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# Communications in Mathematical Sciences

## Volume 11 (2013)

### Number 2

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

Pages: 499 – 521

DOI: http://dx.doi.org/10.4310/CMS.2013.v11.n2.a9

#### Authors

#### Abstract

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.

#### Keywords

Gaussian random field, stochastic elliptic partial differential equation, maximum strain, failure probability

#### 2010 Mathematics Subject Classification

60G15, 60H15