Stochastic fracture mechanics by fractal finite element method

This paper presents stochastic fracture mechanics analysis of linear-elastic cracked structures subjected to mixed-mode (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 mixed-mode 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 accurate in comparison with the Monte Carlo simulation results. Since all gradients are calculated analytically, reliability analysis of cracks can be performed efficiently using FFEM.

[1]  Ted Belytschko,et al.  Fatigue crack growth reliability by probabilistic finite elements , 1991 .

[2]  Ray Kai Leung Su,et al.  Numerical solutions of two-dimensional anisotropic crack problems , 2003 .

[3]  Ray Kai Leung Su,et al.  Two-Level Finite Element Study of Axisymmetric Cracks , 1998 .

[4]  Andrew Y. T. Leung,et al.  Mixed-mode two-dimensional crack problem by fractal two level finite element method , 1995 .

[5]  Edward J. Haug,et al.  Design Sensitivity Analysis of Structural Systems , 1986 .

[6]  K. Choi,et al.  A study of design velocity field computation for shape optimal design , 1994 .

[7]  Juan José Ródenas,et al.  A numerical methodology to assess the quality of the design velocity field computation methods in shape sensitivity analysis , 2004 .

[8]  Andrew Y. T. Leung,et al.  Mode I crack problems by fractal two level finite element methods , 1994 .

[9]  Andrew Y. T. Leung,et al.  Fractal two-level finite element analysis of cracked Reissner's plate , 1996 .

[10]  Andrew Y. T. Leung,et al.  Fractal two-level finite element method for cracked Kirchhoff's plates using DKT elements , 1996 .

[11]  J. Arora,et al.  An active set RQP algorithm for engineering design optimization , 1986 .

[12]  Jasbir S. Arora,et al.  Introduction to Optimum Design , 1988 .

[13]  Sharif Rahman,et al.  A stochastic model for elastic-plastic fracture analysis of circumferential through-wall-cracked pipes subject to bending , 1995 .

[14]  R. Salganik,et al.  Brittle fracture of solids with arbitrary cracks , 1974 .

[15]  Sharif Rahman,et al.  Probabilistic analysis of off-center cracks in cylindrical structures , 2000 .

[16]  M. Grigoriu,et al.  Mixed mode fracture initiation and trajectory prediction under random stresses , 1990 .

[17]  Sharif Rahman,et al.  Probabilistic fracture mechanics: J-estimation and finite element methods , 2001 .

[18]  Ray Kai Leung Su,et al.  Eigenfunction expansion for penny-shaped and circumferential cracks , 1998 .

[19]  Sharif Rahman,et al.  Probabilistic fracture mechanics for nonlinear structures , 2001 .

[20]  G. C. Sih,et al.  The use of eigenfunction expansions in the general solution of three-dimensional crack problems , 1969 .

[21]  A. Leung,et al.  A numerical study of singular stress field of 3D cracks , 1995 .

[22]  Kyung K. Choi,et al.  Shape Design Sensitivity Analysis of Elastic Structures , 1983 .

[23]  R. Su,et al.  Fractal two-level finite-element method for two-dimensional cracks , 1996 .

[24]  Ted Belytschko,et al.  Brittle fracture reliability by probabilistic finite elements , 1990 .

[25]  A. Leung,et al.  Fractal Two-Level Finite Element Method For Free Vibration of Cracked Beams , 1998 .

[26]  Peter Dillström.  ProSINTAP – A probabilistic program implementing the SINTAP assessment procedure , 2000 .

[27]  Andrew Y. T. Leung,et al.  A parametric study on the fractal finite element method for two‐dimensional crack problems , 2002 .

[28]  G. Sih.  Strain-energy-density factor applied to mixed mode crack problems , 1974 .

[29]  Chien H. Wu.  Fracture Under Combined Loads by Maximum-Energy-Release-Rate Criterion , 1978 .

[30]  M. Williams,et al.  On the Stress Distribution at the Base of a Stationary Crack , 1956 .

[31]  R. Su,et al.  Numerical solution of cracked thin plates subjected to bending, twisting and shear loads , 2002 .

[32]  F. Erdogan,et al.  On the Crack Extension in Plates Under Plane Loading and Transverse Shear , 1963 .

[33]  A. Leung,et al.  Mixed mode cracks in Reissner plates , 2001 .

[34]  B. Rao,et al.  Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method , 2008 .

[35]  James W. Provan,et al.  Probabilistic fracture mechanics and reliability , 1987 .

[36]  A. Leung,et al.  Three-dimensional mixed mode analysis of a cracked body by fractal finite element method , 2001 .

[37]  John P. Wolf,et al.  Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method , 2002 .

[38]  Niels C. Lind,et al.  Methods of structural safety , 2006 .