1. Introduction
Due to their excellent properties such as their light weight, magnesium alloys are expected to be candidates for structural materials that can replace conventional materials used in engineering products, including automobiles and aircraft, where environmental friendliness is strongly required [
1]. In particular, since significant
reduction regulations will be imposed on automobiles in the near future, the weight reduction of the car body, which has a direct impact, is considered to be a realistic policy from the viewpoint of cost effectiveness [
2,
3].
In general, magnesium alloys are often used in the manufacture of automotive components by casting, because they can be efficiently cast into complex shapes [
4]. Although there are some features that need to be improved when they are used as sheet metal, such as their anisotropy and low ductility, there is the possibility of achieving a large weight reduction compared to steels and aluminum alloys by promoting their application in panel-like parts [
5]. Therefore, it is of great engineering significance to understand the deformation characteristics of magnesium alloys and to model them appropriately. However, the complex deformation characteristics of magnesium alloys have not been sufficiently investigated, because they are subjected to plastic working less frequently than steels and aluminum alloys.
When conducting forming simulation, it is necessary to select an appropriate material model based on the yield function according to the application. Among such models, the most frequently used is the classical yield function proposed by von Mises [
6]. It is applicable to many isotropic materials and, therefore, still plays an important role. For materials with strong in-plane anisotropy, such as aluminum alloy sheets, Hill’s yield function [
7], which is a modification of the von Mises model, is often used. Various yield functions have subsequently been proposed for more accurate representation. For example, the Yld2000-2d model proposed by Barlat et al. [
8] is often applied to anisotropic materials. Yld2000-2d has eight parameters, which are determined by solving simultaneous equations with multiple variables using eight experimental values. If the stress principal axis coincides with the anisotropic principal axis, the number of parameters decreases to six. In any case, yield stress in equi-biaxial tension as well as in-plane tensile tests are required. Although recent yield functions have shown high accuracy in practical use, the number of material parameters included in them has increased, and the burden of the material testing that they require has become non-negligible. For magnesium alloys, which are the focus of this paper, yield functions that take into account the strength differential (SD) effect are necessary, such as CPB06 [
9] and that proposed by Verma et al. [
10]. Many other yield function models have been proposed, but several advanced material models require not only uniaxial tensile and compression tests, but also biaxial material tests that necessitate specialized equipment and cannot easily be utilized in plastic forming simulations. To be more precise, the presence of friction and bulging in compression tests makes pure compressive material testing difficult. Biaxial testing is often performed using a cruciform testing machine or a hydraulic bulge tester, but these devices are not widespread. In addition, biaxial testing itself has problems to be considered in handling anisotropy and measuring plastic strain. Therefore, if at least some of the difficult-to-perform material tests, such as biaxial tests, could be replaced by numerical simulations, product design based on deformation analysis would be promoted, and the application of magnesium alloys would advance.
To accurately understand the deformation behavior of polycrystalline metals, a simulation model that takes the crystal structure into account is necessary. The phenomenological yield function and work-hardening models are not suitable for this purpose because they ignore microscopic mechanisms. Therefore, in recent years, crystal plasticity models have been frequently used for material modeling and deformation analysis to understand the material properties. There are various methodologies for the homogenization process, such as the visco-plastic self-consistent (VPSC) model, elasto-plastic self-consistent (EPSC) model, and elasto-vicoplastic self-consistent (EVPSC) model based on the self-consistent approach, and the crystal plasticity finite element model (CPFEM) based on finite element discretization. A comprehensive review was provided by Nguyen et al. [
11] concerning research into computational methods based on various crystal plasticity models, from basic theory to recent applications. In modeling based on the finite element method, one crystal grain is discretized by one or more elements, so that the behavior of each crystal and grain boundary can be treated. In addition, because the common finite element code is applicable, there are many application examples. Its applications to numerical material testing by Kraska et al. [
12], Zhang et al. [
13], and Sedighiani et al. [
14] are especially pertinent to this study. They used material test data such as tensile test results to determine material parameters in a virtual material constructed based on the CPFEM and predicted material properties such as biaxial test results. Considering the increasing accuracy requirements for forming simulations and the expanding application range of magnesium alloys in the future, it is necessary to increase the reliability of such numerical material testing methods.
