﻿ 基于单胞有限元的波纹板等效刚度特性<sup>*</sup>
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Equivalent stiffness property of dimpled sheet based on unit cell finite element
WANG Yuan, ZANG Yong, GUAN Ben, QIN Qin
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
Received: 2017-11-13; Accepted: 2017-12-29; Published online: 2018-02-10 14:02
Foundation item: the Fundamental Research Funds for the Central Universities (FRF-TP-16-010A3); Intelligent Manufacturing Comprehensive Standardized and New Model Program of Ministry of Industry and Information 2016
Corresponding author. GUAN Ben, E-mail:guanben@ustb.edu.cn
Abstract: To obtain the macroscopic stiffness property of the dimpled sheet and its relationship with surface morphology structure parameters, the unit cell finite element method finite element method was used to analyze the equivalent stiffness property of the dimpled sheet. Firstly, based on the periodic boundary conditions of unit cell finite element method, the equivalent stiffness of the dimpled sheet with periodic check board pattern was calculated. And then the equivalent stiffness of a specific dimpled sheet was analyzed and verified with the unit cell finite element method. Finally, the effects of structural parameters on the equivalent stiffness of the dimpled sheet were discussed. The analysis results show that with the unit cell finite element method the equivalent stiffness of dimpled sheet can be obtained. The dimpled sheet has higher bending stiffness but lower tension and shear stiffness compared with the base flat sheet. When base sheet's thickness is fixed, the bending stiffness increases as the thickness of dimpled sheet increases, while the tension stiffness and the shear stiffness decrease. As the periodic distance of dimpled sheet increases, the tension stiffness and the shear stiffness increase, while the bending stiffness decreases.
Key words: dimpled sheet     unit cell structure     finite element     periodic boundary condition     equivalent stiffness property

1 单胞有限元模型及等效刚度特性 1.1 单胞的选取及变形假设条件

 图 1 波纹板单胞的选取及结构参数 Fig. 1 Selection and structural parameters of dimpled sheet unit cell

1) 波纹板的厚度远小于宏观结构的长度和宽度。

2) 波纹板的弹性弯曲问题符合小挠度理论。

3) 波纹板在宏观上呈现出均质各向异性的特点。

4) 单胞结构具有相同的应力场和应变场。

5) 波纹板的中面与其几何中性层重合，即图 1(c)中的虚线位置。

1.2 单胞周期性边界条件的有限元实现

1.2.1 变形协调条件的实现

 图 2 单胞结构轮廓示意图 Fig. 2 Schematic diagram of unit cell profile

 (1)

 应变 u(p, y, z)-u(0, y, z) v(p, y, z)-v(0, y, z) w(p, y, z)-w(0, y, z) u(x, p, z)-u(x, 0, z) v(x, p, z)-v(x, 0, z) w(x, p, z)-w(x, 0, z) =1 p 0 0 0 0 0 =1 0 0 0 0 p 0 =1 0 p/2 0 p/2 0 0 =1 pz 0 -p2/2 0 0 0 =1 0 0 0 0 pz -p2/2 =1 0 pz/2 -py/2 pz/2 0 -px/2

1.2.2 应力连续条件的实现

 (2)

1.3 等效拉伸刚度和等效弯曲刚度特性的计算

1.2节实现了波纹板单胞结构有限元模型的建立，本节将基于单胞有限元模型的输出结果建立板材等效刚度特性的计算方法。根据复合材料力学[20]中对刚度特性的定义，波纹板等效刚度特性可写为

 (3)

 (4)

1.4 等效剪切刚度的计算

 (5)
 图 3 波纹板单胞结构组成的梁结构 Fig. 3 Cantilever plate with unit cells of dimpled sheet

 (6)
2 波纹板等效刚度特性的计算及验证

2.1 波纹板形貌特征的提取

 图 4 波纹板单胞形貌的提取 Fig. 4 Unit cell morphology extraction of dimpled sheet

2.2 等效刚度特性计算

2.2.1 等效拉伸刚度和等效弯曲刚度

 图 5 单胞结构在6种变形情况的应力云图 Fig. 5 Stress contours of unit cell in six deformation cases

 (7)
 (8)

 刚度 波纹板 基础薄板 波纹板与基础薄板刚度比值 A11/(N·mm-1) 24 069 38 716 0.622 A12/(N·mm-1) -162 12 776 -0.013 A22/(N·mm-1) 24 069 38 716 0.622 A66/(N·mm-1) 6 558 12 970 0.506 D11/(N·mm) 1 015 806 1.26 D12/(N·mm) 517 266 1.944 D22/(N·mm) 1 015 806 1.26 D66/(N·mm) 1 165 270 4.315

 图 6 单胞在单向拉伸时的应力分布 Fig. 6 Stress distribution of unit cell under uniaxial tension

2.2.2 等效剪切刚度

L=80 mm，P0=1×10-3 MPa时，由式(6)计算得到等效剪切刚度，梁结构自由端挠度值w(L)0可由有限元仿真结果得到，根据已经计算得到的D11可以计算得到等效剪切刚度A44=A55=88N/mm。

2.3 波纹板刚度特性验证

 图 7 等效模型的验证 Fig. 7 Verification of equivalent model

3 结构参数对等效刚度特性的影响

3.1 不同结构参数对波纹板形貌的影响

 图 8 不同结构参数波纹板单胞结构的形貌 Fig. 8 Morphology of unit cell of dimpled sheet with different structural parameters
3.2 相对厚度对刚度特性的影响

 图 9 相对厚度对等效刚度特性的影响 Fig. 9 Effect of h/t on equivalent stiffness property
3.3 相对周期间距对刚度特性的影响

 图 10 相对厚度对刚度特性的影响 Fig. 10 Effect of p/t on equivalent stiffness property
4 结论

1) 通过建立波纹板单胞结构周期性边界条件及应变载荷的计算方法，得到了波纹板的等效刚度特性。该方法计算效率高、数值准确，可以实现以波纹板为基础材料的复杂构件位移响应的计算。

2) 对典型形貌的波纹板等效刚度特性的分析表明，普通薄板经过滚压加工后截面形貌趋于复杂，波纹板相对厚度增加导致其弯曲刚度提升，而在受拉伸载荷时的应力分布不均导致其拉伸刚度降低。

3) 波纹板的相对厚度和波纹相对周期间距都会影响其等效刚度特性。相对厚度的增加会降低拉伸刚度和剪切刚度，提升波纹板的弯曲刚度；波纹相对周期间距的增加会提升拉伸刚度和剪切刚度，降低波纹板的弯曲刚度。

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#### 文章信息

WANG Yuan, ZANG Yong, GUAN Ben, QIN Qin

Equivalent stiffness property of dimpled sheet based on unit cell finite element

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(6): 1230-1238
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0698