In this paper, the meshless local Petrov-Galerkin (MLPG) method is used to analyze the dynamic fracture of an isotropic FGM plate containing a center crack. The dynamic Stress Intensity Factors are studied under the influence of various non-homogeneity ratios. Both the moving least square (MLS) and the direct method have been applied to estimate the shape function and to impose the essential boundary conditions. The enriched weight function method is used to simulate the displacement and Stress field around the crack tip. Normalized dynamic Stress Intensity Factors (NDSIF) are calculated using the path independent integral, J*, which is formulated for the nonhomogeneous material.To validate the method, the homogenous center cracked plate problem is analyzed. The obtained results show good agreement between the analytical solution and the MLPG results for homogenous material. After validation, a center cracked plate made of FGM with two different material gradations (along and normal to the crack length) and three different lengths of FGM zone under the effect of step load are considered, and the following six distinct problems are studied here.

In the current research work, the problem of fracture mechanics in a plate with a central hole under tensile loading is studied. The Stress Intensity Factors are calculated for a finite plate containing two symmetrical hole-edge cracks. The problem is solved by two different methods, namely the finite element method and the FRANC software analysis. At first the finite element method is used and by writing a program in MATLAB software the Stress Intensity Factors at the crack tips are calculated. The same problem is then reanalyzed with the Franc software and the results are compared. The effects of various Factors such as the hole diameter, crack length and crack angle have been investigated on Stress Intensity Factors. The results show that for small crack lengths, the effect of cracks length is more than that of the hole diameter on variation of normalized Stress Intensity Factors, while it is the opposite for large crack lengths, the effect of hole diameter is more than that of the cracks length on variation of normalized Stress Intensity Factors.

In this paper, numerical solutions of multiple cracks problems in an infinite plate are studied. Hypersingular integral equations (hieq) for the cracks are formulated using the complex potential method. For all kernels such as regular or hypersingular kernels, we are using the appropriate quadrature formulas to solve and evaluate the unknown functions numerically. Furthermore, by using this equation the Stress Intensity Factor (SIF) was calculated for crack tips. For two serial cracks (horizontal) and two dissimilar cracks (horizontal and inclined), our numerical results agree with the previous works.

The use of numerical methods, particularly finite element methods in solving different problems are used in abundance. Because these methods are approximate, having a real understanding of the extent and distribution of the errors is extremely important. Studies show that the mesh used in the finite element method will be an essential error rate. So many different ways to find the optimal mesh and minimize the error of the proposed method. These methods are mainly aimed at reducing the error in the Stress field have been obtained. Although one of the aims of reducing errors tension field finite element method is adaptive, but no guarantee is intended to achieve specific issues. These issues can be Craek to the issues noted in the analysis of the elastic parameters of the Stress Intensity Factor will play a key role in determining the direction and Craek the design. The purpose of this study is to measure the Stress Intensity Factor adaptive analysis on the parameter to be modified to accommodate ancillary parameters such as strain, Stress Intensity Factor able to modify the target parameter.

In this paper, the Stress Intensity Factor for a circumferential crack in a thick-walled cylinder is derived analytically and numerically which is subjected to the non-Fourier (hyperbolic) thermal shock. The uncoupled thermoelasticity governing equations for an uncracked cylinder are solved analytically. The weight function method is implemented to obtain the Stress Intensity Factor. The non-dimensional hyperbolic heat equation is solved using finite Hankel transform and separation of variables method. Results show the different behavior of the crack under hyperbolic thermal shock. For relatively short cracks, the maximum Stress Intensity Factor of Fourier and hyperbolic models is closed. But for longer cracks, the Stress Intensity Factor of the hyperbolic model is significantly greater than Fourier model. Moreover, the maximum Stress Intensity Factor in hyperbolic model occurs for a crack the peak of Stress wave reaches to its tip. According to the results, assumption of adequate heat conduction model for structure design under transient thermal loading is critical.

