A four-part boundary value problem in elasticity - indentation by a discontinuously concave punch

A solution is given for the frictionless indentation of an elastic half-space by a flat-ended cylindrical punch with a central circular recess, when the load is large enough to establish a circular region of contact in the recess. The problem is reduced to two simultaneous Fredholm equations using the method of complex potentials due to Green and Collins. Results are presented for the relationship between load, contact radius and penetration for various punch geometries. Introduction In the preceeding paper [1], it was shown how the method of complex potential functions due to Green and Collins provides an elegant solution of a wide range of boundary value problems for the half-space. In the present paper we shall use the method to treat an elastic indentation problem which leads to a four-part boundary value problem the frictionless indentation of an elastic half-space by a rigid, flat-ended cylindrical punch, a central circular region of which is recessed as shown in Fig. 1. The radius of the punch is c and the central recess has radius b and depth e. The punch is pressed into the half-space to a depth d under the action of an axial force P. For small values of P, we anticipate contact only over the annular region b < r _< c and the resulting problem has been discussed by Collins [2], Shiguya et al. [3] and others. For further discussion of this three-part problem and additional references, see Gladwell [4]. In this paper, we concentrate on the situation when the load P is large enough to establish a circular region of contact whose radius we denote by a. Intuitively we anticipate that a ~ b as P ~ c and that in the limit a ~ 0 we shall recover the solution for the annular punch.