Application Of The High-order/extended Finite Element Method In Two Dimensional Fracture Mechanics

The mathematical theories of the p-and the h-p version Finite Element Methods(FEM)have been completely established,which provide a solid theoretical foundation for the numerical simulation of the p-and the h-p version finite element methods.At the same time,the p-and the h-p version finite element methods can not only effectively improve the convergence rate of numerical solutions,but also ensure the accuracy of numerical calculation.There are abundant mathematical researches on the p-and the h-p version FEM.However,Comparing with the classical h-version FEM,there are fewer studies being done on its application in engineering practice,especially the applications in the field of fracture mechanics.It is noted that the convergence of rates of the p-and h-p version are obviously faster than the traditional the h-version FEM.In some cases,such as for singular problems,exponential convergence rates can be obtained.Fracture problems are typically singular(linear elastic fracture)or high gradient(viscous or ductile fracture)problems,so it would be solved more efficiently with the p-or the h-p version FEM being adopted.This paper mainly studies the application of the p-version finite element method in two dimensiondual fracture problems,which is divided into the following two parts:Firstly,the p-version FEM was applied to simulate several classical crack cracking problems,and the cases of cracks in different sizes,different angles and stress concentration regions were analyzed.The p-version FEM is used to handle the crack problems and the Mixed-Mode Stress Intensity Factors(SIFs)is derived based on the displacement field and stress field by using the Contour Integral Method.With fewer meshes,a higher accuracy results can be obtained with a lower Degree of Freedom(DOF)by dividing the grid properly.The numerical solutions of the cracks at different sizes,angles and locations in the stress concentration region show higher accuracy and robustness under similar grid parameters.Comparing with results derived by the Extend Finite Element Method(XFEM)with high order asymptotic displacement field enriched at the crack tip in the literature,SIFs of inclined crack derived in the paper has higher precision and smaller error fluctuation.Secondly,due to the hierarchical shape functions of p-FEM is polynomial,the approximation space is continuous and smooth.When dealing with discontinuities,the grids need to be consistent with the discontinuous geometry.In oder to simulate the discontinuous evolution without remeshing to adapt to the discontinuous interface and save the calculation cost,the eXtended Finite element(XFEM),nowadays which is very popular dealing with the discontinuous problems,is combined in this paper.The discontinuous term which enriched Heaviside function is expanded in the two dimensional quadrilateral hierarchical element.The new quadrilateral hierarchical element realizes describing the discontinuous interface inside the element,and at the same time,the ability of the p-version FEM to improve the calculation accuracy by increasing the interpolation polynomial order.The results show that the p-version FEM needs fewer grids and has higher precision and strong robustness while simulating facture problems.And the p-version FEM combined with extended FEM inherits the advantages of the above two methods in dealing with discontinuous problems and it has better research prospect and application value.