Even though several computational methods have been developed and successfully applied for fracture modelling in recent decades, there are still some issues remaining before further applications in engineering analysis. Due to the properties of enrichment functions in the XFEM, it is problematic to be applied for crack initiation, e.g. when a crack is totally within an element in the XFEM, crack tip enrichments produce no discontinuity in this element. Besides, the XFEM is based on the FEM and there will be issues for large deformation problems. The approximation is obtained through the interpolation of nodes in an element, but when the element is cut by a crack and the nodes at two sides of the crack are far away after deformation, the interpolation between these nodes is unreasonable. In the phase field method for fracture modelling, the solution of an extra PDE for the phase field is required, leading to an increase of computational expense [170]. The EFGM with level sets can provide accurate description of crack geometry, but it meets a dilemma for multiple crack problems. Each crack needs two level set functions, so the computational cost increases with the number of cracks [179]. The NMM makes use of finite cover approximation and is suitable for multiple crack simulation, but the generation of covers and their truncation by cracks is still bothersome [216]. Peridynamics defines “bonds” between particles to describe the deformation of the problem and crack discontinuities are introduced by breaking the connection between particles. This method is suitable for complex crack problems but the accuracy of the crack geometry is only guaranteed by using a very fine distribution of particles which is costly. The CPM has
1.6. Outline of the thesis 29
shown its ability to handle multiple cracks and dynamic fracture but spurious crack results have been found in [204, 213]. Currently a robust method for computational fracture modelling is still needed and is the motivation of this thesis. The CPM is employed and modified to handle complex crack problems in 2D and 3D and to address issues of spurious cracking results, where the calculation efficiency is maintained by using an adaptive approach and this is the goal of this thesis. The contents of each chapter are listed as below.
Chapter 2 demonstrates basic formulations and main features of the EFGM. Although the thesis is focused on fracture modelling, the EFGM for continuum solids is firstly demonstrated, because the CPM is based on the EFGM by introducing crack discon- tinuities into the displacement approximation. It starts from setting up the governing equations of the EFGM, and then the strong form and weak form are introduced. Three techniques of integration required in the weak form are included: nodal integration, inte- gration with background meshes and integration over supports. Since essential boundary conditions cannot be imposed to the weak form directly in the EFGM, four methods in- cluding Lagrange multipliers, penalty method, Nitsche’s method and coupling with the FEM are discussed. An example is used to demonstrate the differences between the EFGM and the FEM.
In Chapter 3, an adaptive CPM is developed for 2D cracks. The development of the CPM is reviewed and issues of the method are discussed. A key contribution of this research is the use of a set of bilinear discontinuous segments centred at particle to approximate a continuous crack path instead of straight discontinuous segments in early CPMs. Cracking angles can be recorded by changing the orientation of those segments, so problems of spurious crack results can be relieved. An adaptive approach is introduced to the CPM to handle the stress gradients around the crack tip, by which particle refinements are added automatically, and different error estimators for adaptivity are included. Several methods to calculate SIFs are mentioned, including an contour integration and a domain interaction integration.
Chapter 4 applies the method developed in Chapter 3 to problems with multiple crack. Different methods to model crack branches and intersections are mentioned, which are through enrichments in basis functions or weight functions. A multi-cracked particle
method is developed which can achieve discontinuities at crack branches in an easier way than through enrichments. Several example with complex crack geometries are used to test the ability of the proposed method.
Chapter 5 contains a discussion of different crack propagation criteria. A review on the maximum principal stress criterion, the minimum strain energy density criterion and the maximum strain energy release rate criterion are included. The theory configurational force for fracture modelling is presented here and its implementation in MMs is also mentioned. A comparison between the use of propagation criteria based on the J-integral and the configurational force is carried out and demonstrated with several crack problems. In Chapter 6, the modified CPM is extended from 2D to 3D cracks. Conventional methods of level sets and triangular meshes to describe 3D crack surfaces are reviewed. The CPM in 3D is combined with triangular facets, where the cracking changes are recorded by modifying the orientation of triangular meshes. Calculation of SIFs in 3D is through an interaction integration method in a local coordinate system decided by triangular meshes. Cracking angle is determined by the maximum principal stress criterion while the crack increment length along a crack front is decided by the Paris’ law. Examples with planar or curved 3D crack surfaces are tested and a comparison between the proposed method and the original CPM is included.
Chapter 7 demonstrates a further development of the proposed method for modelling thermoelastic crack problems. Governing equations of the heat flux are introduced and the implementation of the weak form is included. Two types of cracks under thermal loadings are mentioned: an adiabatic crack and an isothermal crack. An adaptive ap- proach is introduced so enrichment functions can be avoided which are commonly used in current numerical methods. Due to the effect of thermal stress, the interaction integral method needs modification to make it compliant with the assumptions of thermal fracture mechanics.
The thesis finishes with Chapter 8 summarizing all the research work mentioned above and providing an outlook of future directions in terms of MMs and computational fracture mechanics.
Chapter 2
Formulation of the element free
Galerkin method
2.1
Introduction
In this chapter, fundamental formulations of the EFGM are demonstrated before intro- ducing the CPM, since the CPM is based on the EFGM and is specific to crack problems. It starts from the moving least squares approximation, which is used to approximate the displacement field with a set of nodes, while elements are not involved to discretise the problem domain. Then the basic theory of solid mechanics is introduced, precisely the equilibrium of solids under static or quasi-static external loadings. These equilibrium equations are not solved directly but are converted to the weak form, which provides nu- merical benefit of reduced order differentiation of the governing equations. Major issues in meshless methods of integration and imposing essential boundary conditions are dis- cussed and several approaches are provided to address these issues. Finally, an example is used to demonstrate the difference between the FEM and the EFGM.