Correlation functions of stress and strain fields in micro-heterogeneous media

The paper is devoted to the development of a method for calculation of microstructural stresses and strains fields in the multi-phase media based on the calculation of the statistical characteristics of the local stress and strain fields in the components, which takes into account the geometrical and mechanical properties of components. Representative volumes ofstructurally mulit-phase heterogeneous materialswere investigated. It is assumed that the components are homogeneous and isotropic. The internal geometry of the structure as well as the assessment of spatial interaction is described by the moment functions of different orders. The behavior of individual components of the microstructure during loading ofthe representative volume is estimated using the statistical characteristics of the local stress and strain fields. The characteristics of deformation processes are the statistical moment functions of the stress and strain fields in the components of the material. Analytical expressions for the statistical moments and correlation functions of the stress and strain fields are obtainedusing statistical averaging of integral-differential equations that contain moment functions, and derivedfromthe solution of the stochastic boundary value problems in elastic and elastoplastic formulation. Some special cases of typical heterogeneous media with a random microstructure were considered. The correlation functions of stress and strain for sparse structures with spherical and ellipsoidal hollow inclusions in the elastic and elastoplastic cases were built. The study and the selectionof approximating dependences obtained for the correlation functionswere performed. The numerical results can be used to evaluate the mechanical behavior of the inhomogeneous medium microstructural component under different loading conditions and to predict fracture initiation.