Stochastic fracture mechanics using polynomial chaos

Abstract Crack propagation in metals has long been recognized as a stochastic process. As a consequence, crack propagation rates have been modeled as random variables or as random processes of the continuous. On the other hand, polynomial chaos is a known powerful tool to represent general second order random variables or processes. Hence, it is natural to use polynomial chaos to represent random crack propagation data: nevertheless, no such application has been found in the published literature. In the present article, the large replicate experimental results of Virkler et al. and Ghonem and Dore are used to illustrate how polynomial chaos can be used to obtain accurate representations of random crack propagation data. Hermite polynomials indexed in stationary Gaussian stochastic processes are used to represent the logarithm of crack propagation rates as a function of the logarithm of stress intensity factor ranges. As a result, crack propagation rates become log-normally distributed, as observed from experimental data. The Karhunen–Loeve expansion is used to represent the Gaussian process in the polynomial chaos basis. The analytical polynomial chaos representations derived herein are shown to be very accurate, and can be employed in predicting the reliability of structural components subject to fatigue.

[1]  R. Forman,et al.  Numerical Analysis of Crack Propagation in Cyclic-Loaded Structures , 1967 .

[2]  F. Kozin,et al.  Stochastic fatigue, fracture and damage analysis , 1986 .

[3]  R. Rackwitz,et al.  Comparison of analytical counting methods for Gaussian processes , 1993 .

[4]  Igor Rychlik,et al.  Wiener chaos expansions for estimating rain-flow fatigue damage in randomly vibrating structures with uncertain parameters , 2011 .

[5]  W. S. Venturini,et al.  On the performance of response surface and direct coupling approaches in solution of random crack propagation problems , 2011 .

[6]  Sankaran Mahadevan,et al.  Uncertainty quantification and model validation of fatigue crack growth prediction , 2011 .

[7]  B. N. Rao,et al.  Stochastic fracture mechanics by fractal finite element method , 2008 .

[8]  Anne S. Kiremidjian,et al.  Stochastic modeling of fatigue crack growth , 1988 .

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

[10]  Bruno Sudret,et al.  Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach , 2008 .

[11]  G. I. Schuëller,et al.  The fitting of one- and two-dimensional Fatigue Crack Growth laws , 1993 .

[12]  Yibing Xiang,et al.  Application of inverse first-order reliability method for probabilistic fatigue life prediction , 2011 .

[13]  Sharif Rahman,et al.  A dimensional decomposition method for stochastic fracture mechanics , 2006 .

[14]  A. Beck,et al.  Bending of stochastic Kirchhoff plates on Winkler foundations via the Galerkin method and the Askey–Wiener scheme , 2010 .

[15]  P. C. Paris,et al.  A Critical Analysis of Crack Propagation Laws , 1963 .

[16]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[17]  André T. Beck,et al.  Modeling random corrosion processes via polynomial chaos expansion , 2012 .

[18]  Daniel Straub,et al.  Risk based inspection planning for structural systems , 2005 .

[19]  Alaa Chateauneuf,et al.  Random fatigue crack growth in mixed mode by stochastic collocation method , 2010 .

[20]  I. Rychlik,et al.  Rain-flow fatigue damage due to nonlinear combination of vector Gaussian loads , 2007 .

[21]  E. Wolf Fatigue crack closure under cyclic tension , 1970 .

[22]  P. Goel,et al.  The Statistical Nature of Fatigue Crack Propagation , 1979 .

[23]  Roger Ghanem,et al.  Stochastic model reduction for chaos representations , 2007 .

[24]  G. I. Schuëller,et al.  A probabilistic criterion for evaluating the goodness of fatigue crack growth models , 1996 .

[25]  Hisanao Ogura,et al.  Orthogonal functionals of the Poisson process , 1972, IEEE Trans. Inf. Theory.

[26]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[27]  W. S. Venturini,et al.  MULTIPLE RANDOM CRACK PROPAGATION USING A BOUNDARY ELEMENT FORMULATION , 2011 .

