# 接触力学

## 历史

20世纪中期接触力学领域的进步要归功于Bowden和Tabor。Bowden和Tabor首次强调了接触中物体表面粗糙度的重要性。通过对表面粗糙度的研究发现，互相摩擦体间的真实接触面积要小于表面接触面积。这种解释也彻底改变了摩擦学的研究方向。Bowden和Tabor的著作产生了几种粗糙表面的接触力学理论。

## 无黏着弹性接触的经典问题

### 球体和弹性半空间体的接触

${\displaystyle {\frac {1}{E^{*}}}={\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}}$ .

${\displaystyle E_{1}}$,${\displaystyle E_{2}}$ 分别为是接触体的弹性模量，${\displaystyle \nu _{1}}$,${\displaystyle \nu _{2}}$ 是泊松比。

### 两个球体的接触

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$ ,

${\displaystyle p=p_{0}\left(1-{\frac {r^{2}}{a^{2}}}\right)^{1/2}}$ , 式中

${\displaystyle p_{0}={\frac {2}{\pi }}E^{*}\left({\frac {d}{R}}\right)^{1/2}}$ .

### 刚性圆柱体和弹性半空间体的接触

${\displaystyle p=p_{0}\left(1-{\frac {r^{2}}{a^{2}}}\right)^{-1/2}}$ , 式中

${\displaystyle p_{0}={\frac {1}{\pi }}E^{*}{\frac {d}{a}}}$ .

${\displaystyle F=2aE^{*}d{\frac {}{}}}$ .

### 刚性圆锥体和弹性半空间体的接触

${\displaystyle d={\frac {\pi }{2}}a\tan \theta }$ ,

${\displaystyle p(r)=-{\frac {Ed}{\pi a\left(1-\nu ^{2}\right)}}ln\left({\frac {a}{r}}+{\sqrt {\left({\frac {a}{r}}\right)^{2}-1}}\right)}$ .

${\displaystyle F_{N}={\frac {2}{\pi }}E{\frac {d^{2}}{\tan \theta }}}$ .

### 两个中心轴平行的圆柱体间的接触

${\displaystyle F={\frac {\pi }{4}}E^{*}Ld}$ .

${\displaystyle a={\sqrt {Rd}}}$ ,

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$ ,

${\displaystyle p_{0}=\left({\frac {E^{*}F}{\pi LR}}\right)^{1/2}}$ .

## 无黏着赫兹理论

${\displaystyle P_{z}=\int _{S}p(x,y)~\mathrm {d} A~;~~Q_{x}=\int _{S}q_{x}(x,y)~\mathrm {d} A~;~~Q_{y}=\int _{S}q_{y}(x,y)~\mathrm {d} A}$

${\displaystyle M_{x}=\int _{S}y~p(x,y)~\mathrm {d} A~;~~M_{y}=\int _{S}x~p(x,y)~\mathrm {d} A~;~~M_{z}=\int _{S}[x~q_{y}(x,y)-y~q_{x}(x,y)]~\mathrm {d} A}$

### 赫兹理论中的假设

• 应变很小并在弹性范围内；
• 接触物体可以看作是弹性半空间体，也就是说，接触面远小于物体半径；
• 表面是连续的，不确定的 ；
• 表面无摩擦。

## 粗糙表面间的接触

${\displaystyle A={\frac {\kappa }{E^{*}h'}}F}$ ,

${\displaystyle \sigma ={\frac {F}{A}}\approx {\frac {1}{2}}E^{*}h'}$

${\displaystyle \Psi ={\frac {E^{*}h'}{\sigma _{0}}}>2}$

## 黏着接触

${\displaystyle F_{A}=(3/2)\pi \gamma R}$

${\displaystyle F_{A}={\sqrt {8\pi a^{3}E^{*}\gamma }}}$

## 参考文献

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