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摘要: 采用經典彈性力學方法建立了金屬層合板翹曲解析計算力學模型，獲得了厚度方向不均勻延伸與板形翹曲之間的定量關系；并分別建立了在線和離線兩種狀態下金屬層合板翹曲變形的有限元數值模擬模型，對解析計算力學模型進行了驗證；在此基礎上，揭示了金屬層合板產生板形翹曲缺陷的力學根源以及各因素對金屬層合板板形翹曲缺陷演變的影響規律，同時對比分析了雙層和三層結構層合板與均質板的翹曲變形差異以及銅/碳鋼層合板與不銹鋼/碳鋼層合板二者之間的翹曲變形差異。研究表明，金屬層合板翹曲高度與延伸差、厚度比呈正比關系，與厚度呈反比關系，且基層與覆層的切變模量相差越大，厚度比對金屬層合板翹曲變形的影響越大。基于數值模型，模擬研究了層合板在理想均勻分布的初始溫度下，歷經去應力退火過程時，其板形翹曲的變形行為及規律，并與均質板進行比較。最后，在工業生產現場取樣已翹曲層合板，通過測量其彎曲變形量進而反求其初始延伸差，驗證了解析計算力學模型的準確性。Abstract: The layered characteristics of the material in the thickness direction of the metal laminate make it more prone to uneven plastic extension during the thinning, rolling, flattening, and straightening process, resulting in plate-shaped warpage defects and cause the plate-shaped warpage of the metal laminate. The behavior is significantly different from that of a homogeneous metal plate. In this paper, the classical elastic mechanics method was used to establish an analytical computational mechanical model for the warpage of the metal laminate, and the quantitative relationship between the uneven extension in the thickness direction and the warpage of the plate shape was obtained; the online and offline states of the metal laminate were established, respectively. The finite element numerical simulation model of warpage deformation validated the analytical computational mechanics model; based on this, it revealed the mechanical roots of the shape warping defects of metal laminates and the effect of various factors on the shape warpage defects of metal laminates. The influence law of evolution and the difference in warpage deformation between double-layer and three-layer structure laminates and homogeneous plates, as well as the difference in warpage deformation between copper/carbon steel laminates and stainless steel/carbon steel laminates, were compared. Studies have shown that the warpage height of the metal laminate is proportional to the elongation difference and thickness ratio, and it is inversely proportional to the thickness. The greater the difference between the shear modulus of the base layer and the cladding layer is, the larger the effect of the thickness ratio on the warpage deformation of the metal laminate will be. Based on the numerical model, simulation studies were conducted on the deformation behavior and regularity of the plate shape warping of the laminated plate under the ideal uniform distribution of the initial temperature and the stress relief annealing process, and it was compared with that of the homogeneous plate. Finally, a sample of the warped laminate was taken at an industrial production site, and the initial extension difference was reversed by measuring its bending deformation. The result verifies the accuracy of the analytical computational mechanical model.
