Mechanical Properties Simulation of Aluminum Alloy Sheet Using SSP and CPFEM

Abstract

This paper focuses on the mechanical performance of an aluminum alloy with uneven microstructure formed after stamping and brazing. The mechanical behavior of the stamping area is studied through the crystal plasticity finite element method (CPFEM). Firstly, the crystal plastic parameters of the material are obtained by fitting the experimental and simulated results of nanoindentation. Then, a polycrystalline tensile model is established using a Step-by-Step Packing method, with orientation distribution assigned based on EBSD results, followed by polycrystalline tensile simulations using CPFEM. The results demonstrate that CPFEM can effectively simulate the mechanical behavior of the studied aluminum alloy. Additionally, the study reveals significant orientation-dependent mechanical responses.

1. Introduction

Due to the increasing demand for lightweighting in the automotive industry, the design of many automotive components is gradually tending toward thinner sizes. At this point, significant size effects begin to appear in sheet components [1,2]. The size effects can be divided into two types: one is the characteristic size effect, which refers to the thinning of the sample size to a level close to the grain size. The other type is the grain size effect, which refers to the growth of grain size to a level similar to the sample size during processing. The existence of size effect can lead to ununiform distributions of microstructure and texture in materials, resulting in uneven plastic deformation and anisotropy of macroscopic properties [3,4,5,6]. Therefore, traditional finite element methods that use homogeneous materials and isotropy as assumptions are not convenient for predicting the deformation and damage processes of such materials.

In response to this issue, many researchers have combined the crystal plasticity finite element method (CPFEM) with three-dimensional polycrystal models to address issues such as material anisotropy, crystal orientation, and dislocation slip. The advantage of CPFEM lies in its ability to accurately simulate crystal deformation behavior while considering the effects of the microstructure.

For the establishment of three-dimensional polycrystalline models, common methods include the Voronoi method [7,8], the Step-by-Step Packing (SSP) method [9,10], and the method of generating polycrystalline models based on EBSD test images [11,12]. After generating polycrystalline models, the macroscopic mechanical performance of the material could be studied by assigning a single-crystal constitutive relationship to grains, with each grain assigned its corresponding orientation. In engineering, the majority of materials are two-phase or multiphase alloys, with the aim of improving their strength. There are two main methods for considering the influence of reinforcing phase using CPFEM for these alloys. One approach is to study the properties of the matrix and the reinforcing phase separately, and then assign different mechanical constitutive models to different phases [13]. Another approach is to normalize the mechanical properties of the matrix phase and the reinforcing phase, and the obtained material comprehensive constitutive model can simultaneously consider the influence of the matrix phase and the reinforcing phase [14,15]. Both methods can accurately reflect the mechanical properties of the alloys. The material performance testing of the normalization method is easier to conduct, so it is more widely used.

When using CPFEM for simulation, it is necessary to know the elastic–plastic parameters of the crystal. The common method for determining these parameters is mainly through comparing the tensile simulation of polycrystalline models with experimental data. In recent years, the method of obtaining crystal elastic–plastic parameters through single-crystal nanoindentation simulations and experiments has been widely used [16,17]. Firstly, nanoindentation experiments have very few limitations on the shape and size of the specimens. For materials with small dimensions or irregular shapes, parameter measurements can be performed without preparing large-sized or standard tensile specimens. Secondly, compared to traditional tensile tests, nanoindentation can separately study the mechanical behavior of individual grains in polycrystalline materials. By analyzing the mechanical properties of various individual grains, the elastic–plastic parameters of the crystal can be obtained more accurately [18]. Although some studies have been conducted, the majority of CPFEM-based research focuses on materials with a uniform microstructure distribution. Few studies have employed CPFEM to simulate the mechanical response of materials with non-uniform microstructures.

In this paper, CPFEM simulations were conducted on an aluminum alloy with uneven microstructure distribution. For the parameters required by CPFEM, the combination method of nanoindentation simulations and experiments was employed. On this basis, a polycrystalline model of the tensile specimen was generated, and the tensile properties of the aforementioned material were simulated and compared with experimental results.