The majority of crystal plasticity models use formulations based on the strain-rate-dependent constitutive model [
15]. In this model, the strain rate sensitivity is used as a parameter to calculate the strain rate, which assumes that all slip systems are activated at the same time. This situation does not occur in actual material deformation; therefore, the strain rate sensitivity parameter is used to compensate for this inconsistency. In other words, the strain rate sensitivity parameter does not necessarily represent the mechanism of a slip system in a physical sense. Thus, the strain rate sensitivity parameter is not determined by material tests but empirically. To construct a physically accurate alternative model, Takahashi et al. [
16,
17] developed a finite element polycrystal model (FEPM) based on a successive accumulation scheme that directly deals with Schmid’s law. In the FEPM, a criterion introduced by Takahashi et al. [
16] for determining the activity is applied to all slip systems, the slip increment is calculated only for those slip systems that are determined to be active, and the plastic strain increment is calculated as the sum of these. In this method, it is not necessary to determine the activity or inactivity of a slip system in advance; it is just determined as a result of convergence. Therefore, there is no need to introduce physically ambiguous parameters such as the strain rate sensitivity parameter. In addition, an advantage is that the degree of activity of each slip system can be monitored throughout the deformation. Although the successive accumulation-based crystal plasticity model has some advantages, it is not as well validated as the conventional strain-rate-dependent model. The FEPM was recently revalidated by Oya and Araki [
18], and its usefulness in numerical material testing for fcc (face-centered cubic) aluminum alloys was demonstrated. These authors succeeded in predicting unexecuted material test data from known material test data by optimizing the material parameters using a genetic algorithm. This study demonstrated the utility of a crystal plasticity scheme that does not use the common strain-rate-dependent model, but it has not been applied to bcc (body-centered cubic) and hcp materials, which exhibit more complex behavior.
Next, the modeling of magnesium alloys is explained. Magnesium is a material with a hexagonal close-packed (hcp) crystal structure. Different slip and twin modes are considered in the modeling of hcp metals, each of which is assumed to have different critical resolved shear stress (CRSS) and hardening parameters. In modeling plastic deformation behavior, the determination of these microscopic parameters for each deformation mode is usually carried out so that the simulated flow stresses fit the experimental stress–strain curve. The two main deformations at room temperature, when undergoing compression along the basal plane or tension along the basal plan normal, are basal slippage and extension twinning. In fact, these two deformations are characterized by the lowest CRSS. Pyramidal and prismatic slip systems are more likely to be activated during tensile deformation in the basal plane and compression along the basal plane normal, respectively [
19]. To achieve a reliable determination of the parameters of the model, a loading path that would guarantee the activation of all the relevant deformation modes [
20] should be used. It is clear that the parameters are not unique [
21], since only a limited number of stress–strain curves can be used for fitting, and thus the prediction of deformation along other strain paths is inaccurate. The CRSS for each deformation system varies from study to study, because it strongly depends on not only the material properties such as the microstructure, but also the strain path, as well as the model chosen [
22]. The hardening laws of the CRSS for
twinning is surmised to be one of the reasons for this, because it is considered constant in some studies [
22,
23] and decreasing [
24] or increasing [
25] in others. The properties of
extension twins, such as active twin variants, are highly dependent on the strain path.
twinning can be incorporated into the model in a variety of ways, because it introduces partial lattice reorientation and new interactions between twin boundaries and slip modes.
