On the failure probability of one dimensional random material under delta external force

We provide an asymptotic analysis of the small failure probabilities for a piece of elastic random material under a certain external force and boundary condition. The displacement of the material is described by a one dimensional stochastic elliptic differential equation. The differential equation admits random coefficients described by a Gaussian process. Failure is defined as the event that the maximum strain of the material exceeds a certain level. We derive asymptotic approximations of the probability that the strain exceeds a high level b that tends to infinity.