Texture-Based Optimization of Crystal Plasticity Parameters: Application to Zinc and Its Alloy

The macroscopic properties of metallic materials are related to their microstructure, in particular in the case of plastic behavior. However, understanding this relationship is not trivial, especially in the case of the hexagonal close packed (HCP) metals, with many potential slip and twinning systems. The crystal plasticity (CP) theory offers great help in this task. Among HCP metals, the particular attention was recently paid to titanium[1,2] and magnesium[3‐5] due to their high specific strength. Also zirconium (cf. e.g.,[6,7]) was studied extensively thanks to its application as fuel cladding in nuclear power plants.

On the other hand, HCP zinc and its alloys received considerably less attention. However, they are important industrial materials, as steel elements are often coated with zinc for protection against corrosion. It should be also noted that good biocompatibility and optimal corrosion rate have made zinc and its alloys promising candidates for producing bioabsorbable implants e.g., stents.[8,9] Such implants should dissolve gradually in the physiological environment after fulfilling their mission to support damaged tissue during its recovery process. Furthermore, generated corrosion products need to be safely metabolized by human body without any harmful response. Temporary implants could prevent patients from chronic inflammatory, thrombogenicity and endothelial dysfunction. The main disadvantage of zinc and its alloys, limiting their applications for those purposes are low mechanical properties, in particular the yield strength, ultimate tensile strength and total elongation. It was recently reported that improvement in both strength and plasticity of pure zinc can be achieved by alloying with magnesium and by subjecting material to hydrostatic extrusion.[10‐12] Those papers focus mainly on experimental studies relating microstructure with mechanical properties of pure zinc and low-alloyed zinc with addition of 1 wt pct of Mg.

As already noted, there are not many crystal plasticity studies for Zinc and its alloys (see[13] for an extensive overview), especially as compared with other HCP materials. Moreover, the CP parameters provided in the literature are considerably scattered. Therefore, their identification for these materials is particularly challenging. This task in the case of HCP metals and alloys was usually performed either interactively by researchers (a given person would simulate a given test and visually compare the simulated stress–strain curve or texture with the experimental results until obtaining satisfactory agreement) or using some numerical optimization procedure, as e.g., in Reference 14. In the second case the objective function to be minimized was in most cases the difference between the simulated and experimental stress–strain curves.

In the paper of Solas et al.[15] the N-site visco-plastic self-consistent (VPSC) model was applied to model the texture evolution of high purity zinc subjected to channel die compression. In addition the static recrystallization occurring at post-deformation annealing was modeled using the Monte-Carlo method. Although the study is very interesting and the resulting overall texture evolution seems to be consistent with experiments, the crystal plasticity (CP) modeling framework was not very advanced. In particular, very simple linear hardening law was assumed and single linear hardening modulus was taken for each slip system. The power law exponent used was equal to 5. The values of critical resolved shear stress (CRSS) for the three slip system families considered were chosen approximately without clear justification of the particular values. Twinning was not considered.

The crystal plasticity modelling within both Taylor-type and crystal plasticity finite element method (CPFEM) of zinc-based coating was presented in Reference 16. The dislocation slip on basal, prismatic, pyramidal \(\pi _1\) and pyramidal \(\pi _2\) as well as deformation twinning were accounted for. However, it was concluded that the activity of prismatic and pyramidal \(\pi _1\) slip systems was barely present. The parameters were taken from the relevant literature.

Spherical indentation of zinc single crystal was modelled in Reference 17 in the framework of CPFEM. Dislocation slip on basal, pyramidal and prismatic slip was considered, and basal slip was set as much softer than the other two. Twinning was not considered in the cited paper. The set of parameters was chosen based on[16] so as to obtain both agreement with measured uniaxial anisotropic behavior as well as reasonable hardness values.

In Reference 13, the VPSC model was applied to study the behavior of rolled sheets of Zn-Cu-Ti alloy subjected to uniaxial tension. Covariance matrix adaptation evolution strategy (CMA-ES) was used to obtain the set of material parameters based on the experimental stress–strain curves obtained in tension in 3 directions. Contrary to the previous papers, twinning was also considered. Unfortunately, the authors assume the activity of the tensile twinning \(\left\{ 10\bar{1}2\right\} \left\langle 10\bar{1}1\right\rangle \), while for zinc having the c/a ratio equal to 1.856, the \(\left\{ 10\bar{1}2\right\} \left\langle 10\bar{1}\bar{1}\right\rangle \) twinning mode should be considered rather as a compressive twinning (i. e. activated when the c axis of the crystal is being compressed) because its c/a ratio is greater than the ideal value of \(\sqrt{3}\).[18] Taking this into account, it is hard to say if their modeling approach provides reliable results.

