The aim of this chapter is to provide some mathematical tools for the modeling of degradation processes that arise in structures operating under unstable environmental conditions. These uncertainties may come from a lack of scientific knowledge or from the intrinsic random nature of the observed physical phenomenon. The theory of stochastic processes seems to be a good way to handle the randomness of the system, thus it is widely used in many engineering fields that involve all kinds of uncertainties. This is the case for applications in the structural reliability field.
In stochastic analysis dedicated to structural mechanic, the reliability of a structure is often modeled using a classical stress-strength description (see [JOH 88, KOT 03]). That is, a stochastic process describing the "stress" applied on the structure is compared with the "strength", which represents the critical level that must not be overcome by the loads. A slightly different approach is adopted in this chapter: we describe the evolution in time of the level of degradation in a given structure using a stochastic process governed by a differential system. It is compared with a variable describing the failure domain that must not be reached, which will be a real positive constant in the following sections, for the sake of simplicity. It means that there is no renewal possible for the degradation process, that is, the system is not a reparable one. Degradation processes which may be renewed are studied in, for example, [BAG 06]. Associated Chapter written by Julien CHIQUET and Nikolaos LIMNIOS.