Crystal Plasticity Finite Element Simulation Considering Geometrically Necessary Dislocation Distribution for Reproducing Mechanical Anisotropy of Rolled CrMnFeCoNi High-Entropy Alloy

Nomun Gerel-Erdene; Yoshiteru Aoyagi

doi:10.2320/matertrans.MT-MA2024006

Abstract

The current trend in research on the physical properties of high-entropy alloys has been progressively increasing as there are many unknown possibilities for developing high-entropy alloys for advanced applications. This study investigated the effect of microstructures of rolled high-entropy alloy from the viewpoint of crystal orientation and dislocation density distribution to reproduce mechanical anisotropy using crystal plasticity finite element simulation. The crystal orientation and the geometrically necessary dislocation density of the rolled material were quantitatively estimated from experimental data of electron backscatter diffraction. Microstructural observation showed that particular textures were preferably oriented like in typical FCC metals. Even though the simulation results where only the preferred crystal orientation was considered did not show the expected mechanical anisotropy as in the experiment, the computational model with the dislocation density distribution and the preferred orientation showed the same tendency as the experiment.

1. Introduction

Recently, many studies have shown significant attention to high-entropy alloys (HEAs), and the number of studies on their processing, mechanical properties, and modeling properties has been increasing [1]. The functional application of HEAs requires extensive research on their microstructure and mechanical properties. It is reported [2, 3] that simple FCC, BCC, and FCC-BCC phases are mainly observed in HEAs, which makes it simple to compare the material with other FCC alloys with low stacking fault energy (SFE), such as stainless steel and twinning induced plasticity steels [4].

Rolled metals show remarkable mechanical properties depending on only the amount of rolling reduction. For instance, the tensile strength of 20% cross-rolled SAE 304 stainless steel had reached 855.8 with 45.0% elongation without any heat treatment after rolling, and it had been observed that the ductility of the material was enhanced with the rolling reduction. In HEAs, the primary deformation mechanism is associated with the dislocation motion since the plastic deformation is carried by the movement of dislocations rather than twinning in low SFE alloys [5]. The dislocation motion in each slip system depends mainly on crystal orientation—however, the oriented crystal orientation after rolling exhibits mechanical anisotropy in the material. The previous study [6] investigated the influence of crystallographic texture to predict the mechanical anisotropy with analytical models provided with random crystal orientation extracted from electron backscatter diffraction (EBSD) data. Even if the crystal orientations were reproduced, the modeling approach in which the dislocation structure is not characterized could not reproduce the experimental results regardless of modifying mesh size and changing grain shape and size distribution. However, a study [7] on predicting anisotropy of a titanium alloy reported the significance of dislocation-density-dependent crystal plasticity finite element method (CPFEM), showing that considering the dislocation density for the dominant slip system along with the crystal orientation plays a role to exhibit the mechanical anisotropy.

This study aimed to investigate the effect of rolling on the microstructure and to reproduce the mechanical anisotropy of CrMnFeCoNi HEA through experimental observation using EBSD measurement and the crystal plasticity FE simulation. First, we developed methods for selecting the preferred crystal orientation from the EBSD data and for quantification of the relationship between the obtained GN dislocation density and the crystal orientation information. Finally, we investigated whether the mechanical anisotropy was reproduced and compared the analysis and experimental results.

2. Experimental Procedure

2.1 Process and materials

The equiatomic CrMnFeCoNi HEA in this study was prepared by cold-rolling of 20% after arc melting at 1100°C and hot rolling. The specimens were then annealed at 900°C for 1 h and cooled. EBSD observation was conducted to measure the grain size and crystal orientations of the specimens. An area of 750 µm in all directions was scanned with a step size of 2 µm. The microstructure was observed from a rolling direction (RD) and a transverse direction (TD). The boundary of the measurement point where the crystal orientation difference was 15° or more was defined as the grain boundary. The tensile test specimens whose loading direction is parallel to RD and TD were cut from the rolled sheet, respectively, and the uniaxial tensile tests were performed in each direction at a strain rate of 1.0 × 10⁻³ s⁻¹.