2. Experiments

The material used in this study is a 3000 series aluminum alloy, which is part of a cooler manufactured by Valeo. The production processes of the cooler involve stamping and brazing. Figure 1 shows a partial area of the cooler. After undergoing the above production processes, the stamping area has an O-state and uneven microstructure.

2.1. EBSD Testing and Crystal Orientation Acquisition

EBSD testing technology is used to conduct microstructure analysis on the studied stamping area. The cut sample was heat-embedded, and then the surface of the embedded sample was carefully polished to obtain a flat and smooth surface to ensure its suitability for electron beam irradiation. The samples were sequentially polished with 800-, 1200-, and 2000-grit sandpapers, followed by diamond polishing solutions of 3 μm, 1 μm, and 0.25 μm. Finally, 0.05 μm aluminum oxide was applied for 4 h vibration polishing. The schematic diagram of the sample for EBSD testing is shown in Figure 2, with the red box indicating the testing area. The EBSD testing was performed with a step size of 2 μm to ensure that the testing area is sufficiently large.

2.2. Nanoindentation Experiments

Nanoindentation technology is used to measure the mechanical properties within grains at room temperature. The experiment employed a BRUKER TI98 nanoindentation tester (Billerica, MA, USA) with a diamond Berkovich indenter. The centerline-to-face angle of the Berkovich tip is 65.3°. Displacement-controlled loading was used in the nanoindentation test, with a depth of 500 nm and a loading speed of 100 nm/s. When reaching the maximum pressing depth, the pressure was held for 5 s and then unloaded at a speed of 100 nm/s. The aluminum alloy used in the study is a two-phase material with a small-sized reinforcing phase. In order to consider the strengthening effect of the second phase in the simulation, the normalization method introduced in the introduction was adopted. A 3 × 3 indentation matrix was created in each selected grain region. During the nanoindentation test of each grain, the nine selected positions consist of some near and others far from the second phase. Subsequently, the average load–displacement curve of 9 points was calculated as the nanoindentation result of a single grain to consider the strengthening effect of the matrix and second phase. In accordance with ISO 14577-1 [19], the distance from the indentation to the sample boundary should be at least three times the indentation diameter, and the minimum distance between indentations should be at least five times the indentation diameter. In order to meet the above requirements, the spacing between points in the matrix was set to 30 μm.

2.3. Uniaxial Tensile Tests

Due to the limitation of the sample size in this area, standard tensile specimens could not be prepared, and only small rectangular specimens were prepared. The specimen size was 5 mm × 57 mm × 0.9 mm. A Digital Image Correlation (DIC) system (Shenzhen, China) was employed to measure the strain during the experiment, providing precise strain measurements over the specimen surface. During the tensile tests, a tensile rate of 2.5 mm/min was adopted. The cutting position of the tensile specimen and the process of the tensile test with the DIC system are shown in Figure 3.

3. Numerical Modeling

3.1. Single-Crystal Plasticity Constitutive Model

The crystallographic plasticity constitutive model developed in this study follows the framework initially proposed by Rice [20] and Asaro and Needleman [21], which are briefly described below. The deformation of a single crystal is described in a fixed Cartesian coordinate system, where $X$ and $x$ represent the coordinates of any mass point in the initial and current states, respectively. The deformation at this mass point can be represented by the deformation gradient $F$, as follows:

$$F={\displaystyle \frac{\partial x}{\partial X}}$$

(1)

The deformation gradient can be decomposed into the elastic part $F^e$ and the plastic part $F^p$. The elastic parts correspond to elastic deformations and lattice rotations, and the plastic parts represent the plastic shear slip of the lattice. That is,

$$F=F^e\cdot F^p$$

(2)

when the crystal is not deformed, for any slip system α, the unit normal vector of the slip plane of α-slip system is $m^{(\alpha )}$, and the unit vector along the slip direction is $s^{(\alpha )}$, which are orthogonal to each other. After the deformation of crystals, the new slip direction $s^{*(\alpha )}$ and the normal slip plane $m^{*(\alpha )}$ are expressed as follows:

$$\left\{\begin{array}{l}{s}^{*(\alpha )}=F^e\cdot s^{(\alpha )}\hfill \\ m^{*(\alpha )}=m^{(\alpha )}\cdot F^{e^{-1}}\hfill \end{array}\right.$$

