On Phenomenological Failure Loci of Metals under Constant Stress States of Combined Tension and Shear: Issues of Coaxiality and Non-Uniqueness
Abstract
:1. Introduction
2. Review of the Methodology for Failure Characterization using Stress Invariants and the Equivalent Stress or Strain
2.1. Characterization of the Stress State
2.2. Phenomenological Descriptions of the Failure Locus
2.3. MMC Failure Criterion and Assumptions of Proportional Loading
3. Proportional Loading: Coaxial and Non-Coaxial Deformation
3.1. Definition of Proportional Coaxial Loading and Equivalent Strain
- Monotonic loading is applied in the same direction(s) for the duration of the test.
- The components of the stress and cumulative strain tensors can be expressed as constant ratios of one another. Consequently, the equivalent plastic strain is proportional to the principal plastic strain.
- The axes of the principal stress and the cumulative principal strains are aligned (coaxial). The incremental principal strain and the principal stresses are always aligned.
3.2. Shear Loading: Equivalent Stress States with Different Work-Conjugate Equivalent Strains
3.3. Mechanics of Combined Plane Strain Tension and Simple Shear in Plane Stress
3.4. Application of Non-Coaxial Equivalent Strain to the Tension–Torsion Results of Haltom et al. (2013) and Scales et al. (2016) for AA6061-T6
4. Dependence of Failure Loci on the Characterization Tests
4.1. Construction of a Coaxial Failure Locus Based Upon the Major Principal Strain
4.2. Generation of Failure Loci Based Upon the Equivalent Plastic Strain
- 1
- Coaxial path from pure shear to equal biaxial tension;
- 2
- Conventional path from simple shear to uniaxial tension and then equal biaxial tension;
- 3
- Tension–torsion path from simple shear to plane strain tension and then equal biaxial tension;
- 4
- Biaxial path from simple shear to equal biaxial tension; and
- 5
- Shear path from pure shear to simple shear where the stress triaxiality remains zero.
4.3. Characterization of the Plane Stress State for Coaxial and Non-Coaxial Proportional Loading
5. Onset of Diffuse Necking in Combined Tensile and Shear Stress States
6. Discussion and Implications on Phenomenological Fracture Characterization
7. Conclusions
- Stress states composed of a simple shear and superimposed tension, such as constant tension–torsion and butterfly tests, provide a constant stress state in terms of the stress invariants but are not strictly proportional. A constant stress state where the equivalent strain is not proportional to the cumulative principal strains due to the presence of a simple shear component has been denoted as non-coaxial. The presence of normal tensile stress with pure shear (no spin) is coaxial and proportional.
- It is recommended to differentiate between pure and simple shear since only pure shear is proportional and coaxial, which can be important for fracture since the microstructure will evolve differently. For plastic yielding, this distinction may not be as important.
- The failure loci in terms of the equivalent strain and stress triaxiality may depend upon the choice of characterization tests and if non-coaxial or coaxial test data was used in the calibration.
- General loading conditions for failure characterization cannot be uniquely defined by the stress triaxiality and Lode parameters. Thus, the current MMC-type framework may only be unique for coaxial proportional stress states, which appears to exclude the use of simple shear and non-coaxial characterization tests. This is an inconsistency with the fracture characterization tests used to calibrate the models.
- The analytical solution of Hillier [47] for the onset of a tensile instability in a constant tension–torsion test predicts that the onset of localization is delayed or suppressed in combined shear and tensile stress states in comparison with a tensile stress state with the same triaxiality. This is in general agreement with the observations of Scales et al. [26] where the failure strains in CTT tests were significantly higher than those from in-plane tests for the same alloy and no cusp is observed at the triaxiality for uniaxial tension.
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Logarithmic (true) strain tensor | |
Principal strains in descending order | |
Ratio of the incremental normal strain components | |
Ratio of shear stress to normal stress in the applied stress direction | |
Angle of the principal strain directions | |
Work-conjugate Equivalent plastic strain | |
Coaxial equivalent strain or Effective strain computed using cumulative strains | |
Equivalent plastic strain at failure | |
Maximum principal strain at failure | |
True (Cauchy) stress tensor | |
Ratio of the normal stress components | |
Ratio of shear stress to normal stress in the applied stress direction | |
, | Angles to define the normalized shear and normal stress in polar coordinates |
Parameter for the severity of shear loading under plane stress | |
Principal stresses in descending order | |
Maximum principal stress at failure | |
Hydrostatic stress | |
Von mises equivalent stress | |
First invariant of the stress tensor | |
Three invariants of the deviatoric stress tensor | |
Lode parameter: General parameter as there are multiple definitions | |
Lode parameter computed from the deviatoric stress invariants | |
Stress triaxiality | |
Lode parameter computed by normalizing the Lode angle | |
Angle of principal stress directions in plane stress | |
Maximum shear stress | |
Maximum shear stress at fracture |
Flow stress | |
Parameter in the Mohr-Coulomb yield function | |
Parameters in failure locus of Bai and Wierzbicki (2010) | |
Damage parameter | |
Strength coefficient and hardening exponent in the Holloman hardening model | |
Critical sub-tangent for onset of instability |
F | Deformation gradient tensor |
V | Left stretch tensor |
R | Rotation tensor |
B | Left Cauchy-Green stretch tensor |
L | Velocity gradient tensor |
W | Vorticity tensor |
D | Rate of deformation tensor or logarithmic strain rate |
Logarithmic spin tensor | |
Applied shear deformation in deformation gradient | |
Applied extensional stretch in deformation gradient | |
Ratio of the applied shear deformation to the extensional stretch | |
Ratio of the incremental applied shear deformation to the incremental stretch |
CTT | Constant Tension Torsion tests |
MMC | Modified Mohr Coulomb |
FEA | Finite-element analysis |
Appendix A: Mechanics of Coaxial and Non-Coaxial Strain Paths in Proportional Loading
Appendix A.1. Proportional Loading from Simple Shear to Pure Shear
Appendix A.2. Proportional Loading from Simple Shear to Uniaxial Tension
Appendix A.3. Proportional Loading from Simple Shear to Equal Biaxial Tension
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Experiment Data of Scales et al. (2016) | Finite Strain Combined Plane Strain & Simple Shear | |||
---|---|---|---|---|
Triaxiality | Coaxial von Mises Eq. Strain (DIC max.) | Major True Strain | Minor True Strain | Von Mises Work Conjugate Eq. Strain |
0.072 | 1.50 | 1.39 | −1.18 | 1.94 |
0.106 | 1.70 | 1.62 | −1.25 | 2.31 |
0.139 | 1.37 | 1.32 | −0.97 | 1.66 |
0.146 | 1.62 | 1.58 | −1.12 | 2.10 |
0.205 | 1.12 | 1.10 | −0.72 | 1.26 |
0.258 | 0.92 | 0.92 | −0.53 | 0.98 |
0.262 | 1.02 | 1.02 | −0.58 | 1.11 |
0.308 | 0.86 | 0.86 | −0.44 | 0.90 |
0.348 | 0.75 | 0.75 | −0.35 | 0.77 |
0.411 | 0.73 | 0.72 | −0.26 | 0.74 |
0.447 | 0.69 | 0.67 | −0.20 | 0.70 |
0.48 | 0.66 | 0.63 | −0.15 | 0.67 |
0.506 | 0.55 | 0.52 | −0.10 | 0.55 |
0.517 | 0.58 | 0.54 | −0.09 | 0.58 |
0.577 | 0.40 | 0.35 | 0.00 | 0.40 |
Loading Condition | Normal Strain Ratio | Normal Stress Ratio | Shear Stress Ratio | Shear Strain Ratio | Equivalent Strain Ratio |
Pure & Simple Shear | −1 | −1 | |||
Uniaxial Tension & Simple Shear | −0.5 | 0 | |||
Plane Strain Tension & Simple Shear | 0 | 1/2 | |||
Biaxial Tension & Simple Shear | 1 | 1 | |||
Coaxial Loading | 0 | 0 |
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Butcher, C.; Abedini, A. On Phenomenological Failure Loci of Metals under Constant Stress States of Combined Tension and Shear: Issues of Coaxiality and Non-Uniqueness. Metals 2019, 9, 1052. https://doi.org/10.3390/met9101052
Butcher C, Abedini A. On Phenomenological Failure Loci of Metals under Constant Stress States of Combined Tension and Shear: Issues of Coaxiality and Non-Uniqueness. Metals. 2019; 9(10):1052. https://doi.org/10.3390/met9101052
Chicago/Turabian StyleButcher, Cliff, and Armin Abedini. 2019. "On Phenomenological Failure Loci of Metals under Constant Stress States of Combined Tension and Shear: Issues of Coaxiality and Non-Uniqueness" Metals 9, no. 10: 1052. https://doi.org/10.3390/met9101052