Journal of Materials Science & Technology  2019 , 35 (10): 2200-2206 https://doi.org/10.1016/j.jmst.2019.04.030

Orginal Article

Grain-scale deformation in a Mg-0.8 wt% Y alloy using crystal plasticity finite element method

Wenxue Lia, Leyun Wanga*, Bijin Zhoua, Chuanlai Liua, Xiaoqin Zengab

a National Engineering Research Center of Light Alloy Net Forming, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China
b The State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai, 200240, China

Corresponding authors:   *Corresponding author.E-mail address: leyunwang@sjtu.edu.cn (L. Wang).

Received: 2018-12-25

Revised:  2019-03-13

Accepted:  2019-04-10

Online:  2019-10-05

Copyright:  2019 Editorial board of Journal of Materials Science & Technology Copyright reserved, Editorial board of Journal of Materials Science & Technology

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Abstract

Magnesium (Mg) alloys with hexagonal close-packed (HCP) structure usually have a poor ductility at room temperature. The addition of yttrium (Y) can improve the ductility of Mg alloys. To understand the underlying mechanism, crystal plasticity finite element method (CPFEM) was employed to simulate the tensile deformation of a Mg-0.8 wt% Y alloy. The simulated stress-strain curve and the grain-scale slip activities were compared with an in-situ tensile test conducted in a scanning electron microscope. According to the CPFEM result, basal slip is the dominant deformation mode in the plastic deformation stage, accounting for about 50% of total strain. Prismatic slip and pyramidal 〈a〉 slip are responsible for about 25% and 20% of the total strain, respectively. Pyramidal 〈c + a〉 slip and twinning, on the other hand, accommodate much less strain.

Keywords: Magnesium alloys ; Crystal ; Plasticity finite element modeling ; EBSD ; Dislocation ; Mechanical ; Behavior

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Wenxue Li, Leyun Wang, Bijin Zhou, Chuanlai Liu, Xiaoqin Zeng. Grain-scale deformation in a Mg-0.8 wt% Y alloy using crystal plasticity finite element method[J]. Journal of Materials Science & Technology, 2019, 35(10): 2200-2206 https://doi.org/10.1016/j.jmst.2019.04.030

1. Introduction

Magnesium (Mg) alloys with low density and high specific strength can be widely applied in transportation, communication and aerospace industries. As a hexagonal close-packed (HCP) metal, Mg can be deformed by {0001} 〈11 -2 0〉 basal slip, {1 -1 00} 〈11 -2 0〉 prismatic slip, {1 -1 01} 〈11 -2 0〉 pyramidal 〈a〉 slip, {10 -11} 〈11 -2-3 〉 pyramidal 〈c + a〉 slip, and {10 -12} 〈 -1011 〉 twinning [1]. However, at room temperature, Mg mostly relies on basal slip to accommodate the deformation, leading to its poor ductility. This has been a bottleneck so far against its wider applications.

One effective method to improve the ductility of Mg is by adding rare earth (RE) elements such as yttrium (Y) [2], cerium (Ce) [3], gadolinium (Gd) [4], and neodymium (Nd) [5]. Among these RE elements, Y has drawn a lot of attention in recent years. It has been demonstrated that Y can significantly weaken the texture of Mg alloys and reduce the critical resolved shear stress (CRSS) ratio between non-basal slip and basal slip [6], [7]. Sandlöbes et al. [2], [8] identified a high density of pyramidal 〈c + a〉 dislocations in a cold-rolled Mg-3 wt% Y alloy by transmission electron microscopy (TEM). This observation was attributed to the reduction of the stacking fault energy (SFE) by the addition of Y, and the stacking faults can serve as a heterogeneous source for 〈c + a〉 dislocations to nucleate in Mg - Y [9]. Stanford et al. [7] evaluated the effect of Y on the CRSS for various slip modes in Mg using in situ neutron diffraction. Their result suggested that the CRSS of pyramidal 〈c + a〉 slip is still 10 times higher than that of basal slip in a Mg-2.2 wt% Y, which implied that the contribution of 〈c + a〉 slip for the high ductility of Mg - Y alloys may be limited. Wu et al. [10] suggested that Y can reduce the difference in energy per unit length between pyramidal I and II screw dislocations and thus facilitate the cross-slip rather than the dissociation of 〈c + a〉 dislocations. Tsuru and Chrzan [11] found that Y can stabilize the screw 〈a〉 dislocation compact core in Mg and thus enable cross-slip of 〈a〉 dislocations from the basal plane to non-basal planes. This mechanism is supported by our recent experimental investigation on the deformation behavior of Mg-3 wt% Y and Mg-5 wt% Y binary alloys using in situ three-dimensional X-ray diffraction [12], [13], which found that the crystal rotation in many grains was related to the operation of prismatic slip or pyramidal 〈a〉 slip. To fully understand the improved plasticity in Mg - Y alloys, further investigation on the dislocation activity is necessary.

