Mahesh Sivasambu

Monte-Carlo simulations of the fracture of elastic unidirectional model fibre composites are an important tool to understand composite reliability. On account of being computationally intensive, fracture simulations reported in the literature have been limited to simulation patches comprised of a few thousand fibres. While these limited patch sizes suffice to capture the dominant failure event when the fibre strength variability is low (synthetic fibres), they suffer from edge effects when the fibre strength variability is high (natural fibres). On the basis of recent algorithmic developments based on Fourier acceleration, a novel bisection based Monte Carlo failure simulation algorithm is presently proposed. This algorithm is used to obtain empirical strength distributions for model composites comprised of up to 2 20≈ 10 6 fibres, and spanning a wide range of fibre strength variabilities. These simulations yield empirical weakest-link strength distributions well into the lower tail. A stochastic model is proposed for the weakest-link event. The strength distribution predicted by this model fits the empirical distributions for any fibre strength variability.

The stress state in a shear-lag model of a unidirectional fiber composite with an arbitrary configuration of fiber breaks is obtained by the weighted superposition of the stress state due to a single broken fiber. In a periodic patch comprised of N fibers located at the points of a regular lattice, a method to determine the stress state due to a single break was proposed by Landis et al. (J Mech Phys Solids 48(3):621–648, 2000). This method entails the determination of the eigenspace of an N× N matrix, at a computational cost of O(N3). In the present work, an alternative algorithm is proposed. This algorithm exploits the circulant structure of the matrix describing the inter-fiber interactions. The asymptotic computational complexity of the present algorithm equals that of the discrete Fourier transform: O(Nlog N). Run times of the present method with the eigensolution based method are compared, and shown to be very favorable for the present method, even for small N. Power-law scaling of the overloads due to a single break to much larger distances than previously possible has been verified using the present method.

Monte Carlo simulations of the failure of unidirectional fibre composites typically require numerous evaluations of the stress-state in partially damaged composite patches. In a simulated composite patch comprised of N fibres, of which Nb fibres are broken in a common cross-sectional plane transverse to the fibre direction, the stress overloads in the intact fibres are given by the weighted superposition of the unit break solutions associated with each of the breaks. Determining the weights involves solving Nb linear equations, and determining overloads in the intact fibres requires matrix-vector multiplication. These operations require O(Nb3), and O(NNb) floating point operations, respectively. These costs become prohibitive for large N, and Nb; they limit Monte Carlo failure simulations to composite patches of only a few thousand fibres. In the present work, a fast algorithm to determine the overloads in a partially damaged composite, requiring O(Nb1/3NlogN) floating point operations, is proposed. This algorithm is based on the discrete Fourier transform. The efficiency of the proposed method derives from the computational simplicity of weighted superposition in Fourier space. Computations of the stress state ahead of large circular clusters of breaks in composite patches comprised of about one million fibres are used to demonstrate the efficiency of the proposed algorithm.

The strength of metal matrix composites shows wide scatter on account of variability in the strengths of individual fibres. The relationship between the strength distribution of the fibres, and that of the composite is also affected by the non-linear matrix and fibre/matrix interfacial responses. The present study aims to describe the strength distribution of 2D and 3D commercial Ti/SiC composites. This is accomplished by performing Monte Carlo failure simulations of these composites, comprised of up to 128 fibres. A detailed deformation theory based model, developed and validated against experimental data in previous work, is used to calculate load redistribution in the course of each simulation. The empirical composite strength distribution obtained from the simulations follows weakest-link scaling. A stochastic model for the clustered propagation of fibre breaks, akin to a model proposed for polymer matrix composites in the literature, captures the empirical weakest-link strength distribution. A scaling relationship is derived between the composite strength and composite size for a number of reliability levels.