Characterisation and modelling of anisotropic hardening behaviour of cubic and hexagonal close packed polycrystalline metals

ABSTRACT

In this study, anisotropic non-proportional hardening behaviour of the sheet metals with different crystalline structures is investigated. The investigated materials particularly exhibit the Bauschinger effect and complex transient hardening behaviour during loading path change. Furthermore, the finite element simulations are carried out to evaluate the performance of the distortional and multi-surface-kinematic hardening models in flow stress prediction under loading path change. Both models can well reproduce the measured anisotropic hardening, but the distortional hardening concept shows better predictive capability for flow stresses. The improved accuracy of the flow stress prediction with the distortional hardening model compared to the kinematic hardening is attributed to its flexibility in distortion of the yield surface following microstructure deviator independent on the initial yield surface. In contrast, the initial size of the yield surface in the kinematic hardening significantly affects the determination of the re-yielding stress at load reversal.

KEYWORDS:

• Distortional hardening
• Bauschinger effect
• plastic deformation
• kinematic hardening

1. Introduction

When metallic materials experience load reversal, such as tension followed by compression, anisotropic hardening features such as the Bauschinger effect and transient hardening behaviour in the flow stress curves are observed [1–4]. Several efforts have been dedicated to formulating constitutive models capable of reproducing the flow stress under diverse loading conditions.

Under the loading path change condition, the well-known Bauschinger effect, early re-yielding after loading path change, becomes prominent. In order to capture this behaviour, a kinematic hardening law based on the translation of the yield surface has been developed. The linear kinematic hardening model was initially proposed by Prager [5] and Ziegler [6], suggesting different evolution rules for back stress. Subsequently, various researchers introduced nonlinear kinematic hardening models with the isotropic hardening were able to reproduce more realistic flow stress curves under load reversal [7,8]. To date, the multi-surface-based kinematic hardening law proposed by Yoshida and Uemori has proven successful in accurately capturing the complex hardening behaviour for the metallic materials [9,10].

As an alternative to the kinematic hardening model, a concept of a distortion-yield-surface has been proposed [11–14]. In previous studies, this distortion of the yield surface was combined with kinematic hardening law, primarily to capture the Bauschinger effect and transient hardening behaviour during a loading path change. Recently, the homogeneous yield function-based anisotropic hardening (HAH) model was introduced, employing the concept of the distortion-yield surface [15]. By controlling the state variables in the model, the amount of distortion in the yield surface could be managed, allowing the model to accurately capture anisotropic hardening behaviour. Subsequently, extended HAH models were developed, enabling them to predict mechanical responses even when subjected to orthogonal loading path changes [16–18].

Although some research studies provide worthy understanding on predictive capability of the material models for the specific crystal structured sheet metals under strain path changes, there seems to be a lack of comprehensive research regarding the flexibility of these models across various types of crystal structured sheet metals. Thus, this study aims to assess the performance of the constitutive models on the flow stress prediction under loading path change condition for various types of the crystal structured sheet metals. For this purpose, mechanical characteristics of the metallic materials with different crystal structures are investigated when subjected to load reversal, and the distortional hardening and multi-surface-based-kinematic hardening models are employed to reproduce the anisotropic hardening behaviours.

2. Materials and methods

2.1. Materials

In this study, an aluminium alloy (AA) 6111-T4 [19], a high-strength steel DP780 [20], and a titanium alloy Ti64 [21] are used to investigate the anisotropic hardening behaviour of the sheet metals subjected to the load reversal because each sheet metal has different crystal structure. AA 6111-T4 shows a face centred crystal structure (FCC), DP780 has a body centred crystal structure (BCC), and a hexagonal closed packed structure (HCP) of Ti64 is shown.

The mechanical response of the sheet metals such as in-plane anisotropy is characterised from the uniaxial tension tests equipped with the universal tensile machine (UTM). And the anisotropic hardening behaviour of the material subjected to the load reversal is obtained using a specially developed in-plane cyclic loading machine [22–24].