In recent years, some computational tools are employed to understand the deformation mechanisms in HCP metals. Popular computational tools include viscoplastic self-consistent (VPSC) [14], elastoplastic self-consistent (EPSC) [15] and crystal plasticity finite element method (CPFEM). Although VPSC and EPSC models are computationally efficient, they neglect the grain boundary effect and deformation gradients within grains [14], [15], [16], [17]. CPFEM is a more suitable tool to study grain-scale deformation in metals [18]. It typically uses real grain geometry information and grain orientations measured by electron back-scattered diffraction (EBSD) as the input. Therefore, simulation results obtained from CPFEM can be directly compared with the experimental results at both the macroscopic level (i.e. the stress-strain curve) and the mesoscopic level (i.e. deformation behavior inside grains) [19], [20]. Staroselsky and Anand [21] studied the room-temperature deformation of Mg alloy AZ31B using CPFEM to examine the dominant deformation systems during room temperature deformation. They estimated the CRSS of different slip and twin systems. Prakash et al. [22] developed a microstructure-based constitutive model for twinning to study the texture evolution of AZ31 using CPFEM. Guo et al. [23] used CPFEM to study the stress for twin thickening in Mg-6Zn deformed by nanoindentation. Wang et al. [24] employed CPFEM to study the tensile deformation of a commercial purity Ti specimen. By tweaking the hardening parameters and CRSS values, the simulated in-grain slip activities eventually matched the observed surface slip lines.

CPFEM needs to be implemented in a platform. Düsseldorf Advanced Material Simulation Kit (DAMASK) [25], [26] is an open source platform developed by Max-Planck-Institut für Eisenforschung GmbH (MPIE). The crystal plasticity model in DAMASK can solve the elastoplastic boundary problems with a hierarchical structure. Because of its high degree of modularity, it is flexible to import various constitutive frameworks and homogenization schemes in DAMASK. Furthermore, DAMASK can be connected to different finite element solvers and spectral solvers, and it has become a popular platform for crystal plasticity simulations in recent years [27], [28], [29].

In the present study, we performed CPFEM calculation using DAMASK to simulate the deformation of a Mg-0.8 wt%Y sample in an in-situ tensile test. The model gave quantitative estimation of the relative activities of different slip modes, which helped us to understand the role of Y in Mg’s deformation.

2. Material and experiment

Using commercial-purity Mg (99.95 wt%) and a Mg-25 wt% Y master alloy as raw materials, cast Mg-Y alloy was prepared by electric resistance melting in a steel crucible under a gas mixture of SF6 (1 vol.%) and CO2 (99 vol.%) at 730 °C. The chemical composition of the cast billet was measured as Mg-0.8 wt% Y by Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES). The cast billet was homogenized under a protective gas atmosphere at 480 °C for 11 h. A multiple-pass hot rolling process at 400 °C was employed to produce a sheet material of 2.0 mm thickness. The rolled sheet was annealed at 300 °C for 10 h to obtain average grain size of 12 μm (measured by EBSD). A pure Mg sheet with average grain size of 20 μm was prepared by hot rolling at 350 °C and a subsequent annealing at 300 °C for 1 h, which would be used as comparison with the Mg-Y alloy in mechanical behavior.

A tensile specimen of the Mg-Y alloy was prepared using electron discharge machining (EDM). The specimen had a flat dog-bone shape with gauge dimension of 11.0 mm (L) × 4.0 mm (W) × 1.4 mm (T), with its long axis (i.e. tensile axis) being parallel to the rolling direction. The top surface was mechanically polished and then electro-polished in an ethanol solution containing 10 vol.% perchloric at -30 °C for 150 s for the subsequent EBSD measurement.

