Elastic Stresses -Hertzian Contact 弹性应力——赫兹接触
A continuum analysis of contact forces and the deformations that arise from quasi static compression of elastic,elastic-plastic or perfectly plastic bodies can be used to develop a theory of impact for hard bodies composed of rate-independent materials.由弹性体、弹塑性体或者绝对塑性体的准静态压缩引起的接触力与变形的连续分析，能够用来发展由比率独立的材料组成的坚硬体的冲击理论。In this theory deformations are negligible outside a small contact region,and the deforming region acts as a nonlinear inelastic spring between two rigid bodies;the mass of the deforming region is assumed to be negligible.这一理论中小的接触区域之外的变形是微不足道，两个刚体之间的变形区域由一个非线性弹簧替换；变形区域的质量被认为是微不足道的。Hertz (1882)first developed this quasi static theory for elastic deformation localized near the contact patch and applied it to the collision of solid bodies with spherical contact surfaces.Hertz 's theory provides a very good approximation for collisions between hard compact bodies where the contact region remains small in comparison with the size of either body.赫兹首次对接触面附近的弹性变形使用准静态理论，并且将之应用于两球形体表面的接触中。赫兹理论为坚硬紧凑体间的碰撞提供了一个很好的近似值，当然接触区域比起碰撞体本身非常小。
Let nonconforming elastic bodies B and B'come into contact at a point C;in a neighborhood of C the surfaces of the bodies have radii of curvature R B and R B',as described in Fig.6.1.两不相容弹性体B 与B
'接触于C 点；曲率半径分别为R B 和R B'的物体的接触表面C 点附近。If these bodies are compressed by force F =F 3in the normal direction,Hertz showed that the contact region spreads to radius a and within the contact area there is an elliptical distribution of contact pressure 如果物体受法向压力F 作用，接触区域将会是半径为a 的区域，并且接触区域内接触压力为椭圆分布。
a r a r p r p ≤−=,)/1()(2/1220(6.1)
where r is a radial coordinate originating at the center and p 0=p(0)is the pressure at the center of the contact area.式中r 为径向坐标，在接触区中心处压力p 0=p(0)。
This contact pressure generates local elastic deformations and surface displacements that cause initially nonconforming surfaces to touch or conform within a contact area.接触压力导致局部弹性变形和表面位移，而这一位移使得接触区开始不相容的表面开始接触或是重叠。This pressure distribution results in a compressive reaction force F on each body.压力分布导致碰撞体间产生压缩反力。
200322)(a p dr r p F a ππ==∫(6.2)
The mean pressure p is two-thirds of the pressure at the center of the contact circle,3/20p p =.平均压力p 等于接触中心处压力的2/3。For the pressure distribution given in Eq.(6.1),Hertz obtained the norm
reaction massal displacement )(r w i at the surface of body i (i =B,B')from the Boussinesq solution for a force applied normal to the surface of an elastic half space (Timoshenko and Goodier,1970)：如（6.1）压力表达式所示，从弹性半空间体表面施加法向力后得到的Boussinesq 解中，Hertz 获得碰撞体表面的法向位移)(r w i 。
a r a r E ap r w i i i ≤−−=−),/2()1(25.0)(22102πν(6.3)
where compressive displacements are positive.In this expression the elastic moduli of body i are given as Young's modulus E i and Poisson's ratio i ν.式中的压缩量为正值。E i 为弹性模量，i ν为波松比。
The compression of each body i δis equivalent to the relative displacement between the initial contact point C and the center of mass,)0(i i w =δ.物体的压缩量等于接触点C 相对于质量中心的位移，)0(i i w =δ。Thus for axisymmetric bodies with convex contact surfaces of curvature 1−i R ,if the contact area is small in comparison with the cross-section,the radial distribution of the normal displacement can be expressed as 这样对于曲率为1−i R 的外凸面轴对称物体，如果接触区域比起横截面非常小，那么沿径向分布的法向位移就可以表达成：
i
i i R r r w 2/)(2−≈δ
The total indentation from compression B B ′+=δδδcan be related to the pressure magnitude 0p at the center of the contact area by summing the individual effects expressed by Eq.(6.3)：通过表达式（6.3），总的压缩量B B ′+=δδδ与接触中心的压力大小0p 有关系。
*02/E ap πδ=(6.4)
where an effective radius *R and modulus *E have been defined as *R 为等效半径，*E 为等效模量。
111*)
(−−′−+=B B R R R 11212*]
)1()1[(−−′′−−+−=B B B B E E E ννThis size of the contact area can be determined from Eqs.(6.3)and (6.4);the contact radius a is then related to contact force F using Eq.(6.2)：这类型的接触区域可由(6.3)与(6.4)确定；接触半径a 与接触力F 的关系由式(6.2)可得：
*
*2*2*43R aE F R a R ==δRearranging,we obtain 重新整理，有
3/12*
**43(R E F R a =(6.5)3/22
**2*2*)43(R E F R a R ==δ(6.6)3/12**3*2*0)6(23R E F E a F E p ππ==(6.7)
The mean pressure in the contact region p and a compliance relation for interaction force )(δF are obtained from Eqs.(6.6)and (6.7)：接触区的平均压力p 与力相互作用F 可由式(6.6)与(6.7)得出：
**34R E p δπ=,2/3*2**(34R R E F δ=(6.8)
where *R is the effective radius of curvature in the contact area before compression.式中*R 为压缩前接触区的等效半径。This force can be integrated to obtain the work W done by the normal contact force in compressing the small deforming region to any indentation δ，对力积分可以获得法向接触力压缩小变形区域时所作的功W 与压缩量间的关系。
2/5*/0*2***3**)
(158)/()/(*R R d R E R F R E W
R δδδδ∫=′′=(6.9)

版权声明：本站内容均来自互联网，仅供演示用，请勿用于商业和其他非法用途。如果侵犯了您的权益请与我们联系QQ:729038198，我们将在24小时内删除。