The contact problem for a circular plate with a stress-free end face

An axisymmetric contact problem for a circular elastic plate with a stress-free end face into which two symmetrically arranged punches are imbedded, is considered. The problem is solved using a method developed earlier for bodies of finite sizes, which is based on the generalized orthogonality of homogeneous solutions. The problem reduces to a Fredholm integral equation of the first kind in a function describing the displacement of the surface of the plate outside the punch. These functions are sought in the form of a sum of a Schlömilch series and a power function with a root singularity. The ill-posed infinite system of algebraic equations obtained as a result is regularized by the introduction of a small positive parameter. Since the matrix elements of the system, as well as the contact stresses, are defined by poorly converging numerical and functional series, the technique of summation of the residues of these series is used. The distribution of the contact pressure and the dimensionless imbedding force are found. Examples of the calculation of a plane punch are given.