For EBSD measurement, the specimen of the Mg-Y alloy was mounted in a MICROTEST 200 N (Deben, UK) module placed in a scanning electron microscope (SEM, Zeiss Gemini) equipped with an EBSD system (Oxford Instrument, UK). The EBSD scan was performed on a selected area of the specimen prior to the tensile test. Afterwards, the specimen was deformed by tensile test with a load speed of 0.4 mm/min, which is equivalent to a strain rate of 6.0 × 10-4 s-1 at room temperature. The test was paused when the strain reached 2% and another EBSD scan was conducted for the same surface area. Secondary electron images were subsequently taken for analyzing the surface slip activities.

Slip lines in the deformed microstructure are caused by dislocation glide on a particular slip plane. These slip lines should be parallel to the intersection of the slip plane and the specimen surface. Schmid factor (SF) of each slip system can be calculated using the measured grain orientation assuming uni-axial tension. The activated slip systems in individual grains can be inferred by comparing the calculated and the observed slip lines, with SF information as additional information. This is the basic principle of slip trace analysis [30], [31].

3. Simulation theory

The classic phenomenological law [25] is employed in CPFEM to simulate the material deformation. Below is a brief summary.

Total deformation F can be decomposed into two parts: elastic deformation Fe and plastic deformation Fp (F=Fe·Fp) [32]. Fe contains crystal lattice stretching Re and lattice distortion Ue (Fe=Re·Ue). Fp can be described in a rate form [33]:

Ḟp=Lp·Fp (1)

Where, Lp is the plastic velocity gradient, which is related to the slip activities [34], [35]:

Lp= α=1N${\dot{γ}}^{α}$ mα⊗nα (2)

Where, γα˙ is the shear rate on deformation system α, mα and nα are unit vectors defining the slip direction and the slip plane normal of α. For rate-dependent materials, γα˙ depends on initial shear rate γ0˙, resolved shear stress τα and the slip resistance τ0α, as expressed in the following formula:

${\dot{γ}}^{α}$ = ${\dot{γ}}_{0}·|\frac{τ^{α}}{τ^{α}_{0}}|^{n} ·sgn (τ^{α})$ (3)

The values of ${\dot{γ}}_{0}$ and n are set as 0.001 and 20, respectively. To describe strain hardening, $τ^{α}_{0}$ increases with strain and is dictated by [35]:

For slip systems, the Hαβ depends on the following formula:

Hαβ=hαβ∙ $(1-\frac{{τ^{α}_{0}}}{τ^{α}_{sat}})^{a}$ (5)

Where, hαβ is reference hardening modulus between slip-slip systems (set as 1500 MPa) and slip-twin systems (set as 500 MPa). $τ^{α}_{sat}$ is the saturated slip resistance. The hardening model for twinning used in this study follows the scheme in Refs. [26], [33], in which an evolving twin volume fraction ftw is used and the hardening depends on ftw. Deformation systems included in the simulation are basal slip, prismatic slip, pyramidal 〈a〉 slip, pyramidal 〈ac + a〉 slip, tensile twinning (TTW), and compressive twinning (CTW). The material parameters used in the simulation are listed in Table 1.

Table 1   Material parameters used in the present simulation: N is the number of systems in each slip/twin mode, 〈uvw〉 is the slip/twin direction, {hkl} is the slip/twin plane, τ0 is the slip resistance, τsat is the saturated slip resistance.

Slip modeNuvw{hkl}τ0 (MPa)τsat (MPa)
Basal311-2000011428
Prism311-201-10065130
Pyr〈a〉611-201-10165130
Pyr〈c+a611-2-310-11110220
TTW6-101110-1280-
CTW610-1-210-11120-

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Using grain orientation and grain geometry obtained from EBSD as the input, a finite element mesh of the grain structure was established. As EBSD only provides grain geometry on the surface, a quasi-3D model was further constructed by extending the grain geometry on the surface along the Z direction for 15 layers of elements.

4. Results and discussion

{0001} pole figures of the Mg-0.8 wt% Y specimen and the pure Mg specimen are shown in Fig. 1. All the three figures show a strong basal texture, but the texture strength of the Mg-0.8 wt% Y specimen (Fig. 1(a)) is weaker than the pure Mg specimen (Fig. 1(b)), which agrees with the result reported in Ref. [36]. Furthermore, as shown in Fig. 1(c), the texture strength of Mg-0.8 wt% Y specimen decreased from 13.3 mrd to 11.0 mrd after 2% strain. The stress-strain curves of the Mg-0.8 wt% Y specimen and the pure Mg specimen are shown in Fig. 2. The yield strength (σ0.2) of Mg-0.8 wt% Y and pure Mg were around 125 MPa and 100 MPa, respectively. However, the elongation to failure of Mg - Y ($\widetilde{2}$0%) is much higher than that of pure Mg ($\widetilde{5}$%). Therefore, the addition of Y had obviously improved the ductility of Mg. To understand the underlying mechanism, the deformation microstructure was further analyzed.

