PSI - Issue 40

Artyom A. Tokarev et al. / Procedia Structural Integrity 40 (2022) 426–432 Artyom A. Tokarev, Anton Yu. Yants / Structural Integrity Procedia 00 (2022) 000 – 000

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where п is the fourth rank tensor of the crystal grain elastic properties, the components of which are invariable in the MCS. 3. Polycrystal representative volume determination This chapter presents the results of work on the determination of the representative volume of a metal polycrystal, which was carried out using a statistical analysis of volume-average stresses (averaging method, hereinafter referred to simply as stresses) of polycrystalline samples consisting of different numbers of randomly uniformly oriented grains at a fixed moment of the loading process. Statistical analysis implied finding the curve of the average stress over the realizations, as well as their standard deviation. The stresses were obtained by solving a direct boundary value problem. Based on the obtained curves, it is possible to determine the representative volume as the minimum volume, above which the average response of the sample to the same load remains constant with a given accuracy. To estimate the representative volume of the sample, direct boundary value problems were formulated and solved (Trusov and Shveykin (2013)). Polycrystalline samples included k 3 , k = 1..10 cubic grains with an initial orientation distributed by a random uniform law. The samples had unit sizes without specifying the dimension of quantity, since the scale effects were not taken into account in the model. For each given number of grains (except for 1 and 8) R = 20 realizations of loading process with different initial grain orientations were performed. For each set of realizations corresponding to a given number of grains, the mean stresses and their standard deviation were computed when the sample was compressed by 0.1% of the initial height. This percentage of compression is sufficient to activate all slip systems. An estimate (7) based on an unbiased estimate of variance was used to calculate the standard deviation Then the whole polycrystal was divided into finite simplex elements preserving grain boundaries. Partitioning into finite elements (FE) was performed using the gmsh freeware. To solve this problem, a package of programs based on FEM and developed at the MMSP department of PNRPU was used. Using this package, solutions were previously obtained for single-crystal samples of Al (Yants, Trusov and Teplyakova (2017)), which also showed that in the process of deformation a misorientation of individual parts of grains occurs, which are represented by sets of crystallites. The problem of generating the grain structure of a polycrystal and its following FE partition was also solved. The grain structure was generated by dividing a cube, which is the geometry of the entire polycrystal, into equal subcubes, each of which describes the geometry of a particular grain. At the initial moment of time, the orientation of each subcube (grain) was chosen randomly according to a uniform distribution law. The orientations of crystallites (material in finite elements) within the same grain were set the same. Then, the obtained geometry of the polycrystal was partitioned into FEs (tetrahedrons) using gmsh. Thus, at first, all edges are divided into subedges, then the faces into triangles, at the last stage, a volumetric FE partition is performed. gmsh allows you to estimate the quality of the mesh by reporting the values of the minimum and average quality of FEs according to three quality criteria. Based on the quality data of the meshes obtained as a result of experiments with gmsh, it can be stated that using this program it is possible to obtain a mesh of a sufficiently high quality for use in solving of boundary value problems by the FEM. Each grain in each loading process realization consisted of approximately 900 FE. The FE mesh for a polycrystal of 5 3 grains is shown in fig. 1. To obtain the response of the material in each FE (homogeneous crystallite), the elastoviscoplastic model of the physical theory of plasticity (Trusov, Shveykin (2017)), described above, was used. To describe grain boundary hardening the facets of finite elements as crystallite boundaries and neighboring crystallites were explicitly considered (including the orientation of crystallites lattice, shear rates, and other internal variables).   2 1 . 1 x M x R   i R i   (7)