*5.1. Uniaxial Tension*

Uniaxial tension with constant strain rates has been used to experimentally study the rafting behavior in superalloys. Using strain control in the phase-field simulation, we can achieve the tensile mode in solving equilibrium equations. Figure 9 presents the microstructure evolution and plastic strain during the tension of a nickel-based single crystal superalloy. It is evident that there is no significant change in the microstructure, and there is plastic deformation near the γ/γ' interface and the center of the γ' phases. Comparing the numerical results with experimental results, we can determine the mechanical properties of the γ/γ' microstructure and the bulk phase (if two phase structures are not distinguishable) from the input parameters for the numerical calculation. Additionally, we can evaluate the strengthening effect of the γ' phase and determine the effect of the γ' phase on the mechanical properties of nickel-based superalloys on the macroscale.

**Figure 9.** Numerical results of the microstructure and plastic strain at different instants during the tension of a nickel-based single crystal superalloy at a strain rate of 10−<sup>3</sup> per second and 1253 K: (**a1**–**d1**) microstructure, and (**a2**–**d2**) plastic strain (Reproduced with permission from [8] © 2020 Elsevier).

Cottura et al. [67,88] introduced a characteristic plastic length, ζ, in 1D configuration to illustrate the size effect on the plastic deformation of the γ phase and the mechanical properties of the material. Their simulation results suggest that plastic strain is homogeneous in the γ phase if the characteristic plastic length is negligible. Increasing the characteristic plastic length led to the inhomogeneous distribution of the plastic strain and the decrease of the peak value of ζ. For the value of ζ being larger than the width of the γ channels, no plastic deformation would occur in the γ channels. Increasing the characteristic plastic length also increased the flow stress during the tensile deformation.

Wang et al. [8] included the Orowan stress in a visco-plasticity phase-field model to analyze the size effect in a two-dimensional system. Their results show that the flow stress during the tensile deformation decreases with the increase of the γ channel width and the flow stress remains unchanged after the γ channel width reaches a critical value. Zhang et al. [7] also reported similar results for the tension of nickel-based superalloys at high temperature. They pointed out the dependence of the yielding stage during tensile deformation on the penetration of dislocations into γ' precipitates.
