| With the high-speed development of computer technique in recent decades, the effect of numerical calculation in science and engineering has been enhanced gradually, and it has become an important means of scientific research besides theoretical analysis and experimental verification. Functional composite materials are a kind of new material with specific function composited by two or more materials. It has been applied widely to aerospace, constructional engineering, petrochemical industry, electrical engineering, medical and biomimetic engineering, and other high-tech engineerings. However, due to its complexity and heterogeneity, it is hard for traditional numerical methods to be used in functional composite materials. The meshless method as a new numerical method with great prospect developed after traditional numerical methods such as Finite Element Method, Boundary Element Method et al. The most outstanding advantage of meshless method is its independence of meshes. It dispenses with initial partition and reconstruction of meshes. Therefore, the meshless method makes it easy to deal with large deformation of materials in the press forming process of industial materials, crack propagation and fracture analysis, penetration, splash problems of materials in ultra high speed collision process, and flow solid coupling problems et al. thus attracts increasing attentions of researchers. Compared with other meshless methods, element-free Galerkin method is more mature and has been widely used. It has high precision and convergence speed, and good stability without the volume locking phenomenon. The rapid development of element-free Galerkin method has set off hot research interests in the international community of computational mechanics. In this paper, partition of unity method and element-free Galerkin method are combined, enriched element-free Galerkin method based on partition of unity is proposed, and developed through its application in fracture mechanics problemsAt the beginning of the paper, the developmet history of meshless method at home and abroad is overviewed. Several typical meshless methods are reviewed and appraised, and the characteristics, advantages and problems of them are summarized. Applications of meshless method in fracture mechanics problems are introduced. Aimed at fracture problems, meshless method only needs the information of nodes in the calculation, without any grid. Therefore it has the superiority of other methods in dealing with crack problems. Then the study of fracture problems by meshless method becomes a hot research field. In this paper enriched terms are introduced into the approximation function of the conventional element-free Galerkin method to describe the displacement and electric fields near cracks, based on the partition of unity method. With Heaviside function and trigonometric function added as the enhancement function approximation, the specific item is given to enrich the shape function of meshless method. The enriched element-free Galerkin method based on partition of unity is proposed, which can simulate the singularity of the crack tip stress field very well. The discrete of control equations, the structure of the strain-displacement matrix and the assembly of the stiffness matrix are described in details. The handling of the boundary condition is discussed.The enriched element-free Galerkin method based on partition of unity is firstly proposed, and has been applied to fracture mechanics. In the problems of composite material fracture mechanics, the displacement field near the crack tip, the control equations and boundary conditions of composite material statics have been given. The meshless formula of plane fractrure in composite materials is deduced. Because of the particularity of composite materials, the J integral path independence is no longer valid. Therefore, the traditional J integral has been modified, and the modified J integral of composite materials has been put forward. The modified J integral of cracks in different length and oblique cracks has been calculated. For the fracture problems of piezoelectric materials, the control equations have been given. The enriched element-free Galerkin method based on partition of unity has been applied to solve the plane fracture problems of piezoelectric materials. The enriched terms are added in the approximate displacement functions and potential functions to describe the crack displacement and electric field, the stiffness matrix of the electromechanical coupling enriched meshless method is assembled, the meshless formula of piezoelectric materials in plane fracture problems is derived, the enriched element-free Galerkin method based on partition of unity to model piezoelectric materials with cracks is presented. And the general expressions of stress intensity factors and the electric displacement intensity of piezoelectric material fracture problems are derived based on the general solution to piezoelectric material plane fracture problems, the method of calculating energy release rate by displacement and electric potential of the crack surface are proposed. For the plane fracture problems of functionally graded materials, enriched element-free Galerkin method based on partition of unity is applied to solve the plane fracture problems of functionally graded materials for the first time. The enriched meshless formula of the plane fracture problems in functionally graded materials is derived based on the statics control equation and boundary condition of functionally graded materials, and the suitable approximation function, weighting function and penalty function are selected. Due to the particularity of functionally graded materials, the path independence of J integral is no longer valid. Therefore, the traditional J integral is modified. The modified J integral of functionally graded materials is proposed. The crack problems under different boundary conditions, with different crack length and different gradient distribution are solved. For the fracture problems of functionally graded piezoelectric materials, the electromechanical coupling problems of functionally graded piezoelectric material plate with cracks are simulated by enriched element-free Galerkin method based on partition of unity. The enriched meshless formula of the plane fracture problems in functionally graded piezoelectric materials is derived based on the statics control equation of functionally graded materials and fundamental equatiuon of piezoelectric materials. Due to the gradient changes of parameters of functionally graded materials with spatial coordinates as variables, the independence path is no longer valid. Therefore, the traditional J integral is modified. The modified J integral of functionally graded materials is proposed. The modified J integral in crack problems of the functionally graded piezoelectric materials with holes, cracks in different length and different gradient distribution is calculated by using different Gauss integral.No matter in the fracture problems of composite materials, piezoelectric materials, functionally graded materials or functionally graded piezoelectric materials, all the results of numerical examples show that, this method is feasible and effective for problems in fracture mechanics, and the calculation results are good in accuracy. |
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