Similarly to op. cit., also in the present paper the evolutionary algorithm (EA) is used in order to identify the CP parameters. Such an approach is chosen mainly due to considerable number of parameters. The in-house python implementation of the EA based on the one described in Reference 19 was already used to optimize the crystal plasticity hardening parameters using both VPSC[3,4,20] and sequential elasto-viscoplastic self-consistent (SEVPSC)[21] models. However, in each case the cost function was built by summing the values of stress in the discrete strain points. In some cases, also the secants to the stress–strain curve were taken into account. Here we propose another approach. In Reference 22, the least squares error between the simulated and experimental orientation distribution function was used as a cost function in the manual optimization of two critical resolved shear stress values. Here we propose to apply similar technique but combine it with the EA. This way it is possible to establish not just two CRSSs but the entire set of crystal plasticity hardening parameters for three slip and one twinning modes.

The paper is structured as follows. In this introductory section, the properties of Zn and its alloys, their modeling using the CP theory as well as optimization of CP parameters have been briefly reviewed. In the following two sections, the experimental methods and the analysis of experimental orientation and texture data from EBSD and synchrotron measurements are included. The fourth section presents the CP framework and the applied evolutionary algorithm with newly developed texture-based fitness function. In the following two sections, the results are collected and discussed. Finally, the paper presents conclusions. In addition, an appendix contains the MTEX function for numerical comparison of two textures.

Studied materials were high purity zinc (99.99 pct) and high purity zinc alloyed with 1.5 pct wt. of magnesium (99.99 pct), prepared by gravity casting. Ingots were hot extruded at 250\(^\circ \)C with extrusion ratio R = 5.8 and subsequently subjected to multi-pass hydrostatic extrusion (HE) realized in four subsequent passes. As a result, rods with diameter 5 mm and cumulative true strain 3.6 were obtained. Afterwards, samples for microstructural investigation were cut from the received rods so as to observe longitudinal cross-section to extrusion direction. The samples were ground on sand papers with gradation up to 7000 and polished with the use of diamond suspension up to \(\frac{1}{4}\mu \)m. Finally, the samples were electropolished on Struers LectroPol-5 machine at condition 25 V for 15 seconds in C1 Struers electrolyte. Microstructural experimental data analysis was performed using SEM/ EBSD results. Quanta 3D FEG scanning electron microscope equipped with EDAX TSL OIM system was applied for collecting diffraction patterns. EBSD measurements were analyzed using TSL OIM Analysis 7 software. In order to obtain high-quality statistics, EBSD data were acquired with a different step and map size, depending on the materials’ condition. In the case of materials after hot extrusion, a step of \(1\mu \)m and a size of 1280 \(\times \) 1280 \(\mu \)m was selected. A step of 0.2 and 0.1 \(\mu \)m and map size 256 \(\times \) 256 \(\mu \)m and 128 x 128 \(\mu \)m were used, for a material after first and consecutive steps of hydrostatic extrusion, respectively. Analysis was performed only for points indexed as \(\alpha \)-Zn phase. Second phase of Mg2Zn11 particles[10] is marked by black color on EBSD maps. It was assumed that at least 5 measurement points surrounded by continuous grain boundary with misorientation angle 15 degree is required to fulfill the grain definition. The average grain size (AGS) was determined based on the weighted average grain diameter, where the weight is the grain size (average area fraction). Schmid factor maps were obtained based on the acquired orientation data from TSL OIM analysis for each slip system separately.

The texture of the HE processed materials was measured by high-energy synchrotron radiation (\(\lambda \) = 0.142342 Å) using the HGZ beamline (P07B) located at PETRA III at DESY in Hamburg. For investigation the 10 mm long specimens were cut from rods. All surfaces were ground with abrasive paper up to 7000 gradation. In order to ensure a good statistics, the specimens were continuously rotated, i.e., individual measuring points were recorded in the angular range \(\omega \) from \(-2.5\) to 2.5  pct. The obtained Debye–Sherrer rings were transformed into pole figures by StressTexCalculator. For a comparison, EBSD and texture analysis were conducted for pure Zn and Zn-1.5Mg alloy after hot extrusion and after each pass of HE.

