Indentation due to a frictionless rigid flat punch

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Indentation by a plat rigid punch
  • Start with uneven surface profile  .
  • Unsymmetric load  , but sufficient for complete contact over the area  .

Displacement in   direction is

 

where   is a rigid body translation and   is a rigid body rotation.

Rigid body motions can be determined using a statically equivalent set of forces and moments

 

The displacement gradient is

 

Integral is a Cauchy Singular Integral that appears often and very naturally when the problem is solved using complex variable methods.

Note that the only thing we are interested in is the distribution of contact forces  .If we change the variables so that

  and  , then

 

If we write   and   as

 

and do some algebra, we get

 


Flat punch with symmetric load

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In this case,

 

Also,   (origin at the center of  ), hence  . Therefore,

 

At  , the load is infinite, i.e. there is a singularity.