2.2 Experiment results on microstructure and mechanical properties

2.2.1 Microstructure observation by EBSD

Figure 1 shows the crystal orientation distribution of the material after rolling and annealing. In the rolled material, the crystal grains were elongated to RD, and the orientation in the crystal grains was not uniform. Since the texture after rolling is close to that often observed during cold rolling of FCC metal [8], it can be considered that all the crystal structures are FCC structures and caused by cold rolling. Also, the CrMnFeCoNi alloy is known to form FCC structured alloy [1]. The average grain size of the rolled material is about 80 µm, and that of the annealed material was about 30 µm. The overall texture of the annealed material was weaker as it can be seen from the orientation distribution function (ODF) map shown in Fig. 2. The texture evolution after rolling can be investigated from the ODF map. In the φ₂ = 0° and φ₂ = 45° section, the main intensity peak laying along α-fiber is located around G/B ({110}⟨115⟩), Bs ({110}⟨112⟩) and G({110}⟨001⟩) components. On the other hand, Rt-G ({110}⟨110⟩) and Cube ({001}⟨100⟩) components had disappeared after rolling.

Fig. 1

Crystal orientation distribution for normal direction (ND) observed (a) from RD, (b) from TD of rolled HEA, and (c) in annealed HEA.

Fig. 2

ODF maps of (a) annealed and (b) rolled HEAs.

2.2.2 Stress-strain curves measured by uniaxial tensile test

Figure 3 shows the true stress-strain curves of annealed material and rolled material elongated along RD and TD. The true stress and strain were estimated from nominal stress and strain using the general true-nominal translation equations. 0.2% proof stresses were 809 MPa for RD, 856 MPa for TD, and 222 MPa for annealed specimens. The tensile strength of the rolled material was 860 MPa for RD, 899 MPa for TD, and 590 MPa for annealed material. The material in TD tension had higher strength and lower ductility than that of in RD tension, indicating that the rolled material exhibited mechanical anisotropy. Comparing the annealed material with the rolled material, the yield stress and tensile strength were significantly lower, but the ductility was higher. In contrast, a comparison of the true stress-strain curve showed that the stress after 40% true strain in the annealed material increased to the stress level as good as the stress after yielding in RD. This result suggests that the initial dislocation density of the rolled material is considered to correspond to the annealed material at 30% true strain.

Fig. 3

True stress-strain curves for TD and RD of rolled HEA and that for annealed HEA.

3. Crystal Plasticity Finite Element Simulation Considering Preferred Orientation and GN Dislocation Density

3.1 Selecting preferred orientation that reflects rolling texture

This study introduced the method to select the preferred orientation, where crystal orientations closely aligned. Using a parameter called “orientation degree”, each measurement point was weighted based on the number of points whose misorientation angles are 5° or less representing the lower limit of low-angle-grain boundaries [9]. Higher orientation degree indicated closer crystal alignment. The arranged data was then divided equally into the number of grains in the computational model, with one orientation randomly selected per grain. Figure 4 shows the ODF map where the selected orientations for ten combinations with 64 grains are plotted along with EBSD data of rolled HEA.

Fig. 4

Distribution of crystal orientation selected by the proposed method in ODF map with experimental data.

3.2 Crystal plasticity model

The elastoplastic constitutive equation for crystal plasticity theory in the rate form considering slip system is expressed as follows.

\begin{equation} \overset{\nabla}{\boldsymbol{T}} = \boldsymbol{C}^{\text{e}}:\boldsymbol{D} - \boldsymbol{C}^{\text{e}}:\sum_{\alpha}\dot{\gamma}^{(\alpha)}(\boldsymbol{s}^{(\alpha)} \otimes \boldsymbol{m}^{(\alpha)})_{s} \end{equation}

(1)

where $\overset{\nabla}{\boldsymbol{T}}$, C^e, and D are the corotational rate of Cauchy stress, the anisotropic elastic modulus tensor, and the deformation rate tensor, respectively, and $\dot{\gamma }^{(\alpha )}$, s⁽^α⁾, and m⁽^α⁾ are the slip rate, the slip direction vector, and the slip plane normal vector of the slip system α, respectively. The slip rate $\dot{\gamma }^{(\alpha )}$ is defined as the following hardening law [10].

