Elastic Contact for a Flat Punch

For a cylindrical flat punch indenting an elastic half-space with the flat side parallel to the plane of the half-space, the contact area is simply given by na, where a is the radius of the cylinder. For an applied load P, the indentation 8 (i.e., the change in distance of the center points beyond the first contact of the bodies) of the punch becomes 

E and v are the Young s modulus and Poisson s ratio of the half space material. The indentation of a flat punch on an elastic half-space is proportional to the applied load. The contact area stays constant independent of load. We obtain a linear relation between load and displacement with an effective spring constant of E a/(l — v ) = E a. The vertical surface displacement Az(r) of the elastic half-space is given by 

2) Heinrich Hertz, 1857-1984 German physicist, professor in Karlruhe and Bonn. 

Since we have assumed frictionless contact, the surface of the elastic half-space is free to slide below the indenting punch. The radial displacement of the surface within the contact area is given by 

The radial displacement outside the punch area is given by l-2v) l + v)Fi, 

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