# Low-Cycle Fatigue Behavior of Hot-Bent Basal Textured AZ31B Wrought Magnesium Alloy

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## Abstract

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## 1. Introduction

## 2. Material and Methods

#### 2.1. Hot-Bending Process

#### 2.2. Microstructural Investigations

#### 2.3. Uniaxial Quasi-Static and Cyclic Tests at Room Temperature

## 3. Numerical and Experimental Verification of the Novel, Hot-Bent Uniaxial Specimen

## 4. Results and Discussion

#### 4.1. Microstructural Analyses

#### 4.2. Quasi-Static Tests

#### 4.3. Low-Cycle Fatigue Tests

#### 4.3.1. The Concept of Highly Strained Volume

#### 4.3.2. Fatigue Modeling

`scipy.optimize.curve_fit`from SciPy© is used to fit the experimental data to the double power function

## 5. Conclusions

- Numerical and experimental results of the hot-bent specimen show that a homogeneous and uniaxial stress state with a tolerable low superimposed bending stress of 5% in the gauge area can be realized.
- The hot-bending process leads to changes in the microstructure with a smaller average grain size and deformed grain boundaries. Furthermore, it is observed that the c-axes become inclined from the normal direction (ND) towards the transverse direction (TD), which is more pronounced on the compression layer compared with the tension layer due to the formation of $\left\{10\overline{1}2\right\}$ tension twins. Hence, the Schmid factor for basal slip in RD and TD increases.
- The uniaxial tension and compression tests reveal that the anisotropic and asymmetric elasto-plastic material behavior persists after the hot-bending process. In addition, lower Young’s moduli are observed for the hot-bent material compared with the as-received material. Compression tests show that even in the hot-bent specimens, the massive formation of tension twins causes macroscopic bands of twinned grains (BTGs). Within the BTGs, the compressive strain $|{\epsilon}_{11,\mathrm{BTG}}|$ is considerably higher compared with the areas outside the BTGs.
- Finally, the study proves that the recently proposed concept of highly strained volume (CH$\epsilon $V) can accurately estimate the lifetime of as-received and hot-bent AZ31B magnesium alloy, even by combining the two materials in one model. This allows the CH$\epsilon $V to be applied to geometrically complex hot-bent components in order to estimate their lifetime.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BTG | Band of twinned grains |

CH$\epsilon $V | Concept of highly strained volume |

DIC | Digital image correlation |

EBSD | Electron backscatter diffraction |

FEM | Finite Element Method |

HSR | Highly strained region |

HSRA | Highly strained region with high strain amplitudes |

LLL | Lower load level |

Mg | Magnesium |

ND | Normal direction |

PTFE | Polytetrafluoroethylene |

RD | Rolling direction |

SEM | Scanning electron microscope |

TD | Transverse direction |

ULL | Upper load level |

${V}_{\epsilon}$ | Highly strained volume |

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**Figure 1.**Manufacturing process of the hot-bent uniaxial specimen: (

**a**) taper of the planar twin-roll cast magnesium sheets; (

**b**) hot-bent process; (

**c**) assembly for mechanical testing with the clamping jaws, the anti-buckling device, the extensometer, and the hot-bent uniaxial specimen.

**Figure 2.**Hot-bent uniaxial specimen: (

**a**) Geometry details of the hot-bent uniaxial specimen and (

**b**) the test setup for quasi-static and cyclic tests.

**Figure 3.**Verification of the hot-bent uniaxial specimen: (

**a**) normal stress field ${\sigma}_{11}({x}_{1},{x}_{2},{x}_{3})$ of a FEM analysis of a quarter model of the hot-bent specimen; (

**b**) plot of the stress ${\sigma}_{11}$ over the path ${L}_{{x}_{1}}$; (

**c**) plot of the stress ${\sigma}_{11}$ over the path ${L}_{{x}_{3}}$; (

**d**) strain field ${\epsilon}_{11}({x}_{1},{x}_{2},{x}_{3})$, measured with DIC, of the hot-bent specimen with marked gauge area at an extensometer strain ${\epsilon}_{11,\mathrm{ext}}=0.45\%$.

**Figure 4.**EBSD orientation maps with a view in the ND: (

**a**) the as-received material with a scanning step of $0.085$ $\mathsf{\mu}$$\mathrm{m}$; (

**b**) the hot-bent material of the tension layer; (

**c**) the compression layer. The scanning step of the EBSD maps of the hot-bent material from the tension layer and the compression layer is $0.07$ $\mathsf{\mu}$$\mathrm{m}$ and $0.045$ $\mathsf{\mu}$$\mathrm{m}$, respectively.

