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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。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.
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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)
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
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
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
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
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
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
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 x方向 y方向 x方向 y方向 P1 0 0 0 0 P2 αux 0 αk11ux/l αk21ux/l P3 αux αuy $\alpha \left( {{k_{11}}{u_x} + {k_{12}}{u_y}} \right)/l$ $\alpha \left( {{k_{21}}{u_x} + {k_{22}}{u_y}} \right)/l$ P4 0 αuy αk12uy/l αk22uy/l 
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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 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
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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，σY/MPa 鋼材 210 0.3 280
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表 4 模型所用面板單層材料力學參數
Table 4. Material properties of single layer
Ex/GPa Ey/GPa Gxy/GPa μxy μyz Tx/MPa Ty/MPa Sxy/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 ux=0.033 uy=?0.01 2 對稱約束 對稱約束 對稱約束 ux=?0.01 uy=0.0208
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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，σY/MPa 鋼材 200 0.3 360 鋁合金 72 0.33 100
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中文字幕在线观看表 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 ux=0.01 uy=?0.001998 2 對稱約束 對稱約束 對稱約束 ux=?0.003732 uy=0.01
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參考文獻
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