PSI - Issue 40

N. Kondratev et al. / Procedia Structural Integrity 40 (2022) 239–244 N. Kondratev et. al. / Structural Integrity Procedia 00 (2022) 000 – 000

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"grow" to polyhedra that completely fill the space without overlaps and voids. The weighting factors of polyhedra determine the shape of the grains (in the statistical sense), which may differ from the equiaxial shape. In classical models of the Taylor – Bishop – Hill type, based on the statistical approach, the spatial discretization of the model variables and the geometry of the grain structure isn’t performed. In this case, it is possible to set independently for each grain the geometric characteristics, using the statistical distribution laws of the corresponding quantities (grain volumes, facet areas, etc.). Wherein, the question about the “compatibility” of the grain structure built in this way inevitably arises. It isn’t possible to match geometric characteristics obtained by statistical methods with the grain structure in three-dimensional space without voids and overlaps. In the considered modified statistical model, it is assumed that each grain is surrounded by certain neighboring grains (Fig. 1, (a)). The grain structure for the modified statistical model is determined by specifying the volume and shape for each grain indicating the boundary facets and adjacent grains. The grain boundary facets are determined by their normals n i and areas s i ( i is the facet number index). To implement this class of models, data on the current state of neighboring grains, described by model internal variables, including parameters of the stress-strain state, defective structure, and geometric characteristics are required. In this regard, the grain structure constructing procedure for the statistical model using the free software Neper is proposed (Quey R. & Renversade L. (2018)). The Neper package allows creating a grain structure of polycrystals using the Laguerre approach and corresponding statistical experimental data. Experimental data are used to define the main grain structure parameters, one of which is the average grain size d . More precisely, the grain structure is specified by the grain sizes statistical distribution. The grains shape is determined much less frequently in experiments. A possible parameter for describing the shape of grains is “sphericity” S , which is defined as the ratio of a sphere surface area (with a volume equal to a volume of a grain) to the grain surface area. The maximum value of the parameter S is 1 for a grain in the form of a sphere (Quey R. & Renversade L. (2018)). Experimentally established geometric characteristics of grains are input parameters in the Neper software, on this basis the polyhedral grain structure is constructed. It is possible to specify statistical distribution laws for both parameters: grain size d and sphericity S . To form a structure corresponding to the experimental data, Neper solves the optimization problem using the Praxis method (Brent R. P. (2013)). This allows the structure to be updated locally, providing better performance in generating the required structure. The variables in the optimization problem are coordinates of grains centers and weight coefficients. The objective function is constructed on the basis of the sphericity distribution laws S and the relative grain size d eq : d eq = d /< d >, where d is the grain size, defined as radius of an equivalent volume sphere, < d > is the average grain size. The optimization problem is given in (Quey R. & Renversade L. (2018)), numerical implementations of algorithms for its solution is built into Neper. As a result, by Neper for each grain following parameters are determined: grain volume v , facets areas s i and facets normals n i , neighbor grains, which also have their own values v , s i , n i . This information represents the initial data on the grain structure for the statistical model, which are transmitted into the program module of the latter to solve the problem of deformation process description. A grain structure formed in this way for the statistical model (a set of parameters v , s i , n i with indication of neighbor grains) has an image in the three-dimensional Euclidean space (Fig. 1, (b)). In a process of plastic deformation, a distribution of variables d eq and S changes, which are calculated in the statistical model. With a significant evolution of d eq and S , the grain structure is re-formed in Neper and transferred back to the statistical model.