Continuum approach to high-cycle fatigue. The finite life-time case with stochastic stress history

In this paper, we consider continuum approach for high-cycle fatigue in the case where life-time is finite. The method is based on differential equations and all basic concepts are explained. A stress history is assumed to be a stochastic process and this leads us to the theory of stochastic differential equations. The life-time is a quantity, which tells us when the breakdown of the material happens. In this method, it is naturally a random variable. The basic assumption is, that the distribution of the life-time is log-normal or Weibull. We give a numerical basic example to demonstrate the method.