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

陳曉敏(CHEN,XIAOMIN)

指導教授 ： 王仲宇

國立中央大學/工學院/土木工程學系/碩士(2014年)

摘要

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。

關鍵字

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

並列摘要

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.

並列關鍵字

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

參考文獻

[1] Cundall, P A and O D L Strack, A Discrete Numerical Model for Granular Assemblies, Geotechnique, Vol. 29, No. 1, pp. 47-65 ,1979.

[2] Shi, G H., Discontinuous Deformation Analysis, A new Numerical Model for the Static and Dynamics of Block systems, Ph. D. Thesis, Department of Civil Engineering, University of California at Berkeley ,1988.

[3] Mindlin, R. D., Compliance of Elastic Bodies in Contact. J. of Appl. Mech., 16, pp. 259-268, 1949.

[4] Haug, E. J., Wu S C, Yang S M. Dynamics of mechanical systems with coulomb friction, stiction, impact and constraint addition-deletion—I Theory. Mechanism and Machine Theory, 21(5): 401-406, 1986.

[5] Moreau, J. J., Evolution problem associated with a moving convex set in a Hilbert space. Journal of Differential Equations, 26(3): 347-374, 1977.