基于均勻化理論的復合材料安定性分析方法

秦方; 張樂樂; 黃松華; 陳耕

doi:10.13374/j.issn2095-9389.2019.01.06.001

基于均勻化理論的復合材料安定性分析方法

doi: 10.13374/j.issn2095-9389.2019.01.06.001

秦方¹,
張樂樂^1, ,,
黃松華¹,
陳耕²

1.
北京交通大學機械與電子控制工程學院，北京 100044
2.
德國亞琛工業大學機械工程材料應用研究所，亞琛 52062

基金項目: 國家自然科學基金資助項目（51475036）

詳細信息

通訊作者:
E-mail：llzhang1@bjtu.edu.cn

中圖分類號: O344.5
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出版歷程
- 收稿日期: 2019-01-06
- 刊出日期: 2019-12-01

Shakedown analysis method for composites based on homogenization theory

1.
School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China
2.
Institute for Materials Applications in Mechanical Engineering, RWTH Aachen University, Aachen 52062, German

More Information

Corresponding author: E-mail: llzhang1@bjtu.edu.cn

摘要

摘要: 周期性非均質復合材料具有微觀結構特征，需要均勻化理論進行宏觀和微觀的多尺度分析來研究其性能表現。針對其耐久強度性能，應用塑性極限安定下限定理，特別分析了其在長期交變載荷下的安定狀態。結合工程應用目標，提出一種全新的代表性單元邊界條件，結合圓錐二次優化算法進行數值計算，可以從材料微結構和組分性能出發，經過彈性應力場求解確定位移邊界載荷數值，最終由優化求解得到復合材料板材的面內塑性性能容許域。所求得的應力域以單向應力為基，可根據結構宏觀的單向應力狀態變化幅值直接進行安定狀態與否的判定。通過文中的多個算例，驗證了所編寫的軟件及計算流程的可行性及數值準確性，展示了該方法在工程模型中的應用場合和工程實踐意義。
- 復合材料 /
- 極限安定分析 /
- 均勻化理論 /
- 代表性單元 /
- 圓錐二次優化
Abstract: Direct methods of plastic analysis are widely used in composites analysis to determine material strength for safety assessment or lightweight optimization design. Multi-scale processing of periodic heterogeneous composite material is needed due to its existing of microstructure. The standard method is to determine the macroscopic properties from the calculation results of microcosmic representative volume elements (RVEs) by using the homogenization theory. However, in current practice, there are some disadvantages of transforming the micro strain domain to the macro stress shakedown domain when considering multiple external loads. The domain cannot fully demonstrate the shakedown condition, and it is impossible to evaluate a known loading combination only from the knowledge of whether the load leads to the shakedown state. To overcome this disadvantage, a new comprehensive approach was proposed to enhance endurance limit strength of composites under variable loads for long term. Considering the example of in-plane strength analysis, for microcosmic RVEs, a new set of boundary condition was defined in the form of uniform strain. The boundary condition was derived from the elastic response under unit loads by using Hook’s law and stiffness matrix. The resulting elastic stress field was used later for plastic shakedown analysis. Based on the lower bound theorem of plastic mechanics, optimization programming for load factor was performed, and after proper mathematical reformulation, the conic quadratic optimization problem could be solved efficiently. Macro-stress shakedown domain can be obtained after scale-transformation of the RVE results. The bases of this stress domain are unidirectional stress in geometry space. The stress amplitude of a structure can be evaluated by this domain for determining the shakedown state in a simple and practical manner. Further, changes in the boundary condition of RVE do not affect the limit and elastic analysis. Finally, few numerical examples were presented for verification and illustration. This approach can be expanded to three dimensions and employed for more complex structures.
- composite material /
- limit and shakedown analysis /
- homogenization theory /
- representative volume element /
- conic quadratic optimization

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圖 1 RVE邊界條件示意圖

Figure 1. Boundary condition of RVE

下載: 全尺寸圖片幻燈片

圖 2 微觀位移載荷域(a) 和對應宏觀應力域(b) 對應示意圖

Figure 2. Displacement load in micro-scale (a) and the corresponding stress domain in macro-scale (b)

下載: 全尺寸圖片幻燈片

圖 3 以f、g函數為基的微觀位移載荷域

Figure 3. Micro-scale displacement load with function f, g as bases

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圖 4 圓孔方板算例. (a)模型; (b)安定計算結果

