A strength reliability model of unidirectional fiber-reinforced ceramic matrix composites by Markov process

We propose a stochastic process analysis for predicting the strength and reliability of a unidirectional fiber-reinforced ceramic matrix composite. The analysis is based on a Markov process, in which it is assumed that a state of damage in the composite is developed with each fiber breakage. When the Weibull distribution is used to describe the strength distribution of the fiber, the probability of being in each state can be solved analytically in a closed form. Using the solutions of the probabilities, a discussion about damage tolerance of the composites is quantitatively developed, from the viewpoint of materials reliability engineering. To compare the proposed stochastic process analysis with previously proposed strength models in the basis of classical bundle theory, we obtained the expected value and variance in the composite stress from solutions of the probabilities of being in the states. The expected value and variance both consist of two terms, equivalent to the effects of both bundle structure and stress recovery in broken fibers. These are surprisingly in agreement with the solutions analyzed by Phoenix and Raj [9]. The effect of stress recovery in broken fibers produces a positive in the expected value and a negative in the variance and is thus a significant mechanism for increasing the strength and reliability of the composite. In addition we predicted the expected values of composite strengths and variances from the solutions. The corresponding value of normalized fiber stress to obtain the expected value in strength and the variance agreed relatively well with the value predicted by Hui et al. [13]. We further verified that the composite strength obeys a normal distribution, which has the expected value and standard deviation predicted above. Finally, we predicted the probabilities in strength of the composites with various sizes and concluded that the ceramic matrix composite is quite reliable in strength when the number of fibers corresponds to that in practical use.