B. N. Rao

Mesh-free analysis of cracks in isotropic functionally graded materials

This paper presents a Galerkin-based meshless method for calculating stress-intensity factors (SIFs) for a stationary crack in two-dimensional functionally graded materials of arbitrary geometry. The method involves an element-free Galerkin method (EFGM), where the material properties are smooth functions of spatial coordinates and two newly developed interaction integrals for mixed-mode fracture analysis. These integrals can also be implemented in conjunction with other numerical methods, such as the finite element method (FEM). Five numerical examples including both mode-I and mixed-mode problems are presented to evaluate the accuracy of SIFs calculated by the proposed EFGM. Comparisons have been made between the SIFs predicted by EFGM and available reference solutions in the literature, generated either analytically or by FEM using various other fracture integrals or analyses. Agood agreement is obtained between the results of the proposed meshless method and the reference solutions. 2002 Elsevier Science Ltd. All rights reserved.

A coupled meshless-finite element method for fracture analysis of cracks

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with the traditional finite element method (FEM) for analyzing linear-elastic cracked structures subject to mode-I and mixed-mode loading conditions. The EFGM was used to model material behavior close to cracks and the FEM in areas away from cracks. In the interface region, the resulting shape function, which comprises both EFGM and FEM shape functions, satisfies the consistency condition thus ensuring convergence of the method. The proposed method was applied to calculate mode-I and mode-II stress–intensity factors (SIFs) in a number of two-dimensional cracked structures. The SIFs predicted by this method compare very well with the existing solutions obtained by all-FEM or all-EFGM analyses. A significant saving of computational effort can be achieved due to coupling in the proposed method when compared with the existing meshless methods. Furthermore, the coupled EFGM–FEM method was applied to model crack propagation under mixed-mode loading condition. Since the method is partly meshless, a structured mesh is not required in the vicinity of the cracks. Only a scattered set of nodal points is required in the domain of interest. A growing crack can be modeled by simply extending the free surfaces, which correspond to a crack. By sidestepping remeshing requirements, crack-propagation analysis can be dramatically simplified. A number of mixed-mode problems were studied to simulate crack propagation. The agreement between the predicted crack trajectories with those obtained from existing numerical simulation and experiments are excellent.

Probabilistic Analysis Using High Dimensional Model Representation and Fast Fourier Transform

This paper presents an efficient probabilistic analysis method for predicting component reliability of structural/mechanical systems subject to random loads, material properties, and geometry. The proposed method involves High Dimensional Model Representation (HDMR) for the limit state/performance function approximation and fast Fourier transform for solving the convolution integral. The limit state/performance function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at most probable point. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full-scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through five examples involving explicit/implicit performance functions.

An element-free Galerkin method for probabilistic mechanics and reliability

A stochastic element-free Galerkin method was developed for reliability analysis of linear-elastic structures with spatially varying random material properties. A random field representing material properties was discretized into a set of random variables with statistical properties obtained from the statistical properties of random field. In conjunction with meshless formulations, the first-order reliability method was employed to predict the full probabilistic characteristics of a structural response. Unlike the stochastic finite element method, the stochastic mesh-free method does not require a structured mesh, instead only a scattered set of nodal points is required in the domain of interest. As well, there is no need for fixed connectivities between nodes. Numerical examples show good agreement between the results of the developed method and Monte Carlo simulation. Furthermore, the stochastic meshless method provides convergent solutions of the probability of failure. Since mesh generation of complex structures can be far more time-consuming and costly effort than solution of a discrete set of equations, the developed meshless method provides an attractive alternative to finite element method for solving stochastic-mechanics problems.

Factorized high dimensional model representation for structural reliability analysis

Purpose – To develop a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry.Design/methodology/approach – High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable correlations are weak and if the response function is dominantly of additive nature, allowing the physical model to be captured by the first few lower order terms. But, if multiplicative nature of the response function is dominant then all right hand side components of HDMR must be used to be able to obtain the best result. However, if HDMR requires all components, which means 2N number of components, to get a desired accuracy, making the method very expensive in practice, then factorized HDMR (FHDMR) can be used. ...

Stochastic meshfree method for elasto-plastic damage analysis

This paper presents a stochastic meshfree method for solving boundary-value problems in damage mechanics under elasto-plastic conditions. Isotropic ductile damage evolution law is used to model the coupled elasto-plastic damage growth. Uncertainty associated with initial damage in materials is considered as random field. Moving least squares shape function method and Karhunen Loeve expansion method are used for random field discretization. Statistical parameters of the response quantities are computed using perturbation method. The proposed method involves a new stochastic stress update procedure to solve the nonlinear equations in terms of discretized random variables arising from perturbation of equilibrium equations system. Numerical examples comprising of one and two dimensional problems are presented to illustrate the effectiveness of proposed method.

Numerical Simulation of Fracture Process in Cement Concrete Using Unit Cell Approach.

Modeling concrete at lower length scale is useful to predict the failure processes taking place in the material. In the present work, the deformation and failure of concrete is studied using the unit cell model. A unit cell simulates the deformation and failure in a representative volume element and postulates the material response curve based the response of a unit cell. A parametric study has been conducted to study the effect of aggregate size on the overall behavior of concrete. Uniaxial compression tests were simulated numerically. Interfacial transition zone (ITZ) was modeled using plane strain elements with brittle cracking model. Nonlinear stress-strain curves for various aggregate sizes were obtained using current unit cell approach. Simulation results indicate that the ITZ does not play a major role in determining the overall failure strength of concrete.

Continuum shape sensitivity analysis of a mixed-mode fracture in functionally graded materials

Abstract This paper presents two new methods for conducting a continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic functionally graded material. These methods involve the material derivative concept from continuum mechanics, domain integral representation of interaction integrals, known as the M -integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed 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 are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. Three numerical examples are presented to calculate the first-order derivative of the stress–intensity factors. The results show that first-order sensitivities of 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.

High dimensional model representation for structural reliability bounds estimation under mixed uncertainties

This paper presents an efficient uncertain analysis method for estimating the bounds on the reliability of structural systems in the presence of mixed uncertain variables. The proposed method involves high dimensional model representation (HDMR) for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through numerical examples.

Detection of crack location and size in structures using improved damaged finite element

In this paper two-dimensional finite element with an embedded edge crack proposed by Potirniche et al [1] is improved further for crack depth ratios ranging up to 0.9 h (h is the element depth) and for predicting natural frequency of a cracked beam more accurately. The element is implemented in the commercial finite element code ABAQUS as user element (UEL) subroutine. The accuracy of the UEL is verified by comparing the first natural frequency for the bending mode for several beam cases with different damage locations with available experimental data. Subsequently a methodology to detect crack location and size in conjunction the proposed improved cracked element with is presented for singularity problems like a cracked beam. The frequency response functions, function of crack location and size, are approximated by means of surface-fitting techniques. Measured natural frequencies are used in a crack detection process and the crack location and size can be identified by finding the point of intersection of three frequency contour lines. The experimental data from beams studied by other researchers is employed to verify the accuracy of the proposed methodology in the diagnosis of structural crack faults.