(3)

The rate of deformation can be represented by the velocity gradient $L$, which can also be decomposed into the elastic part $L^e$ and the plastic part $L^p$ as follows:

$$L=\stackrel{·}{F}\cdot F^{-1}=\stackrel{·}{F^e}\cdot F^{e^{-1}}+\stackrel{·}{F^p}\cdot F^{p^{-1}}=L^e+L^p$$

(4)

The crystal plasticity theory is based on the assumption that the primary mechanism of plastic deformation is dislocation slip, and attributes the macroscopic plastic deformation of metallic materials to the accumulation of shear deformation in each slip system. The plastic velocity gradient $L^p$ is determined by the sum of the shear strains of all slip systems. The formula is as follows:

$$L^p={\displaystyle \sum _{\alpha =1}^N}{\stackrel{·}{\gamma }}^{(\alpha )}{s}^{*(\alpha )}{m}^{*(\alpha )}$$

(5)

where $N$ is the total number of crystal slip systems. For aluminum, which has a face-centered cubic (FCC) crystal structure, $N$= 12. ${\stackrel{·}{\gamma }}^{(\alpha )}$ is the plastic slip rate for α-slip system. The plastic part of the deformation rate tensor $D^p$ and the plastic part of the spin tensor $W^p$ are given as follows:

$$D^p={\displaystyle \frac{1}{2}}[L^p+{(L^p)}^T]={\displaystyle \sum _{a=1}^N{P}^{(\alpha )}}{\stackrel{·}{\gamma }}^{(\alpha )}$$

(6)

$$W^p={\displaystyle \frac{1}{2}}[L^p-{(L^p)}^T]={\displaystyle \sum _{a=1}^N{\omega }^{(\alpha )}}{\stackrel{·}{\gamma }}^{(\alpha )}$$

(7)

where

$$\left\{\begin{array}{l}{P}^{(\alpha )}={\displaystyle \frac{1}{2}}(s^{*(\alpha )}{m}^{*(\alpha )}+m^{*(\alpha )}{s}^{*(\alpha )})\hfill \\ {\omega }^{(\alpha )}={\displaystyle \frac{1}{2}}(s^{*(\alpha )}{m}^{*(\alpha )}-m^{*(\alpha )}{s}^{*(\alpha )})\hfill \end{array}\right.$$

(8)

Assuming that crystal slip follows Schmid’s law, the resolved shear stress ${\tau }^{(\alpha )}$ for α-slip system is given by the following formula:

$${\tau }^{(\alpha )}=\sigma :P^{(\alpha )}$$

(9)

where $\sigma$ represents the Cauchy stress. The expression for the plastic flow rate is as follows:

$${\stackrel{·}{\gamma }}^{(\alpha )}={\stackrel{·}{\gamma }}_0{\left|{\displaystyle \frac{{\tau }^{(\alpha )}}{g^{(\alpha )}}}\right|}^{n}sign\left({\displaystyle \frac{{\tau }^{(\alpha )}}{g^{(\alpha )}}}\right)$$

(10)

where ${\stackrel{·}{\gamma }}_0$ is the reference flow rate, n is the rate-sensitivity exponent, and $g^{(\alpha )}$ is the current strength of α-slip system. For this model, the hardening effect is typically expressed in the following incremental form:

$${\stackrel{·}{g}}^{(\alpha )}={\displaystyle \sum _{\beta }{h}_{\alpha \beta }}\left|\stackrel{·}{{\gamma }_{\beta }}\right|$$

