Numerical Investigations of Shape Memory Alloy Fatigue
Abstract
:1. Introduction
2. Constitutive Model
3. Numerical Simulations
3.1. Axial Loading Test
3.2. Torsional Loading Test
3.3. Thermal Loading Test
3.4. Shape Memory Effect
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
Austenite phase | |
Parameters to control the evolution of functional damage | |
Parameters to control the evolution of structural damage | |
Equivalent critical damage | |
State at which reaches a critical value | |
Internal variable associated with structural damage | |
Internal variable associated with functional damage | |
Elastic moduli for austenite and martensite phases | |
Elastic tensor | |
Elastic tensor of austenite and martensite phases | |
Yield surface | |
Kinematic hardening modulus | |
Kinematic hardening modulus of austenite and martensite phases | |
Plastic modulus | |
Plastic modulus of austenite and martensite phases | |
Parameters related to the critical stress due to positive (+) and negative (−) detwinned martensite phase | |
Parameters related to the critical stress due to the austenitic phase | |
Twinned martensite | |
Positive (+) and negative (−) detwinned martensite | |
Parameters associated with the TRIP effect | |
The reference value of the parameter | |
Parameters related to the saturation effect | |
Number of cycles until failure | |
General formulation of the parameters , , , , e | |
Auxiliary variables | |
Second-order tensor defined from the loading history | |
The maximum value of the mechanical loading | |
Temperature | |
Reference temperature in a stress-free state | |
Temperature above which the austenitic phase is stable | |
Reference temperature for the determination of the yield stress for high temperatures | |
Temperature below which the martensitic phase is stable | |
Temperature below which the TRIP strain does not occur | |
Parameter that controls the height of the stress–strain hysteresis loop | |
Normal and shear components of the tensor | |
Fourth-order tensor that controls the stress–strain hysteresis loop width | |
Volume fraction associated with the martensitic variants, , and the austenite phase, A | |
Volume fraction associated with the starting of the phase transformation process | |
Plastic multiplier | |
Equivalent strain field | |
Equivalent strain field to the initial state | |
Equivalent stress field | |
Kronecker delta | |
Elastic strain tensor | |
Deviatoric strain tensor | |
Plastic strain tensor | |
TRIP strain tensor | |
Parameters employed to represent the martensite and austenite structural fatigue strength | |
Parameters associated with the internal dissipation | |
Parameters associated with the internal dissipation | |
Parameters associated with the internal dissipation | |
Parameter that defines the coupling between the phase transformation and isotropic hardening | |
Parameter that defines the coupling between the phase transformation and kinematic hardening | |
Parameter related to the tensor | |
Sub-differential of the convex set | |
Sub-differential of the convex set | |
Temperature functions that define the critical stress value for the phase transformation | |
Phase transformation values for austenite and martensite phases | |
Internal variable associated with isotropic hardening | |
Saturation variables for each phase | |
Yield stress | |
Yield stress of the martensitic phase | |
Yield stress of the austenitic phase at temperatures and | |
Stress tensor | |
Deviatoric stress tensor | |
Internal variable associated with kinematic hardening | |
