ELEMENTARY EXACT METHOD FOR SOLVING MIXED BOUNDARY-VALUE-PROBLEMS OF POTENTIAL-THEORY, WITH APPLICATION TO HALF-PLANE CONTACT AND CRACK PROBLEMS

A new method is presented for the exact solution in closed form of a m ixed boundary-value problem of potential theory when the potential is prescribed on one half-plane (say, y greater-than-or-equal-to 0, z = 0 ) and the charge-density distribution is prescribed on the other half- plane (y < 0, z = 0). The method is based on a new integral representa tion for the reciprocal of the distance between two points. Its substi tution into the simple layer distribution leads to an integral equatio n which can be solved exactly, with no integral transforms or series e xpansions involved. The general results are applied to solving a punch problem and a half-plane crack problem. A complete solution for the f ields of stresses and displacements is given in closed form and in ter ms of elementary functions.