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Key words:
- metal laminate /
- warpage /
- longitudinal extension /
- analytical method /
- finite element
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圖 1 金屬層合板結構模型。（a）雙層結構的層合板；（b）三層結構的層合板
Figure 1. Structural model of metal laminate: (a) double-layer structure laminate; (b) three-layer laminate
圖 2 金屬層合板翹曲有限元模型。（a）工業在線有張力帶狀層合板翹曲模型；（b）離線裁切后塊狀層合板翹曲模型
Figure 2. Finite element model for warping of metal laminates: (a) industrial online warpage model of tensioned ribbon laminate; (b) warping model of block laminate after offline cutting
圖 3 金屬層合板翹曲模態。（a）C翹翹曲模態；（b）L翹翹曲模態
Figure 3. Warping mode of metal laminate: (a) C warping mode; (b) L warping mode
圖 4 延伸差對不同翹曲模態變形的影響。（a）C翹；（b）L翹
Figure 4. Effect of extension difference on the warpage height of C warping (a) and L warping (b)
圖 5 厚度對不同翹曲模態變形的影響。（a）C翹；（b）L翹
Figure 5. Effect of thickness on warpage height of C warping (a) and L warping (b)
圖 6 厚度比對不同翹曲模態變形的影響。（a）C翹；（b）L翹
Figure 6. Effect of the thickness ratio on warpage height of C warping (a) and L warping (b)
圖 7 金屬層合板不同翹曲后界面處應力分布情況。（a）C翹；（b）L翹
Figure 7. Stress distribution at the interface of metal laminate after C warping (a) and L warping (b)
圖 8 層合板與均質板冷卻過程熱變形行為對比。（a）工業在線有張力帶狀層合板和均質板對比；（b）離線裁切后塊狀層合板和均質板對比
Figure 8. Comparison of the thermal deformation behavior of laminated and homogeneous plates during cooling: (a) comparison of industrial online tension band laminate and homogeneous board; (b) comparison of block laminate and homogeneous board after offline cutting
圖 9 不銹鋼?碳鋼雙金屬層合板翹曲模態。（a）工業在線有張力層合板翹曲模態；（b）離線裁切后塊狀層合板翹曲模態
Figure 9. Stainless steel?carbon steel bimetal laminate warpage mode: (a) warping mode of industrial online tension laminate; (b) warping mode of block laminate after offline cutting
圖 10 終冷溫度對C翹（a）和四角翹（b）翹曲變形的影響
Figure 10. Effect of the final cooling temperature on warpage height of C warpage (a) and four corners warpage (b)
圖 11 冷卻后層合板界面處應力分布。（a）工業在線有張力時帶狀金屬層合板；（b）離線裁切后塊狀金屬層合板
Figure 11. Stress distribution at the interface of the laminate after cooling: (a) strip metal laminate with tension in the industrial line; (b) bulk metal laminate after offline cutting
表 1 金屬層合板力學性能
Table 1. Mechanical properties of metal laminates
Metal laminates Material E / GPa G/GPa $\nu $ Copper / carbon steel T2/ Q235 108/210 41/81 0.32/0.3 Stainless steel/carbon steel 304/ Q235 202/210 78/81 0.3/0.3 
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表 2 解析計算與有限元計算C翹翹曲高度結果對比
Table 2. Comparison between analytical calculation and finite element calculation of C warpage height
Metal laminate Laminate structure Extension difference / 10?5 Finite element results / mm Analytical calculation results / mm Relative error / % Copper / carbon steel Double layer 50/100 7.41/14.82 7.23/14.46 2.43/2.43 Three layers 50/100 5.20/10.41 5.28/10.55 1.52/1.33 Stainless steel / carbon steel laminate Double layer 50/100 8.40/16.80 8.36/16.72 0.48/0.48 Three layers 50/100 7.77/15.53 8.20/16.40 5.24/5.31
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表 3 解析計算與有限元計算L翹翹曲高度結果對比
Table 3. Comparison between analytical calculation and finite element calculation of L warpage height
Metal laminate Laminate structure Extension difference /
10?5Finite element results / mm Analytical calculation results / mm Relative error / % Copper / carbon steel Double layer 50/100 93.73/186.65 94.34/188.68 0.65/1.08 Three layers 50/100 66.57/132.92 68.37/136.75 2.63/2.80 Stainless steel / carbon steel laminate Double layer 50/100 111.77/222.35 111.44/222.87 0.30/0.23 Three layers 50/100 105.45/210.17 109.36/218.72 3.58/3.91
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中文字幕在线观看表 4 L翹翹曲高度測量值與計算值對比
Table 4. Comparison of the measured values and calculated values of the L warpage height
Working condition Sampling parameters / mm Measurement results / mm Calculation results /
mmRelative error /
%d1 d2 h1 h2 1 299.8 300.2 15.8 12.5 14.15 13.40 5.3 2 399.6 400.4 35.7 31.2 33.45 31.84 4.8 3 499.4 500.3 65.9 63.1 64.50 62.18 3.6
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