  Random microstructures of heterogeneous materials play a crucial role in the
material macroscopic behavior and in predictions of its effective properties. A
common approach to modeling random multiphase materials is to develop so-called
surrogate models approximating statistical features of the material. However,
the surrogate models used in fatigue analysis usually employ simple
microstructure, consisting of ideal geometries such as ellipsoidal inclusions,
which generally does not capture complex geometries. In this paper, we
introduce a simple but flexible surrogate microstructure model for two-phase
materials through a level-cut of a Gaussian random field with covariance of
Mat\'ern class. Such parametrization of the covariance function allows for the
representation of a few key design parameters while representing the geometry
of inclusions in a more general setting for a large class of random
heterogeneous two-phase media. In addition to the traditional morphology
descriptors such as porosity, size and aspect ratio, it provides control of the
regularity of the inclusions interface and sphericity. These parameters are
estimated from a small number of real material images using Bayesian inversion.
An efficient process of evaluating the samples, based on the Fast Fourier
Transform, makes possible the use of Monte-Carlo methods to estimate
statistical properties for the quantities of interest in a given material
class. We demonstrate the overall framework of the use of the surrogate
material model in application to the uncertainty quantification in fatigue
analysis, its feasibility and efficiency, and its role in the microstructure
design.
