Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half -plane containing the inclusions (which can have the different elastic prop erties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's f undamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homo geneous plane is extended on case of a half plane. The unknown functions ar e approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips ar e taken into account. Only two types of the boundary elements (straight seg ments and circular arcs) are used to approximate the boundaries. All the in tegrals involved in CHSIE are evaluated in a closed form. A wide range of e lasticity problems for a half-plane with cracks, openings and inclusions ar e solved numerically.