1. Introduction

Development of novel materials improves performance and efficiency of the structures, and also leads to development of advanced and sophisticated structures. This complex process of materials, structures and technology has led to the development of composite materials. Strength and stiffness plays a key role in evaluating the worth of the material. These characteristics provide strength to the structure to retain its desired shape and size under loading or any other external action.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Cracks/flaws are inevitable in all engineering materials. Loading under severe environmental conditions may either initiate new cracks or may cause the propagation of pre-existing cracks in the structures. Theoretically, fracture can be defined as the breaking or rupturing of a material resulting into its separation into two or more pieces.

Fatigue and quasi-static fracture are two forms of crack growth phenomenon. Fatigue fracture refers to the slow propagation of cracks under cyclic loading conditions where the stress intensity factors are below the fracture toughness of the material. Quasi-static fracture is observed near the end of the fatigue life when the increased crack length leads to stress

Fatigue Fracture of Functionally Graded Materials Under Elastic-Plastic Loading Conditions Using Extended…

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In many cases, multiple cracks may exist in the components. There interaction resulting in the variation of stress intensity factor, stress distribution and propagation direction of the major crack. In the past, the failure of structures was analyzed in the presence of multiple cracks [4]. Some efforts have been made using analytical, experimental and simulation techniques to

Although, many analytical [8, 9] and experimental methods [10, 11] have been explored for the calculation of fracture parameters even then the drawbacks associated with experimental investigation and scarcity of analytical solution have impelled the analysts towards alternative

Many numerical methods are available to simulate the problems of fatigue failure in materials. These include finite element method (FEM), boundary element method, hybrid boundary node method [12, 13], meshfree methods [14–18] and extended finite element method [19, 20]. Out of these methods, FEM has been widely used for solving a wide variety of engineering and industrial problems [21–26]. It has achieved a remarkable success in solving various linear and non-linear problems [27–34]. Despite its numerous advantages and unparalleled success, it is not well-suited for solving the problems involving crack propagation. In crack growth problems, element edges provide natural lines along which a crack can grow. This is advantageous if the crack path is known a priori, but in most of the fracture phenomenon, the crack path is unknown. Thus, FEM requires a conformal mesh and re-meshing to ensure that the element boundaries coincide with the moving discontinuities (crack). Moreover, crack tip singularity cannot be accurately modeled by standard finite element approximation. Therefore, the modeling of crack growth becomes quite tedious and time consuming due to the modification in mesh topology at each stage of crack propagation. To overcome this difficulty, a new method known as extended finite element method (XFEM) has been developed to model arbitrary discontinuities without a need of conformal mesh or re-meshing. Level set method (LSM) is used in conjunction with XFEM for defining as well as tracking the geometry of cracks and other discontinuities like holes and inclusions. To cope up with these problems, XFEM has been adopted as a tool for the analysis of fatigue crack propagation in FGM.

A domain based interaction integral approach can be used for calculating the stress intensity factors for homogeneous, bi-layer and functionally graded materials under thermal as well as mechanical loading. In this chapter, interaction integral approach will be extended to calculate the SIFs for FGM and bi-layered FGM under mechanical loads. The interaction integral is calculated based on J-integral. The J-integral for an elastic body subjected to thermo-

intensity factors which are above the fracture toughness [2, 3].

analyze the effect of interaction among multiple cracks [5–7].

techniques. Numerical methods hold the promise in this regard.

2. Calculation of SIF for FGM

mechanical load is given as,

Composite materials manifested in the middle of the twentieth century. Composites are naturally occurring or engineered materials made from two or more constituents with different chemical or physical properties distinct boundary among constituents. Lightweight composite materials with high strength to weight and stiffness to weight ratios have been used successfully in aircraft industry and other engineering applications. Under high temperature conditions the strength of the metal is deteriorated whereas, ceramics have excellent resistance to heat.

