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The interaction between a straight crack and a circular inhomogeneity with interface stiffness and energy is considered. The Gurtin and Murdoch model is adopted, wherein the interface between the inhomogeneity and the matrix is regarded as a material surface that possesses its own mechanical properties and surface tension. The elastostatics problem is decomposed into two complimentary problems for (1) a circular disk with unknown distributions of traction and displacements along its boundary and (2) a loaded isotropic plane containing a circular hole with unknown distributions of traction and displacements along its boundary. The unknown distributions are determined through the application of the constitutive relations at the material surface. For selected values of the dimensionless parameters that quantify the geometry, material properties and applied loading, the stress field, stress intensity factors and energy release rates are calculated using a complex boundary integral equation approach. Particular attention is paid to crack-tip shielding and anti-shielding that develops as a result of the stresses introduced by the material surface. An illustrative example involving a perforated plate loaded in tension suggests that the material surface produces a modest level of expected effective toughening.

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), 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 fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.

Growth of pressure-induced fractures originating from a wellbore at an arbitrary angle to the direction of far field stresses is considered. A parametric study of hydraulic fracturing process from the point of view of fracture mechanics is presented in conditions of slow and fast pressurization rate. The study is based on the linear elastic fracture mechanics formulation that involves a complex hypersingular equation (CHSIE) for a plane with a circular opening and a system of arbitrary curvilinear cracks. Stepwise method is used to model fracture propagation. Dimensionless parameters influencing the fracture path are defined.

Numerical algorithm to simulate 2-D smooth crack is presented. The stepwise method based on local criteria of propagation is used. Two crack propagation criteria are employed. At the first stage of propagation, the maximum tensile stress criterion is used to take into account the abrupt change in tangent direction. At subsequent stages, the assumption that the stress intensity factor (SIF) K_2= 0 at the current crack tip is exploited. The analytical formulae for calculating SIFs are given. The displacement discontinuities (DD) involved in these formulae are found from the numerical solution of a complex hypersingular integral equation (CHSIE) for a piecewise homogeneous plane with curvilinear cracks. The new mechanism of smooth approximation of the crack path by circular arcs at each propagation stage is suggested. Numerical results are given. They confirm the efficiency of the algorithm suggested.