# Parameter Estimation and Application of Anisotropic Yield Criteria for Cylindrical Aluminum Extrusions: Theoretical Developments and StereoDIC Measurements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background Theory for Yield Functions and Plastic Strain

## 3. Experimental Investigations and Parameters Determination

#### 3.1. Material Properties and Microstructure

#### 3.2. Anisotropic Yield Function and Hardening Parameters

_{o}

^{2}− r

_{i}

^{2}), with outer tube radius, r

_{o}, and inner tube radius, r

_{i}.

_{o}, and will be used to normalize results for both MB1 and MB2. As shown in Figure 3a, σ

_{o}= 287 Mpa for MB1 and Figure 4a, σ

_{o}= 342 Mpa for MB2). The effective stress, ($\overline{\sigma}$), from the LDD measurements is considered to be the “reference” stress state.

^{o}to relate the applied axial stress in the rotated specimen to the stresses in the X-Y-Z coordinate system, and (iii) Equation (5) to define an equation for g. Finally, to determine the parameters f and h, the data in Figure 3b and Figure 4b for the RDD specimens shows that there is circumferential symmetry in the longitudinally extruded rod material. Consistent with this observation, the authors assume that the yield stress due to shear are equal in both the X-Y plane and the Y-Z planes, so that f = h. Following this procedure, the six yield function parameters for the two 28.6 mm diameters extruded Al 6061 rods were determined, and the results are shown in Table 3 for both the MB1 and MB2 extruded aluminum bar materials.

## 4. Theoretical Prediction and Experiments for Simple Torsion Case

## 5. Discussion of Results

_{x}/σ

_{y}≈ 0.5, with the Barlat criteria predicting yielding for lower stresses (15% for MB1, 30% for MB2) than the Von Mises criteria. Thus, the Barlat yield criteria provides substantially improved accuracy in the prediction of yielding for those applications where such differences are truly important.

## 6. Summary and Conclusions

- Uniaxial tension and simple torsion experiments were performed on a series of specially machined tubular and dog-boned rectangular specimens extracted from two longitudinally extruded Al6061-T6 cylindrical bars to obtain the required material parameters for anisotropic yield criteria. Here, the yield function Yld91 developed by Barlat et al., with six measured material parameters, is used to model the anisotropic response of the extruded aluminum material and an isotropic yield criterion using the Von Mises criteria is employed to predict an isotropic response.
- Since the extrusion process is circumferentially symmetric for our longitudinal extrusions, yield stresses in all radial directions are expected to be similar, a condition that was confirmed from a series of radially oriented specimen experiments.
- Assuming isotropic strain hardening beyond yielding, the Von Mises isotropic and Barlat Yld91 parameters yield functions with power-law hardening and incremental theory of plasticity are used to develop and then implement a constitutive model for elastic-plastic material behavior and predict the response of the extruded MB1 and MB2 material. These constitutive models are implemented in a relatively simple numerical analysis platform to predict the stress-strain response for both materials undergoing torsional loading.
- Theoretical results indicate that results obtained using the anisotropic yield function are in excellent agreement with simple torsion experiments for both materials. Conversely, results obtained using an isotropic yield function underestimate, and then over-estimate, the stress-strain response of the MB1 and MB2 specimens, respectively.
- Direct comparison of the experimental and theoretical results indicates that both extruded Al6061-T6 materials are significantly anisotropic, with the longitudinal yield stress deviating from the radial yield behavior in the extruded materials by over 20%. Furthermore, for biaxial loading cases, the Barlat yield criteria provides improved accuracy in the prediction of yielding for those applications where isotropic yielding is not adequate.
- Predictions of the stress-strain response for the MB1 and MB2 tubular specimens using the Barlat Yld91 anisotropic yield criterion are in excellent agreement with experimental measurements, with differences less than 5% for both specimens, clearly demonstrating the importance of yielding anisotropy for accurate prediction of material response when undergoing complex manufacturing processes, including longitudinal extrusion.
- Results also show that the nominally similar extrusion processes for mother bars MB1 and MB2 are not the same, with substantial differences in the stress-strain behavior for the two materials being consistent with microstructural features that indicate the MB2 extrusion process was more severe.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Sample Design and Preparation

**Figure A1.**(

**a**) Tension and tension torsion test specimens’ orientations as excised from both the longitudinal and radial directions (LD and RD) of the mother bars MB1 and MB2. Geometries are (

**b**) LDT, (

**c**) LDD and (

**d**) RDD0, RDD45, RDD90 for radial specimens. (All units in mm).

#### Appendix A.2. Experimental Setup

**Figure A2.**Experimental setup for simple torsion loading of LDT specimens using Stereovision system and electromechanical tension-torsion load cell.

**Figure A3.**Experimental setup (

**a**) for uniaxial tensile loading of RDD0, RDD45, RDD90 specimens using microscope Stereovision and micro-tensile test system (

**b**) close view of micro-tensile test grip.

Parameter # | Vic-3D | Vic-3D Stereo-Microscope |
---|---|---|

Cameras and Lenses | Grasshopper3 GS3-U3-91S6M (8 bits, 3376 × 2704) Schneider XENOPLAN 1.9/35-0511 | 5 MP CMOS PointGrey camera (2448 × 2048) |

Lighting | White LED Lighting | LEO with a linear polarizing film |

Calibration | 14 × 10 dot grid, 5 mm dot size (H95-00-03), 70 stereo calibration image pairs | 15 × 15 dot grid, 0.28 mm dot size, 70 stereo calibration image pairs |

Lens distortion | 1st order radial distortion correction | 10 stereo distortion image pairs |

Subset size | 29 × 29 pixels^{2} | 35 × 35 pixels^{2} |

Step size | 9 pixels | 11 pixels |

Filter type | Center-weighted Gaussian filter | |

Shape function | Affine | |

Strain filter size | 5 × 5 | |

Strain measurement | Lagrangian large strain tensor definition for all strain components | |

Average speckle size | 0.25 mm | 0.013 mm |

#### Appendix A.3. Elastic Modulus, Poisson’s Ratio and Hardening Parameter for Extruded Rods

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**Figure 1.**Microstructure of (

**a**) Al6061-T6 MB1, and (

**b**) Al6061-T6 MB2 longitudinally extruded round bar specimens. Images obtained at 1000X using a Keyence microscope. The axes X and Y represent the radial and longitudinal directions, respectively.

