B Nageswara Rao

01-11-2008, Reddy, R. M., B Nageswara Rao

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.

01-08-2023, Sidharth, P. C., B Nageswara Rao

In the recent decade, there has been a growing interest in using the phase-field approach to model fracture processes in various materials. Conventional phase-field implementations can simulate fracture processes using bi-linear finite element (LFE) shape functions but at the expense of a very fine mesh. In contrast, exponential finite element (EFE) shape functions can predict sharp gradients in solution variables with coarse meshes due to their exponential nature. A potential advantage lies in reducing the number of elements in the problem without losing accuracy in the solution. However, EFE shape functions do not yield a good approximation unless they are oriented relative to the expected crack propagation path. This study uses an approximate analysis using LFE shape functions to orient the EFE shape functions before the computations. Computational advantages are reported in terms of accuracy in predicted load responses and the computational times incurred.

01-10-2008, B Nageswara Rao, Kuna, M.

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials. Conservation integrals of J-type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples. © 2008 Elsevier Ltd. All rights reserved.

01-01-2006, B Nageswara Rao, Rahman, S.

This paper presents a new continuum shape sensitivity method for calculating mixed-mode stress-intensity factors for a stationary crack in two-dimensional, linear-elastic, isotropic FGMs with arbitrary geometry. The method involves the material derivative concept taken from continuum mechanics, the mutual potential energy release rate, and direct differentiation. Since the governing variational equation is differentiated prior to discretization, resulting sensitivity equations are independent of approximate numerical techniques, such as the finite element method, boundary element method, mesh-free method, or others. The discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires multiplication of displacement vectors and stiffness sensitivity matrices. By judiciously selecting the velocity field, the method only requires displacement response in a subdomain close to the crack tip, thus making the method computationally efficient. Seven finite-element based numerical examples, which comprise mode-I and mixed-mode deformations and/or single or multiple interacting cracks, are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method. Comparisons have been made between stress-intensity factors predicted by the proposed method and available reference solutions in the literature, generated either analytically or numerically using various other fracture integrals or analyses. Excellent agreement is obtained between the results of the proposed method and previously obtained solutions. Therefore, shape sensitivity analysis provides an attractive alternative to fracture analysis of cracks in homogeneous and non-homogeneous materials.

26-10-2023, Sidharth, P. C., B Nageswara Rao

This study introduces a novel implementation of exponential finite element (EFE) shape functions within the phase field model for predicting fracture responses in functionally graded materials (FGMs). The proposed approach utilizes an effective fracture toughness concept to analyze load–displacement responses and crack paths in various examples of FGMs. To optimize computational efficiency, a mixed scheme combining both linear finite element (LFE) and EFE shape functions is employed. Specifically, only the critical elements are corrected using EFE shape functions. Comparative analysis of simulation results against converged outcomes demonstrates the superiority of the EFE scheme, even when employing coarser meshes. The EFE approach accurately predicts load–displacement responses and crack paths for the evaluated examples. The proposed implementation offers a reliable and efficient tool for studying fracture behavior in FGMs. Despite the higher integration schemes and orientation requirements associated with EFE shape functions, the additional computational effort is found to be negligible. Implementation aspects include a staggered iteration scheme, a hybrid tension–compression splitting scheme, and automatic orientation of EFE shape functions.

01-01-2010, B Nageswara Rao, Kuna, M.

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. © 2009 Elsevier Ltd. All rights reserved.

29-10-2010, Rajesh, K. N., B Nageswara Rao

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.

01-10-2005, B Nageswara Rao, Rahman, Sharif

This paper presents a new continuum shape sensitivity method for calculating the mixed-mode stress-intensity factors of a stationary crack in two-dimensional, linear-elastic, orthotropic functionally graded materials with arbitrary geometry. The method involves the material derivative concept taken from continuum mechanics, the mutual potential energy release rate, and direct differentiation. Since the governing variational equation is differentiated prior to discretization, resulting sensitivity equations are independent of approximate numerical techniques, such as the finite element method, boundary element method, mesh-free method, or others. The discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires multiplication of displacement vectors and stiffness sensitivity matrices. By judiciously selecting the velocity field, the method only requires displacement response in a subdomain close to the crack tip, thus making the method computationally efficient. Three finite-element based numerical examples, which comprise mode-I and mixed-mode deformations, are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method. Comparisons have been made between stress-intensity factors predicted by the proposed method and available reference solutions in the literature, generated either analytically or numerically using various other fracture integrals or analyses. Excellent agreement is obtained between the results of the proposed method and previously obtained solutions. Therefore, shape sensitivity analysis provides an attractive alternative to fracture analysis of cracks in homogeneous and non-homogeneous orthotropic materials. © 2005 Elsevier Ltd. All rights reserved.

01-07-2008, Reddy, R. M., B Nageswara Rao

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.

01-01-2008, B Nageswara Rao, Kuna, M.

This paper presents the domain form of interaction integrals based on three independent formulations for computation of stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors for cracks in functionally graded magnetoelectroelastic materials. Conservation integrals of J-type are derived based on the governing equations for magnetoelectroelastic media and the crack tip asymptotic fields of homogeneous magnetoelectroelastic medium as auxiliary fields. Each of the formulations differs in the way auxiliary fields are imposed in the evaluation of interaction integrals and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded magnetoelectroelastic medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples. © Springer Science+Business Media B.V. 2008.