Figure 4. Plate with hole: (a) model; (b) shakedown results

下載: 全尺寸圖片幻燈片

圖 5 層合板模型. (a)層合結構示意圖; (b)纖維角度定義

Figure 5. Lamination model: (a) laminated structure; (b) orientation angle of fiber

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圖 6 層合結構分析結果. (a)微觀容許位移場; (b)宏觀容許應力場

Figure 6. Analysis results: (a) micro-scale admissible displacement domain; (b) macro-scale admissible stress domain

下載: 全尺寸圖片幻燈片

圖 7 多孔薄壁材料示意圖. (a)結構模型；(b)微觀模型；(c)RVE模型

Figure 7. Sketch maps of porous material: (a) structure model; (b) microscopic model; (c) RVE model

下載: 全尺寸圖片幻燈片

圖 8 所計算多孔材料宏觀容許應力域結果對比

Figure 8. Comparison of macro-scale admissible stress domain of porous material

下載: 全尺寸圖片幻燈片

表 1 由微觀位移載荷域推導宏觀應力域

Table 1. Derivation of macro-scale stress domain from micro-scale displacement boundary

點	微觀位移載荷域		宏觀應力域
點	x方向	y方向	x方向	y方向
P₁	0	0	0	0
P₂	αu_x	0	αk₁₁u_x/l	αk₂₁u_x/l
P₃	αu_x	αu_y	$\alpha \left( {{k_{11}}{u_x} + {k_{12}}{u_y}} \right)/l$	$\alpha \left( {{k_{21}}{u_x} + {k_{22}}{u_y}} \right)/l$
P₄	0	αu_y	αk₁₂u_y/l	αk₂₂u_y/l

下載: 導出CSV

表 2 由宏觀應力域推導對應的微觀位移載荷域

Table 2. Derivation of micro-scale displacement boundary from macro-scale stress domain

點	宏觀應力域		微觀位移載荷域
點	x方向	y方向	x方向	y方向
P₁	0	0	0	0
P₂	Σ₁	0	s₁₁Σ₁l	s₂₁Σ₁l
P₃	Σ₁	Σ₂	s₁₁Σ₁l+s₁₂Σ₂l	s₂₁Σ₁l+s₂₂Σ₂l
P₄	0	Σ₂	s₁₂Σ₂l	s₂₂Σ₂l

下載: 導出CSV

表 3 圓孔方板模型幾何和材料參數

Table 3. Geometry and material parameters of plate with hole

長度，l/mm	孔徑，D/mm	厚度，h/mm	D/l
100	20	2	0.2

材料	彈性模量，E/MPa	泊松比，υ	屈服強度，σ_Y/MPa
鋼材	210	0.3	280

下載: 導出CSV

表 4 模型所用面板單層材料力學參數

Table 4. Material properties of single layer

E_x/GPa	E_y/GPa	G_xy/GPa	μ_xy	μ_yz	T_x/MPa	T_y/MPa	S_xy/MPa
146.9	11.4	6.18	0.3	0.4	1370.6	66.5	133.8

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表 5 修改后層合板RVE邊界條件

Table 5. Modified boundary condition of lamination RVE mm

單位載荷編號	x-0-y平面	x-0-z平面	y-0-z平面	平面x=10	平面y=10
1	對稱約束	對稱約束	對稱約束	u_x=0.033	u_y=?0.01
2	對稱約束	對稱約束	對稱約束	u_x=?0.01	u_y=0.0208

下載: 導出CSV

表 6 多孔薄壁材料參數

Table 6. Material parameters of porous materials

材料	彈性模量，E/GPa	泊松比，υ	屈服強度，σ_Y/MPa
鋼材	200	0.3	360
鋁合金	72	0.33	100

下載: 導出CSV

表 7 修改后多孔材料RVE邊界條件

Table 7. Modified boundary condition of porous material RVE mm

單位載荷編號	x-0-y平面	x-0-z平面	y-0-z平面	x=4平面	y=3平面
1	對稱約束	對稱約束	對稱約束	u_x=0.01	u_y=?0.001998
2	對稱約束	對稱約束	對稱約束	u_x=?0.003732	u_y=0.01

下載: 導出CSV

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