**Figure 5.**(0002) Pole figures: (

**a**) from the as-received material; (

**b**) the hot-bent material of the tension layer; (

**c**) the hot-bent material of the compression layer. The data for the pole figure is obtained by an EBSD measurement with a scan size of $540\mathsf{\mu}\mathrm{m}$ × $225\mathsf{\mu}\mathrm{m}$ (TD × RD).

**Figure 6.**EBSD measurement to detect $\left\{10\overline{1}2\right\}$ tension twins: (

**a**) EBSD orientation map with view in ND from the hot-bent material of the compression layer with a scanning step of $0.056$ $\mathsf{\mu}$$\mathrm{m}$; (

**b**) band contrast with detected $\left\{10\overline{1}2\right\}$ tension twins (misorientation angle (86.3 ± 5)°, marked with blue color.

**Figure 7.**Schmid factor maps for basal slip (

**a**) from the as-received material with respect to a possible load in RD and (

**b**) TD; (

**c**) of the hot-bent material from the tension layer with respect to a possible load in RD and (

**d**) TD; (

**e**) of the hot-bent material from the compression layer with respect to a possible load in RD and (

**f**) TD.

**Figure 8.**Engineering stress–strain curves from strain-controlled uniaxial tests with the as-received and the hot-bent material in RD and TD: (

**a**) engineering stress strain curves of the uniaxial tension tests; (

**b**) engineering stress strain curves of the uniaxial compression tests.

**Figure 9.**Strain field ${\epsilon}_{11}({x}_{1},{x}_{2})$ of a hot-bent uniaxial specimen with an extensometer strain ${\epsilon}_{11,\mathrm{ext}}=-0.45\%$ measured with DIC. The BTG, the anti-buckling device, and the extensometer are marked. The length scale starts at the beginning of the transition radius.

**Figure 10.**3D and 2D ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$-${V}_{\epsilon}$-${N}_{\mathrm{f}}$ fatigue diagrams including the results from the fatigue tests of the hot-bent specimen, the regression surface (green color), and the two 2-times error planes (red color). For the fatigue diagrams, the mean effective strain amplitude ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$ is used.

**Figure 11.**3D and 2D ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$-${V}_{\epsilon}$-${N}_{\mathrm{f}}$ fatigue diagrams including the results from the fatigue tests of the as-received specimen, the regression surface (green color), and the two 2-times error planes (red color). For the fatigue diagrams, the mean effective strain amplitude ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$ is used.

**Figure 12.**3D and 2D ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$-${V}_{\epsilon}$-${N}_{\mathrm{f}}$ fatigue diagrams including the results from the fatigue tests of the hot-bent and as-received specimens, the regression surface (green color), and the two 2-times error planes (red color). For the fatigue diagrams, the mean effective strain amplitude ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$ is used.

Mg | Al | Zn | Mn | Cu | Si | Fe | Ni | Ca | Other Impurities |
---|---|---|---|---|---|---|---|---|---|

balance | 2.75 | 1.08 | 0.368 | 0.00262 | 0.0187 | 0.00282 | 0.00038 | 0.00041 | <0.004 |

**Table 2.**Results of the strain-controlled tension test, in which the strain at the outer radius ${\epsilon}_{11,\mathrm{outer}}$ and inner radius ${\epsilon}_{11,\mathrm{inner}}$ in the gauge area are measured with strain gauges to determine the superimposed bending stress ${\sigma}_{11,\mathrm{B}}$.

${\mathit{\epsilon}}_{11,\mathbf{ext}}$ (%) | ${\mathit{\sigma}}_{11}$ (MPa) | ${\mathit{\epsilon}}_{11,\mathbf{outer}}$ (%) | ${\mathit{\epsilon}}_{11,\mathbf{inner}}$ (%) | ${\mathit{\sigma}}_{11,\mathbf{B}}$ (MPa) |
---|---|---|---|---|

0.100 | 41.1 | 0.100 | 0.0900 | 2.05 |

0.200 | 82.1 | 0.210 | 0.180 | 6.16 |

0.300 | 123 | 0.320 | 0.290 | 6.16 |

**Table 3.**Results of the yield stresses ${\sigma}_{\mathrm{Y}}$ and the Young’s moduli E from the uniaxial tension and compression tests. These tests are carried out with the as-received and the hot-bent material in RD and TD.

