The Application Of Hypersigular Integral Equation Method In Several Kinds Of Fracture Mechanics Problems

A large number of physical problems and engineering problems can be described by the hypersingular integral equations,but it is very diffuclt to get the analytical solution.There-fore,the researchers in related fields have turned their attention to the study of numerical solutions of hypersigular integral equations.However,hypersigular integral equation are di-vergent in common sense and cauchy principal value sense,and are only solved under the definition of Hadamard finite-part integral,thus the numerical solution of the hypersingular integral equation is very difficult to get.Compared with Cauchy integral equation,the hyper-singular integral equation can more accurately describe the actual physical and engineering problems.Therefore,studying its high-precision numerical solution method is a hot topic,which has important scientific significance and application prospects.In this paper,numerical solutions of hypesigular integral equation and its applications in several kinds of fracture mechanics problems are discussed.Firstly,the traditional piece-wise linear collocation method is improved to eliminate the singularity at the interpolation point.According to the different selection of basis functions in the modified piecewise linear collocation method,a continuous piecewise function at the interpolation node is constructed to replace the basis function of piecewise linear collocation,then the hypersingular integral equation is solved by the definition of the Hadamard finite part integral,and numerical ex-amples are used to verify the feasibility of the modified piecewise linear collocation method in solving the hypersingular integral equations.Secondly,based on the homotopy pertur-bation method for solving hypersingular integral equations of second order singularity,it is extended to solve the hypersingular integral equation with higher order singularities and use numerical examples to verify the high efficiency of solving the hypersingular integral equation with homotopy perturbation method.Finally,the coplanar crack problem of vari-ous homogeneous elastomers and the crack problems in functionally graded materials with strain gradient effects are discussed,the hypersingular integral equation derived from the crack problem is solved by homotopy perturbation method,and the value of the stress inten-sity factor at the crack endpoint is calculated.