B Nageswara Rao

This paper presents fractal finite element based continuum shape sensitivity analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates. © 2009 Elsevier Ltd. All rights reserved.

In this paper, a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions, is presented. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Three numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study. © 2008 Elsevier B.V. All rights reserved.

This paper presents a coupling technique for integrating the element free Galerkin method (EFGM) with the fractal finite element method (FFEM) to analyze unbounded problems in the half space. FFEM is adopted to model the far field of an unbounded domain and EFGM is used in the near field. In the transition region interface elements are employed. The shape functions of interface elements which comprise both the element free Galerkin and the finite element shape functions, satisfy the consistency condition thus ensuring convergence of the proposed coupled EFGM FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary. The numerical results show that the proposed method performs extremely well converging rapidly to the analytical solution. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the weight function, the scaling parameter and the number of transformation terms, on the quality of the numerical solutions.

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with the fractal finite element method (FFEM) to analyze unbounded problems in the half-space. FFEM is adopted to model the far field of an unbounded domain and EFGM is used in the near field. In the transition region interface elements are employed. The shape functions of interface elements which comprise both the element-free Galerkin and the finite element shape functions, satisfy the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary. The numerical results show that the proposed method performs extremely well converging rapidly to the analytical solution. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the weight function, the scaling parameter and the number of transformation terms, on the quality of the numerical solutions. © 2011 Elsevier Ltd.

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, KI and KII, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or KI and KII. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study. © 2008 Elsevier Ltd. All rights reserved.

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two-dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress-intensity factors (SIFs) and T-stress. The numerical results show that SIFs and T-stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed-mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM-FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack-propagation analysis can be dramatically simplified. © 2010 John Wiley & Sons, Ltd.

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with fractal the finite element method (FFEM) for analyzing homogeneous, anisotropic, and two dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element-free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T-stress. The numerical results based on all four orthotropic cases show that SIFs and T-stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. © 2010 Springer Science+Business Media B.V.

This paper presents a coupling technique for integrating the fractal finite element method (FFEM) with element free Galerkin method (EFGM) for analyzing homogeneous, isotropic, and two dimensional linear elastic cracked structures subjected to mode I loading condition. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled FFEM-EFGM. The proposed method combines the best features of FFEM and EFGM, in the sense that no structured mesh or special enriched basis functions are necessary and no post processing (employing any path independent integrals) is needed to determine fracture parameters such as stress intensity factors (SIFs) and T-stress. The numerical results show that SIFs and T-stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions.

This paper presents a coupling technique for integrating the fractal finite element method (FFEM) with element-free Galerkin method (EFGM) for analyzing homogeneous, isotropic, and two-dimensional linearelastic cracked structures subjected to Mode I loading condition. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the element-free Galerkin and the finite element shape functions, satisfy the consistency condition thus ensuring convergence of the proposed coupled FFEM-EFGM. The proposed method combines the best features of FFEM and EFGM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path-independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T-stress. The numerical results show that SIFs and T-stress obtained using the proposed method, are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack-length to width ratio, on the quality of the numerical solutions. Copyright © 2010 by Institute of Fundamental Technological Research,.

This paper presents probabilistic fracture-mechanics analysis of linear-elastic cracked structures subjected to mixedmode (modes I and II) loading conditions using fractal finite element method (FFEM). The method involves FFEM for calculating fracture response characteristics; statistical models of uncertainties in load, material properties, and crack geometry; and the first-order reliability method for predicting probabilistic fracture response and reliability of cracked structures. The sensitivity of fracture parameters with respect to crack size, required for probabilistic analysis, is calculated using continuum shape sensitivity analysis. Numerical examples based on mode-I and mixedmode problems are presented to illustrate the proposed method. The results show that the predicted failure probability based on the proposed formulation of the sensitivity of fracture parameter is in good agreement with the Monte Carlo simulation results. Since all gradients are calculated analytically, reliability analysis of cracks can be performed efficiently using FFEM.