2.2. Constitutive models and finite element modelling

In this paper, the distortional hardening model, namely, homogeneous yield function based anisotropic hardening (HAH) model [15,16], is adopted to reproduce the reverse loading behaviour for the sheet metals. The model can account for the anisotropic hardening properties such as the Bauschinger effect, transient hardening, and permanent softening behaviours by controlling the yield surface distortion. A brief formulation of the model is as follows.

The material model is formulated as a combination of two components, a stable function $\varphi$ and fluctuating function ${\varphi }_h$, which represent the material anisotropy and distortion of the yield surface, respectively. (1) $\mathrm{\Phi }\left(\sigma ,{\mathrm{f}}_1,{\mathrm{f}}_2,{\hat{\mathbf{h}}}^{\mathbf{s}}\right)=\{{\varphi }^{\mathrm{q}}\left(\sigma \right)+{\varphi }_h\left(\sigma ,{\mathrm{f}}_1,{\mathrm{f}}_2,{\hat{\mathbf{h}}}^{\mathbf{s}}\right){\}}^{1/q}={\overline{\sigma }}_{\mathrm{IH}}(\overline{ϵ}),$(1) where $\mathrm{\Phi }$ is the equivalent stress, $\sigma$ is the Cauchy stress, s is the deviatoric stress. The exponent q is a material coefficient to control the sharpness of the distorted yield surface, $\overline{ϵ}$ is the equivalent strain, and ${\overline{\sigma }}_{\mathrm{IH}}\left(\overline{ϵ}\right)$ is the reference flow stress function to be determined by the uniaxial tension tests. A special tensor ${\hat{\mathbf{h}}}^{\mathbf{s}}$ is introduced to memorise the material deformation history and decide the direction of the yield surface distortion. The hat symbol ($\stackrel{ˆ}{}$) above h represents the normalised quantity of the tensor as ${\hat{\mathrm{x}}}_{\mathrm{ij}}={\mathrm{x}}_{\mathrm{ij}}/\sqrt{8/3\cdot {\mathrm{x}}_{\mathrm{kl}}{\mathrm{x}}_{\mathrm{kl}}}$, the initial ${\hat{\mathbf{h}}}^{\mathbf{s}}$ is equivalent to the normalised quantity of the deviatoric stress s when the first plastic deformation occurs. The detailed calculation of the special tensor ${\hat{\mathrm{h}}}^{\mathrm{s}}$ can be found in Appendix A.

Anisotropic hardening properties can be modelled by the yield surface distortion, which is controlled by the fluctuating function ${\varphi }_h$. The fluctuating function is composed of the two state variables f₁ and f₂, and they are additionally expressed by the plastic state variable g₁, and g₂. (2) ${\varphi }_h\left(\sigma ,{\mathrm{f}}_1,{\mathrm{f}}_2,{\hat{\mathbf{h}}}^{\mathbf{s}}\right)={\mathrm{f}}_1^{\mathrm{q}}|{\hat{\mathbf{h}}}^{\mathbf{s}}:\mathbf{s}-\left|{\hat{\mathbf{h}}}^{\mathbf{s}}:\mathbf{s}\right|{|}^{\mathrm{q}}+{\mathrm{f}}_2^{\mathrm{q}}|{\hat{\mathbf{h}}}^{\mathbf{s}}:\mathbf{s}+\left|{\hat{\mathbf{h}}}^{\mathbf{s}}:\mathbf{s}\right|{|}^{\mathrm{q}},$(2) (3) ${\mathrm{f}}_{\mathrm{k}}={\left[{\displaystyle \frac{1}{{\mathrm{g}}_{\mathrm{k}}^{\mathrm{q}}}-1}\right]}^{\frac{1}{\mathrm{q}}},\mathrm{and}\mathrm{k}=1,2.$(3) The evolutionary functions of the state variables in the model can be found in the papers [15,16], and the HAH model has several material coefficients (k₀, k₁, k₂, k₃, k₄, and k₅) to be determined from the flow stresses subjected to the load reversal.