In this study, a new numerical material testing method for magnesium alloys with a hcp structure was developed by applying the FEPM as a crystal plasticity solver. This paper presents the first application of the FEPM to a numerical material testing method for hcp-structured materials and introduces a modeling method and a parameter determination method using a genetic algorithm (GA) for its construction. In
Section 2, a brief description of the FEPM and its application to hcp materials is presented. In
Section 3, we explain the process of using the GA to determine the material parameters included in the hardening law of the slip system in the crystal plasticity model, which are unknown parameters, and the proposed numerical material testing method is used to predict the unknown material properties, employing experimental data of cast and rolled sheets of magnesium alloy AZ31 obtained from the literature. The validation results of the proposed method are discussed in
Section 4, and the conclusions are presented in
Section 5.
4. Discussion
The following is a discussion of the investigations that were conducted on cast and rolled AZ31 specimens. The difference between the two materials was that the cast material was isotropic, i.e., had a random texture, while the rolled material was anisotropic and had a strongly oriented texture. In the case of either material, since it is not easy to determine the multiple microscopic parameters of a polycrystalline model that are not directly measurable, some learning or optimization process was required. Even with a genetic algorithm, there was no certainty that a perfect optimization could be achieved, and the reliability of the results was highly dependent on the initial values and optimization scheme. In addition, a sufficient number of experimental data must be available to perform reliable optimization, but material tests other than uniaxial tensile tests are not common. This problem is particularly acute for hcp materials, where several different types of slip systems are active. In addition to the usual uniaxial tensile tests, biaxial tensile and compression tests may also be required, making material testing even more difficult. Therefore, if the numerical material testing method proposed in this paper can predict experimental data for material tests that are difficult to perform, the benefits would be especially great for hcp materials.
Although this study dealt with two types of materials, cast and rolled, from the viewpoint of the importance of deformation analysis, the main target was the prediction of the material properties of rolled sheet materials, which are subjected to various press forming operations. The crystal plasticity model used in this study contained 12 hardening parameters. It would be very difficult to determine all of them to fit the behavior of rolled AZ31. Therefore, we first estimated the parameters of cast AZ31, whose material properties are considered isotropic. Although the initial optimization variables were obtained by trial and error, judging from the stress–strain curves, the determined parameters were reasonable. Therefore, cast AZ31, although an hcp material, had relatively simple properties and was an appropriate first optimization target.
Next, using the parameters obtained for cast AZ31 and applying them directly to rolled AZ31, errors were naturally observed. Therefore, we designed another optimization scheme and performed optimization using the initial parameters separately prepared. In this scheme, pseudo-anisotropy data obtained by numerical rolling were used for the crystallographic orientation. In addition, the treatment of twinning systems in this study should also be explained. As was already described, latent hardening coefficients were considered necessary to explain indirect hardening due to crystal rotation by twins. However, in doing so, the stresses were overestimated. This resulted in significantly lower CRSS values for pyramidal and prismatic slips and the underestimation of stresses during TTC. From these results, it could be said that the hardening effect of twinning on the other slips in the proposed model could be fully explained only by the lattice rotation during deformation. These efforts improved the simulation results of all the stress–strain curves so that they were close to the experimental results in the range of up to . These results indicate that the polycrystalline model and its microscopic parameter optimization scheme have physical validity and are reliable as a numerical material testing method.
It should also be noted that the strain-rate-dependent coefficient was not used in the verification process described above. This meant that in the deformation analysis conducted in this study and in the optimization process using it, the activity of the slip systems was evaluated without any discrepancy between the analytical and experimental results, which is an example that proves the usefulness of the FEPM with less physical ambiguity.
However, errors were not entirely absent, and the largest gap was in the ND-T prediction curve. Pyramidal slip behavior is not very relevant in uniaxial stress, but it becomes a major slip system under biaxial stress conditions, such as thickness direction compression (NT-D). Therefore, one of the reasons for this error may be that there is no teaching curve that can optimize only the parameters related to it. In addition, the treatment of latent hardening requires further validation. Since the direct experimental verification of latent hardening is very difficult, the validity of the proposed method could be enhanced by applying it to more macroscopic material test data.