\begin{equation} \dot{\gamma}^{(\alpha)} = \dot{\gamma}_{0}^{(\alpha)}\left(\frac{\tau^{(\alpha)}}{g^{(\alpha)}}\right)\left|\frac{\tau^{(\alpha)}}{g^{(\alpha)}}\right|^{\frac{1}{m} - 1}\dot{\gamma}_{0}^{(\alpha)} \end{equation}

(2)

where, τ⁽^α⁾, g⁽^α⁾, and m are the reference slip rate, the resolved shear stress, the flow stress, and the strain rate sensitivity, respectively. The resolved shear stress τ⁽^α⁾ for the slip system α can be expressed using the Cauchy stress tensor T as,

\begin{equation} \tau^{(\alpha)} = (\boldsymbol{s}^{(\alpha)} \otimes \boldsymbol{m}^{(\alpha)}):\boldsymbol{T} \end{equation}

(3)

The flow stress g⁽^α⁾ for slip system α is expressed using the Bailey-Hirsch equation extended to polycrystal structures.

\begin{equation} g^{(\alpha)} = \tau_{0}{}^{(\alpha)} + \sum_{\beta} a\mu b\sqrt{\varOmega_{\textit{SS}}^{(\alpha\beta)}\rho_{\textit{SS}}^{(\beta)} + \varOmega_{\textit{GN}}^{(\alpha\beta)}\rho_{\textit{GN}}^{(\beta)}} \end{equation}

(4)

where τ₀⁽^α⁾, a, μ, and b are the reference flow stress, numerical parameter, shear modulus, and magnitude of the Burgers vector, respectively. $\rho_{\textit{SS}}^{(\beta )}$ and $\rho_{\textit{GN}}^{(\beta )}$ represent the statistically stored (SS) dislocation density and the GN dislocation density, respectively, and the dislocation interaction matrices between the slip system α and slip system β are denoted by $\varOmega_{\textit{SS}}^{(\alpha \beta )}$ and $\varOmega_{\textit{GN}}^{(\alpha \beta )}$, respectively [11, 12]. This parameter could differ depending on the stacking fault energy (SFE) of types of HEAs. The HEAs in this study have low stacking fault energy, and the smaller SFE is the most likely latent hardening in the material. Therefore, the dislocation interactions were considered weaker than those of high SFE materials so that the dislocation could move more independently. This dislocation strengthening or dislocation interaction is considered in the interaction matrix in the equation. In CPFEM, it is difficult to directly influence the atomic-level fluctuations of HEAs on the constitutive equation-level parameters because the element size is about submicron. Such parameters should contain information that already reflects atomic-level fluctuations, and such simplification does not preclude the reproduction of the singularity of HEAs.

3.3 Estimating GN dislocation density from EBSD data

Dislocation accumulation during deformation alters mechanical properties in crystalline materials. Thus, considering the rolling-induced dislocation density distribution is crucial for reproducing the rolling texture. In some studies [13–15], two main approaches were employed to recover the GN dislocation from the EBSD measurement data because the measurement can obtain the local crystal orientation and the lattice distortion. The first approach [16], based on the strain gradient model, utilizes kernel average misorientation (KAM) as a measure of the GN dislocation content. The second approach mainly relies on the dislocation density tensor. Calcagnotto et al. [17] demonstrated methods, including the KAM, to calculate the GN dislocation density from 2D-EBSD and 3D-EBSD data, yielding consistent results. However, the KAM value depends on the distance of selected neighboring points, drastically changing with increasing neighboring rank.

In this study, the calculation of GN dislocation density from EBSD is based on the second method, in which the lattice curvature is assumed to be the content of GN dislocation density. Since only the curves caused by the dislocation can be measured via EBSD, the dislocation can be resolved from the dislocation density tensor expressed by its form of orientation gradient [18]. A difference between two orientations is defined by the rotation and described by the misorientation angle, which can be derived in many substitutive ways. However, Pantleon et al. [19] calculated the misorientation angle from the orientation matrix as follows.

\begin{equation} \Delta\theta_{k} = -e_{kij}\Delta g_{ij}\frac{\Delta\theta}{2\sin \Delta \theta} \end{equation}

(5)

where Δθ = arccos[(Δg_ii − 1)/2]. Thus, from the misorientation between two neighboring points distanced by the step size (Δx_l), the lattice curvature is expressed as,

\begin{equation} k_{kl} = \frac{\partial \theta_{k}}{\partial x_{l}} \approx \frac{\Delta \theta_{k}}{\Delta x_{l}} \end{equation}

(6)

The accuracy of this prediction method is affected by many things, including the accuracy of the EBSD measurement (mainly the clarity of the crystal lattice). It also detects orientation differences due to elastic lattice distortion. In this case, the dislocation density is overestimated, but if rolled specimens are observed, as in this study, they are considered negligible compared to dislocations introduced by rolling.

