Modeling method for the static crack problems of functionally graded materials with general properties

In this chapter, the modeling methods for the crack problems of functionally graded materials (FGMs) with general (or arbitrary) properties are reviewed and discussed. The focus of this chapter is on the analytical and numerical modeling methods for the crack problem of FGMs with general properties. Firstly, the basic problem (a "particular" crack problem of a functionally graded layered structure with exponential properties and a crack crossing the interface) for establishing the piecewise-exponential model (PE model) is presented. Then the PE model for FGMs with general mechanical properties is introduced. In the PE model, the FGM is divided into some nonhomogeneous layers along the gradient direction of the properties, with each layer's properties varying exponentially. By this method, the real material properties can be approached by a series of exponential functions. Moreover, considering the significance of the investigation of the thermal fracture behavior for FGMs, the PE model is extended to the crack problems of FGMs with general thermomechanical properties. It should be mentioned that: in past decades, there are very limited literatures on the analytical study of crack problems in the FGMs with non-exponential properties. However, for the FGMs with exponential properties, there have been lots of investigations regarding many types of crack problems. These achievements will provide a wide support for the PE model to be applied in solving many types of crack problems in FGMs with general properties. On the other hand, the analytical modeling methods for the crack problems of FGMs with general properties are very difficult to consider the influences of the complex internal structures (such as the component particles and interfaces) on the fracture behavior of FGMs. Therefore, numerical modeling methods for the crack problems of FGMs with general properties should be developed together with the analytical methods. The key problem in the numerical modeling methods lies in how to obtain the crack tip quantities with high efficiency. An interaction integral and its associated domain form without any derivatives of the material properties are introduced. It can be used to analyze the crack problems of FGMs with general properties, considering the effects of the component particles and internal interfaces on the fracture behavior.