عنوان مقاله [English]
Fracture mechanics is a vast and growing field of mechanics concerned with the study of crack propagation in materials. In order to study the behavior of the system with varying properties˓ using analytical solutions, denote exact results. Unfortunately, very few practical systems lead to analytical solutions thereupon using numerical approach have been considered to find the close answer to practical results. The stiffness matrix is an inherent feature of a numerical method. Its condition has a great influence on numerical calculation and the stability of the solution. Since the application of numerical methods such as the standard finite element method, the extended finite element method and the isogeometric analysis approach in the problems of fracture mechanics has been approved, in this contribution, a theoretical comparison between stiffness matrices derived from these numerical methods for a cracked body, has been conducted. For this purpose, a computer code is prepared to make a comparison based on stiffness matrix geometry and mathematical properties such as matrix sparsity, stiffness index, bandwidth, number of independent rows and columns, zero and non-zero elements, symmetric/ nonsymmetric and hermitian/ nonhermitian stiffness matrix are investigated. In order to simulate crack, the finite element method uses a special 8-node singular element. And in case of extended finite element, the principles of enrichment of the interpolation functions of finite element and the application of the partition of unity method considered to apply discontinuities implicitly into the domain. Also in the isogeometric analysis approach repetition of two different control points between two patches can create a discontinuity and also demonstrates a singularity in the stiffness matrix. In addition, the NURBS of order 3 is utilized as the basis functions to approximate the geometry and the solution. By comparing the stress distribution in all three methods, the accuracy of the calculations and the smoothness of the results are investigated. And it is found that stiffness matrix obtained from the isogeometric analysis method is non-diagonal in the fracture problems. Extended finite element stiffness matrix in comparison with other methods has a better condition.