Regular and well-posed formulation of the boundary integral method for a singular biharmonic problem (Q1909376)

The function \(u(r, \psi)= \sum^\infty_{j= 0} a_j v^*_j+ b_j \widetilde v_j\) \((- \pi< \psi< \pi)\), where \[ v^*_j= r^{j+ 2} [\cos j \psi- \cos (j+ 2) \psi],\;\widetilde v_j= r^{j+ 3/2} \Biggl[ {\cos (j- 1/2) \psi\over j- 1/2}- {\cos (j+ 3/2) \psi\over j+ 3/2}\Biggr], \] is the Airy stress function for an elastic plate with a crack along a segment \(0< r< r_0\), \(\psi= \pm \pi\). The upper half of the boundary is denoted by \(L\), without the slit \(\psi= \pm \pi\). The authors present the calculation of the functions \(\Delta u\) and \(\partial \Delta u/\partial n\) on \(L\).