基于分层法的功能梯度三明治壳线性弯曲无网格分析

陈卫; 汤智宏; 彭林欣

doi:10.21656/1000-0887.440262

基于分层法的功能梯度三明治壳线性弯曲无网格分析

doi: 10.21656/1000-0887.440262

陈卫^1,,
汤智宏¹,
彭林欣^2, ,

1.
南华大学土木工程学院，湖南衡阳 421001
2.
广西大学土木建筑工程学院，南宁 530004

基金项目:

国家自然科学基金 12162004

国家自然科学基金 11562001

详细信息

作者简介:
陈卫(1991—)，男，讲师，博士，硕士生导师(E-mail: chenwei@usc.edu.cn)

通讯作者:
彭林欣(1977—)，男，教授，博士，博士生导师(通讯作者. E-mail: penglx@gxu.edu.cn)

中图分类号: O342
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- 文章访问数: 213
- HTML全文浏览量: 86
- PDF下载量: 31
- 被引次数: 0
出版历程
- 收稿日期: 2023-08-29
- 修回日期: 2023-12-15
- 刊出日期: 2024-05-01

Linear Bending Analysis of Functionally Graded Sandwich Shells With the Meshless Method Based on the Layer-Wise Theory

1.
School of Civil Engineering, University of South China, Hengyang, Hunan 421001, P.R.China
2.
School of Civil Engineering and Architecture, Guangxi University, Nanning 530004, P.R.China

摘要

摘要: 基于3D连续壳理论和一阶剪切变形理论，采用分层法，提出了一种求解功能梯度三明治壳线性弯曲问题的移动最小二乘无网格法. 通过映射技术，将随动坐标系上的二维无网格节点信息映射到三维壳中，并在随动坐标系上形成移动最小二乘近似的形函数. 因基于3D连续壳理论的壳数值解答无法像特定壳一样给出其厚度方向的显式表达式，该文将功能梯度三明治材料壳结构中材料参数变化的部分划分成若干层，得到每层的材料参数为常数. 利用最小势能原理，推导出了功能梯度三明治壳线性弯曲的无网格控制方程. 通过引入一个厚度方向的线性变换，使得每层厚度方向的Gauss积分均在-1至1区间内，不违背一阶剪切变形理论. 采用完全转化法施加本质边界条件. 以功能梯度三明治板、柱壳、双曲扁壳经典几何形状壳为例，讨论了不同梯度系数、径厚比和曲率半径等对数值结果的影响，并将计算结果与文献解对比. 研究表明，该方法在求解不同形状的功能梯度三明治壳线性弯曲问题时，具有收敛性好、计算精度高的特点.
- 分层法 /
- 无网格法 /
- 映射技术 /
- 功能梯度三明治壳 /
- 线性弯曲
Abstract: Based on the 3D continuous shell theory and the 1st-order shear deformation theory, a moving least squares meshless method was proposed for solving the linear bending problem of functionally graded sandwich shells with the layered method. With the mapping technology, the 2D meshless node information on the convected coordinate system was mapped on the 3D shell, and the shape function of moving least squares (MLS) approximation was formed on the convected coordinate system. Due to the inability of shell numerical solutions based on the 3D continuous shell theory to provide an explicit expression for the thickness direction of a specific shell, the portion of material parameter changes in the functionally graded sandwich material shell structure was divided into several layers, and the material parameters of each layer were obtained as constants. The governing meshless equation for linear bending of functionally graded sandwich shells was derived under the principle of minimum potential energy. Through introduction of a linear transformation in the thickness direction, the Gaussian integral in the thickness direction of each layer was bounded within the range of -1 to 1, without violation of the 1st-order shear deformation theory (FSDT). The essential boundary conditions were employed with the complete transformation method. Finally, the effects of different gradient coefficients, diameter to thickness ratios, and curvature radii on numerical results were discussed through examples of functionally graded sandwich plates, cylindrical shells, and hyperbolic shallow shells with classical geometric shapes. The calculated results were compared with the literature solutions. The results show that, the proposed method has the characteristics of good convergence and high computation accuracy in solving linear bending problems of functional graded sandwich shells with different shapes.
- layer-wise method /
- meshless method /
- mapping technology /
- functionally graded sandwich shell /
- linear bending

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图 1 曲壳无网格几何模型及映射技术

Figure 1. The meshless geometric model for the curved shell and the mapping technique

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图 2 3类功能梯度壳

Figure 2. Three types of functionally graded shells

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图 3 不同节点数下方板挠度分析结果

Figure 3. Analysis results of plate deflections under different node numbers

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图 4 不同宽厚比下方板中点收敛性分析

Figure 4. Convergence analysis of the midpoint of the plate under different width-to-thickness ratios

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图 5 功能梯度方板受正弦荷载示意图

Figure 5. Schematic diagram of a functionally graded square plate under uniformly distributed load

