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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1. Introduction Polycrystals mechanical properties substantially depend on the material structure state (Beyerlein I. J. & Knezevic M. (2018), Trusov P. V. & Shveikin A. I. (2019)). Effective properties used in solving problems of plasticity, creep, fatigue, and others are defined by the grain structure (Armstrong R. W. 1970, Fan H. et al. (2015), Cruzado A. et al. (2018)). One of the most famous grain structure influence on material properties is the dependence of a yield stress on an average grain size (Hall-Petch effect) (Hall E. O. (1951), Petch N. J. (1953), Armstrong R. W. (1970), Kozlov E.V. et al. (2004)). Besides the grain size, important characteristics determining the mechanical properties (especially anisotropy) are parameters describing grain structure morphological features (primarily the lattice orientations distribution, the shape and boundaries of the grains) (Hirth J. P. (1972), Dillon S. J. et. al. (2016), Yan F. et. al. (2017), Quey R. & Renversade L. (2018)). This is clearly demonstrated in polycrystalline materials with a lamellar, columnar, acicular, dendritic structure and is greatly enhanced by a presence of residual nonequilibrium phases (for example, martensite) (Yan F. et. al. (2017), Kumar P. et. al. (2018), Sohrabi M. J. et. al. (2020), Zou Z. et al. (2021)). A significant change (usually grain size reduction) in the grain structure occurs in a result of severe plastic deformation. This is closely related to the processes of fragmentation and subsequent grains refinement (Kozlov E. V. et al. (2004), Valiev R. Z. & Langdon T. G. (2006), Koneva N. A. et al. (2008), Ostanina T. V. et. al. (2020)). Most of technological processes of metal forming such as rolling, drawing, etc. usually occur at elevated temperatures, which leads to activation of high-temperature processes accompanying plastic deformation. In this case, recrystallization, recovery and phase transitions have a special influence on a grain structure state (Rollett A. et. al. (2017), Nasiri Z. et. al. (2020)). As a result of intense high-temperature deformation grain and defect structures are developed, that leads to changing in material macro-properties. Modern methods of mathematical modeling make it possible to explore both material stress-strain state and structure (the grain one and the defect one) evolution. The effective and flexible tool for this purpose is multilevel models with internal variables, which are based on detailed consideration of inelastic deformation processes at various structural-scale levels (Trusov P. V. & Shveikin A. I. (2019))). There are two main types for the models of this class: direct one and statistical one (Trusov P. V. & Shveykin A. I. (2013a), Trusov P. V. & Shveykin A. I. (2013b)). Direct models are based on a spatial consideration of field variables (stresses, strains, and other internal variables) for each grain included in the considered representative macrovolume, taking into account its orientation, shape, and boundaries. Statistical models consider a representative macrovolume of material (“macropoint”) as a set of individual grains combined into a polycrystalline aggregate using one of the hypotheses: Reuss, Voigt or Kroner. Direct models are more precision and detailed, but they are more resource-intensive than statistical ones. Within the framework of the multilevel approach, the modified statistical model of inelastic deformation is proposed, where the state of neighbor grains is explicitly taken into account by their interaction along contacting boundaries. In this case, the key element of model development is an initial grain structure formation according to the experimental data and its subsequent restructuring under inelastic deformation. The statistical model allows realizing random distribution (according to a certain law) grain structure elements (grain volume, number of neighbor grains, orientation and area of grain boundary flat sections (facets)), but in this case it isn’t possible to ensure that it corresponds to the real geometry of the grain structure. The paper considers the procedure for describing the initial grain structure formation, its restructuring within the framework of modified statistical multilevel models that take into account the state of neighbor structural elements. 2. The procedure for grain structure construction Currently, the most commonly used methods for grain structures construction are the geometric methods of Voronoi and Laguerre (Fan H. et al. (2004), Redenbach C. (2009)). A structure obtained by the Voronoi method consists of an array of non-concave, completely filling the space and non-overlapping polyhedron. However, this method generates structures using average number of faces per polyhedron exceeding that in real structures (Matzke E. B., & Nestler (1946), Kumar P. et. al. (1992), Liu G. et. al. (2002)). Volumes of Voronoi polyhedra obey the gamma distribution (Kumar P. et. al. (1992)), although in real materials the distribution law can be arbitrary. The generalization of the Voronoi is the Laguerre method (Suzudo T. & Kaburaki H. (2009), Morfa C. R. et al. (2018)). In this method, the polyhedral structure is generated from a set of initial "embryos" with given weights, which later