As for purpose of comparison, the multi-surface-kinematic hardening model (denoted as Y-U model) proposed by Yoshida and Uemori [9,10,25] is also adopted to reproduce the anisotropic hardening behaviours for the sheet metals. The evolution of total backstress α consists of the translating rate of the yield surface ${\alpha }_{\ast }$ and bounding surface β, and each evolution law is summarised as follows: (4) $\mathrm{d}\alpha =\mathrm{d}{\alpha }_{\ast }+\mathrm{d}\text{β},$(4) (5) $\mathrm{d}{\alpha }_{\ast }=\mathrm{C}\left({\displaystyle \frac{\mathrm{B}+{\mathrm{R}}_{\mathrm{sat}}-\mathrm{Y}}{\mathrm{Y}}\left(\sigma -\alpha \right)-\sqrt{{\displaystyle \frac{\mathrm{B}+{\mathrm{R}}_{\mathrm{sat}}-\mathrm{Y}}{\varphi \left({\alpha }_{\ast }\right)}}}{\alpha }_{\ast }}\right)\mathrm{d}\overline{ϵ},$(5) (6) $\mathrm{d}\text{β}=\mathrm{m}\left({\displaystyle \frac{\mathrm{d}}{\mathrm{Y}}\left(\sigma -\alpha \right)-\text{β}}\right)\mathrm{d}\overline{ϵ},$(6) where ${\alpha }_{\ast }$ is the relative kinematic motion of the yield surface with respect to the bounding surface with centre β. The detailed formulation of the Y-U model can be found in the literatures [9,10,25], and the material parameters such as Y, C, B, m, R_sat, b, h can be also identified from the experimental data.

Finite element (FE) simulations are conducted to validate the mechanical response of the sheet metals with a commercial finite element software ABAQUS/Standard [26], and the constitutive models are implemented into FE software through the user-defined material subroutine (UMAT) [27,28]. One element of a reduced 4-node shell type (S4R) with 5 integration points through the thickness direction is used, and displacement boundary conditions, which are equivalent to the experiments, are applied. It has been well known that reduced integration of the shell type element in the ABAQUS commercial finite element software usually generates more accurate results and significantly reduces running time, especially in three dimensions [26]. It is worth noting that in-plane anisotropy of the sheet metals is reproduced by using the non-quadratic anisotropic plastic yielding function Yld2000-2d model [29], which is characterised with 8 material parameters, α₁₋₈, and the exponent a (Appendix B).

3. Results and discussion

3.1. Flow stress prediction for FCC metal

The non-quadratic anisotropic function Yld2000-2d with the exponent a = 8 is used to consider the crystal structure property. Isotropic elasticity with 72 GPa of Young’s modulus and 0.33 of Poisson’s ratio is assumed. The isotropic hardening parameters in Equation (1) are identified from the uniaxial tension results and optimised using the Voce hardening law ${\overline{\sigma }}_{\mathrm{IH}}\left(\overline{ϵ}\right)=\mathrm{A}-\mathrm{B}\cdot e^{-\mathrm{C}\overline{ϵ}}$. The determined material constants of the constitutive models for AA6111-T4 sheets are listed in Table 1.

Table 1. Material coefficients of the constitutive models for AA6111-T4 sheet.

Yld2000-2d
α1	α2	α3	α4	α5	α6	α7	α8	a
0.879	1.136	0.952	1.048	1.009	0.952	1.034	1.237	8.0
HAH model
k0	k1	k2	k3	k4	k5	A	B	C
30.0	200.0	90.0	0.7	1.0	0.0	285.58	137.21	20.72
Y-U model
Y	C	B	Rsat	b	m	h
148.0	562.5	178	189.1	24.6	8.1	0.001

Figures 1 and 2 show the comparison between experimental and simulated results of the flow stress under tension followed by compression, using HAH and Y-U models, respectively. Both material models can reproduce the anisotropic hardening behaviour, especially the Bauschinger effect and transient hardening property. In order to quantify the Bauschinger effect, the Bauschinger ratio (B_r) can be defied as. (7) ${\mathrm{B}}_r={\displaystyle \frac{{\sigma }_f-{\sigma }_r}{2{\sigma }_f},}$(7) where ${\sigma }_f$ denotes the flow stress at the start of unloading and ${\sigma }_r$ is the initial stress during reverse loading.