**Figure 2.**Coordinate system used throughout the developments. The X-axis is an arbitrarily selected radial direction that approximately corresponds to the bisector between the pair of stereo-cameras used for surface strain measurements (see Appendix A). The Y-axis is in the longitudinal direction. The Z axis is another radial direction that is orthogonal to X and Y.

**Figure 3.**MB1 specimen results for (

**a**) true stress vs. true strain measurements for tension loading of the LDD, RDD0, RDD45, and RDD90 specimens and (

**b**) shear stress vs. shear strain for torsion of LDT specimens.

**Figure 4.**MB2 specimen results for (

**a**) true stress vs. true strain measurements for tension loading of the LDD, RDD0, RDD45, and RDD90 specimens and (

**b**) shear stress vs. shear strain for torsion of LDT specimens.

**Figure 5.**Predicted normalized yield surfaces for biaxial stress states with Barlat and Von Mises yield function for two Al 6061-T6 bar.

**Figure 6.**Flow chart theoretical prediction of total shear strain using isotropic and anisotropic yield criteria.

**Figure 7.**Torque vs. time data acquired from electro-mechanical system load cell during simple torsion experiments.

**Figure 8.**Comparison of experimental and theoretical shear stress vs. total shear strain response for Al6061-T6, MB1.

**Figure 9.**Comparison of experimental and theoretical shear stress vs. total shear strain response for Al6061-T6, MB2.

**Table 1.**Mechanical properties and chemical composition of Al6061-T6 MB1 and MB2 tubes given by the manufacturer (McMaster-Carr).

Rod Stock | Mechanical Properties | Chemical Composition | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Alloy | Dia (mm) | Ultimate Strength (MPa) | Yield Strength (MPa) | Elongation (%) | Si | Fe | Cu | Mn | Mg | Cr | Zn | Ti | |||

Min | Max | Min | Max | Min | Max | ||||||||||

6061-T6 MB1 | 28.575 | 317.2 | 327.5 | 286.1 | 299.4 | 16.5 | 18 | 0.71 | 0.28 | 0.33 | 0.05 | 0.89 | 0.05 | 0.02 | 0.02 |

6061-T6 MB2 | 28.575 | 341.3 | 375.8 | 319.3 | 355.8 | 15.8 | 19.5 | 0.76 | 0.37 | 0.33 | 0.11 | 0.90 | 0.11 | 0.06 | 0.03 |

Trial No | Mode | Load Cell | Strain Measurement Approach | Specimen | Material Al 6061-T6 | Number of Experiments |
---|---|---|---|---|---|---|

1 | Tension | MTS | Extensometer | LDD | MB1 | 2 |

MB2 | 2 | |||||

2 | Tension | Psylotech micro- tensile tester | VIC 3D with stereo microscope | RDD0 | MB1 | 2 |

MB2 | 2 | |||||

RDD45 | MB1 | 2 | ||||

MB2 | 2 | |||||

RDD90 | MB1 | 2 | ||||

MB2 | 2 | |||||

3 | Torsion | Electromechanical TestResources frame with torsion load cells | VIC 3D software with standard cameras | LDT | MB1 | 2 |

MB2 | 2 |

6061-T6 Al Tube | m | a | b | c | f | g | h |
---|---|---|---|---|---|---|---|

MB1 | 8 | 1.0000 | 1.1516 | 1.000 | 0.8750 | 1.0835 | 0.8750 |

MB2 | 8 | 1.0000 | 1.4452 | 1.000 | 1.2069 | 1.3059 | 1.2069 |

Material Properties | 6061-T6 Al Tube (MB1) | 6061-T6 Al Tube (MB2) |
---|---|---|

Modulus of Elasticity | 69 Gpa | 69 Gpa |

Poisson’s ratio | 0.33 | 0.33 |

Hardening Parameter, n | 13.51 | 16.47 |

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**MDPI and ACS Style**

Yasmeen, F.; Sutton, M.A.; Deng, X.; Ryan, M.; Reynolds, A.P.
Parameter Estimation and Application of Anisotropic Yield Criteria for Cylindrical Aluminum Extrusions: Theoretical Developments and StereoDIC Measurements. *Appl. Sci.* **2021**, *11*, 9701.
https://doi.org/10.3390/app11209701

**AMA Style**

Yasmeen F, Sutton MA, Deng X, Ryan M, Reynolds AP.
Parameter Estimation and Application of Anisotropic Yield Criteria for Cylindrical Aluminum Extrusions: Theoretical Developments and StereoDIC Measurements. *Applied Sciences*. 2021; 11(20):9701.
https://doi.org/10.3390/app11209701

**Chicago/Turabian Style**

Yasmeen, Farzana, Michael A. Sutton, Xiaomin Deng, Megan Ryan, and Anthony P. Reynolds.
2021. "Parameter Estimation and Application of Anisotropic Yield Criteria for Cylindrical Aluminum Extrusions: Theoretical Developments and StereoDIC Measurements" *Applied Sciences* 11, no. 20: 9701.
https://doi.org/10.3390/app11209701