Probabilistic fracture mechanics by Galerkin meshless methods ? part II: reliability analysis

Abstract

This is the second in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper, a stochastic meshless method is presented for probabilistic fracture-mechanics analysis of linear-elastic cracked structures. The method involves an element-free Galerkin method 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 a virtual crack extension technique described in the companion paper [1]. Numerical examples based on mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that the predicted probability of fracture initiation based on the proposed formulation of the sensitivity of fracture parameter is accurate in comparison with the Monte Carlo simulation results. Since all gradients are calculated analytically, reliability analysis of cracks can be performed efficiently using meshless methods.Keywords Probabilistic fracture mechanics, Stochastic meshless method, Element-free Galerkin method, Stressintensity factor, J-integral, Probability of failure IntroductionProbabilistic fracture mechanics (PFM) is becoming increasingly popular for realistic evaluation of fracture response and reliability of cracked structures. Using PFM, statistical uncertainties can be incorporated in engineering design and evaluation, a long-standing need in the stochastic-mechanics community. The theory of fracture mechanics provides a mechanistic relationship between the maximum permissible load acting on a structural component to the size and location of a crack -either real or postulated -in that component. Probability theory determines how the uncertainties in crack size, loads, and material properties, when modeled accurately, affect the integrity of cracked structures. PFM, which blends these two theories, accounts for both mechanistic and stochastic aspects of the fracture problem, and hence, provides a more rational means to describe the actual behavior and reliability of structures than traditional deterministic methods [2].While development is ongoing, a number of methods have been developed or implemented for estimating statistics of various fracture response and reliability. Most of these methods are based on linear-elastic fracture mechanics (LEFM) and a finite element method (FEM) that employs the stress-intensity factor (SIF) as the primary crack-driving force [2][3][4][5][6][7]. For example, using SIFs from an FEM code, Grigoriu et al.[3] applied first-and secondorder reliability methods (FORM/SORM) to predict the probability of fracture initiation and a confidence interval of the direction of crack extension. The method can account for random loads, material properties, and crack geometry. However, the r...

An enriched meshless method for non‐linear fracture mechanics

“…Indeed, the enriched basis in Equation (13) has been successfully used in meshless analysis of linear-elastic cracked structures [9,[12][13][14][15], including analysis of cracks in functionally graded materials [16]. However, the HRR ÿeld is di erent from the LEFM crack-tip ÿeld.…”

“…The latter guess resulted in a well-conditioned matrix, even though the accuracy in the approximation ofũ i (Â; n), using either sin(Â=2) sin 2Â and cos(Â=2) sin 2Â or sin(Â=2) sin 3Â and cos(Â=2) sin 3Â as additional terms, is observed to be the same. To evaluate both Types I and II approximations, Shih's [23] HRR ÿeld data ofũ i (Â; n), i = 1; 2, obtained by solving the eigenvalue problem numerically, were ÿtted with Equations (14) and (15). Note that Shih's [23] HRR ÿeld data were reported in polar, co-ordinate system, so a polar to rectangular co-ordinate system transformation was performed before ÿtting the HRR data using Equations (14) and (15).…”

Probabilistic fracture mechanics by Galerkin meshless methods ? part I: rates of stress intensity factors

“…Numerical examples based on mode-I and mixed-mode problems are presented to illustrate the proposed method. The probabilistic meshless analysis of cracks using first-order sensitivities of SIFs is described in the companion paper (Part II) [31].…”

Stochastic meshless analysis of elastic?plastic cracked structures

“…By sidestepping remeshing requirements, crack-propagation analysis can be significantly simplified. However, most mesh-free development in fracture analysis to date has been focused on either deterministic [10][11][12][13][14][15][16] or some probabilistic [17,18] LEFM problems. Research in nonlinear fracture mechanics using meshless methods has not been widespread and is only currently gaining attention.…”

“…In recent years, various Galerkin-based meshless methods have been developed or investigated to solve fracturemechanics problems without the use of a structured grid [10][11][12][13][14][15][16][17][18][19]. These gridless or meshless methods employ moving least-squares (MLS) approximation of a function that permits the resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes.…”