The Hertz problem : rigid cylindrical punch

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Hertz indentation
  • The contact length   depends on the load  .
  • There is no singularity at  .
  • The radius of the cylinder ( ) is large.

We have,

 

Hence,

 

and

 

Therefore,

 

and

 

Plug back into the expression for   to get

 

This expression is singular at   and  , unless we choose

 

Plugging   into the equation for  ,

 

Two deformable cylinders

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If instead of the half-plane we have an cylinder; and instead of the rigid cylinder we have a deformable cylinder, then a similar approach can be used to obtain the contact length  

 

and the force distribution