---

Functionally graded materials have been taken into consideration by many researchers in the last two decades. Gradual changes of mechanical properties in FGMs decrease stress concentration, crack initiation and propagation and delamination. Many of the present and potential applications of FGM contain contact loading.This kind of loading causes surface crack initiation which is followed by subcritical crack propagation.Thus, propagation of surface cracks is one of the most important failure mechanisms in FG structures. In this article two dimensional sliding contact of a rigid flat punch on a homogeneous substrate with an FGM coating is studied. Plane strain condition is considered in this problem. The Properties of the substrate and the FGM layer are assumed to be elastic and the Poisson’s ratio is assumed to be constant. The modulus of elasticity in the graded layer is calculated based on TTO model approximation. This model defines a parameter q which considers the microstructural interactions. The governing equations are solved by Finite Difference method by means of MATLAB software. The influence of different parameters such nonhomogeneity,q, the dimensions of the punch, the thickness of the graded layer and the coefficient of friction on the mode I and II stress intensify factors are investigated.

---

In this paper, the eXtended Finite Element Method is implemented to model the effect of the mechanical and thermal shocks on a cracked 2D orthotropic media. The uncoupled thermoelasticity equations are considered. Isoparametric four-node and eight-node rectangular elements are used to discrete governing equations. The dynamical stress intensity factors are computed by the interaction integral method. The Newmark and the Crank–Nicolson time integration schemes are used to numerical solve the spatial-discretized elastodynamic and thermal equations, respectively. A MATLAB code is developed to carry out all stages of the calculations from mesh generation to post-processing. Several elastic and thermoelastic numerical examples are implemented, to check the accuracy of the results and to investigate the effect of the orthotropic direction on the stress intensity factors.

---

To investigate, understanding and predicting dynamic fracture behavior of a cracked body, dynamic stress intensity factors (DSIFs) are important parameters. In the present work interaction integral method is presented to compute static and dynamic stress intensity factors for three-dimensional cracks contained in the functionally graded materials (FGMs), and is implemented in conjunction with the finite element method (FEM). By a suitable definition of the auxiliary fields, the interaction integral method which is not related to derivatives of material properties can be obtained. For the sake of comparison, center, edge and elliptical cracks in homogeneous and functionally graded materials under static and dynamic loading are considered. Then material gradation is introduced in an exponential form in the two directions in and normal to the crack plane. Then the influence of the graded modulus of elasticity on static and dynamic stress intensity factors is investigated. It has been shown that, material gradation has considerable reduce influence on DSIFs of functionally graded material in comparison with homogenous material. While, static stress intensity factors can decrease or increase, depend on the direction of gradation material property.

---

In the present study, in order to evaluate the elastic displacement field and subsequently the fracture parameters within the isotropic homogeneous elastic solids with the edge or interior cracks, the extended finite element method with level set technique was used to avoid the disadvantages associated with the standard finite element method. An overdeterministic least squares method was utilized to determine the crack stress intensity factors as well as the coefficients of the higher order terms in the Williams' asymptotic series solution for structures containing crack in various modes of failure by fitting the series solution of displacement fields around the crack tip to a large number of nodal displacements obtained from the extended finite element method. For validating the results, several cracked specimens subjected to pure mode I, pure mode II, and mixed modes I/II loading were performed. Comparisons with results available from the literature obtained by the other formulations reveal the efficiency and the simplicity of the proposed method and demonstrate the capability of it to capture accurately the crack stress intensity factors and the coefficients of higher order terms.

---

This paper concerns the effect of auxiliary fields and distance of contours from the crack tip on the accuracy of stress intensity factors of Functionally Graded Materials (FGMs), using the interaction integral method. In the first step, defining auxiliary fields of displacement, strain, and stress appropriately, the interaction integral is derived which is independent of derivatives of properties of the materials. Actual and auxiliary fields of displacement, strain and stress are used to compute the interaction integral. Actual fields are obtained by isoparametric finite element method, while auxiliary fields are constructed by use of the crack tip properties on the basis of William’s solution. These auxiliary fields are not appropriate, except near the crack tip. Therefore, different non-equilibrium and incompatibility formulations are used to consider the changes in non-homogeneous material. Considering the changes in FGMs as an exponential function, the results will be obtained from these formulations and are compared with others recorded in the literature. Furthermore, considering different contours, the effect of distance of contours from the crack tip on the stress intensity factors of FGMs is examined. The results confirm that the solutions using the incompatibility and constant constitutive tensor are more accurate. In contrast the non-equilibrium method is not proper for contours which are placed far away from the crack tip and presents less accuracy.

---

Static and dynamic stress intensity factors are important parameters in the fracture behavior of the cracked bodies. In the present study the displacement correlation technique (DCT) is presented to calculate static and dynamic stress intensity factors of functionally graded materials (FGMs). The displacement field is obtained using finite element method (FEM) and ABAQUS software. To consider the variation of material properties, a subroutine is prepared in the UMAT subroutine of the software. Eight-node singularity elements are used in the FEM. As ABAQUS software is not able to calculate stress intensity factors of FGMs, so a MATLAB code is developed to obtain these factors. By analyzing an example under dynamic load, dynamic fracture behavior of orthotropic FGMs and effect of non-homogeneity parameter are investigated for two cases of material properties variation directions which are perpendicular to each other. To verify presented method, a center crack in a plate of homogeneous and FGM materials are analyzed under static and dynamic loads, the results are compared with data of literatures. The results show that, if the material properties vary parallel to the crack direction, the mode I dynamic stress intensity factor at the crack tip located in the stiffer part increases with increasing of non-homogeneity parameter, while for variation in the normal direction to the crack, this factor first increases and then decreases.

---

In this paper, the extended Finite Element Method (XFEM) is implemented to compute the Stress Intensity Factors (SIFs) for rectangular media subjected to a hygrothermal loading. In governing hygrothermoelasticity equations, the cross coupled of temperature and moisture fields and temperature-dependent diffusion in some cases are considered. Furthermore, an interaction integral for hygrothermal loading is developed to compute the stress intensity factors. The non uniform mesh of isoparametric eight-nod rectangular element is used in XFEM to decrease the absolute error in SIFs computations. In order to numerical results validation, the SIF of mode I is obtained analytically. The coupled governing equations are firstly decoupled in terms of new variables and then solved by the separation of variable method. According to the results, the moisture concentration gradient has a significant effect on the SIFs so should be considered in the model. Up to reaching temperature to its steady state, the cross coupled of temperature and moisture synchronies their time variation which affects on the time variation of SIF. At early time of thermal shock, the SIF for shorter cracks is not necessarily lesser than the longer ones. Also, the mode I SIF for longer and inclined cracks is smaller. On the other hand, considering the moisture concentration as a temperature function increases the time required to reach the moisture steady state.