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图 6 功能梯度柱壳示意图

Figure 6. Schematic diagram of a functionally graded cylindrical shell

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图 7 双曲壳几何模型

Figure 7. Geometry of a doubly-curved functionally graded shell

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图 8 梯度指数为2和R_x/a=5的四边简支铝/氧化铝不同形状壳挠度云图(EFG)

Figure 8. Deflection neghogram for the simply supported Al/Al₂O₃ different shape shells with gradient indices p=2 and R_x/a=5(EFG)

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表 1 功能梯度材料组成元素

Table 1. Properties of the FGM components

property	Al	ceramic
property	Al	Al₂O₃	ZrO₂
E/GPa	70	380	200
ν	0.3	0.3	0.3

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表 2 分层数及节点数对铝/氧化铝方板中点归一化挠度w¹的影响(p=10，Type B)

Table 2. Effects of the numbers of layers and nodes on central deflectionw¹ of the Al/Al₂O₃ square plate (p=10, Type B)

layers	present
layers	5×5	9×9	13×13
1/4/1	0.829 7	0.826 1	0.826 1
1/8/1	0.864 8	0.861 0	0.861 0
1/12/1	0.861 9	0.858 0	0.858 0
1/16/1	0.860 8	0.857 0	0.857 0

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表 3 不同梯度指数p下，铝/氧化铝方板中点归一化挠度w¹(Type B)

Table 3. Normalized central deflection w¹ for the Al/Al₂O₃ square plate with different gradient indices p(Type B)

method	p
method	0	0.5	1	4	10
Nguyen et al.^[35](ITSDT)	0.374 4	0.524 5	0.634 5	0.833 1	0.880 7
Neves et al.^[34](quasi-3D)	0.371 1	0.523 8	0.630 5	0.819 9	0.864 5
present	0.374 9	0.523 1	0.632 2	0.819 8	0.858 0

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表 4 不同层状厚度与不同梯度指数p下，铝/氧化锆方板中点归一化挠度w² (Type C)

Table 4. Normalized central deflection w² for the Al/ZrO₂ square plate with different layer thicknesses and gradient indices p(Type C)

p	theory	1-0-1	2-1-2	1-1-1	2-2-1	1-2-1
1	Nguyen et al. ^[35](ITSDT)	0.323 5	0.306 2	0.291 9	0.280 8	0.270 9
	Zenkour^[36](FSDT)	0.324 8	0.307 5	0.293 0	0.281 7	0.271 7
	Neves et al. ^[34](quasi-3D)	-	0.307 0	0.292 9	0.282 0	0.272 2
	present	0.324 3	0.307 1	0.292 7	0.281 4	0.271 5
5	Nguyen et al. ^[35](ITSDT)	0.409 1	0.391 7	0.371 3	0.349 5	0.334 7
	Zenkour^[36](FSDT)	0.411 2	0.394 2	0.373 6	0.351 2	0.336 3
	Neves et al. ^[34](quasi-3D)	-	0.390 5	0.370 5	0.349 0	0.334 7
	present	0.410 8	0.393 7	0.372 2	0.350 9	0.336 0
10	Nguyen et al. ^[35](ITSDT)	0.417 5	0.403 9	0.385 4	0.362 0	0.348 2
	Zenkour^[36](FSDT)	0.419 2	0.406 6	0.387 9	0.364 0	0.350 0
	Neves et al. ^[34](quasi-3D)	-	0.402 6	0.384 3	0.361 2	0.348 0
	present	0.418 9	0.406 2	0.387 5	0.363 6	0.349 7

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表 5 分层数及节点数对四边简支铝/氧化锆柱壳中点挠度w³的影响(p=1，Type A)

Table 5. Effects of the numbers of layers and nodes on central deflection w³ of the simply supported Al/ZrO₂ cylindrical shell (p=1, Type A)

number of layers	present
number of layers	5×5	9×9	13×13	17×17
4	0.060 76	0.060 07	0.060 10	0.060 12
8	0.060 97	0.060 29	0.060 30	0.060 34
12	0.061 01	0.060 33	0.060 36	0.060 38

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表 6 不同边界条件与不同梯度指数p下，铝/氧化锆柱壳中点归一化挠度w³(Type A)

Table 6. Normalized central deflection w³ for the Al/ZrO₂ cylindrical shell with different boundary conditions and gradient indices p(Type A)

B.Cs	method	p
B.Cs	method	0	0.2	0.5	1	2	5
SSSS	kp-Ritz^[24]	0.042 67	0.048 07	0.054 25	0.060 72	0.066 58	0.072 35
SSSS	present	0.042 47	0.047 69	0.053 80	0.060 30	0.066 37	0.072 46
CCCC	kp-Ritz^[24]	0.013 47	0.015 16	0.017 11	0.019 15	0.021 02	0.022 89
CCCC	present	0.013 71	0.015 39	0.017 36	0.019 47	0.021 45	0.023 47