In the different applications of thin plats in engineering industrial, some holes are created in the structure that can have different shapes such as circular, elliptical, and quasi-square. When the plate is subjected to loading, Stress concentration around the hole causes the crack initiation in these areas that can results in a catastrophic failure. In this paper, mode II Stress Intensity Factor (SIF) for two unequal aligned cracks emanating from a circle or a quasi-square hole in an infinite plane was investigated. The complex variable theory of Muskhelishvili and conformal mapping method were used. To obtain mapping function, Schwarts Christoffel integral was combined with some simple mapping functions. Accordingly, a new mapping function is presented and approximated to the sum of series expansion. Using this approximate mapping, SIF is calculated with high accuracy. Surfaces of the cracks and hole are traction-free. The plane is subjected to the pure shear at infinity. The analytical results are in good agreement with the literature. The obtained Stress Intensity Factors have good accuracy for small cracks. The equation presented in this paper is applicable to the length of the different cracks and calculates the Intensity coefficients of mode II for very small cracks with high accuracy. Results show that the shape of the hole is important only for the small cracks.

In this paper, the Stress Intensity Factor (SIF) in the opening mode I is investigated in cracked beams with the I-shaped cross sections under axial loading. In cracked cross sections, a couple is made due to shifting the two centroids, or in the other word due to a misalignment between the axes of the axial force. The analysis is carried out through two analytical and numerical approaches. In analytical approach, a mathematical model for SIF is proposed via the theory of the energy release rate in the region around the crack. This model is adapted by considering the moment of the couple. It is presented in two situations in terms of the crack location including the crack in a part of the flange, and the crack in the flange and a part of the web. In numerical approach, geometric and material characteristics and type of loading are modeled by using Abaqus software; then SIF values of the I-shaped cracked beam are determined. In the presented numerical solution, two methods of C-integral and X-FEM are employed to model the crack. The validity of proposed equations is confirmed by the comparison between the presented results of numerical and analytical approaches.

This study shows how to create different types of crack and discontinuities by using isogeometric analysis approach (IGA) and extended finite element method (XFEM). In this contribution, two unique features of isogeometric analysis approach are utilized to create discontinuous zones. Discontinuities consist of crack and cohesive zone. In isogeometric analysis method NURBS is used to approximate both geometry and primary variable. NURBS can create quadratic shapes exactly. Also, Stress Intensity Factors are calculated in the vicinity of the crack tips for two dimensional problems and are compared with corresponding analytical and numerical counterparts. Extended finite element method is the other numerical method which is used in this work. The enrichment procedure is utilized in extended finite element method to create discontinuities. The wellknown path independent J-integral approach is used in order to calculate the Stress Intensity Factors. Also, in mixed mode situation, the interaction integral (M-integral) is considered to calculate the Stress Intensity Factors. Results show that isogeometric analysis method has desirable accuracy as it uses lower degree of freedoms and consequently lower computational efforts than extended finite element method. In addition, creating the internal cohesive zone as one of the most important issues in computational fracture mechanics is feasible due to the special features of isogeometric analysis. The present study demonstrates the capability of isogeometric analysis parametric space to control the inter-element continuity and create the cohesive zone.

Numerical methods, especially the Finite Element method, are increasingly being used for solving different problems. Due to fact that these methods are approximate, having a good understanding and judgment about the errors and their distribution is very important.Hence education of users of engineering analysis and design software is necessary and inattention to it may result in catastrophe. As regards this issue, the problem of determining the Stress Intensity Factor in a cracked plate under tensile Stresses, by using the finite element method together with error estimation and adaptivity, is the subject of this article. For this purpose, an academic FORTRAN code has been developed which is able to estimate the finite element solution error by using the superconvergent patch Stress recovery method. In addition, an adaptive solution with remeshings in each step is carried out to improve the quality of the employed finite element mesh. Comparing the obtained results with the analytical solution, as well as the ANSYS commercial software, it is observed that the employed algorithm for error estimation has a better performance and can be used for determination of the Stress Intensity Factor in complex structures with arbitrary cracks.