Contact problem is one kind of mixed boundary value problems. Most of the analytical formula for contact problems can be found in the books written by Galin (1961), Gladwell (1980), and Johnson (1985). Combining the complex variable formulation with the method of analytical continuation, Muskhelishvili (1954) and England (1971a) provided solutions for several types of punch problems on isotropic elastic bodies. Usually, the mathematical model of elasticity can be divided into two parts. One is the basic equations which include equilibrium equations, constitutive laws, and kinematic relations. The other is the boundary conditions which can be distinguished into traction, displacement, and mixed boundary value problems. Once a problem is formulated based upon the basic equation, its solvability is usually dependent on the boundary conditions. As to the same boundary geometry, the mixed boundary value problems are more complicated than the traction or displacement boundary value problems. Therefore, the usual step to deal with the elasticity problems is from simple geometry to complicated geometry and then from traction (or displacement) boundary to mixed boundary. This is exactly the step we take in this book.