However, the components of the curvature tensor in the third axis cannot be obtained from the EBSD data that was measured in only two directions because the curvatures are determined only in the investigating plane. Since the components of the dislocation density tensor cannot be represented by the dislocation types solely, the total dislocation density tensor is expressed as the sum of each dislocation density ρ⁽^α⁾ from a different dislocation system α, with the Burgers vector b⁽^α⁾, and dislocation line vector l⁽^α⁾.

\begin{equation} {\boldsymbol{\alpha}} = \sum_{\alpha}\boldsymbol{b}^{(\alpha)} \otimes \boldsymbol{l}^{(\alpha)}\rho^{(\alpha)} \end{equation}

(7)

There are 18 possible configurations of dislocations in FCC crystals (12 for edge dislocation and 6 for screw dislocation) with Burgers vector b = a/2⟨110⟩. For this indeterminate linear system, the total dislocation density is minimized by the linear optimization programming to recover the content of GN dislocation density, assuming that the weights for all dislocation types are equal. The details of this approach are given in the study [19].

3.4 Quantitative estimation of relationship between crystal orientation and GN dislocation density

This study investigated the effect of the crystal orientation and the GN dislocation density for the mechanical anisotropy from the viewpoint of the Schmid Factor. The starting model, which reflects the dislocation density distribution and crystal orientation after rolling, is established from the relationship between the GN dislocation density and the Schmid factor. Since any slip system with a higher Schmid factor is susceptible to slip, the GN dislocation density resolved from the slip system with maximum Schmid factor is considered for its relationship to the crystal orientation. The loading direction to calculate the Schmid factor was TD. Figure 5 shows the correspondence of the moving average of the GN dislocation density to the Schmid factor (to TD). By approximating, the quantitative relationship between the dislocation density and the crystal orientation can be expressed as follows.

\begin{equation} \rho_{\textit{GN}} = \frac{c_{1}}{1 + e^{-c_{2}(f - c_{3})}} + \rho_{0} \end{equation}

(8)

Here, c₁ = 5.3 × 10¹⁴, c₂ = −30.0, c₃ = 0.49, and ρ₀ = 4.3 × 10¹⁴.

Fig. 5

Estimated relationship between GN dislocation density obtained from EBSD and Schmid factor to TD.

3.5 Tensile simulation considering GN dislocation density distribution and preferred crystal orientations

The tensile simulations along RD and TD were performed to investigate whether the mechanical anisotropy was reproduced. The computational model with second-order elements has 5,282 elements and 64 grains with an average grain size of 80 µm (see Fig. 6). Periodic boundary conditions were applied in all directions. In order to compare the effectiveness of the proposed method, ten different combinations of crystal orientation were considered in three simulations: (i) conventional - orientations were selected randomly from EBSD data, (ii) only the preferred orientations were considered without the GN dislocation density distribution, (iii) both the preferred orientations and the GN dislocation density distribution were considered.

Fig. 6

The crystal orientation distribution of a computational model with second-order tetrahedral elements.

4. Simulation Results and Discussion

Figure 7 illustrates the ratio of 0.2% proof stress of the models subjected to two directions. In essence, the ratio is $\sigma_{y}^{\textit{TD}}/\sigma_{y}^{\textit{RD}}$. The graph shows the experimental results indicating that the actual material exhibited the anisotropy along with the analysis results. While the model with randomly selected orientations (diamond) failed, having some combinations where the ratio is less than one to yield the same tendency as the experimental results, considering the preferred orientations (triangle) led to an improvement. However, relying solely on crystal orientations might only partially acquire the material’s texture. When the GN dislocation density distribution and the preferred orientations are considered, the same trend as the experiment is revealed for all combinations in this model (square). This result suggests that the proposed method effectively reproduced the mechanical anisotropy observed in the rolled material. The average 0.2% of the proof stresses of the model with preferred orientation was 656 MPa for RD and 667 MPa for TD, whereas the model with the dislocation density distribution yielded 956 MPa for RD and 986 MPa for TD on average. One contributing factor to the higher yield stress observed in the analytical model may arise from the potential of overestimating the GN dislocation density content from EBSD data since the estimation depends on the measurement resolution.