In general, hydrostatic extrusion (HE) could be modeled including the entire geometry of the process. However, here we assume that deformation of the material point near the axis of specimen’s symmetry can be approximated using the constant velocity gradient with the following components in the basis \(\mathbf{i} _k\) with the direction \(\mathbf{i} _3\) along the extrusion direction (ED) (see Figure 1 for the coordinate system):One HE pass corresponds to 1.0 cumulative strain¹, two HE passes to 2.0 strain and four HE passes to 3.6 strain. In the following, the Schmid factor (SF) maps calculated based on the experimental results for pure Zn will be shown for the initial state and after 1 and 4 HE passes. The orientation maps are also presented. In the case of Zn-1.5Mg alloy only the SF maps before HE and after first and second passes will be shown. The dominant mechanism of microstructure evolution after more than two HE passes was significantly changed for this alloy and investigation of this particular phenomenon is outside the scope of the paper.

Figures 2 and 3 present EBSD data analysis using TSL OIM ANALYSIS 7 software. SF maps were obtained for the longitudinal cross sections (LCS) to the extrusion direction using orientation data and the stress deviator of the same form as the strain rate tensor given by Eq. [1]. For each map average grain size was calculated. In the case of pure zinc, the largest reduction of the grain size was observed after the first pass and microstructure with equiaxed grains was obtained. After next passes only slight difference was observed in the size and shape of grains. The initial microstructure of Zn-1.5Mg alloy is considerably different, namely, two phases are present: \(\alpha \)-Zn and eutectic composed of \(\alpha \)-Zn and Mg2Zn11 intermetallic phase. The grains are elongated in the extrusion direction. The \(\alpha \)-Zn grains are larger than eutectic grains but they are much smaller than the grains present in pure Zinc. Further reduction of grain size can be seen in the second HE pass. The small equiaxed grains of \(\alpha \)-Zn are separated by eutectic bands, which are also getting narrower after additional passes of extrusion.

The partition fraction of Schmid factor value exceeding 0.4 for each slip system is presented on the graphs in Figures 2(b) and 3(b) for each pass for pure Zn and Zn-1.5Mg alloy, respectively. The correlation between AGS, elongation to fracture \(\epsilon _f\) and Schmid factor for different slip systems is also presented. In the case of hydrostatic extrusion the calculated Schmid factor on slip systems other than basal is high. This can lead to their activation provided that also the critical value of the resolved shear stress (RSS) at a given temperature for those systems is sufficiently low. In the case of pure Zn, the compressive twinning presents significant Schmid factor in the first HE pass but it decreases in subsequent passes. The pyramidal <c+a> \(\left\{ 11\bar{2}2\right\} \left\langle \bar{1}\bar{1}23\right\rangle \) and prismatic slip systems can be characterized as possessing significantly high values of the Schmid factor, especially in the case of Zn-1.5Mg alloy. However, according to the literature (cf.[13,15‐17] and references cited therein), the critical resolved shear stress for basal slip for Zn and its alloys is the lowest. Taking into account both the value of SF and the CRSS, it seems that in the case of pure zinc the basal slip should be the most active system after second pass.

Figure 4 shows the pole figures obtained from the texture measured for the initial texture (after hot extrusion but before HE) of (a) pure Zn and (b) Zn-1.5Mg alloy. These pole figures (and all the other pole figures presented next) were plotted using the MTEX software.[24] One can see that the initial texture is close to the ring texture with the basal planes being parallel to the extrusion direction. However, the circular symmetry is not observed, as some localization of the basal planes can be seen. The prismatic planes show even more localization. Judging from our previous results for \(\alpha \)-Ti[1] it can be surmised that the prismatic slip could be the primary deformation mechanism in the course of the initial hot extrusion. Verfication of this observation is however outside the scope of the present paper.