Figure 1. Flow stress prediction for FCC metal with HAH model and corresponding evolution of the yield surface during plastic deformation. Experimental data are reproduced from [19].

Figure 2. Flow stress prediction for FCC metal with Y-U model and corresponding evolution of the yield surface during plastic deformation.

The Bauschinger ratio for both experimental and simulated data are as follows:

(1) Experiment – 0.8 (2%), 0.78 (5%), 0.76 (7%), (2) HAH model – 0.828 (2%), 0.754 (5%), 0.729 (7%), (3) Y-U model – 0.753 (2%), 0.634 (5%), 0.573 (7%). The Bauschinger ratio results indicate that the Bauschinger effect becomes larger as more plastic deformation is subjected, and the Y-U model over-estimates the Bauschinger effect although the predicted flow curves of Y-U model are agreement with the experimental data.

Figures 1 and 2 also present the evolution of the yield locus during the plastic deformation. The locus of the HAH model is distorted in the opposite direction of the loading, enabling it to reproduce the flow stress curve under loading path changes. In contrast, the Y-U model uses the concept of the translation of the yield surface without shape change of the yield surface. Owing to the pre-determined initial yield stress in the Y-U model, the Bauschinger effect may be overestimated as aforementioned results.

3.2 Flow stress prediction for BCC metal

The exponent a = 6 for the non-quadratic anisotropic function Yld2000-2d is recommended for the BCC metal. Isotropic elastic property with 198 GPa of Young’s modulus and 0.33 of Poisson’s ratio is assumed. The Swift hardening law ${\overline{\sigma }}_{\mathrm{IH}}\left(\overline{ϵ}\right)=\mathrm{A}(\mathrm{B}+\overline{ϵ}{)}^{\mathrm{C}}$ in Equation (1) is used for the strain hardening model. The optimised material constants of the constitutive models for DP780 sheets are listed in Table 2.

Table 2. Material coefficients of the constitutive models for DP780 sheet.

Yld2000-2d
α1	α2	α3	α4	α5	α6	α7	α8	a
0.928	1.024	0.962	0.988	1.004	0.917	1.004	1.032	6.0
HAH model
k0	k1	k2	k3	k4	k5	A	B	C
30.0	100.0	90.0	0.4	0.9	30.0	1280.0	0.0008	0.146
Y-U model
Y	C	B	Rsat	b	m	h
508.4	83.3	824.2	265.7	87.2	4.01	0.001

Figures 3 and 4 illustrate the plot results between experimental and simulated data of flow stress during tension and subsequent compression, employing the HAH and Y-U models, respectively. Both models successfully capture the anisotropic hardening characteristics, particularly the Bauschinger effect and transient hardening behaviour. Furthermore, the quantified data for Bauschinger effect for both experimental and simulated data are as follows:

Figure 3. Flow stress prediction for BCC metal with HAH model and corresponding evolution of the yield surface during plastic deformation. Experimental data are reproduced from [20].

Figure 4. Flow stress prediction for BCC metal with Y-U model and corresponding evolution of the yield surface during plastic deformation.

(1) Experiment – 0.662 (2%), 0.637 (6%), 0.611 (7%), (2) HAH model – 0.706 (2%), 0.632 (6%), 0.619 (7%), (3) Y-U model – 0.702 (2%), 0.627 (6%), 0.588 (7%). The finding indicates that the Bauschinger ratio in DP780 sheets reaches saturation rapidly during initial plastic straining. And the Bauschinger ratio results predicted by the constitutive models are good agreements with experimental data. Nevertheless, owing to the inherent nature of the kinematic hardening law, the Bauschinger ratio predicted with the Y-U model cannot capture the saturation observed with increased plastic straining.

The corresponding evolving yield loci for each material model during load reversal are also illustrated in Figures 3 and 4. In the case of the monotonic loading, the amount of the plastic hardening affects the size of the yield locus for the HAH model and the distorted yield surface of the HAH model effectively reproduces anisotropic hardening behaviour. It is noteworthy that the re-yielding stress predicted by the HAH model under load reversal is always a negative stress value. Conversely, the Y-U model may predict a positive re-yielding stress after load reversal, influencing springback predictions in the metal forming process [30].

