# Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material
## 2 Material Model and Crystal Plasticity Model
### RVE Polycrystalline Material Model  
Polycrystalline metal has a structure with randomly arranged grains, when studying its fatigue failure mechanism, the inhomogeneous deformation caused by the anisotropy properties and crystal slip of grain should be taken into consideration. To model the material, referring to the literature [30,31,34], the Voronoi RVE is constructed using a Voronoi polyhedron aggregation, see Figure 2. The RVE consists of 216 grains, divided as 8000 eight-node hexahedral elements, and has 9261 nodes. In RVE, the shape, size and crystal orientation of the grains is numerically generated in a random manner. The model does not include the factors of cracks, micro holes, other metallurgical defects, and their evolution as well. The boundary of adjacent element sets is regarded as the grain boundary, which is a geometric interface fully connected. The RVE is set as a material element, and its apparent response is the same as that of the uniform deformation section of the specimen under strain control. It needs to point that, considering the very large amount of calculation in the fatigue cycle simulation, so it is impossible to set too many grains and elements in the model. Therefore, the model does not consider the reinforcement phase and phase interface inside the grain but uses the approximately equiaxed ideal FCC crystal to simulate the grain. The size effect and the strain gradient influence, which are related to the complex microstructure, are neglected. The overall response of RVE is approximately isotropic, and Cauchy stress tensor \(\Sigma\) and logarithmic strain tensor \(E\) separately equaled to the mean of the local Cauchy stress \(\sigma\) and logarithmic strain \(\varepsilon\) over the RVE, approximately, and are, respectively, calculated by the load per unit area and displacements of the surfaces. The reference coordinate system chosen is identical to the both macro and micro stress strain tensor components, \(\Sigma_{ij}\) and \(E_{ij}\) with \(\sigma_{ij}\) and \(\epsilon_{ij}\). Figure 1: Steady stress–strain loops of GH4169 at 500 \({}^{\circ}\)C (according to the test data from ref. [33]). Figure 2: Local and overall stresses and strains of the representative Voronoi volume element (RVE). In this paper, the boundary condition is designed as the deformation of RVE surfaces macroscopically consistent with the specimen and maintaining uniaxial stress state. Under the tensile-compressive cyclic loading at specified strain amplitude, the changes in macroscopic strain of the specimen are controlled by the applied loading. The displacement vector \(\mathbf{u}\) on the RVE boundaries satisfies the equation. \[\mathbf{u}=\overset{\text{-}}{\mathbf{F}}\cdot\mathbf{X} \tag{1}\]  
where \(\mathbf{F}\) is the overall deformation gradient of the material, and \(\mathbf{X}\) is the position vector. In Figure 2, the three surfaces of RVE whose outer normal is opposite to the coordinate axis are, respectively, fixed in normal, and the nodes of other surfaces are, respectively, constrained linearly to ensure that the deformed surface keeping planar and the outer normal remains unchanged. The peak values of the measured longitudinal strain \(E_{a}\) and stress \(\Sigma_{a}\) are approximately constant, respectively. The stretching (longitudinal) direction is parallel to the direction of coordinate axis 3. The macroscopic stress components: \(\mathbf{\Sigma}_{ij}=0\), \(i\
eq j\); and \(\Sigma_{ii}=0\), \(i\
eq 3\) (where the subscript index drawn with underline means no sum). If the periodic boundary condition is used instead of plane assumption, the displacement vector on the boundary must satisfy the equation. \[\mathbf{u}=\overset{\text{-}}{\mathbf{F}}\cdot\mathbf{X}+\mathbf{v} \tag{2}\]  
where \(\mathbf{v}\) is a periodic fluctuation vector. To achieve this, we need to change the surface node constraints mentioned above into the linear constraints of the corresponding nodes of the positive and negative surfaces. It allows the parallel surface of the RVE to fluctuate parallelly.