卡普斯坦方程的一个推广及其与Kikuchi和Oden模型在库仑静摩擦定律上的比较
A Generalisation of the Capstan Equation and a Comparison Against Kikuchi and Oden's Model for Coulomb's Law of Static Friction
AI总结：
本文推广卡普斯坦方程至非圆形几何，推导闭式解，并与Kikuchi和Oden模型在曲线坐标下比较，发现摩擦力与杨氏模量无关，随曲率、泊松比增大或厚度减小而增大，且隐含摩擦系数依赖模型。
摘要：
在本文中，我们将卡普斯坦方程推广到非圆形几何形状。我们推导了一个闭式解，适用于具有零泊松比的膜（或具有任意泊松比的弦），该膜由刚性棱柱支撑（在极限平衡情形和稳态平衡情形下）或由刚性一般锥体支撑（仅在极限平衡情形下）。我们的模型表明，弹性体的应力分布取决于刚性障碍物在接触区域的曲率变化。作为比较，我们将Kikuchi和Oden的库仑静摩擦定律模型推广到曲线坐标，并进行数值实验，以检验Kikuchi和Oden模型在曲线坐标下以及我们推广的卡普斯坦方程所隐含的库仑静摩擦定律的性质。我们的数值结果表明，对于固定的摩擦系数，摩擦力与弹性体的杨氏模量无关，而增加曲率、增加泊松比以及减小弹性体的厚度会增加摩擦力。我们的分析还表明，对于给定的张力比，每个模型推断出的摩擦系数不同，表明隐含的摩擦系数依赖于模型。
英文摘要：
In this article, we extend the capstan equation to non-circular geometries. We derive a closed form solution for a membrane with a zero Poisson's ratio (or a string with an arbitrary Poisson's ratio) supported by a rigid prism (at limiting-equilibrium case and in steady-equilibrium case) or supported by a rigid general cone (at limiting-equilibrium case only). Our models indicate that the stress profile of the elastic body depends on the change in curvature of the rigid obstacle at the contact region. As a comparison, we extend Kikuchi and Oden's model for Coulomb's law of static friction to curvilinear coordinates, and conduct numerical experiments to examine the properties of Coulomb's law of static friction implied by Kikuchi and Oden's model in curvilinear coordinates and implied by our generalised capstan equation. Our numerical results indicate that for a fixed coefficient of friction, the frictional force is independent of the Young's modulus of the elastic body, and increasing the curvature, increasing the Poisson's ratio and decreasing the thickness of the elastic body increases the frictional force. Our analysis also shows that for given a tension ratio, the coefficient of friction inferred by each model is different, indicating that the implied-coefficient of friction is model dependent.
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