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.

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