Fig. 1.   EBSD-measured {0001} pole figures of (a) the Mg-0.8 wt% Y specimen before deformation, (b) the as-rolled pure Mg specimen before deformation, and (c) the Mg-0.8 wt% Y specimen after 2% strain. The texture strength is shown for each pole figure.

Fig. 2.   Engineering stress-strain curves of the Mg-0.8 wt% Y specimen (average grain size = 12 μm) and the pure Mg specimen (average grain size = 20 μm).

Fig. 3(a) shows the SEM image of a selected area of the Mg-Y specimen after being deformed. Slip traces were found in some grains. The inverse pole figure (IPF) of the same area measured by EBSD with grain orientation represented by hexagonal unit cells before deformation is shown in Fig. 3(b). Grain shapes slightly changed from the initial condition (Fig. 3(b)) to the deformed condition (Fig. 3(a)). Using the grain orientations, we could compute the directions of all possible slip planes in the sample coordinate system and draw their intersections with the surface plane. Comparing the computational result with the observed slip lines allowed us to infer the activated slip system. The identified slip systems in grains 1-9 are summarized in Table 2 in bold font. It indicates that the slip traces in grains 1-7 were all from basal slip, whose theoretical slip traces are represented by red lines in Fig. 3(a).

Fig. 3.   (a) SEM image of Mg-0.8 wt% Y specimen after 2% strain, the red lines represent the direction of basal slip traces; (b) inverse pole figure (IPF) measured by EBSD with grain orientation represented by hexagonal unit cells before deformation.

Table 2   Orientation and slip activities identified in each grain by slip trace analysis (bold font) and CPFEM simulation only (non-bold font).

Grain IDEuler angles (°)Slip modeSlip SystemSF
1(217, 121, 341)Basal(0001) [2-1-10]0.481
2(139, 51, 6)Basal(0001) [-12-10]0.452
3(130, 134, 304)Basal(0001) [-1-120]0.368
Prism(10-10) [-12-10]0.363
4(214, 160, 23)Basal(0001) [-12-10]0.260
Prism(-1100) [-1-120]0.458
5(221, 143, 18)Basal(0001) [-12-10]0.407
Prism(-1100) [1-1-20]0.345
6(235, 33, 266)Basal(0001) [-12-10]0.292
7(142, 153, 301)Basal(0001) [-12-10]0.326
8(247, 17, 259)Prism(01-10) [2-1-1-0]0.451
Prism(-1100) [-1-120]0.398
9(109, 20, 26)Prism(011-0) [2-1-10]0.494

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Based on the grain orientation and grain geometry measured by EBSD in the same area with the SEM images, a finite element mesh was established by DAMASK, as shown in Fig. 4(a). Different colors in Fig. 4(a) represent different second Euler angles (Φ) in all grains. The mesh was then computationally stretched at a strain rate of 0.001 s-1 in the RD direction. A simulated stress-strain curve was finally obtained using the parameters in Table 1, which matches the experimental result very well, as shown in Fig. 4(b).

Fig. 4.   (a) A finite element mesh established by DAMASK using the grain orientation and grain geometry measured by EBSD, the color represents the second Euler angles (Φ) in all grains; (b) the experimental stress-strain curve of the Mg-0.8 wt% Y specimen and the simulation result.

In addition to the macroscopic stress-strain curve, the accumulated shear of basal slip, prismatic slip, and pyramidal 〈a〉 slip in individual grains after 2% strain in grain 1-9 can also be obtained from the simulation, as shown in Fig. 5. It is obvious that basal slip was the main deformation mode in grains 1-7. It is worth noting that basal slip shear is not homogeneously distributed in these grains. In grain 4, for example, basal slip is concentrated in the upper right part, which is consistent with the observed slip traces in this grain (Fig. 3(a)). Note that the accumulated shear of basal slip could reach 18.9%, which is much higher than the macroscopic strain (2%) in the specimen. In addition to basal slip, prismatic slip was also activated in grains 5, 8, and 9. These grains did not show apparent prismatic slip traces, perhaps because the c-axis of these grains were almost perpendicular to the surface, and the prismatic slip had a very small component along the surface normal.

