The Extended Reproducing Kernel Interpolation Method For Fracture Problems

The meshless methods require only nodes and a description of the external and internal boundaries and there is no need totally or partly for element connectivity.The meshless methods can effectively eliminate complex meshing and disadvantageous influence of mesh distortion.In addition,meshless methods possess many advantages such as high continuity of approximation,good accuracy,and easy implementation of adaptive algorithms.The reproducing kernel interpolation method(RKIM)constructs shape function by a coupling of a simple function with Kronecker delta property and an enhanced function with format of the reproducing kernel particle shape function.Therefore the RKIM possesses interpolation property on any point and no less than the kernel function high order smoothness.In order to solve the elastodynamic problems in three-dimensional axisymmetric solids more effectively,a novel numerical method based on the RKIM is presented and discussed in detail.Because of axial symmetry of geometry and boundary conditions,only a set of discrete nodes on a cross section are required in the computation and therefore the preprocessing of this method is very simple.The Newmark algorithm is employed for time integration.Numerical examples showed that the proposed method for solving axisymmetric elastodynamic problems possesses the advantages of meshless methods and high precision.An extended reproducing kernel interpolation method(XRKIM)is proposed for the linear elastic fracture problems.Based on the partition of unity,both enriched functions,including jump function and crack tip singularity function,are added in the approximation of RKIM for the purpose of representing the discontinuous displacement field along crack face and stress singularity around the crack tip.The shape functions of the present method possess the properties of Kronecker delta functions and therefore the essential boundary conditions can be implemented easily.The theory of level set,construction of discontinuous approximation function and the discrete format of the governing equation are introduced in detail.Furthermore,the interaction integral method is utilized to numerically calculate the stress intensity factor(SIF)which is one of the important parameters to characterize the strength of stress field at the crack tip.Analyses of numerical examples demonstrate that the extended reproducing kernel interpolation method can effectively solve fracture problem.