PSI - Issue 25
Aleksandr Shalimov et al. / Procedia Structural Integrity 25 (2020) 386–393 Author name / Structural Integrity Procedia 00 (2019) 000–000
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To create models of interpenetrating structures, methods based on the analytical definition of surfaces that separate the two phases are used. The geometry models of this type can be periodic or random. For models with periodic structure, analytical expressions containing the sum of products of periodic functions ( sin and cos ) are used. The representative volume elements created in this way have symmetry of crystallographic groups, such as cubic, tetragonal, rhombohedral and rhombic. Typically, materials which microstructure that would totally comply with models of periodic structure are rare, and such models are more often used to approximate and simplify existing random environments. A more common case is when the microstructure of materials is characterized by the randomness of the mutual arrangement of phases. In this paper, emphasis was placed on the study of such type of media. One of the methods to develop random interpenetrating structures is to determine the boundary between two phases by using the values of the random Gaussian fields function (Bargmann et al., 2018). In this case the structure can be created in two steps: (1) generation of a random field in two or three dimensional space on the basis of a Gaussian random function represented by a Fourier series containing random values; (2) performing level set for the generated field on the basis of the condition by which the points of the random field belong to the first or second phase. The following function can be used to define interpenetrating structures using Gaussian random fields (Berk, 1987; Cahn, 1965): 1 1 2 N i i i i f x c Cos k x a N (1) Here, is position of radius vector, is size of RVE, sets the number of harmonics, wave phases � are evenly distributed on the interval �0,2 � , wave directions � are taken equal. The coefficient � is randomly selected. Different phases of a representative volume are determined by assigning points of space according to the following conditions: the point belongs to phase 1 if � � � and phase 2 if � � � , where the parameter defines the separation surface. The examples of resulting open-cell microstructure models are presented on Fig. 1.
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Fig. 1. (a) RVE model with pores volume fraction 25%; (b) RVE model with pores volume fraction 50%
For the possibility of application of FE analysis to the microstructure RVEs, algorithms of discretization of the derived geometrical regions using meshes with different element configurations have been developed and implemented. For the purpose of the present study, RVE geometry was meshed using four-node tetrahedral elements. The advanced mesh controls were used to reach mesh consistency on the borders between two phases. The meshed geometry for the above illustrated models is shown on Fig. 2.
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