FGMs can be referred as multiphase composite materials in which the composition or microstructure or both are spatially varied which lead to a certain gradation in the local material properties. FGMs can be defined as multi-phase composites. FGMs are synthesized such that they own continuous variations in volume fractions of their components in space to return a pre-established composition. FGMs possess continuously varying properties in one or more than one direction and the form non-homogeneous macrostructure due to these variations. By gradually varying the volume fraction of the constituents, FGMs exhibit a smooth and continuous change from one surface to another, thus reducing interface problems, and minimizing thermal stress concentrations. The ceramic phase of FGMs provides a good resistance to heat, while the metal phase provides a strong mechanical performance and hence reduces the possibility of catastrophic failure.

The major advantages of FGM over conventional materials are firstly, FGM satisfies the working conditions for which it is specifically developed. Secondly, it is economical as it reduces material costs for particular engineering applications. Thirdly, it can reduce the magnitude of residual and thermal stresses generated under working conditions. Finally, FGMs exhibit better fracture toughness and bond strength. This is normally achieved by using a ceramic layer connected with a metallic layer. FGMs have wide area of engineering applications like in the computer circuit and aerospace industries. FGMs have typical applications is in aircraft and automotive industries as thermal barrier coatings (TBCs).

In general, all structural components are subjected to thermo-mechanical cyclic load. The fatigue life of these components is generally predicted without considering the effect of defects/discontinuities present in component. However, FGMs are commonly made by sintering process, which are porous in nature. These discontinuities at the vicinity of a major crack tip lead to increase the effective SIF at the major crack tip due to which the life of the components get depreciated. Hence, the analysis of FGMs in the vicinity of discontinuities becomes very important from the design point of view. To widen the spectrum of applications of FGMs, the fatigue/fracture behavior should be properly evaluated.

Over the years, greater understanding of fracture mechanics has undoubtedly prevented a significant number of structural failures. Fracture mechanics approach for the design of structures includes flaw size as one of the key variables. Fracture toughness replaces strength of material as a relevant material attribute, and its evaluation is mainly done in composites using the J-integral approach [1]. Failure of FGM has always been a trending domain of research for scientists and engineers due to the wide spectra of their engineering applications.

Fatigue and quasi-static fracture are two forms of crack growth phenomenon. Fatigue fracture refers to the slow propagation of cracks under cyclic loading conditions where the stress intensity factors are below the fracture toughness of the material. Quasi-static fracture is observed near the end of the fatigue life when the increased crack length leads to stress intensity factors which are above the fracture toughness [2, 3].

In many cases, multiple cracks may exist in the components. There interaction resulting in the variation of stress intensity factor, stress distribution and propagation direction of the major crack. In the past, the failure of structures was analyzed in the presence of multiple cracks [4]. Some efforts have been made using analytical, experimental and simulation techniques to analyze the effect of interaction among multiple cracks [5–7].

Although, many analytical [8, 9] and experimental methods [10, 11] have been explored for the calculation of fracture parameters even then the drawbacks associated with experimental investigation and scarcity of analytical solution have impelled the analysts towards alternative techniques. Numerical methods hold the promise in this regard.

Many numerical methods are available to simulate the problems of fatigue failure in materials. These include finite element method (FEM), boundary element method, hybrid boundary node method [12, 13], meshfree methods [14–18] and extended finite element method [19, 20]. Out of these methods, FEM has been widely used for solving a wide variety of engineering and industrial problems [21–26]. It has achieved a remarkable success in solving various linear and non-linear problems [27–34]. Despite its numerous advantages and unparalleled success, it is not well-suited for solving the problems involving crack propagation. In crack growth problems, element edges provide natural lines along which a crack can grow. This is advantageous if the crack path is known a priori, but in most of the fracture phenomenon, the crack path is unknown. Thus, FEM requires a conformal mesh and re-meshing to ensure that the element boundaries coincide with the moving discontinuities (crack). Moreover, crack tip singularity cannot be accurately modeled by standard finite element approximation. Therefore, the modeling of crack growth becomes quite tedious and time consuming due to the modification in mesh topology at each stage of crack propagation. To overcome this difficulty, a new method known as extended finite element method (XFEM) has been developed to model arbitrary discontinuities without a need of conformal mesh or re-meshing. Level set method (LSM) is used in conjunction with XFEM for defining as well as tracking the geometry of cracks and other discontinuities like holes and inclusions. To cope up with these problems, XFEM has been adopted as a tool for the analysis of fatigue crack propagation in FGM.
