PSI - Issue 31

Artyom Chirkov et al. / Procedia Structural Integrity 31 (2021) 80–85

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Artyom Chirkov et al. / Structural Integrity Procedia 00 (2019) 000–000

3

(a) (c) Fig. 2. Simulated distribution of grains in the computational domain (a), the log-normal fit of distribution (b), the dependence of initial yield stress on grain size (c). this work, we used this method to obtain the polycrystalline sample which is illustrated in Fig. 2a. Dimensions of the computational domain are also shown in Fig. 2a. The polycrystalline aggregate consists of 500 grains having di ff erent sizes. The grain’s size distribution is given in Fig. 2b and approximated by the log-normal law. The dependence of yield stress on grain size is illustrated in Fig. 2c. A total number of mesh elements (voxels) is 499 along the Z-axis, 100 along the Y-axis, and 20 along the X-axis. Table 1 summarizes all physical-mechanical properties of polycrystalline sample which are used in the modeling. (b)

Table 1. Physical-mechanical properties of the material. ρ , g / cm 3 K, GPa µ , GPa

Y

0 , MPa

∆ σ , MPa

h , MPa

7.846

172

79.2

see Fig. 2c

40

300

The Finite-di ff erence method (FDM) is employed to simulate the plastic flow of sample. Numerical simulation is carried out in a 3D formulation. Comprehensive formulation of the method applied, as well as basic equations, could be found in Wilkins (1999).

3. Mathematical formulation of the problem

System of solid mechanics equations is used as a basis for mathematical model Romanova et al. (2011); Makarov and Peryshkin (2017); Eremin and Makarov (2019); Balokhonov et al. (2020). We assume that all grains are isotropic. Moreover, for the sake of simplicity, all grains have the same values of elastic moduli. The boundaries between grains represent an ideal mechanical contact. Therefore, the only inhomogeneity of samples is due to the dependence of individual grains yield stress on their size. Details of the ”up-down-up” constitutive equation are discussed below. Here, we assign the linear hardening law of material in point. The dependence of yield stress on plastic strain is described by Eq. 1. Y =   Y up = Y low + ∆ σ, γ P = 0 Y low + h γ P , γ P > 0 , (1) wherein Y up is an upper yield stress, Y low = Y 0 in Fig. 2c is a lower yield stress, and ∆ σ is a stress drop in a local point of material. J 2 -plasticity is employed as a part of the constitutive response and is given by Eq. 2.

P ) = τ −

2 3

f ( σ i j , γ

Y

(2)