PSI - Issue 6

Ekaterina L. Alekseeva et al. / Procedia Structural Integrity 6 (2017) 128–133 E.A. Alekseeva et al. / Structural Integrity Procedia 00 (2017) 000–000

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Fig. 3. The distribution of the Mises equivalent plastic strain field for the case of average axial strain of the sample is 20%.

The dependence of the yield stress on the average grain size in a polycrystalline metal is described by the empirical Hall-Petch law cf. Petch (1953); Hall (1951). Hall and Petch explained the dependence they discovered by the fact that the growth conditions of the microcrack inside the grain depend on the grain size. The e ff ect of surface tension forces on the yield strength was taken into account in modeling the experimental data (see Kudinova (2016)). These forces depend on the grain size, both due to the curvature of its surface, and due to the ratio of bulk elastic forces to surface forces. Based on this assumption, it is possible to model an isotropic polycrystalline material as a set of single-crystal grains having di ff erent mechanical characteristics. Because of the huge number of grains, an analytical description of the material is possible only with the help of stochastic di ff erential equations. This approach adds to the model the possibility of stochastic instability, which makes it di ffi cult to directly verify the initial assumptions. Consider a model material in which the elastoplastic deformation of each grain obeys a bilinear hardening law. The grain sizes have, in the first approximation, a normal distribution with a relatively small dispersion. Therefore, it can be assumed that the distribution of the yield strengths of grains also satisfies a normal distribution with a satisfactory accuracy. We consider the problem of quasistatic tension of a sample in the form of a thin strip from the described material. Simulation is performed with help of finite element program ANSYS. We shall model the microstructure of the material in such a way that each finite element is one grain. In order to simplify the discrete model, we set the geometric sizes of all finite elements to the same. The dynamic problem is solved for the three-dimensional case. The boundary conditions have the form: u z | z = 0 = 0 , u z | z = L = vt The properties of the material used in the calculation are shown in Table 1.

Table 1. Properties of material. Property

Value

E, GPa

425 0.3 850

ν

M[ σ T ], MPa D[ σ T ], MPa

72.25

H, GPa

2

The distribution of the Mises equivalent plastic strain field on the surface of the sample is shown in Fig. 3 for the case of average axial strain of 20%. There is a characteristic inhomogeneity of strain in the form of strips directed at an angle to the direction of the load action. As the total deformation of the sample increases, new bands of plastic strain localization appear. In this case, the old ones either stabilize or become wider. This e ff ect can be interpreted as ”jumps” of regions of plastic strain localization, when they are suddenly formed at points not connected with the already existing ones. The dependence of the plastic strain on the coordinate on the sample surface along the center line (see Fig. 3) for the average deformation of the sample of 20% is shown in Fig.4.