Figures 5 and 6 show the pole figures obtained from the textures after HE measured for pure zinc and the Zn-1.5Mg alloy, respectively. One can see that the texture of pure Zn after the 1st HE pass (Figure 5(a)) can be characterized by the basal plane normals being inclined about 70 degree to the rod axis. Contrary to the initial texture, the poles are distributed quite uniformly around ED. In addition, there is a local (0001) maximum corresponding to grains having the basal plane normal nearly aligned with ED. It seems to result from the activity of twinning and this view will be discussed later. The texture after second (Figure 5(b)) and fourth (Figure 5(c)) HE passes is similar but lacks the central maximum. The origin of its disappearance is not obvious and shall be discussed later. The texture for Zn-1.5Mg (Figure 6) is similar, but the two main fibres are present in different proportions. Clearly, after the first HE pass (Figure 6(a)) the central (0001) maximum is more intense than the 70 degree inclined ring. This changes after second pass where the ring fibre is more intense. However, the central component does not disappear as it was the case for pure Zn.

The first question to be discussed is the origin of the \(\left\langle 0001\right\rangle \)||ED fibre. To show that its appearance results from the reorientation related to twinning, the simulation with the same parameters but without considering twinning was carried out for Zn-1.5Mg alloy. The resulting texture after one HE pass is presented in Figure 12. It can be clearly seen that in such a case, this component is not present. Thus, its twinning-related origin is confirmed.

The main disagreement of the presented modeling results as compared to the experimental results is too strong \(\left\langle 0001\right\rangle \)||ED fibre after 2nd and subsequent passes. The probable reason for its disappearance observed in experiments shall be discussed in the following. In the paper of Solas et al.[15] the static recrystallization of high purity zinc was studied both experimentally and using the Monte Carlo approach. The authors concluded that the nucleation occurs in highly deformed grains and then it proceeds by growth into the less deformed grains. The grains with their ’c’ axis nearly perpendicular to the channel die compression axis are hard to deform by the easy basal slip. As a result they do not deform much, their stored energy is relatively low and they are consumed by other growing crystallites. Here, the central (0001) maximum corresponds to grains whose ’c’ axis is nearly aligned with ED. The Schmid factor of an ideal \(\left\langle 0001\right\rangle \)||ED orientation for basal slip with respect to the stretching force along ED is:Therefore, the analyzed component is also the “hard” orientation in the understanding of Solas et al. Here, we do not apply any direct recrystallization modeling. In addition, op. cit. deals with static recrystallization occurring during the post-deformation annealing. However, as long as the conclusions drawn there are correct, the most probable reason behind the disappearance of the \(\left\langle 0001\right\rangle \)||ED maximum seems to be the discontinuous dynamic recrystallization. This does not mean that dynamic recrystallization does not lead to other phenomena important for texture evolution. Detailed investigation of the effect of dynamic recrystallization on microstructure evolution is, however, outside the scope of the present paper.

As the paper presents the optimization of material parameters by comparing only the simulated texture, it is interesting to investigate the sensitivity of the approach to different parameter values. This issue was investigated in the Electronic Supplementary Material. It was revealed that the proposed fitness function is sensitive to such parameters as \(\tau _0\), \(\tau _{sat}\), \(f_{sat}\) and \(\mu \), so they could be identified with a good accuracy. On the other hand, it is not much sensitive to \(\beta \), \(h_0\) and latent hardening parameters. Therefore, to assess the accuracy of these parameters more experimental data would be necessary. To this end the proposed fitness function can be also combined with the classically used functions based on mechanical testing, cf. e.g., Reference 3.

It would be interesting to validate the optimized sets of parameters by simulating other plastic deformation process. However, a true validation was not possible due to lack of additional experimental data for the investigated materials of precisely the same composition and deformation history. Nevertheless, we have simulated rolling of pure zinc with the same amount of strain as in Reference 16 and compared the resulting texture (Figure 13) with the experimental one as presented in Figure 6(d) in op. cit. Reasonable agreement between the simulated and experimental textures has been found.

In the paper, the evolutionary algorithm supplied with a newly developed texture-based fitness function was applied in order to establish the crystal plasticity parameters of pure Zn and Zn-1.5Mg alloy subjected to hydrostatic extrusion. The active deformation modes were thus revealed and discussed. It was shown that the predicted slip and twinning system activities are consistent with Schmid factors calculated based on EBSD measurements. In addition to the best of the authors knowledge, this is the first study where the texture evolution of pure Zn with as much as 3.6 strain (4 HE passes) was simulated and where the reorientation due to \(\left\{ 10\bar{1}2\right\} \left\langle 10\bar{1}1\right\rangle \) compressive twinning was taken into account.

Based on the acquired results, it can be concluded that: