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
4. Theoretical Prediction and Experiments for Simple Torsion Case
5. Discussion of Results
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
Appendix A.2. Experimental Setup
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 pixels2 | 35 × 35 pixels2 |
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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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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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
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 StyleYasmeen, 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