(11)

where $h_{\alpha \beta }$ is the slip hardening modulus. The hardening modulus reflects the characteristic that the slip resistance gradually increases with the increase in deformation. $h_{\alpha \beta }$ (α = β) is called the self-hardening modulus, which measures the effect or hardening caused by the deformation of a particular slip system on itself. $h_{\alpha \beta }$ (α ≠ β) is called the latent hardening modulus, which reflects the degree of interaction and influence between different slip systems. In this study, the hyperbolic function proposed by Peirce et al. is used to describe the hardening law:

$$h_{\alpha \beta }=h(\gamma )[q+(1-q){\delta }_{\alpha \beta }]$$

(12)

$$h(\gamma )=h_0{\mathrm{sech}}^2\left|{\displaystyle \frac{h_0\gamma }{{\tau }_s-{\tau }_0}}\right|$$

(13)

$$\gamma ={\displaystyle \sum _{\alpha }{\displaystyle {\int }_0^t\left|{\stackrel{·}{\gamma }}^{(\alpha )}\right|}}dt$$

(14)

where $\gamma$ is the accumulated plastic shear strain. $q$ is the interaction constant between potential slip systems, also known as the latent hardening parameter. ${\delta }_{\alpha \beta }$ is the Kronecker delta symbol. $h_0$ is the initial hardening modulus. ${\tau }_s$ and ${\tau }_0$ are the saturated flow stress and the critical resolved shear stress, respectively.

3.2. Selection of Crystal Plasticity Constitutive Parameters

It is necessary to provide the single-crystal elastic–plastic parameters of the crystal plasticity model before simulation. The aluminum alloy has a typical FCC structure, which contains 12 slip systems. Due to the symmetry of the crystal structure, FCC crystals have three independent elastic constants [6], $C_{11}$, $C_{12}$, and $C_{44}$. The crystal elastic constants of the material can be obtained through molecular dynamics simulations. According to elastic constants obtained by molecular dynamics simulation, the Young’s modulus can be calculated using the Voigt–Reuss–Hill model [22]:

$$\begin{array}{l}E={\displaystyle \frac{9BG}{3B+G}}\\ \nu ={\displaystyle \frac{3B-2G}{2(3B+G)}}\\ B_V=B_R={\displaystyle \frac{(C_{11}+2C_{12})}{3}}\\ G_V={\displaystyle \frac{(C_{11}-C_{12}+3C_{44})}{5}}\\ G_R={\displaystyle \frac{5(C_{11}-C_{12})C_{44}}{4C_{44}+3(C_{11}-C_{12})}}\end{array}$$

(15)

where $B$ is the bulk modulus and $G$ is the shear modulus. The calculated Young’s modulus is compared with that obtained from nanoindentation testing.

The plasticity parameters that need to be provided in the crystal plasticity model include ${\tau }_0$, ${\tau }_s$, $h_0$, $q$, etc. The crystal plasticity parameters are mainly obtained by fitting the simulation and experimental results of nanoindentation. The range of crystal plasticity parameters for aluminum alloys, as shown in

Table 1. The values range of crystal plasticity parameters.

Parameter	Value Range
${\tau }_0$/Mpa	20~200
${\tau }_s$/Mpa	50~700
$h_0$/Mpa	90~900
$q$	1~3

, was derived from published references [23,24,25,26,27,28]. During simulation, parameters are adjusted within these defined ranges. The selected optimal parameters will be used in subsequent polycrystalline model simulations.

The nanoindentation process was simulated using ABAQUS/Standard with a three-dimensional model, as shown in Figure 4. The model comprises a deformable single-crystal matrix and a rigid indenter. The single-crystal matrix model has dimensions of 50 μm × 50 μm × 20 μm, containing 96,000 elements. The type of element is C3D8R, with reduced integration and applied enhanced hourglass control to improve the accuracy of the simulation and mitigate potential convergence issues. In order to enhance simulation accuracy, a refined mesh is applied to the region near the contact between the single-crystal matrix and the indenter. The maximum element size of the single-crystal matrix finite element model is 2 μm, and the minimum element size is 0.167 μm. The mesh has been thoroughly tested for convergence to ensure reliable results. The indenter model was simplified as a rigid body, with an element size of 0.005 mm and element type C3D8R. The simulation is controlled through displacement. A displacement along the negative z-axis is applied to the indenter, with a value of 500 nm. These settings are consistent with the actual experimental conditions. The depth of indentation is less than one-tenth of the model thickness, which can avoid the impact of fixed constraints on the bottom [29]. Fixed constraints are applied to the bottom and side surfaces of the single-crystal model. For contact interaction, a surface-to-surface contact mode with hard contact conditions is used to prevent penetration between the contact surfaces, allowing infinitely large normal forces. Additionally, [30] indicates that the effect of friction on the load–displacement curve is negligible. Therefore, tangential contact is set to a finite sliding mode, with no friction between the contact surfaces.