3.3. Flow stress prediction for HCP metal

The in-plane anisotropy for the HCP crystal structure metals has been assessed utilising a specialised plastic yielding model, such as the CPB 06 yield function, capable of capturing both in-plane anisotropy and the strength differential effect [31–34]. However, it has been confirmed that the plastic yielding behaviour of Ti64 sheets could be reproduced by using Yld2000-2d model with the exponent a = 12 [21]. The material is assumed to possess isotropic elastic properties, including a Young’s modulus of 115 GPa and a Poisson’s ratio of 0.33. The strain hardening model adopts the Swift hardening law ${\overline{\sigma }}_{\mathrm{IH}}\left(\overline{ϵ}\right)=\mathrm{A}(\mathrm{B}+\overline{ϵ}{)}^{\mathrm{C}}$ in Equation (1) is used for the strain hardening model. The material constants identified for the constitutive models of Ti64 sheets are provided in Table 3.

Table 3. Material coefficients of the constitutive models for Ti64 sheet.

Yld2000-2d
α1	α2	α3	α4	α5	α6	α7	α8	a
1.053	0.994	1.074	0.959	0.955	0.978	1.064	0.918	12.0
HAH model
k0	k1	k2	k3	k4	k5	A	B	C
30.0	130.0	85.0	0.25	1.0	0.0	1360.0	0.0114	0.047
Y-U model
Y	C	B	Rsat	b	m	h
1120.1	100.0	1125.2	101.1	196.1	13.8	0.001
Extended Y-U model
Y	C1	C2	B	Rsat	b	m	h
750.0	40000	170.0	1125.2	101.1	196.1	13.8	0.001

Figures 5 and 6 provide the predicted flow stress under cyclic loading for HAH and Y-U models, respectively, and their comparison with experimental data. As shown in Figure 5, the flow stress predicted by the HAH model exhibits the Bauschinger effect and transient hardening property. However, the Y-U model could not reproduce the anisotropic hardening behaviour of Ti64 sheets. The underestimation of the Bauschinger effect is evident for the Y-U model, as indicated in Figure 6. Comparable results are also observed concerning the Bauschinger ratio.

Figure 5. Flow stress prediction for HCP metal with HAH model and corresponding evolution of the yield surface during plastic deformation. Experimental data are reproduced from [21].

Figure 6. Flow stress prediction for HCP metal with Y-U model and corresponding evolution of the yield surface during plastic deformation [21].

(1) Experiment – 0.738 (4%), 0.678 (−4%), (2) HAH model – 0.635 (4%), 0.614 (−4%), (3) Y-U model – 0.927 (4%), 0.883 (−4%).

The Y-U model's inaccurate flow stress prediction can be elucidated by examining the evolution of the yield surface. As shown in Figure 5, the HAH model distorts the yield locus during plastic deformation despite the substantial size of the initial yield surface. However, the Y-U model maintains the size of the yield surface but translates the yield locus over a short distance, as depicted in Figure 6. The higher yield stress Y leads to the lower level of the re-yielding stress ${\sigma }_f-2\mathrm{Y}$. Consequently, the Y-U model cannot capture any of the complex reverse loading characteristics.

A more accurate prediction of the flow stress under cyclic loading can be achieved through the extended Y-U model. In this model, the material parameter C assumes two different values C₁ and C₂ for forward and reverse loading, respectively, as provided in Table 3. And a lower initial yield stress Y is selected, resulting in a smaller size of the yield surface.

Figure 7 shows the flow stress prediction under cyclic loading using the extended Y-U model, demonstrating comparably accurate results. Moreover, the Bauschinger ratio is more realistically estimated at 0.618 (4%) and 0.589 (−4%), as reflected in the evolution of the yield locus. Consequently, the extended Y-U model reproduce effectively anisotropic hardening characteristics at lower strain levels, particularly for metals with high initial yield stress.

Figure 7. Flow stress prediction for HCP metal with extended Y-U model and corresponding evolution of the yield surface during plastic deformation.