Pseudo-potential of dissipation | |
Helmholtz free energy density | |
Auxiliary variable | |
Thermal expansion coefficients for austenite and martensite phases | |
Tensor related to the thermal expansion coefficients | |
Tensor related to the thermal expansion coefficients of the austenite and martensite |
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EA (GPa) | EM (GPa) | ΩA (MPa/K) | ΩM (MPa/K) | (MPa) | (MPa) |
58.0 | 36.0 | 0.74 | 0.17 | 0.028 | 100.0 |
(MPa) | (MPa) | (MPa) | (MPa) | ||
6.00 | 5.00 | 0.60 | 105.50 | 0.10 | 0.80 |
TM (K) | TA (K) | (K) | (GPa) | ||
0.10 | 0.80 | 266.0 | 297.0 | 300.0 | 0.30 |
(GPa) | (GPa) | KA (GPa) | KM (GPa) | HA (GPa) | HM (GPa) |
1.15 | 0.60 | 1.4 | 0.4 | 4.0 | 1.1 |
TF (K) | |||||
−0.01 | −0.01 | 423.0 | 0.85 | 0.05 | 0.85 |
n | |||||
0.05 | 297.0 | 1.0 × 10−4 | 0.1 | 0.4 | 0.7 |
5.0 × 10−6 | 5.5 × 10−12 | 1.58 | 5.0 × 10−12 | 8.9 × 10−12 | 1.58 |
3.38 | 0.87 | 0.16 |
Frequency (Hz) | Variation (%) | |||
---|---|---|---|---|
0.25 | 750 | 2308 | 2284 | 1.0 |
EA (GPa) | EM (GPa) | ΩA (MPa/K) | ΩM (MPa/K) | (MPa) | (MPa) |
45.0 | 29.0 | 0.74 | 0.17 | 0.014 | 20 |
(MPa) | (MPa) | (MPa) | (MPa) | ||
1.00 | 15.00 | 0.10 | 26.50 | 1.10 | 0.70 |
TM (K) | TA (K) | (K) | (GPa) | ||
1.20 | 0.60 | 223.0 | 278.0 | 300.0 | 0.5 |
(GPa) | (GPa) | KA (GPa) | KM (GPa) | HA (GPa) | HM (GPa) |
1.5 | 1.0 | 1.4 | 0.4 | 4.0 | 1.1 |
TF (K) | |||||
−0.01 | −0.01 | 423.0 | 0.12 | 0.05 | 0.12 |
n | |||||
0.05 | 290.0 | 0.02 | 0.12 | ||
1.58 | 1.58 | ||||
1.20 | 0.10 | 0.53 |
Frequency (Hz) | Variation (%) | |||
---|---|---|---|---|
0.1 | 260 | 7769 | 7755 | 0.2 |
0.1 | 346 | 10,549 | 10,263 | 2.7 |
EA (GPa) | EM (GPa) | ΩA (MPa/K) | ΩM (MPa/K) | (MPa) | (MPa) |
22.0 | 22.0 | 0.74 | 0.17 | 0.037 | 110.0 |
(MPa) | (MPa) | (MPa) | (MPa) | ||
1.00 | 10.00 | 10.20 | 115.00 | 4.00 | 1.50 |
TM (K) | TA (K) | (K) | (GPa) | ||
4.00 | 1.50 | 310.0 | 395.0 | 303.0 | 0.50 |
(GPa) | (GPa) | KA (GPa) | KM (GPa) | HA (GPa) | HM (GPa) |
1.50 | 1.00 | 1.4 | 0.4 | 4.0 | 1.1 |
TF (K) | |||||
−0.01 | −0.01 | 480.0 | 0.34 | 0.03 | 0.34 |
n | |||||
0.03 | 310.0 | 0.046 | 0.18 | 0.2 | 2.0 |
5.0 × 10−6 | 8.0 × 10−12 | 1.5 | 3.0 × 10−6 | 1.0 × 10−10 | 1.2 |
1.20 | 0.10 | 0.53 |
EA (GPa) | EM (GPa) | ΩA (MPa/K) | ΩM (MPa/K) | (MPa) | (MPa) |
50.0 | 25.0 | 0.74 | 0.17 | 0.042 | 150.0 |
(MPa) | (MPa) | (MPa) | (MPa) | ||
0.50 | 1.00 | 0.50 | 150.00 | 105.00 | 10.00 |
TM (K) | TA (K) | (K) | (GPa) | ||
105.00 | 10.00 | 296.0 | 322.0 | 295.0 | 1.00 |
(GPa) | (GPa) | KA (GPa) | KM (GPa) | HA (GPa) | HM (GPa) |
1.50 | 1.30 | 1.4 | 0.4 | 4.0 | 1.1 |
TF (K) | |||||
−0.01 | −0.01 | 480.0 | 8.1 | 8.1 | 8.1 |
n | |||||
8.1 | 290.0 | 0.90 | 0.01 | 0.01 | 2.0 |
10.0 × 10−11 | 7.0 × 10−11 | 1.5 | 10.0 × 10−11 | 10.0 × 10−11 | 1.5 |
11.00 | 1.00 | 0.75 |
Variation (%) | ||||
---|---|---|---|---|
1000 | 0 | 74 | 73 | 1.3 |
800 | 0 | 196 | 129 | 34.2 |
600 | 0 | 275 | 246 | 10.5 |
400 | 0 | 518 | 487 | 6.0 |
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Dornelas, V.M.; Oliveira, S.A.; Savi, M.A.; Pacheco, P.M.C.L. Numerical Investigations of Shape Memory Alloy Fatigue. Metals 2021, 11, 1558. https://doi.org/10.3390/met11101558
Dornelas VM, Oliveira SA, Savi MA, Pacheco PMCL. Numerical Investigations of Shape Memory Alloy Fatigue. Metals. 2021; 11(10):1558. https://doi.org/10.3390/met11101558
Chicago/Turabian StyleDornelas, Vanderson M., Sergio A. Oliveira, Marcelo A. Savi, and Pedro Manuel Calas Lopes Pacheco. 2021. "Numerical Investigations of Shape Memory Alloy Fatigue" Metals 11, no. 10: 1558. https://doi.org/10.3390/met11101558