The crystal plasticity finite element model is calculated using Huang Y’s UMAT subroutine in ABAQUS 2022 [31]. This subroutine assumes that the plastic deformation of the material originates solely from dislocation slip within the crystal, without considering other deformation mechanisms caused by diffusion, twinning, and grain boundary sliding. In this study, the reason for selecting this subroutine is that the main deformation mechanism of aluminum alloys is dislocation slip. Grain orientations are assigned by inputting the Miller indices into the subroutine.

3.3. Development of a Polycrystalline Model

A geometric model is established based on the actual dimensions of the tensile specimen, and a discretized mesh is generated using the C3D8R element type. To balance computational accuracy and speed, the element size was set to 0.1 mm × 0.1 mm × 0.1 mm, resulting in 256,000 elements. As shown in Figure 5 one end of the specimen is completely fixed, with all degrees of freedom restricted. The other end is coupled to a reference point, where a loading rate along the positive z-axis is applied. The coupled nodes move with the reference point. The time period was set to 150 s, and the time increment was set to 0.3 s, which reduces the amount of data while ensuring data accuracy.

Polycrystalline models were generated using the SSP method [9]. The main idea of this method is to progressively pack a pre-meshed model into multiple sets based on a specific growth formula. First, some elements are randomly selected from the meshed model, and the coordinates of their center points are used as seed points. Each selected element is assigned a non-zero digital index (DI), while the DI of other elements remains zero. After the random seeding, a specific growth equation is applied to grow the seeds. The growth rate of the seeds in different directions can be specified according to the growth equation. Ellipsoid, sphere, and cylinder equations are fundamental analytical functions that allow us to construct typical microstructures of various materials. As per this function, the volume around the seeds increases at each step of the SSP process. During the growth process, the growth step length is set to one-third of the grid length. Figure 6 illustrates the process of generating a tensile polycrystal model using the spherical growth equation based on the SSP method. As the volume of all crystals increases, for every element that does not belong to any crystal, the coordinates of its center point are checked to see if they fall within the volume of any crystal. If this is the case, the zero DI of the element is replaced by the corresponding seed’s DI, and the element is assigned to the corresponding crystal. The SSP process is repeated until all elements have a non-zero DI.

4. Results and Discussion

4.1. Microstructure and Crystal Orientation

The Inverse Pole Figure (IPF) [001] obtained from the EBSD test for the stamping area is shown in Figure 7. According to the test results, the grain size in this area is uneven, with an average size of approximately 1.32 mm for large grains. The size ratio along the x-direction to the y-direction is 0.2. The measurement of another surface shows that the ratio of the x-direction to the z-direction is also close to 0.2. Based on the EBSD results, the Orientation Distribution Function (ODF) was analyzed and is shown in Figure 8, and the overall texture strength is relatively low, at only 3.755. Orientation distribution analysis shows that the orientation in

Table 2. Detailed information of grain orientation with the highest proportion.

H	k	l	u	v	w	Proportion
−0.737	−0.177	0.653	0.671	−0.068	0.739	27.16%

accounts for approximately 27.16%, which is referred to as the “main orientation”. This main orientation is close to the {011} [0–11] orientation. The proportions of other orientations are relatively small.

