Stochastical model of microstructural failure based on restoration of distributions laws for random stress and strain fields in microheterogeneous media

A common direction in the micromechanics of heterogeneous media with a random structure is related to the methods of stochastic mechanics. Microstructure of a heterogeneous material has a significant effect on the distribution of stress and strain fields. Methods and tools of statistical analysis allow to take into account interactions within a many-particle system and investigate fields distributions from an analytical point of view, which is used to create stochastic models describing the mechanical behavior of a material. This paper presents an approach to the analysis of the distribution of stress and strain fields in representative volumes of inhomogeneous media based on the restoration of their distribution laws. The techniques of finding the parameters of the distribution laws are described. In the framework of the stochastic model, the central moments of the random variable obtained for the stress and strain fields in individual phases of the material are used for this. An algorithm for calculating the fracture probability using the laws of distribution of stress and strain fields is proposed on the basis of the probability representation of the failure criteria. Based on the analysis of the probability of failure of phases of a representative volume, a stochastic model of a progressive failure is presented. Some numerical results were derived for the particular case of a non-homogeneous structure; the estimates obtained with the help of the stochastic model are compared with the results of finite element modeling. The different types of parametric distribution laws used to reflect a real distribution of stress fields in a representative volume are compared on the basis of the obtained finite element analysis and the calculation of the moments in the framework of the stochastic model. For the discussed case study, a technique for calculating the fracture probabilities at static loading of a representative volume is implemented.