Fig. 5.   Accumulated shear of basal slip, prismatic slip, and pyramidal 〈a〉 slip in grains 1-9 after 2% strain from the simulation result. The most active slip system(s) are given for each grain.

Slip activities in individual grains are strongly affected by the SFs. Fig. 6(a-c) uses different colors to display the SFs of basal slip, prismatic slip and pyramidal 〈a〉 slip in different grains. Fig. 6(d-f) shows the accumulated shear of basal slip, prismatic slip, and pyramidal 〈a〉 slip in all grains after 2% strain. By comparing Fig. 6(a) with (d), the grains with high SF of basal slip usually exhibit basal slip activities, e.g. grains 1 and 2. But conversely, some grains with obvious basal slip traces do not have high SFs. For example, grain 4 has a relatively low SF of basal slip but exhibited a high accumulated shear of basal slip. The accumulated shear of basal slip in grain 4 has similar direction as that in grain 1. Thus, basal slip in grain 4 is likely the result of slip transfer from grain 1. Fig. 6(b) and (e) shows the SF and accumulated shear of prismatic slip in all grains. Only those grains with high SF of prismatic slip and low SF of basal slip developed prismatic slip activities, e.g. grains 8 and 9. Two prismatic slip systems were activated in grain 8, i.e. (01-1 0) [2 -1-1 0]in the right part and (-1 100) [-1-1 20] in the left part. Grain 9 developed (01 -1 0) [2 -1-1 0] prismatic slip. Pyramidal 〈a〉 slip activities were much lower than prismatic slip. By comparing simulation results in Fig. 6(d-f) and the SEM image (Fig. 3(a)), it turns out that the simulation revealed more details of the mesoscale deformation, especially the activation of non-basal slip in some grains.

Fig. 6.   SFs of (a) basal slip, (b) prismatic slip, and (c) pyramidal 〈a〉 slip in all grains; the accumulated shear of (d) basal slip, (e) prismatic slip, and (f) pyramidal 〈a〉 slip in all grains after 2% strain.

Fig. 7 shows the variation of von Mises Stress in the grain aggregate after 2% strain. Comparing Fig. 7 with Fig. 6(d-f), the areas of stress concentration often happened to be the areas with prismatic slip and pyramidal 〈a〉 slip (e.g. grains 8 and 9). In those grains where basal slip was dominating, the stress was relatively low (e.g. grains 1, 2, 3, and 7). Grain 4 had lower stress in the top right part (where basal slip was concentrated) than the rest part. Therefore, the activation of prismatic slip and pyramidal 〈a〉 slip is not only controlled by the SF but also influenced by the stress concentration.

Fig. 7.   Variation of von Mises Stress in different grains after 2% strain according to the simulation result.

Relative fraction of various slip modes during the tensile test can also be extracted from the simulation result, as shown in Fig. 8. Initially, basal slip dominated the deformation. When the tensile test entered the plastic deformation stage, the relative fraction of basal slip decreased rapidly, while the fraction of prismatic slip and pyramidal 〈a〉 slip increased. In the end, basal slip accounted for about 50% of the deformation while non-basal 〈a〉 slips such as prismatic slip and pyramidal 〈a〉 slip were responsible for about 25% and 20% of the total deformation, respectively. On the other hand, the fraction of pyramidal 〈c + a〉 slip was very low.

Fig. 8.   Development of relative fraction of various slip modes during simulated deformation.

Based on the above results, it can be concluded that adding small amount of Y (e.g. 0.8 wt% Y addition in this work) promotes non-basal 〈a〉 slip activities, which agrees with some recent experimental work [12], [13]. A higher concentration of Y may be necessary to activate more 〈c + a〉 slip.

5. Conclusions

In this work, the grain-scale deformation mechanism of the rolled Mg-0.8 wt% Y alloy has been studied using in-situ tensile test and CPFEM. The stress-strain curve from simulation matches the experimental result. At the grain scale, active slip systems can be identified by both slip trace analysis and accumulated shear from simulation. CPFEM allows us to identify slip activity that cannot be identified by surface slip trace. CPFEM can also predict the relative fraction of various slip modes during the specimen’s deformation.

In the Mg-0.8 wt% Y specimen, basal slip was the dominant deformation mode, especially when the strain was low. Prismatic slip and pyramidal 〈a〉 slip were activated in later deformation. Pyramidal 〈c + a〉 slip and twinning were barely activated. The activation of non-basal slip modes is dependent on not only SF but also stress concentration.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Nos. 51631006, 51671127 and 51825101).


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