Based on the known microstructure, a polycrystalline model as shown in Figure 9a was generated using SSP method. According to the average grain size, there should be 800 grains in the specimen model. Therefore, 800 seeds were randomly selected from all elements and the grown ratio was set as 5:5:1 along the x-, y-, and z-axes. The EBSD results indicate that the grains with orientations similar to that shown in Table 2 are mostly large grains. Therefore, among the 800 grains in the polycrystalline model, grains with larger sizes were assigned this orientation, and the cumulative volume proportion was 27.16%. The remaining grains were assigned random orientations. After redistributing the orientation, the polycrystalline model is shown in Figure 9b. It can be seen that the generated polycrystalline model also has an uneven microstructure, which is similar to the real one.

Based on EBSD testing, three grains with different orientations in Figure 8 were selected to obtain their orientation information. These orientations were then applied to nanoindentation simulations.

Table 3. The orientation of three selected grains shown in Figure 8 for nanoindentation experiments.

Normalized Miller Indices	Grain 1	Grain 2	Grain 3
h	−0.473	0.637	−0.390
k	0.426	−0.209	−0.664
l	0.771	0.742	0.638
u	0.584	0.693	0.806
v	−0.503	−0.267	0.088
w	0.637	−0.670	0.585

shows the orientation information of the three grains. The orientation of Grain 2 shown in Table 3 is similar but not identical to the main orientation shown in Table 2. The main orientation shown in Table 2 is obtained from the grain orientation distribution of the material’s overall region. Therefore, in subsequent simulations, the two similar grain orientations will be used independently based on the simulation requirements.

4.2. Results of Nanoindentation Simulation

Studies indicate that intermetallic compounds and precipitate phases significantly influence strength, but generally have minimal effect on the elastic modulus [32]. Therefore, the elastic constants of the studied aluminum alloy were primarily obtained through molecular dynamics simulations of pure aluminum. The elastic constants obtained from the simulations are shown in

Table 4. Elastic constants of pure aluminum obtained from molecular dynamics simulations.

${\mathit{C}}_{\mathbf{11}}$/MPa	${\mathit{C}}_{\mathbf{12}}$/MPa	${\mathit{C}}_{\mathbf{44}}$/MPa
115,600	62,400	32,700

. Based on Table 4 and Equation (15), the Young’s modulus of this material can be calculated and is equal to 80.26 GPa.

Additionally, the Young’s modulus was determined from the unloading segment of the displacement–load curve obtained from the nanoindentation experiments [33]. For each grain, the Young’s modulus was calculated by averaging the values obtained from nine test points. The Young’s moduli for the three grains are presented in

Table 5. Young’s modulus of the three grains measured by nanoindentation experiments.

	Grain 1	Grain 2	Grain 3	Average
Young’s modulus/GPa	79.83	80.34	80.21	80.13

. The average Young’s modulus of the material was then computed to be 80.13 GPa. The relative error between the simulation and experimental results for the Young’s modulus is only 0.16%. Therefore, the elastic constants listed in Table 4 are deemed valid and will be utilized in the subsequent simulations.

To determine the plastic parameters, a series of simulations were performed with a rough mesh, and the resulting numerical load–displacement curves were compared. It was found that ${\tau }_0$ and ${\tau }_s$ have a significant impact on the simulated load–displacement curve, while other parameters have very little effect on the curve. According to Figure 10a,b, as ${\tau }_0$ increases, the peak load rises, while the final residual indentation height decreases. In the hardening model proposed by Peirce D, ${\tau }_0$ is defined as the minimum shear stress required for slip systems to initiate sliding. Therefore, ${\tau }_0$ is directly related to the yield strength of the material. The smaller the value of ${\tau }_0$, the more likely the material is to undergo plastic deformation. As shown in Figure 10c,d, the increase in ${\tau }_s$ leads to an increase in peak load and final residual indentation height. ${\tau }_s$ represents the saturated flow stress and determines the ultimate value of stress during plastic deformation. Therefore, a larger ${\tau }_s$ may bring higher strength limits to the material.

The calibrated crystal plasticity constitutive parameters are presented in

Table 6. Crystal plasticity constitutive parameters determined.