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表 7 不同径厚比R/h与不同梯度指数p下，四边简支铝/氧化锆柱壳中点归一化挠度w³(Type A)

Table 7. Normalized central deflection w³ for the simply supported Al/ZrO₂ cylindrical shell with different radius-to-thickness ratios R/h and gradient indices p(Type A)

p	R/h	method
p	R/h	FSDT^[37]	CST^[37]	analytical^[38]	kp-Ritz^[24]	present
1	50	0.004 24	0.004 08	0.004 30	0.004 28	0.004 25
	100	0.060 56	0.060 02	0.060 91	0.060 72	0.060 30
	200	0.725 84	0.724 70	0.727 10	0.728 30	0.722 21
2	50	0.004 64	0.004 46	0.004 70	0.004 69	0.004 67
	100	0.066 40	0.065 78	0.066 79	0.066 78	0.066 37
	200	0.803 07	0.801 73	0.805 60	0.805 70	0.801 18

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表 8 分层数及节点数对四边简支Type B铝/氧化铝柱壳中点挠度w_c×10^-11的影响，p=1，R/h=1 000(单位: m)

Table 8. Effects of the numbers of layers and nodes on central deflection w_c×10^-11 for the simply supported Al/Al₂O₃ cylindrical shell, p=1, R/h=1 000, Type B (unit: m)

layers	present
layers	31×3	51×5	71×7	91×9
1/4/1	3 946.5	4 155.7	4 172.5	4 173.7
1/8/1	3 950.7	4 159.2	4 176.2	4 177.1
1/12/1	3 951.5	4 159.9	4 176.9	4 177.3

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表 9 不同径厚比R/h与不同梯度指数p下，四边简支Type B铝/氧化铝柱壳中点挠度w_c×10^-11(单位: m)

Table 9. Normalized central deflection w_c×10^-11 for the simply supported Al/Al₂O₃ cylindrical shell with different radius-to-thickness ratios R/h and gradient indices p, Type B (unit: m)

p	method	R/h
p	method	4	10	100	1 000
1	CST^[39]	0.004 6	0.066 1	55.428	4 223.3
	FSDT^[39]	0.065 9	0.209 9	56.530	4 224.5
	present	0.064 8	0.208 7	56.420	4 176.2
5	CST^[39]	0.006 1	0.086 4	73.651	6 578.3
	FSDT^[39]	0.102 0	0.312 9	75.437	6 582.7
	present	0.098 9	0.310 5	75.120	6 579.8

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表 10 分层数及节点数对四边简支铝/氧化铝柱壳中点挠度w⁴的影响

Table 10. Effects of the numbers of layers and nodes on central deflection w⁴ of the simply supported Al/Al₂O₃ cylindrical shell

layers	present
layers	5×5	9×9	13×13	17×17
4	8.873 9	8.762 6	8.759 9	8.760 0
8	9.010 0	8.897 0	8.894 3	8.894 4
12	9.036 3	8.923 0	8.920 2	8.920 4
16	9.045 6	8.932 1	8.929 4	8.929 5

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表 11 不同梯度指数p下，四边简支铝/氧化铝不同形状壳中点归一化挠度w⁴

Table 11. Normalized central deflectionw⁴ for the simply supported Al/Al₂O₃ different shells with various gradient indices p

shell type	method	p
shell type	method	0	1	5	10	∞
cylindrical shell (R_x/a=5, R_y/b=∞)	ESDT^[37]	4.526 5	8.964 8	13.942 0	15.460 0	24.572 0
	FSDT^[37]	4.492 1	8.907 2	13.683 0	15.152 0	24.385 0
	present	4.525 5	8.920 2	13.827 4	15.401 0	24.567 1
spherical shell (R_x/a=5, R_y/b=5)	ESDT^[37]	4.157 1	8.119 3	12.816 0	14.333 0	22.567 0
	FSDT^[37]	4.128 5	8.072 9	12.601 0	14.071 0	22.412 0
	present	4.161 1	8.073 9	12.699 3	14.268 6	22.588 9
hyperbolic paraboloid shell (R_x/a=5, R_y/b=-5)	ESDT^[37]	4.664 6	9.286 8	14.362 0	15.876 0	25.322 0
	FSDT^[37]	4.627 8	9.224 6	14.086 0	15.550 0	25.122 0
	present	4.653 9	9.239 8	14.250 7	15.815 4	25.263 9
elliptical paraboloid shell (R_x/a=5, R_y/b=7.5)	ESDT^[37]	4.300 3	8.444 2	13.253 0	14.772 0	23.344 0
	FSDT^[37]	4.269 4	8.393 7	13.021 0	14.493 0	23.177 0
	present	4.303 0	8.399 1	13.136 1	14.709 9	23.359 2

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