4. Conclusions

In this study, characterisation and modelling of the anisotropic hardening behaviour are investigated for AA6111-T4, DP780, and Ti64 sheets, each featuring distinct crystal structures. The presented experimental data on flow stress under load reversal indicate that all materials exhibit the Bauschinger effect and transient hardening behaviour. A numerical study utilising the HAH and Y-U models is conducted to assess their predictive capabilities in anisotropic hardening. Both models demonstrate superior performance in predicting the flow stress response under loading path change for AA6111-T4 and DP780 sheets. However, during cyclic loadings of Ti64 sheets, the Y-U model generates inaccurate flow stress predictions. It is found that the size of the initial yield surface in the Y-U model is influential factor to predict anisotropic hardening behaviour.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Myoung-Gyu Lee appreciates support from the NRF of the Korean government (Grant No. 2022R1A2C2009315), and MOTIE of Korea (Project No.1415185590, 20022438). Jinwoo Lee is grateful for the support of the “Regional Innovation Strategy” (RIS) through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (MOE) (2021RIS-003).

Notes on contributors

Jinwoo Lee

Jinwoo Lee is an Assistant Professor in the School of Mechanical Engineering, University of Ulsan, Republic of Korea.

Hyung-Rim Lee

Hyung-Rim Lee is a post-doc fellow in the Department of Materials Science and Engineering, Seoul National University, Republic of Korea.

Myoung-Gyu Lee

Myoung-Gyu Lee is a Professor in the Department of Materials Science and Engineering, Seoul National University, Republic of Korea.

Appendices

Appendix A: Calculation of the special tensor ${\hat{\mathbf{h}}}^{\mathbf{s}}$ for various loading conditions

The special tensor ${\hat{\mathbf{h}}}^{\mathbf{s}}$ in Equation (1) can be calculated under different loading conditions. The deviatoric stress s and special tensor ${\hat{\mathbf{h}}}^{\mathbf{s}}$ when the first loading is the uniaxial tension along the rolling direction (RD) are as follows: (A1) $\mathbf{s}=\left[\begin{array}{ccc}\frac{2{\sigma }_t}{3}& 0& 0\\ 0& -\frac{{\sigma }_t}{3}& 0\\ 0& 0& -\frac{{\sigma }_t}{3}\end{array}\right]\mathrm{and}{\hat{\mathbf{h}}}^{\mathbf{s}}=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& -\frac{1}{4}& 0\\ 0& 0& -\frac{1}{4}\end{array}\right]$(A1) where ${\sigma }_t$ is the magnitude of the applied stress.

The deviatoric stress s and corresponding tensor ${\hat{\mathbf{h}}}^{\mathbf{s}}$ for the various loading conditions are as follows: Case (1) uniaxial compression in RD; Case (2) uniaxial tension in TD. (A2) $\mathrm{Case}1:\mathbf{s}=\left[\begin{array}{ccc}-\frac{2{\sigma }_t}{3}& 0& 0\\ 0& \frac{{\sigma }_t}{3}& 0\\ 0& 0& \frac{{\sigma }_t}{3}\end{array}\right]\mathrm{and}{\hat{\mathbf{h}}}^{\mathbf{s}}=\left[\begin{array}{ccc}-\frac{1}{2}& 0& 0\\ 0& \frac{1}{4}& 0\\ 0& 0& \frac{1}{4}\end{array}\right]$(A2) (A3) $\mathrm{Case}2:\mathbf{s}=\left[\begin{array}{ccc}-\frac{{\sigma }_t}{3}& 0& 0\\ 0& \frac{2{\sigma }_t}{3}& 0\\ 0& 0& -\frac{{\sigma }_t}{3}\end{array}\right]\mathrm{and}{\hat{\mathbf{h}}}^{\mathbf{s}}=\left[\begin{array}{ccc}-\frac{1}{4}& 0& 0\\ 0& \frac{1}{2}& 0\\ 0& 0& -\frac{1}{4}\end{array}\right]$(A3)

