Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling

Abstract

SUMMARYFracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind, but the potential benefits of no meshing (particularly in 3D) prompt continued research into their development. In methods where the crack face is not explicitly modelled (as the edge of an element for instance), two procedures are instead used to associate the displacement jump with the crack surface: the visibility criterion and the diffraction method. The visibility criterion is simple to implement and efficient to compute, especially with the help of level set coordinates. However, spurious discontinuities have been reported around crack tips using the visibility criterion, whereas implementing the diffraction method in 3D is much more complicated than the visibility criterion. In this paper, a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved. The formulation is based on the use of level set coordinates and the element-free Galerkin method, and is generally applicable for single or multiple crack problems in 2D or 3D. The paper explains the formulation and provides verification of the method against a number of 2D crack problems.

A meshless sub-region radial point interpolation method for accurate calculation of crack tip fields

“…As has been in shown [25] the number of integration points required for results of acceptable accuracy is typically about 10 times the total number of nodes, and is heighten for fracture modelling in 2D to 11~12 times for single crack problem in 2D [19]. The integration of the present method is slightly different from that of the EFGM or the original RPIM in that it performs two independent integration over V p and V c as shown in Figure 4.…”

“…Since the stress terms in Eqs. (18) and (19) use the Williams stress expansion as the basic solution, the SIFs for both mode I and II at the crack tip can be directly found as…”

“…Recently, much effort has been directed towards the application of meshless methods to crack problems to overcome the difficulties in traditional numerical methods [9][10][11][12][13][14][15][16][17][18][19]. Despite clear general progress with these methods, there are still some technical issues in their application to fracture problems, for instance, it is often awkward and an expensive task to refine the nodal arrangement near the crack tip in order to increase the solution accuracy, since the stress results tend to be oscillatory near the crack tip.…”

“…However, introducing such an enriched basis in meshless approximations can lead to ill-conditioning of the global stiffness matrix, and special treatments [12,17] have to be used to alleviate this problem. Thirdly, many meshless methods employ the J-integral or contour integral scheme for the calculation of SIF, which is performed as a post-processing step applied to the stress results, such as in the formulations using the FEM described in [15][16][17][18][19] and partition of unity enriched boundary element method (PU-BEM) [21,22]. This is unlike the case with the isoparametric FEM or sub-region mixed variational principle based FEM where the SIF can be directly obtained as part of the solution [3][4][5].…”

An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems

“…Meshless methods remain of interest in the computational mechanics community, in particular the element-free Galerkin method (EFGM) [1], because in these methods only a set of nodes is required for the problem discretisation, making them ideal for modelling problems with large deformation, material damage, projectile penetration, fragmentation, crack growth and moving boundaries, the details of which can be found in [2,3,4,5,6,7,8,9,10,11,12]. However it is well-known that they tend to be more computationally expensive than finite element methods (FEMs) set the same job.…”