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We study the problem of estimating small failure probabilities for elastic random material described by a one dimensional stochastic elliptic differential equation with certain external forcing and boundary conditions. Gaussian random functions are used to model the spatial variation of the material parameters. The failure event of the bulk material is simply characterized by the exceeding of certain thresholds for the maximum strain in the material. Using large deviation heuristics, we provide an intuitive description of the most probable realization of the random material parameters leading to critical situations of material failure. An efficient Monte Carlo method to compute such probabilities is presented.