PSI - Issue 5

Mikhail Tashkinov et al. / Procedia Structural Integrity 5 (2017) 608–613 Mikhail Tashkinov/ Structural Integrity Procedia 00 (2017) 000 – 000

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structurally heterogeneous materials is commonly studied using the concept of a representative volume element, when the size of microstructural inhomogeneities is much smaller than the size of the volume itself. In structurally heterogeneous materials deformation and fracture processes depend on the peculiarities of local stress fields, for calculation of which fluctuations and deviations from the averaged values, caused by inhomogeneities at the microscale, are important. A common direction in micromechanics of materials with random structure is the methods of statistical mechanics for many-particle systems when multi-point statistics for stresses, deformations and displacement fields are used to describe the interaction of inhomogeneities. The characteristics of the fields presented in this way allow taking into account randomness of elements arrangements in the structure, as well as dispersion of the properties of the components. A large number of papers have been devoted to the description of structurally inhomogeneous media using tools of statistics and probability theory, among which works of Baniassadi et al. (2012), Jiao et al. (2008), Liu and Ghoshal (2014), Sheidaei et al. (2013), Yeong and Torquato (1998) were devoted to the methodology of describing and identifying the microstructure of inhomogeneous media by means of correlation functions; the works of Aboudi et al. (2013), Hyun and Torquato (2001), Kpobie et al. (2012), Torquato (2010) present the results of analysis of the properties of the microstructure and their relationship to the mechanical and physical properties of the medium; the influence of microstructure parameters on effective properties was also studied by Matveeva et al. (2014), Melro et al. (2012), Rasool and Böhm (2012), Yu et al. (2013). Nomenclature failure probability stress tensor strain tensor dist random variable distribution law Young’s modulus Poisson’s ration RVE representative volume elemts The heterogeneity of the microstructure has a significant effect on the distribution of stress and strain fields in representative volume elements. Methods and tools of statistical analysis make it possible to investigate these distributions from the analytical point of view. In this case, for example, the characteristics of failure of representative volume and its components can be expressed in probabilistic quantities. The approach presented in this paper allows to investigate probability of fracture initiation for the RVE phases basing on statistical representation of fracture criteria and recovery of the stress and strain distribution functions in phases. 2. Failure probability based on fields distribution There are several approaches for obtaining distribution of stress and strain fields in RVE. The first and most common approach is to create a finite element model and process the data calculated for sets of elements belonging to the matrix or inclusions. In this case, the result depends on mesh, the density of which increases with the complexity of the geometry of the model. Another method is based on formalization of information about microstructure by means of correlation functions and by solving an integral-differential equation containing the Green's function to obtain the fluctuations of displacements, strain and stress fields. This approach is described by Buryachenko (2007), Chen and Ang (2014), Tashkinov (2015), Xu et al. (2009). The accuracy of distribution laws reconstruction in this case depends on order of the correlation functions and order of the statistical moments obtained on the basis of the integral differential equation solution. In this work implementation of the methodology was performed using finite element analysis based on a mesh with voxel-type elements. For instance, let’s consider the following procedure for restoration of stress distribution. For each component of the stress tensor obtained for the set of finite elements corresponding to the phase, a sample of pairs of stress values and the corresponding probability (weight coefficient) is constructed. These samples can be analyzed using the moments of a random variable or the procedure for maximizing the log-likelihood function (Cam