[28]  P. H. Wirsching,et al.  Fatigue Under Wide Band Random Stresses Using the Rain-Flow Method , 1977 .

[29]  Wen-Fang Wu,et al.  A study of stochastic fatigue crack growth modeling through experimental data , 2003 .

[30]  Michael Havbro Faber,et al.  Sensitivities in Structural Maintenance Planning , 1996 .

[31]  Alaa Chateauneuf,et al.  Coupled reliability and boundary element model for probabilistic fatigue life assessment in mixed mode crack propagation , 2010 .

[32]  P. H. Wirsching,et al.  Advanced fatigue reliability analysis , 1991 .

[33]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[34]  Igor Rychlik,et al.  Uncertainty in fatigue life prediction of structures subject to Gaussian loads , 2009 .

[35]  Robert E. Melchers,et al.  Overload failure of structural components under random crack propagation and loading - a random process approach , 2004 .

[36]  A. Beck,et al.  DESIGN-POINT EXCITATION FOR CRACK PROPAGATION UNDER NARROW-BAND RANDOM LOADING , 2013 .

[37]  George E. Karniadakis,et al.  Time-dependent generalized polynomial chaos , 2010, J. Comput. Phys..

[38]  Hamouda Ghonem,et al.  Experimental study of the constant-probability crack growth curves under constant amplitude loading , 1987 .

[39]  Bruno Sudret,et al.  Efficient computation of global sensitivity indices using sparse polynomial chaos expansions , 2010, Reliab. Eng. Syst. Saf..

[40]  Gerhart I. Schuëller Reliability – Statistical methods in fracture and fatigue , 2007 .

[41]  Hiroshi Ishikawa,et al.  Reliability assessment of structures based upon probabilistic fracture mechanics , 1994 .

[42]  Mircea Grigoriu,et al.  On the accuracy of the polynomial chaos approximation for random variables and stationary stochastic processes. , 2003 .

[43]  S. Winterstein,et al.  Random Fatigue: From Data to Theory , 1992 .

[44]  Christian Soize,et al.  Reduced Chaos decomposition with random coefficients of vector-valued random variables and random fields , 2009 .

[45]  Y. K. Lin,et al.  On fatigue crack growth under random loading , 1992 .

[46]  P. Beaurepaire,et al.  Reliability-based optimization of maintenance scheduling of mechanical components under fatigue , 2012, Computer methods in applied mechanics and engineering.

[47]  Roger Ghanem,et al.  Simulation of multi-dimensional non-gaussian non-stationary random fields , 2002 .

[48]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[49]  N. Wiener The Homogeneous Chaos , 1938 .

[50]  André T. Beck,et al.  Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach , 2011 .

[51]  Alfonso Fernández-Canteli,et al.  A new probabilistic model for crack propagation under fatigue loads and its connection with Wöhler fields , 2010 .

[52]  Anne S. Kiremidjian,et al.  Time series analysis of fatigue crack growth rate data , 1986 .

[53]  Gerhart I. Schuëller,et al.  Design of maintenance schedules for fatigue-prone metallic components using reliability-based optimization , 2010 .

[54]  G. Schuëller,et al.  Time variant structural reliability analysis using diffusive crack growth models , 1989 .

[55]  B. Sudret,et al.  An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis , 2010 .

[56]  Y. K. Lin,et al.  A stochastic theory of fatigue crack propagation , 1985 .

[57]  S. D. Manning,et al.  A simple second order approximation for stochastic crack growth analysis , 1996 .

[58]  M. Nagode,et al.  A general multi-modal probability density function suitable for the rainflow ranges of stationary random processes , 1998 .

[59]  Asok Ray,et al.  A stochastic model of fatigue crack propagation under variable-amplitude loading , 1999 .

[60]  Akira Tsurui,et al.  Application of the Fokker-Planck equation to a stochastic fatigue crack growth model , 1986 .