臺灣博碩士論文加值系統

本文係建構在完全潤滑理論下，結合金屬箔片冷軋之板塊模型，利用二維雷諾方程式及參考N.A. Fleck對庫倫摩擦冷軋所建立之變形機制，來探討及分析在軋延過程中軋輥、箔片及油膜厚三者尺度變形之相依關係，並計算油壓、箔片應力及剪應力分佈。由於油膜厚與油壓間敏感的相互影響，更增加了完全潤滑架構分析的困難及不確定性。因此，為獲得更穩定的結果，學者過去通常採用混合潤滑模式來分析，以確保軋輥與具粗度峰谷工件二者型廓之相依關係。由於整個軋延加工區域中，粗度峰直接受到軋輥壓制，而維持完全契合之形態。故即使考量潤滑因素，其作用亦僅局限在粗度波谷區域，這使得在數值計算過程中能有效掌握其變形，同時更具穩定性。
在完全潤滑之架構下，由於軋輥與工件間被油膜區隔，使得三者間之數學關係具高度非線性，故難以建構完全偶合之制御方程式。此外，箔片各變形區段之平衡式與降伏準則皆互有差異，更增加了迭代過程中，不斷修正邊界位置之覆雜性與困難度。
研究中發現，低承載時變形區域並無明顯之平坦黏滯區域及壓力峰，且軋輥與箔片間僅具一中性點。當承載逐漸增加，壓力曲線局部高點，亦將往出口方向移動，但尚未能超越軋輥中心位置，同時其減縮率仍無法明顯增加。因此，為使減縮率提高，在高承載時，將中性區域近似為赫茲壓力曲線，並反覆修正與迭代運算進口區及出口區之壓力及油膜厚，使得箔片在出口端發生第二次塑性減縮，藉以提高減縮率，其結果與Fleck庫倫摩擦冷軋之壓力曲線類似。在高減縮率時，由於潤滑油黏度與壓力之相互作用，而產生高於庫倫摩擦冷軋之摩擦係數，因此，潤滑冷軋之承載及力矩亦將大於庫倫摩擦冷軋。
在實例分析中，先以Fleck摩擦冷軋及Sutcliffe混合潤滑冷軋之相同案例，來比較及驗證本研究理論及電腦程式之正確性。同時，並藉由其他實例演算結果，來綜合討論軋延速度、潤滑油黏度、軋輥半徑、箔片材料強度及前後張力等因素，對軋延效果之影響。同時由結果中顯示，低減縮率及高減縮率之油壓力分佈，分別近似於液動潤滑及彈性液動潤滑（EHD），此亦符合物理現象。
本研究除了證實箔片完全潤滑冷軋之可行性外，同時亦顯示潤滑油黏度是造成摩擦力及油壓力大幅攀升之主要因素，此將嚴重影響軋延效率，故選用適當黏度之潤滑油，將是箔片潤滑冷軋之重要課題。

In this thesis, the lubrication problem by numerical simulation of foil rolling process is presented. We analyze the corresponding relationships between roll, foil and film shape which uses the two-dimensional Reynolds equation and refers to the mechanism of cold rolling of Coulomb friction established by Fleck. Meanwhile, we compute the distributions of pressure, strip stress and shear stress, which is based on the full lubrication configuration combined with the slab model. The mutually sensitive influence between film thickness and pressure adds to the difficulty and uncertainty of full-lubrication analysis. In order to obtain more stable results, scholars have generally adopted a mixed-lubrication regime in the past to guarantee the mutually dependent profiles between roll and foil. Due to asperities suppressed by the roll in the bite, it maintains the complete profile match. Even though the lubricant is presented, it is confined only in the valley. The foil deformation is grasped and the computation has comparatively good stability in the numerical simulation process.
Under full lubrication, high non-linearity appears and the completely coupled governing equation cannot be constructed because the roll and the foil are still separated by the lubricant. In addition, the diverse equilibrium equations and yielding criteria in discrete regions add to the complexity of unceasingly revising the boundary position during the iterative process.
In the study, the numerical results show that the flat zone and pressure hill are not obvious when the load is low. In addition, there exists only one neutral point between roll and foil. The maximum pressure peak position moves toward the outlet but cannot surpass the center of roll while increasing the initial load. Simultaneously, the reduction was also unable to increase apparently. In order to enhance the reduction, the pressure distribution of neutral zone is approximated as Hertz pressure when high load is applied. Then we repeatedly correct and iterate the pressure and film thickness at the inlet and outlet to obtain the second plastic reduction at the outlet. In this way, the reduction is enhanced and the final pressure curve is similar to Fleck’s Coulomb friction solution. Because of the interaction between the viscosity and the pressure at high reduction, it produces a friction coefficient higher than the Coulomb friction. Therefore, the load and torque are higher.
In the example analysis, we first compare the results of the same cases by Fleck and Sutcliffe and confirm the correctness of theory and computer program adopted in this research. At the same time, in other example calculation results, we completely discuss the related effects on rolling quality with the rolling speed, the viscosity of lubricating oil, the roll radius, the material strength of foil and forth and back tension. Furthermore, through the results in this research, it shows that pressure distribution of low and high reductions are distinctly approximated to hydrodynamic and elastohydrodynamic(EHD) lubrications respectively, which also corresponds with physical phenomenon.
Apart from confirming the practicability of cold rolling of full lubricated foil, this research also shows that the viscosity is the primary factor of apparent enhancement of friction shear stress and normal pressure, which will seriously affects rolling efficiency. Therefore, selecting lubricant of suitable viscosity will be an important topic of cold rolling of lubricated foil.