Appendix B: Anisotropic plastic yielding model − yld2000-2d

The effective stress in Yld2000-2d model [29] is formulated as. (B1) $\varphi \left(\sigma \right)={\left({\displaystyle \frac{{\xi }^{\mathrm{\prime }}+{\xi }^{\mathrm{\prime \prime }}}{2}}\right)}^{1/a}={\overline{\sigma }}_{\mathrm{e}},$(B1) (B2) ${\xi }^{\mathrm{\prime }}=|{\mathrm{S}}_1^{\left(1\right)}-{\mathrm{S}}_2^{\left(1\right)}{|}^{\mathrm{a}}\mathrm{and}{\xi }^{\mathrm{\prime \prime }}=|{2\mathrm{S}}_2^{\left(2\right)}+{\mathrm{S}}_1^{\left(2\right)}{|}^{\mathrm{a}}+|{2\mathrm{S}}_1^{\left(2\right)}+{\mathrm{S}}_2^{\left(2\right)}{|}^{\mathrm{a}},$(B2) where ‘a’ represents an exponent of the yield function, recommended as 8 for FCC metals and 6 for BCC metals. ${\overline{\sigma }}_{\mathrm{e}}$ is the effective stress, ${\mathrm{S}}_{1,2}^{\left(\mathrm{i}\right)}$ (i = 1, 2) represent the principal stresses calculated from the tensors S⁽ⁱ⁾ (i = 1, 2), and the tensors are obtained with the two linear transformations from the Cauchy stress σ defined as. (B3) ${\mathbf{S}}^{\left(1,2\right)}={\mathbf{L}}^{\left(1,2\right)}\cdot \sigma ,$(B3) (B4) $\left[\begin{array}{c}{\mathrm{S}}_{\mathrm{xx}}^{\left(\mathrm{i}\right)}\\ {\mathrm{S}}_{\mathrm{yy}}^{\left(\mathrm{i}\right)}\\ {\mathrm{S}}_{\mathrm{xy}}^{\left(\mathrm{i}\right)}\end{array}\right]=\left[\begin{array}{ccc}{\mathrm{L}}_{11}^{\left(\mathrm{i}\right)}& {\mathrm{L}}_{12}^{\left(\mathrm{i}\right)}& 0\\ {\mathrm{L}}_{21}^{\left(\mathrm{i}\right)}& {\mathrm{L}}_{22}^{\left(\mathrm{i}\right)}& 0\\ 0& 0& {\mathrm{L}}_{66}^{\left(\mathrm{i}\right)}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\mathrm{xx}}\\ {\sigma }_{\mathrm{yy}}\\ {\sigma }_{\mathrm{xy}}\end{array}\right]\mathrm{with}\mathrm{i}=1,2$(B4)

(B5) $\left[\begin{array}{c}{\mathrm{L}}_{11}^{\left(1\right)}\\ {\mathrm{L}}_{12}^{\left(1\right)}\\ {\mathrm{L}}_{21}^{\left(1\right)}\\ {\mathrm{L}}_{22}^{\left(1\right)}\\ {\mathrm{L}}_{66}^{\left(1\right)}\end{array}\right]=\left[\begin{array}{ccc}2/3& 0& 0\\ -1/3& 0& 0\\ 0& -1/3& 0\\ 0& 2/3& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\\ {\alpha }_7\end{array}\right]\mathrm{and}\left[\begin{array}{c}{\mathrm{L}}_{11}^{\left(2\right)}\\ {\mathrm{L}}_{12}^{\left(2\right)}\\ {\mathrm{L}}_{21}^{\left(2\right)}\\ {\mathrm{L}}_{22}^{\left(2\right)}\\ {\mathrm{L}}_{66}^{\left(2\right)}\end{array}\right]={\displaystyle \frac{1}{9}\left[\begin{array}{cccc}-2& 2& 8& -2\\ 1& -4& -4& 4\\ 4& -4& -4& 1\\ -2& 8& 2& -2\\ 0& 0& 0& 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 9\end{array}\right]\left[\begin{array}{c}{\alpha }_3\\ {\alpha }_4\\ {\alpha }_5\\ {\alpha }_6\\ {\alpha }_8\end{array}\right].}$(B5)