Fig. 7

The ratio of 0.2% proof stress of TD to RD, corresponding to the model with random crystal orientation (diamond), preferred orientation (triangle), GN dislocation density and preferred orientation (square), and experiment (dot).

In the study [20], the contribution of yield stress in cold-rolled AlMgCuMn alloy with FCC structure was investigated using a strength model to estimate the dislocation density. Despite determining crystal information and other parameters through EBSD and TEM, their result exhibited consistency with the Taylor model. However, the dislocation density estimated by TEM was higher than that predicted by the Taylor model, which ranged with 2∼6 × 10¹⁴ m⁻² between 10% and 40% reduction. This result suggests that the GN dislocation density we obtained holds promise. Adams et al. [14] investigated the relationship between the minimum dislocation density and EBSD step size using the conventional method (Hugh method), which is similar to the KAM method, while considering the minimum dislocation density as one dislocation per Burgers circuit. The dislocation density range calculated through these methods ranges between 1 × 10¹²∼10¹⁵ m⁻² at step size of 1 µm. Although these variations may seem large for our analytical model, which consists of only 64 grains with uniform crystal orientation within each grain, the range of GN dislocation density distribution we obtained falls within the previously mentioned values. Moreover, the dislocations estimated by the presented method corresponds to dislocations with the same sign between EBSD measurement points. Therefore, the estimated GN dislocation density is always equal or less than the actual dislocation density.

Figure 8 shows distributions of the Schmid factor to TD and GN dislocation density. The total values of all 12 slip systems represent the GN dislocation density. The initial GN dislocation density had relatively larger values at the grain with a higher Schmid factor. In eq. (8), when the Schmid factor is less than 0.4, the GN dislocation density remains constant. This implies that despite fluctuations in Schmid factor values, the dislocation density distribution among most grains tends to maintain uniformity. However, despite this uniformity, the model subjected to tension along TD showed higher dislocation density values in contrast to tension along RD, resulting in higher yield stress in TD. Therefore, it can be inferred that the approximate expression based on the Schmid factor and the dislocation density provides a reasonable method to determine the initial dislocation density distribution.

Fig. 8

Result distributions of (a) Schmid factor to TD, (b) the initial GN dislocation density, and the GN dislocation density at 0.1% strain of tensile simulation (c) to RD, and (d) to TD. A fluctuation of up to 10% was given to the initial GN dislocation at each integration point, reflecting material heterogeneity [21].

5. Conclusion

The mechanical properties and microstructural observation of 20% cold-rolled equiatomic CrMnFeCoNi HEA were investigated by tensile test and EBSD. The rolled material exhibited mechanical anisotropy, having a higher yield stress in TD due to the rolling-induced texture. In order to reproduce the mechanical anisotropy in the computational method from the viewpoint of crystal orientation and dislocation density distribution, the preferred crystal orientations in the material were selected by the order of parameter called orientation degree. The conclusions obtained are summarized as follows:

(1) Based on the crystal orientation selected as the preferred texture, tensile simulations were performed without the GN dislocation density distribution induced by rolling. Although the model exhibited improved results than the conventional model, the model was not sufficient to yield the experimental result.
(2) In order to assign the GN dislocation density to the computational model with respect to the crystal orientation, the relationship between the dislocation density and the crystal orientation information (Schmid factor to TD) was obtained quantitatively.
(3) The initial GN dislocation density distribution established from the crystal orientation was reflected in the computational model. The model with the initial GN dislocation density distribution resulted in an overall increase in yield stress compared to the conventional model and the model with the preferred orientations. Notably, the yield stress in TD exceeded that in RD for all ten combinations. Hence, it is essential to consider the dislocation density distribution along with the crystal orientation to reproduce the mechanical anisotropy of rolled materials.

Acknowledgments

This work was supported by JSPS KAKENHI Grant Number JP18H05453.

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