第一章 緒 論...........................................1
1.1 軋延技術簡介........................................1
1.2 冷軋之定義..........................................1
1.3 軋延箔片之應用......................................2
第二章 文獻回顧及研究架構...............................3
2.1 軋延相關文獻摘要....................................3
2.2 箔片冷軋與傳統厚板冷軋之比較.......................10
2.3 本研究與其他箔片冷軋研究之差異性...................11
2.4 研究架構...........................................12
第三章 基本理論........................................13
3.1 基本假設...........................................13
3.2 軋輥、箔片及油膜厚之幾何關係.......................14
3.3 軋輥之彈性變形.....................................14
3.3.1 變形量計算.......................................14
3.3.2 應變計算.........................................15
3.3.3 軋延方向之應力計算...............................16
3.3.4 軋延表面速度計算.................................16
3.4 箔片之彈-塑性變形..................................17
3.4.1 平衡方程式.......................................17
3.4.2 剪應力...........................................18
3.4.3 應變.............................................19
3.4.4 箔片軋延表面速度.................................20
3.4.5 降伏準則.........................................20
3.4.6 箔片軋延變形區分.................................22
3.5 潤滑油膜公式.......................................26
3.5.1 雷諾方程式.......................................26
3.5.2 油膜厚度計算.....................................27
3.5.3 壓力與黏度關係...................................28
3.5.4 潤滑油流體速度...................................28
3.6 關聯方程式之整合及應用.............................29
3.6.1 幾何協調性.......................................29
3.6.2 流體連續性.......................................29
3.6.3 壓力及剪應力一致性...............................29
3.7 邊界條件...........................................30
3.7.1 油膜壓力.........................................30
3.7.2 箔片.............................................30
第四章 數值分析........................................31
4.1 赫茲(Hertz)壓力公式................................31
4.2 初始參考膜厚之應用公式.............................31
4.3 軋輥變形及應變矩陣.................................32
4.3.1 變形矩陣.........................................32
4.3.2 應變矩陣.........................................33
4.4 統御方程式之無因次化...............................35
4.4.1 Reynolds方程式...................................35
4.4.2 流體之摩擦剪應力.................................36
4.4.3 平衡方程式.......................................36
4.4.4 降伏準則.........................................36
4.4.5 軋延承載.........................................37
4.4.6 軋延力矩.........................................37
4.4.7 變形及應變矩陣...................................37
4.5 數值運算...........................................38
4.5.1 節點劃分.........................................38
4.5.2 求解Reynolds方程式...............................38
4.5.3 彈性或抑制塑性區之張應力公式.....................39
4.5.4 塑性縮減區之張應力公式...........................39
4.5.5 數值積分.........................................39
4.6 電腦模擬分析.......................................41
4.6.1 電腦程式運算步驟說明.............................41
4.6.2 電腦流程圖.......................................43
第五章 實例應用........................................44
5.1 研究之比較及驗證...................................44
5.2 軋延初始條件.......................................45
5.3 結果說明...........................................46
5.4 討論...............................................50
第六章 結論............................................56
參考文獻 ..............................................57
附 圖 ..............................................61
附 表 .............................................139
附錄A. 變形及應變矩陣推導...........................146
附錄B. 參數無因次化.................................149
附錄C. 解油膜三次方程式.............................151
附錄D. Coulomb摩擦軋延塑性減縮區平衡式..............153

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