Research On Extended Finite Element Method And Some Issues In Its Application

There are two kinds of discontinuous problems in solid mechanics.One is weak discontinuity which is caused by the mutation of material properties,such as bi-material and inclusion.The other one is strong discontinuity which is caused by geometry mutations from the object interior,such as crack.Finite Element Method(FEM)has been successfully used in the field of continuum mechanics.But FEM has obvious limitations because of its dependence on the mesh grid.Extended Finite Element Method(XFEM)shows great advantage in dealing with discontinuity.Based on the concept of partition of unity,XFEM reconstructs the local discontinuous displacement field by the enrichment of node displacement,making the displacement approximation function independent of the physical grid partition.But,how to choose suitable enrichment function for the accurate construction of discontinuous displacement field,and how to improve the computational efficiency and precision of XFEM still need further research.Aiming to solve these two problems,XFEM and generalized extended finite element method(GXFEM)which based on the XFEM are studied,and the applications of XFEM and GXFEM in discontinuity problems such as crack,interface crack,inclusion are researched in this thesis.The main content includes two aspects as follows:1.Fundamental research on XFEM and GXFEM.Compared with the general principle of FEM,several new FEMs such as XFEM and GXFEM are briefly studied.XFEM can reconstruct some characteristics of discontinuity problem by improving the solution space through the enlargement of enrich function,and determine the geometric features of discontinuity through level set function.The construction of enrichment function in crack and particle inclusion is given.The discrete equation and stiffness matrix of XFEM with crack and particle inclusion are deduced based on the principle of virtual work.Problems in XFEM such as element blending,element enriching,crack tip strengthening,and the calculation accuracy of J integral are researched.Finally,the influence of the grid density on the calculation accuracy for XFEM is analyzed by numerical example.Eight-node isoparametric element has more application values than linear element in engineering,but there are little researches on eight-node.In this paper,formulas of XFEMwith eight-node are derived,integral scheme of the stiffness matrix is given,and MATLAB program is developed to calculate the crack tip stress intensity factor of a plate with crack.Problems such as the selection of enrichment area,the differentiation between grid reconstruction,element decomposition,and the numbering of enriched node are researched.The crack tip stress intensity factors of classical cracked plate are calculated and comparative research is performed between eight-node XFEM and linear XFEM.The influence of the mesh parameters on the XFEM results is further studied.Simulations verify the validity and effective of the formulas and program of eight-node XFEM in this thesis.In view of the advantages that the XFEM does not need to remesh the physical grid in processing the crack problem,MATLAB program for analyzing fatigue crack propagation is written,where XFEM is used to calculate the crack tip stress intensity factor and the crack propagation angle is judged by the maximum circumferential tensile stress criterion.Finally,the crack propagation life is calculated by the fatigue crack propagation increment model.XFEM has great advantages in dealing with discontinuous problems,but it has no significantly improvement in computation accuracy.GXFEM can improve the computation accuracy by generalizing node degree of freedom and enhancing the order of the node interpolation function.Combined with the advantages of XFEM and GXFEM,the basic principle of GXFEM is given,and the corresponding formulas are derived in this thesis.The basis function of the singular field at the crack tip of Westergaard is used as the interpolation function of the node displacement to achieve higher calculation precision.The numerical integral method for element stiffness matrix and the method for calculating the stress intensity factor of the crack tip are given,and a new method for analyzing the stiffness matrix of GXFEM is also presented.Only the related part stiffness matrix of the new crack tip needs to be calculated under crack propagation case.The global stiffness matrix is calculated by Cholesky factorization,which greatly improves the calculation efficiency.The program of GXFEM is written.Through the calculation of typical cracked plate,it is showed that the accuracy of the stress intensity factor is higher,without meshing the structure densely.2.Research on the application of XFEM and GXFEM in crack,interface crack and inclusion.It is of great importance to study the effect of particles on the growth of matrix crack for the design and application of particle reinforced composite materials.In this thesis,the interaction integral expression for the stress intensity factor of the crack tip is derived.And extend theapplication of the interaction integral by defining an appropriate auxiliary field.Then,by using XFEM and interaction integral together,the displacement approximation equation,the interaction integral method and the integral method of the element stiffness matrix are given for particulate reinforced composite XFEM.Simulations of the crack propagation path of side-crack and hole edge crack are carried out.The influence of the elastic modulus and the position of the particles on the stress intensity factor and the energy release rate are studied.Currently,the GXFEM is still limited in isotropic material,applications in the bimaterial interface crack have not been reported as far as we know.In this thesis,bimaterial interface crack is studied by using GXFEM.The displacement approximation equation of bimaterial interface crack is given.The integral method of the interaction integral and the integral strategy of element stiffness matrix are presented.A new enrichment function of bimaterial interface crack tip is proposed.The elements of enrichment function are reduced from 12 to 6 by triangular transformation,resulting in the improvement of the computational efficiency.Numerical examples indicating that calculating the stress intensity factor of the crack in bimaterial interface by the use of GXFEM is successful and effective.