In this thesis, a general finite element method for multiple growing discontinuities is presented. More specifically, multiple cracks are grown using the eXtended Finite Element Method (X-FEM), where growth and coalescence of cracks and percolation can be treated without remeshing. The cracks are described by vector level sets in brittle elastic media that can be homogeneous or inhomogeneous and contain bi-materials or holes. First, brittle materials are studied in the framework of linear fracture mechanics; a stability analysis is developed when multiple cracks are competing to grow at the same time. The purpose of this thesis is to develop a multiple crack growth model for unit cells at the micro-scale that could be eventually coupled later with a macro-scale model. The method does not limit the number of cracks, and is applied to unit cells with up to ten growing cracks; it is also used to model fatigue fracture described by a Paris law with up to 50 cracks. Preliminary results of multiple cracks growing by a cohesive crack growth law are presented at the end of the thesis.

Novel techniques for modeling 3D cracks and their evolution in solids are presented. Cracks are modeled in terms of signed distance functions (level sets). Stress, strain and displacement field are determined using the extended finite elements method (X-FEM). Non-linear constitutive behavior for the crack tip region are developed within this framework to account for non-linear effect in crack propagation. Applications for static or dynamics case are provided.

Introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics Explores the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation. Covers numerous applications of XFEM including fracture mechanics, large deformation, plasticity, multiphase flow, hydraulic fracturing and contact problems Accompanied by a website hosting source code and examples

Volume is indexed by Thomson Reuters BCI (WoS). Crack growth is a complex phenomenon and difficult to model without assumed simplifications. In recent years, there has been an increasing realization that advanced computational methods such as the Finite Element Method, the Boundary Element Method and Mesh Free Methods can be used to simulate crack growth in complex structural parts. This special issue contains original papers written by active researchers in the field of computational fracture mechanics. The aim is to inform those in the fracture mechanics community who might not be at the forefront of computational research of the recent developments and techniques available and the wide range of applications for which solutions are now possible.

An overview of the virtual crack closure technique is presented. The approach used is discussed, the history summarized, and insight into its applications provided. Equations for two-dimensional quadrilateral elements with linear and quadratic shape functions are given. Formula for applying the technique in conjuction with three-dimensional solid elements as well as plate/shell elements are also provided. Necessary modifications for the use of the method with geometrically nonlinear finite element analysis and corrections required for elements at the crack tip with different lengths and widths are discussed. The problems associated with cracks or delaminations propagating between different materials are mentioned briefly, as well as a strategy to minimize these problems. Due to an increased interest in using a fracture mechanics based approach to assess the damage tolerance of composite structures in the design phase and during certification, the engineering problems selected as examples and given as references focus on the application of the technique to components made of composite materials.

This book treats computational modeling of structures in which strong nonlinearities are present. It is therefore a work in mechanics and engineering, although the discussion centers on methods that are considered parts of applied mathematics. The task is to simulate numerically the behavior of a structure under various imposed excitations, forces, and displacements, and then to determine the resulting damage to the structure, and ultimately to optimize it so as to minimize the damage, subject to various constraints. The method used is iterative: at each stage an approximation to the displacements, strains, and stresses throughout the structure is computated and over all times in the interval of interest. This method leads to a general approach for understanding structural models and the necessary approximations.

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