Finite element method with account of singularity for mixed mode cracks

The paper deals with obtaining an analytical solution for stiffness matrix coefficients at a crack tip area for mixed mode cracks in plane strain conditions. The numerical study is focused on an infinite plate with a straight-through central crack under mixed loading. Analytical solutions are obtained as kinematic boundary conditions for plane strain. We analyzed distribution features of the stress-strain state fields and stress intensity coefficients at the top of the crack area, determined using the finite element method taking into account the singularity. The analytical formulas are obtained which set the kinematic conditions for a general and special case of loading a plate with a defect in the elastic setting for the case of plane deformation. The comparative analysis of the numerical results is presented for two cases of forming the design diagram of the top of the crack: the traditional method of creating a mathematical cut and the finite element method taking into account the singularity. The advantage of using the finite element method considering the singularity is found. We used an example of a plate with a through straight rectilinear central crack with the equal biaxial tension to show that setting the boundary conditions at the top of the crack taking into account the singularity allows one to significantly reduce dimensions of a calculation scheme of the finite element method and keep the calculation accuracy. It is concluded that such a formulation can be applied in an elastic-plastic formulation. The comparison between the classical finite element solution and finite element with singularity is presented. The convenience of the finite element method with singular boundary conditions is demonstrated.