METHOD OF HYPER-SINGULAR INTEGRAL EQUATIONS IN BOUNDARY VALUE PROBLEMS OF FRACTURE MECHANICS
DOI:
https://doi.org/10.32782/mathematical-modelling/2023-6-2-5Keywords:
hypersingular integral equation, boundary element method, finite element method, stress intensity factorAbstract
The purpose of this study is to develop an effective numerical method for analysing stress-strain state of structures with crack-like defects. The novelty of the proposed approach is in applying hypersingular integral techniques for solving the benchmark test problem of calculating the stress intensity factor. The reference problem consists in determining the stress-strain state of a sample with a circular crack under uniform tension. Finite and boundary element methods are used to solve this problem. When applying the finite element method, the sample in the form of a parallelepiped containing the central circular crack is considered. A mesh of finite elements is used, which thickens near the crack. When using the boundary element methods, the crack is considered in an infinite three-dimensional space. The boundary value problem of the elasticity theory for a cracked body is reduced to hypersingular integral equations using the potential theory methods. At the same time, the calculation of the stress intensity factors is reduced to the jump of movement determination along the crack contour. The determination of this jump is carried out by solving the hypersingular integral equation on the circular domain. The paper proposes the use of one-dimensional and two-dimensional hypersingular equations for this simulation. Analytical formulas are obtained for calculating finite parts according to Hadamard of the singular integrals, which are the matrix elements of the solving system of linear algebraic equations, using both types of hypersingular integral equations. Using the specified methods, the numerical values of the stress intensity factors were obtained. A comparison of the results obtained by different methods was made. The dimensions of solving systems of linear algebraic equations are also compared. The advantages and disadvantages of the applied methods are clarified.
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