卵形顆粒法向與切向接觸之等效線性彈簧值之推導與驗證

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姓名

陳曉敏(CHEN,XIAOMIN) 查詢紙本館藏

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土木工程學系

論文名稱

卵形顆粒法向與切向接觸之等效線性彈簧值之推導與驗證
(Development and Verification of the Normal and Tangential Equivalent Linear Contact Springs of Egg Shape Particles)

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摘要(中)

本論文針對圓球形顆粒以及卵球形顆粒間的接觸，基於Hertz法向接觸理論和Mindlin的切向接觸勁度理論，進行了等效線性的彈簧勁度值之研究。通過研究表明，Hertz接觸理論近似解的物理實質就是可以用等效的線性彈簧來模擬顆粒間法向碰撞接觸之非線性行為；法向等效的線性彈簧值與顆粒初始法向相對速度，顆粒的彈性常數和接觸區域幾何形態相關；而切向接觸彈簧值與法向擠壓量和顆粒的彈性常數密切相關。
卵形顆粒之形狀是利用四圓弧逼近法及座標旋轉的觀念，卵形顆粒表面可以分為兩個球面以及一個曲面，顆粒間的接觸分成球面、曲面接觸為主。卵形顆粒表面屬於自旋生成的曲面，曲面上任意一點所對應子午線的半徑與平行圓的半徑就是我們所需要的主曲率半徑，這樣就能快速確定主曲率半徑，簡化計算。
將推導出來的等效線性彈簧值運用于顆粒運動的數值模擬之程式，進行顆粒正碰，斜碰之模擬，並將結果與理論結果進行比較與驗證。

摘要(英)

In this thesis, equivalent linear contact spring constants are developed for the contact between deformable particles such as spherical particles and egg-shaped particles, based on the Hertz contact theory and Mindlin contact theory of tangential stiffness. The physical significance of this study is that we can adopt an equivalent linear spring stiffness to model the non-linear contact behavior of Hertz’s contact; It’s shown that the theory normal equivalent linear contact spring stiffness is related to the initial relative velocity of the two contact particles, elastic constant and geometry of contact area; tangential stiffness is related to the normal mutual approach distance of the two particles and elastic constant.
The newly developed egg-shaped particles are designed by the revolution of an ellipse, formed by four connected arcs, about the major axis passing through its centroid. Surface of egg-shaped particles can be divided into two spherical surfaces and a curved surface. Mainly contact can be divided into spherical surface contact and curved surface contact. The surface of egg-shaped particles belongs to rotating surface, so that we can quickly determine the principal radius of curvatures in the two contacted curves.
Numerical simulations for granular assemblies used equivalent linear contact spring stiffness, simulate the behavior of direct impact and oblique impact and compare the result with theoretical solution.

關鍵字(中)

★ 顆粒
★ Hertz接觸理論
★ Mindlin接觸理論
★ 等效
★ 線性

關鍵字(英)

★ particles
★ Hertz contact theory
★ Mindlin contact theory
★ equivalent
★ ,linear

論文目次

摘要 I
ABSTRACT II
致謝 III
目录 IV
表目錄 VI
圖目錄 VIII
第1章緒論 1
1-1 研究背景 1
1-2 研究動機與目的 2
1-3 研究方法 3
1-4 論文內容 3
第2章文獻回顧 4
2-1 衝量動量法 4
2-2 連續接觸力模型 6
2-3 有限元素法 9
第3章顆粒法向接觸彈簧勁度值分析 11
3-1 球體的法向接觸分析 12
3-1-1 球體與球體的法向接觸分析 12
3-1-2 球體與平面的法向接觸分析 18
3-2 卵球的法向接觸分析 19
3-2-1 卵球與卵球的法向接觸 22
3-2-2 卵球與平面的法向接觸 30
第4章顆粒切向接觸彈簧勁度值分析 33
4-1 圓球的切向接觸彈簧值分析 33
4-1-1 圓球與圓球的切向接觸彈簧值分析 33
4-1-2 圓球與平面的切向接觸彈簧值分析 36
4-2 卵球的切向接觸彈簧值分析 37
第5章數值算例驗證 41
5-1 圓球的碰撞驗證與測試 41
5-1-1 圓球與圓球碰撞—正向接觸 41
5-1-2 圓球與圓球碰撞—斜向接觸 44
5-1-3 圓球與平面碰撞—正向接觸 53
5-1-4 圓球與平面碰撞—斜向接觸 59
5-1-5 圓球碰撞之法向接觸彈簧值變化 64
5-2 卵球的碰撞測試 68
5-2-1 卵球正碰 68
第6章結論與建議 73
6-1 結論 73
6-2 建議 74
參考文獻 76
附錄A：非球形曲面之HERTZ接觸理論[12] 79
附錄B：剛體碰撞狀況的分類判別法 85
附錄C：剛體平面斜碰之求解 91
附錄D 卵球之接觸判斷[33] 97

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指導教授

王仲宇(WANG, CHUNG-YUE)

審核日期

2014-7-28

推文