. 工程力学, 2020, 37(8): 202 − 212. doi: 10.6052/j.issn.1000-4750.2019.09.0541 ZHAO Weidong, GAO Shiwu, HUANG Yongyu. Nonlinear stability analysis of functionally graded shallow spherical shells in thermal environments [J]. Engineering Mechanics, 2020, 37(8): 202 − 212. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.09.0541
[19]	刘涛, 汪超, 刘庆运, 等. 基于等几何方法的压电功能梯度板动力学及主动振动控制分析[J]. 工程力学, 2020, 37(12): 228 − 242. doi: 10.6052/j.issn.1000-4750.2020.04.0266 LIU Tao, WANG Chao, LIU Qingyun, et al. Analysis for dynamic and active vibration control of piezoelectric functionally graded plates based on isogeometric method [J]. Engineering Mechanics, 2020, 37(12): 228 − 242. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.04.0266
[20]	REDDY J N. Analysis of functionally graded plates [J]. International Journal for Numerical Methods in Engineering, 2000, 47: 663 − 684.
[21]	TOURATIER M. An efficient standard plate theory [J]. International Journal of Engineering Science, 1991, 29(8): 901 − 916. doi: 10.1016/0020-7225(91)90165-Y
[22]	ZHAO S, ZHAO Z, YANG Z, et al. Functionally graded graphene reinforced composite structures: A review [J]. Engineering Structures, 2020, 210: 110339.
[23]	NGUYEN-XUAN H, TRAN L V, NGUYEN-THOI T, et al. Analysis of functionally graded plates using an edge-based smoothed finite element method [J]. Composite Structures, 2011, 93(11): 3019 − 3039. doi: 10.1016/j.compstruct.2011.04.028
[24]	LEE Y Y, ZHAO X, LIEW K M. Thermoelastic analysis of functionally graded plates using the element-free kp-Ritz method [J]. Smart Materials and Structures, 2009, 18(3): 35007. doi: 10.1088/0964-1726/18/3/035007
[25]	LEISSA A W. Conditions for laminated plates to remain flat under inplane loading [J]. Composite Structures, 1986, 6(4): 261 − 270. doi: 10.1016/0263-8223(86)90022-X