${\mathit{h}}_{\mathbf{0}}$/MPa	${\mathit{\tau }}_{\mathbf{0}}$/MPa	${\mathit{\tau }}_{\mathit{s}}$/MPa	$\mathit{q}$
290	22	56	2.6

. Figure 11 compares the nanoindentation experimental results with simulation results for three grains with different orientations. The analysis shows good agreement between the two sets of results, with errors of 1.02%, −1.18%, and −0.93%, respectively. Furthermore, Figure 12 shows the indentation morphology from the nanoindentation experiments. The average indentation width from the experiments was compared to the simulated indentation width, and the corresponding errors were calculated to further assess the validity of the parameters. The results are provided in

Table 7. The indentation width from the experiments and the simulations.

Indentation Width	Grain 1	Grain 2	Grain 3
EXP-average indentation width/μm	3.45	3.49	3.46
SIM- indentation width/μm	3.51	3.52	3.51
Errors/%	1.74	0.72	1.45

. Both experimental and simulation data indicate that, at the same indentation depth, the peak load for Grain 1 is the highest, while Grain 2 exhibits the lowest peak load.

During the nanoindentation experiment, when the indenter slowly presses into the surface of the material, plastic deformation occurs, and complex stress states are generated in the indentation area. These stresses cause the material to flow and deform. Figure 13 and Figure 14 depicts the Mises stress and true strain of three different orientations of grains during nanoindentation simulation. Due to the anisotropy of crystal materials and the activation of different slip systems, there are variations and asymmetry in the distribution of stress and strain under different crystal orientations. Among the three grains, Grain 1 experiences the highest Mises stress during indentation, while its true strain is the lowest. Figure 15 shows the indentation pile-up morphology and height of grains with different orientations during the indentation process. Due to the fact that the maximum pile-up height in the simulation occurs on the indentation diagonal, the coordinates of the nodes on the indentation diagonal shown in Figure 15 were extracted to observe the sinking and pile-up height. It is evident that the pile-up height of Grain 2 is the highest. The above phenomenon indicates that Grain 1 may have the higher structural stability and strength, while Grain 2, which exhibits softer material properties, undergoes more significant plastic strain. This indicates that grain orientation significantly affects the local deformation of the indentation area.

The phenomenon described above can be explained by converting the nanoindentation test results of three grains with different orientations into stress–strain curves. The elastic–plastic properties of metallic materials are typically described by power-strengthening models, and their stress–strain ($\sigma$-$\epsilon$) relationship is given by:

$$\sigma =\left\{\begin{array}{c}\begin{array}{cc}E\epsilon & (\sigma \le {\sigma }_y)\end{array}\\ \begin{array}{cc}{\sigma }_y{(1+{\displaystyle \frac{E}{{\sigma }_y}}{\epsilon }_p)}^n& (\sigma >{\sigma }_y)\end{array}\end{array}\right.$$

(16)

where $E$ represents the Young’s modulus, ${\sigma }_y$ is the yield strength, and $n$ is the strain hardening index. The values of ${\sigma }_y$ and n of the single crystals with three different orientations were obtained using the method proposed by Dao et al. [34], and shown in

Table 8. Parameters of the stress–strain relationship for three grains with different orientations.

	Grain 1	Grain 2	Grain 3
${\sigma }_y$/MPa	29.88	28.96	29.81
$n$	0.4012	0.3876	0.3987

. The stress–strain curves for the three grains with different orientations were plotted using Equation (16) and the data in Table 8 and shown in Figure 16. It can be observed that Grain 1 exhibits the highest strain hardening exponent $n$, while Grain 2 has the lowest. Similarly, Grain 1 has the highest yield strength ${\sigma }_y$, while Grain 2 has the lowest. For the same strain, Grain 1 requires the highest stress, whereas Grain 2 requires the lowest. Thus, in the nanoindentation experiments and simulations of the three grains, when the same indentation depth is applied, Grain 1 requires a larger load to achieve the corresponding deformation, while Grain 2 needs a smaller load.

