This paper presents a polynomial dimensional decomposition method for calculating the probability distributions of random crack-driving forces commonly encountered in elastic-plastic fracture analysis of ductile solids. The method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier-polynomial expansion of component functions, and an innovative dimension-reduction integration for calculating the expansion coefficients. Unlike the previous development, the new decomposition does not require sample points, yet it generates a convergent sequence of lower-variate estimates of the probability distributions of crack-driving forces. Numerical results, including the probability of fracture initiation of a through-walled-cracked pipe, indicate that the decomposition method developed provides accurate, convergent, and computationally efficient estimates of the probabilistic characteristics of the J-integral.

1.
Madsen
,
H. O.
,
Krenk
,
S.
, and
Lind
,
N. C.
, 1986,
Methods of Structural Safety
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
2.
Rackwitz
,
R.
, 2001, “
Reliability Analysis—A Review and Some Perspectives
,”
Struct. Safety
0167-4730,
23
(
4
), pp.
365
395
.
3.
Rubinstein
,
R. Y.
, 1981,
Simulation and the Monte Carlo Method
,
Wiley
,
New York, NY
.
4.
Niederreiter
,
H.
, and
Spanier
,
J.
, 2000,
Monte Carlo and Quasi-Monte Carlo Methods
,
Springer-Verlag
,
Berlin
.
5.
Rahman
,
S.
, 2006, “
A Dimensional Decomposition Method for Stochastic Fracture Mechanics
,”
Eng. Fract. Mech.
0013-7944,
73
, pp.
2093
2109
.
6.
Efron
,
B.
, and
Stein
,
C.
, 1981, “
The Jackknife Estimate of Variance
,”
Ann. Stat.
0090-5364,
9
, pp.
586
596
.
7.
Rabitz
,
H.
, and
Alis
,
O.
, 1999, “
General Foundations of High Dimensional Model Representations
,”
J. Math. Chem.
0259-9791,
25
, pp.
197
233
.
8.
Sobol
,
I. M.
, 2003, “
Theorems and Examples on High Dimensional Model Representations
,”
Reliab. Eng. Syst. Saf.
0951-8320,
79
, pp.
187
193
.
9.
Rahman
,
S.
, 2008, “
A Polynomial Dimensional Decomposition for Stochastic Computing
,”
Int. J. Numer. Methods Eng.
0029-5981,
76
, pp.
2091
2116
.
10.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, 1972,
Handbook of Mathematical Functions
,
9th ed.
,
Dover
,
New York, NY
.
11.
Xu
,
H.
, and
Rahman
,
S.
, 2004, “
A Generalized Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics
,”
Int. J. Numer. Methods Eng.
0029-5981,
61
, pp.
1992
2019
.
12.
Anderson
,
T. L.
, 2005,
Fracture Mechanics: Fundamentals and Applications
,
3rd ed.
,
CRC
,
Boca Raton, Florida
.
13.
2008, ABAQUS Version 6.8, User’s Guide and Theoretical Manual,
Simulia-Dassault Systèmes
, Providence, RI.
14.
Rahman
,
S.
, 2001, “
Probabilistic Fracture Mechanics: J-Estimation and Finite Element Methods
,”
Eng. Fract. Mech.
0013-7944,
68
, pp.
107
125
.
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