Jianhui Liao

Chapter 4 – Fundamental Concept and Formula of X-FEM

The fundamental concept and formula of the extended finite element method (X-FEM) are introduced in this chapter, like partition of unity and the level set method, as well as the technical process of program implementation. The expressions for enriched shape functions of the element are provided, which include the displacement discontinuity crack, called “strong discontinuity”, and strain discontinuity interface, called “weak discontinuity”, such as inclusions and interfaces of bimaterials. The essential theory and formula of fracture mechanics are used to establish the enriched shape functions at the crack tip, which are described in Chapter 2 for steady-state cracks and in Chapter 3 for the dynamic fracture situation respectively. The deduction of a weak form of X-FEM is carried out in order to generate the finite element governing equations. Finally, fracture in one dimension is considered to illustrate the basic concept of X-FEM.

X-FEM Simulation of Two-Phase Flows

A simulation method of transient immiscible and incompressible two-phase flows is proposed in this chapter, which demonstrates how to deal with the multi-phase flow problem by applying X-FEM methodology. In particular, the projection term consisting of discontinuity and precise integration is introduced to solve the Navier–Stokes equation. A number of numerical examples describe the interface tracking and capturing schemes used to deal with interfacial discontinuity to model incompressible two-phase flows.

Chapter 6 – X-FEM on Continuum-Based Shell Elements

The two-dimensional X-FEM theory model, computational formula, and numerical examples were given in Chapters 4 and 5 Chapter 4 Chapter 5 respectively. In this chapter, a novel theory formula and computational method for X-FEM is developed for three-dimensional (3D) continuum-based (CB) shell elements to simulate arbitrary crack growth in shells using the concept of enriched shape functions. Due to the advantages of CB shell elements, the shell thickness variation and surface connection can be addressed during deformation. The stress intensity factors of a crack in the CB shell element are calculated using the equivalent domain integral method for a 3D arbitrary nonplanar crack. The maximum energy release rate is adopted as propagation criterion. Numerical examples of different fracture shells are presented.

Chapter 1 – Overview of Extended Finite Element Method

The study of computational fracture mechanics is of great importance for both scientific research and engineering applications. Since being proposed in 1999, the extended finite element method (X-FEM) has become an efficient tool for solving crack arbitrary propagation problems. The basic idea and recent progress in the development of X-FEM are reviewed. The structure of this book is also given at the end of the chapter.

Chapter 3 – Dynamic Crack Propagation

The problems of quasi-steady-state fracture mechanics are described in Chapter 2 . In this chapter, the problem of dynamic fracture mechanics is first presented to explain that simulation of crack arbitrary propagation in structures and materials is a very difficult issue. Next, node force release technology is introduced to simulate crack propagation in the conventional finite element method, which is in fact to assign the crack route and growth speed along the element edge. This technology differs from the extended finite element method (X-FEM), because cracks may experience arbitrary propagation in the element. Obviously, one of the advantages of X-FEM is that it can be used to illustrate a natural selection route of crack growth. Finally, as an example of engineering applications of dynamic fracture mechanics, rapid crack propagation or arrest in a pressurized gas pipeline is presented.

Chapter 5 – Numerical Study of Two-Dimensional Fracture Problems with X-FEM

The fundamental concept and formula of X-FEM were introduced in Chapter 4 . Based on the program developed by the authors and colleagues, numerical studies of two-dimensional fracture problems are given in this chapter to further demonstrate the capability and efficiency of the algorithms and programs of X-FEM in applications of strong and weak discontinuity problems.

Chapter 7 – Subinterfacial Crack Growth in Bimaterials

The algorithm is discussed and a program is developed based on X-FEM in this chapter for simulating subinterfacial crack growth in bimaterials. Numerical analyses of crack growth in bimaterials give a clear description of the effect on fractures made by the interface and loading. Computational results are compared with experiment data, which shows that X-FEM is more powerful in capturing the actual crack path than standard FEM. Further results show that there is an equilibrium state of mode I cracks in bimaterials if the effects of material inhomogeneity and loading asymmetry counteract each other.

Chapter 10 – Research Progress and Challenges of X-FEM

In the preceding chapters, the developments and applications of X-FEM in traditional mechanics research areas, like macro-scale fracture mechanics, plate and shell fractures, heterogeneous and composite materials, two-phase flows, etc., have been reviewed. According to recent work in the author’s group in the field of micro- and nano-scale research, we will review the broad progress and challenges of X-FEM in this chapter in micro- and nano-scale mechanics, multi-scale computation, and other new fields.

Chapter 2 – Fundamental Linear Elastic Fracture Mechanics

Fundamental linear elastic fracture mechanics are described in this chapter. The concept of energy release and balance is introduced during crack growth. Stress fields are provided for mode I, II, and III cracks at the crack tip location. The concepts of stress intensity factor and material fracture toughness are introduced. An analytical method is given to determine the stress intensity factor and energy release rate. Three kinds of complex fracture theories are discussed: the maximum circumference tension stress intensity factor theory, the minimum strain energy density stress intensity factor theory and the maximum energy release rate theory. The crack orientation angle, fracture criterion, and stress intensity factor of complex mode cracks are found using the analytical method. The interaction integral method is described to solve the stress intensity factor under quasi-steady-state conditions.

Chapter 8 – X-FEM Modeling of Polymer Matrix Particulate/Fibrous Composites

A method for representing discontinuous material properties in a heterogeneous domain using the extended finite element method (X-FEM) has been applied to study static and dynamic properties of polymer matrix particulate/fibrous composites. Representative volume elements of the composite material microstructure are generated by the random sequential adsorption (RSA) algorithm, where level set fields represent the matrix/inclusion interfaces within the domain. The equations of motion are integrated explicitly in time with mass lumping on the nodal and enriched degrees of freedom. This method shows improved agreement of the wave attenuation coefficient (WAC) from the experimental measurements of ultrasonic, longitudinal waves in particulate composites compared with analytical methods, especially for the high volume fraction. The WAC is also computed for fiber composites, including random and aligned fiber orientations.