Improved Complex Variable Element Free Galerkin Method For Linear Elastic Crack Growth Problems

Meshless method is a new class of numerical method.Compared with traditional numerical methods,it can get rid of constraints of elements or grids and only needs relevant information of nodes when constructing shape functions.Meshless method obtains advantages of wide application range and high accuracy.Due to the absence of grid singularity and distortion,meshless method has been widely used to solve problems of fracture,large deformation,elastoplasticity and so on.So far meshless method has become one of the research hotspots for engineering and scientific researchers,and it also becomes the trend of science and engineering computing.In order to avoid the shortcomings of moving least-squares method,such as a large amount of calculation and ill-conditioned matrix,a complex variable element-free Galerkin method which combined a complex variable moving least-squares method(based on an approximation of a vector function)and an element-free Galerkin weak form was introduced.In addition,two improved complex variable element-free Galerkin methods were introduced to give the approximation function explicit mathematical and physical meanings,the first one used a new functional to approximate vector functions,while the second one brought conjugate basis functions.Those complex variable methods reduce computational cost by reducing the number of basis function terms so as to decrease undetermined coefficients in trial function.Discontinuous fields near cracks and singular fields at crack tips are two basic topics which need to be dealt with in meshless fracture analyses.The former problem always needs to introduce certain kind of discontinuous criteria such as the visibility criterion,the diffraction method,and the transparency method,while the latter one usually uses enriched basis functions or enhanced trial functions.In order to simulate the discontinuous field at crack tip of multiple cracks or branch cracks,a new discontinuous criterion,the modified weight function method,was introduced and improved in this paper.The local coordinate system of the modified weight function method was reconstructed to simplify the derivation process of the modified weight function formula,and correcting strategies as well as schemes for different calculation points in multi-crack computational domain were proposed to optimize computational processes.The calculation results showed that the modified weight function method can easily deal with multi-crack problems and had high calculation accuracy.In this paper,two improved complex variable element-free Galerkin methods were applied to fracture mechanic problems of elastic bodies with cracks.By applying Galerkin's weak formula of fracture mechanic problems and applying essential boundary conditions by penalty function method,an improved complex variable element-free Galerkin method for fracture mechanics was established.Crack discontinuity field was treated by four discontinuity criteria in meshless method.Numerical analyses of single crack and multiple cracks in elastic bodies were carried out respectively,displacement fields,stress fields and stress intensity factors at the crack tips were given and compared.Numerical results showed that compared with complex variable element-free Galerkin method,the improved complex variable element-free Galerkin method has higher accuracy in calculating stress field,displacement field and stress intensity factors at crack tip.The accuracy of diffraction method or transparency method is better than that of visibility criterion,while the modified weight function method can obtain stress intensity factors with higher accuracy,and fit the singular field of multiple cracks better.At present,researchers at home and abroad have not yet adopted complex variable element-free Galerkin method to deal with crack propagation problems.In this paper,two improved complex variable element-free Galerkin methods were applied to linear elastic crack growth problems.By applying Galerkin's weak formula of fracture mechanic problems,the improved complex variable element-free Galerkin method for crack growth problem was established.Crack propagation conditions,criteria for computing the angle of crack propagation and the implementation process of crack propagation were given.Crack propagation analyses of two-dimensional beams,slabs with cracks or holes were carried out respectively.Crack growth trajectories and stress intensity factor curves were given.The crack paths calculated by different sizes of M-integral domain,steps and different complex variable basis functions were ploted and analyzed.These paths match well with those in literature,which verifies the correctness and validity of referred methods in this paper.