Boussinesq's problem for a debonded boundary (Q1825068)

In the cylindrical coordinates system the following axially-symmetric mixed boundary-value problem is consideed: \(u_ z(r,0)=\Delta\), \(0\leq r\leq a\); \(\sigma_{rz}(r,0)=0\), \(0<r<b\); \(\sigma_{zz}(r,0)=0\), \(a<r<b\); \(u_ r(r,0)=0\), \(b\leq r<\infty\); \(u_ z(r,0)=0\), \(b\leq r<\infty\). The presented solution aims to deal with the physical problem as follows: 1. An elastic isotropic half-space is bounded symmetrically along the boundary \(b\leq r<\infty\) and debonded in the region \(0\leq r\leq b\)- a crack region. 2. A rigid cylindrical punch of the radius a is applied symmetrically in the crack plane. A rather complicated solution involving the Hankel transforms technique and integral equations is presented, with well known weaknesses at the boundaries \(r=a\) and \(r=b\) (oscillatory singularity and overlapping displacement). The main result - the derived load-displacement relationship seems to be of a sufficient accuracy and of some practical use.