4.3. Polycrystalline Tensile Simulation

The stress–strain curve obtained from the tensile simulation of the above polycrystalline model is shown in Figure 17. Figure 17 presents the results from two different simulation models. In one model, grains accounting for 27.16% of the total volume were assigned a main orientation (shown in Table 2), while the remaining grains were given random orientations. In the other model, all grains were assigned random orientations. It can be observed that the stress–strain curve obtained from the model with 27.16% main orientation shifts downward relative to the latter model and aligns better with the experimental results. This indicates that the calibrated crystal plasticity constitutive parameters and the polycrystalline model with reassigned orientations can effectively describe the mechanical behavior of the material. The difference between the simulation results of the two models is mainly attributed to the main orientation in the former model, which is similar to the orientation of Grain 2 shown in Table 3 and exhibits a lower yield strength and strain hardening exponent. The yield strength and strain hardening exponent of the grain with main orientation were determined to be 29.02 MPa and 0.3904, respectively, through nanoindentation simulations and the method [34] described earlier. These grains with the main orientation made a significant contribution to the plastic deformation of the specimen during the tensile process.

Additionally, there are some differences between the simulated and the experimental curves in the strain range of 0.0025–0.031 (gray area in Figure 17). We speculate that the observed differences mainly arise from the limitations of the hardening model. The plastic region of the stress–strain curve obtained from the tensile test is close to a power-law curve. The crystal plasticity model used in the study describes the stress state of the crystal based on Cauchy stress. This model does not take into account the strengthening effect of grain boundaries. Without incorporating a grain boundary strengthening model, the calculated stress–strain curve may deviate from the power-law form. Meanwhile, the hardening model in crystal plasticity theory uses a function related to the accumulated shear strain, which slightly differs from the traditional power-law function in local regions.

The stress and strain distributions during the tensile simulation are shown in Figure 18a,b. It can be observed that, when using the crystal plasticity constitutive model for tensile simulation, the stress and strain distributions are uneven. The black lines in Figure 18 outline the boundaries of the grains with the main orientation mentioned in Table 2. It can be observed that the areas with high strain are primarily concentrated in these grains. Figure 18c shows the specimen in the tensile test, and it can be observed that the stress concentration positions in the experiment and simulation are quite similar. Meanwhile, the stress concentration areas are not located at the center of the specimen, which is attributed to the inhomogeneous microstructure.

5. Conclusions

Simulations of single-crystal nanoindentation and polycrystalline tensile behavior were conducted using CPFEM and EBSD results. The simulation outcomes were compared with the experimental data. The main conclusions drawn from this study are as follows:

Firstly, the crystal plasticity constitutive parameters for the studied aluminum alloy were obtained by fitting the load–displacement curves from nanoindentation experiments and numerical simulations.

Secondly, based on the EBSD results, a polycrystalline tensile simulation model was established using the SSP method. The orientation with the highest proportion was assigned proportionally to the polycrystalline model, effectively restoring the true microstructure of the polycrystalline specimen. The crystal plasticity constitutive parameters obtained from nanoindentation were used for polycrystalline tensile simulation, achieving simulation of the tensile process and acquisition of stress–strain curves. The resulting stress–strain curves align well with the experimental results, indicating that the generated polycrystalline model and the calibrated parameters are well suited for the material studied in this work.

Finally, both the experimental and simulation results of single-crystal nanoindentation demonstrate that the mechanical properties are orientation-dependent. Furthermore, the polycrystalline tensile simulations confirmed that the distribution of grain orientations also influences the macroscopic mechanical properties of the material. In the polycrystalline tensile simulation, significant differences were observed between the stress–strain curves obtained from the model with 100% random orientations and the model with 27.16% main orientation (near {011} [0–11]). This can be attributed to the fact that the grains with the main orientation exhibit relatively low strain hardening exponent and yield strength, resulting in weaker hardening effects. Therefore, compared to the model with 100% random orientations, the model with 27.16% main orientation has a lower overall hardening index, and stress increases more slowly with strain during plastic deformation. The main orientation plays a certain role in the plastic deformation of this material.