Uniaxial Tension Tests | Uniaxial Compression Tests | |||
---|---|---|---|---|

ID | ${\mathbf{\sigma}}_{\mathbf{Y}}$ (MPa) | $\mathit{E}$ (MPa) | $|{\mathbf{\sigma}}_{\mathbf{Y}}|$ (MPa) | $\mathit{E}$ (MPa) |

hot-bent, RD | 177.1 | 41,067 | 128.3 | 40,862 |

as-received, RD | 178.1 | 42,673 | 122.8 | 43,880 |

hot-bent, TD | 142.5 | 38,945 | 124.9 | 38,422 |

as-received, TD | 150.6 | 42,034 | 116.4 | 42,879 |

**Table 4.**Applied extensometer strain amplitude ${\epsilon}_{\mathrm{a},\mathrm{ext}}$, extensometer strain ratio ${R}_{\epsilon}$, test frequency f, highly strained volume ${V}_{\epsilon}$, mean effective strain amplitude ${\tilde{\overline{\epsilon}}}_{11,\mathrm{a}}$, and the numbers of cycles to failure ${N}_{\mathrm{f}}$ of the strain-controlled cyclic tests. The tests are performed on the hot-bent specimens.

Specimen ID | ${\mathit{\epsilon}}_{\mathbf{a},\mathbf{ext}}$ (%) | ${\mathit{R}}_{\mathit{\epsilon}}\phantom{\rule{3.33333pt}{0ex}}$(−) | f (Hz) | ${\mathit{V}}_{\mathit{\epsilon}}$ (mm^{3}) | ${\tilde{\overline{\mathit{\epsilon}}}}_{11,\mathbf{a}}$ (%) | ${\mathit{N}}_{\mathbf{f}}$ (−) |
---|---|---|---|---|---|---|

UG12-060 | 0.35 | −1 | 1 | 132 | 0.344 | 14,377 |

UG12-061 | 0.35 | −1 | 1 | 523 | 0.334 | 15,075 |

UG12-062 | 0.45 | −1 | 0.5 | 43.0 | 0.469 | 2890 |

UG12-063 | 0.45 | −1 | 0.5 | 23.3 | 0.470 | 4619 |

UG12-065 | 0.55 | −1 | 0.2 | 24.4 | 0.595 | 807 |

UG12-071 | 0.55 | −1 | 0.2 | 81.7 | 0.532 | 1645 |

UG12-067 | 0.8 | −1 | 0.1 | 46.5 | 0.885 | 435 |

UG12-072 | 0.8 | −1 | 0.1 | 16.0 | 0.822 | 535 |

UG12-068 | 1.2 | −1 | 0.1 | 32.2 | 1.21 | 163 |

UG12-070 | 1.2 | −1 | 0.1 | 4.99 | 1.15 | 224 |

**Table 5.**Determined material values ${C}_{1}$, ${d}_{1}$, and ${d}_{2}$ for the hot-bent material, the as-received material, and the combination of the hot-bent and as-received material. In addition, the coefficient of determination ${r}^{2}$ is given to estimate the quality of the regression.

Specimens | ${\mathit{C}}_{1}$ (%) | ${\mathit{d}}_{1}$ (−) | ${\mathit{d}}_{2}$ (−) | ${\mathit{r}}^{2}$ (−) |
---|---|---|---|---|

hot-bent | 6.13 | −0.310 | −0.0189 | 0.968 |

as-received | 6.30 | −0.279 | −0.0460 | 0.942 |

hot-bent & as-received | 4.94 | −0.266 | −0.0231 | 0.944 |

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**MDPI and ACS Style**

Nischler, A.; Denk, J.; Saage, H.; Klaus, H.; Huber, O.
Low-Cycle Fatigue Behavior of Hot-Bent Basal Textured AZ31B Wrought Magnesium Alloy. *Metals* **2021**, *11*, 1004.
https://doi.org/10.3390/met11071004

**AMA Style**

Nischler A, Denk J, Saage H, Klaus H, Huber O.
Low-Cycle Fatigue Behavior of Hot-Bent Basal Textured AZ31B Wrought Magnesium Alloy. *Metals*. 2021; 11(7):1004.
https://doi.org/10.3390/met11071004

**Chicago/Turabian Style**

Nischler, Anton, Josef Denk, Holger Saage, Hubert Klaus, and Otto Huber.
2021. "Low-Cycle Fatigue Behavior of Hot-Bent Basal Textured AZ31B Wrought Magnesium Alloy" *Metals* 11, no. 7: 1004.
https://doi.org/10.3390/met11071004