Extended Finite Element Method For Several Kinds Of Fracture Mechanics Problems

The Extended Finite Element Method(XFEM)was proposed by Belytschko and Black of Northwestern University in 1999.It is an effective numerical method for solving discontinuous problems.Based on the partition of unity,an enrichment function is added to the traditional finite element method,which reflect the discontinuity of crack.XFEM does not need to re-mesh in the calculation process,which can save the calculation cost effectively and improve the calculation efficiency.Due to this unique advantage,the XFEM has developed in many fields rapidly and widely,such as dynamic crack propagation,cohesive crack model,inhomogeneity and shear band evolution.Firstly,this paper introduces some basic theoretical knowledge of fracture mechanics and extended finite element.The ramp function is introduced into the displacement function,and the interacting numerical integration method is used to derive the related formulas of the extended finite element method for solving the fracture problem in detail.The stress intensity factors of several classical plane cracks are calculated,and the correctness of the improved method is verified by comparing the calculated results with the theoretical values.Secondly,aiming at the more common multi-crack problem in engineering structure,this paper uses the extended finite element method to calculate the stress intensity factor of multiple cracks under different distribution conditions and analyzes the effect of some factors,such as the crack length relative to the plate and the distance of crack.The numerical results in this paper have certain reference value for predicting structural damage.Finally,for the curve crack problem,the stress intensity factor of the arc crack which have analytical solution are solved by the extended finite element method,and compared with the analytical solutions.The feasibility of the method is verified.On this basis,the Stress intensity factor of cosine crack is solved.