Application of an Artificial Neural Network to Develop Fracture Toughness Predictor of Ferritic Steels Based on Tensile Test Results
Abstract
:1. Introduction
2. Materials and Methods
2.1. Selection of Machine Learning Model
2.2. Overview of Multilayer Perceptron in an Artificial Neural Network
2.3. Goodness Valuation of Constructed Learning Model
2.4. Dataset
2.5. Fracture Toughness Prediction by the Constructed Learning Model
3. Discussion
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
B | test specimen thickness |
J | J-integral |
KJc | fracture toughness |
T | temperature (°C) |
T0 | ASTM E1921 MC reference temperature (°C) for a 25 mm thick specimen with a fracture toughness of 100 MPa·m1/2 |
W | specimen width |
σYS, σB | yield (0.2% proof) and tensile strength |
σ0ZA | yield stress at the temperature T (°C) described by the Zerilli equation (i.e., Equation (11)) |
R2 | coefficient of determination |
Xi | input value of MLP |
aj | activation unit of MLP |
n | number of input value |
k | number of activation unit |
f(X) | output value of MLP |
connection weight between input value Xi and activation unit aj | |
activation function | |
connection weight between activation unit aj and output value f(X) | |
Y | teaching data |
E | loss function |
α | regularization strength of L2 norm term |
w(t) | connection weight at timestep t in Adam |
m(t) | exponential moving averages of the gradient at timestep t in Adam |
v(t) | exponential moving averages of the squared gradient at timestep t in Adam |
bias-corrected first moment estimates at timestep t in Adam | |
bias-corrected second raw moment estimates at timestep t in Adam | |
η | learning rate in Adam |
hyper parameter for numerical stability in Adam | |
β1 | hyper parameter for in Adam |
β2 | hyper parameter for in Adam |
μY | average value of the true objective values |
Abbreviations | |
ASTM | American Society for Testing and Materials |
BCC | body-centered cubic |
C(T) | compact tension; specimen type |
DBT | ductile-to-brittle transition |
MC | master curve |
nT | notation used to indicate specimen thickness, where n is expressed in multiples of 25 mm |
RPV | reactor pressure vessel |
RT | room temperature |
SE(B) | single-edge notched bend bar; specimen type |
Z–A | Zerilli–Armstrong |
SED | strain energy density |
PCCV | pre-cracked Charpy V-notch; specimen type |
MLP | multiplayer perceptron |
ANN | artificial neural network |
ReLU | rectified linear unit |
Adam | adaptive moment estimation |
References
- James, P.M.; Ford, M.; Jivkov, A.P. A novel particle failure criterion for cleavage fracture modelling allowing measured brittle particle distributions. Eng. Fract. Mech. 2014, 121–122, 98–115. [Google Scholar] [CrossRef]
- ASTM E1921-10. Standard Test Method for Determination of Reference Temperature, T0, for Ferritic Steels in the Transition Range; American Society for Testing and Materials: Philadelphia, PA, USA, 2010. [Google Scholar]
- Odette, G.R.; He, M.Y. A cleavage toughness master curve model. J. Nucl. Mater. 2000, 283, 120–127. [Google Scholar] [CrossRef]
- Ritchie, R.O.; Knott, J.F.; Rice, J.R. On the relationship between critical tensile stress and fracture toughness in mild steel. J. Mech. Phys. Solids 1973, 21, 395–410. [Google Scholar] [CrossRef]
- Curry, D.A.; Knott, J.F. The relationship between fracture toughness and microstructure in the cleavage fracture of mild steel. Met. Sci. 1976, 10, 1–6. [Google Scholar] [CrossRef]
- Ishihara, K.; Hamada, T.; Meshii, T. T-scaling method for stress distribution scaling under small-scale yielding and its application to the prediction of fracture toughness temperature dependence. Theor. Appl. Fract. Mech. 2017, 90, 182–192. [Google Scholar] [CrossRef]
- Wallin, K. Irradiation damage effects on the fracture toughness transition curve shape for reactor pressure vessel steels. Int. J. Press. Vessels Pip. 1993, 55, 61–79. [Google Scholar] [CrossRef]
- ASTM E1921-19b. Standard Test Method for Determination of Reference Temperature, T0, for Ferritic Steels in the Transition Range; American Society for Testing and Materials: Philadelphia, PA, USA, 2019. [Google Scholar]
- Kirk, M.T.; Natishan, M.; Wagenhofer, M. Microstructural limits of applicability of the master curve. In Fatigue and Fracture Mechanics; Chona, R., Ed.; American Society for Testing and Materials: Philadelphia, PA, USA, 2001; Volume 32, pp. 3–16. [Google Scholar]
- Kirk, M.T. The Technical Basis for Application of the Master Curve to the Assessment of Nuclear Reactor Pressure Vessel Integrity; ADAMS ML093540004; United States Nuclear Regulatory Commission: Washington, DC, USA, 2002.
- Wallin, K. The size effect in KIC results. Eng. Fract. Mech. 1985, 22, 149–163. [Google Scholar] [CrossRef]
- Dodds, R.H.; Anderson, T.L.; Kirk, M.T. A framework to correlate a/W ratio effects on elastic-plastic fracture toughness (Jc). Int. J. Fract. 1991, 48, 1–22. [Google Scholar] [CrossRef]
- Kirk, M.T.; Dodds, R.H.; Anderson, T.L. An approximate technique for predicting size effects on cleavage fracture toughness (Jc) using the elastic T stress. In Fracture Mechanics; Landes, J.D., McCabe, D.E., Boulet, J.A.M., Eds.; American Society for Testing and Materials: Philadelphia, PA, USA, 1994; Volume 24, pp. 62–86. [Google Scholar]
- Rathbun, H.J.; Odette, G.R.; He, M.Y.; Yamamoto, T. Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition—A model based analysis. Eng. Fract. Mech. 2006, 73, 2723–2747. [Google Scholar] [CrossRef]
- Meshii, T.; Lu, K.; Takamura, R. A failure criterion to explain the test specimen thickness effect on fracture toughness in the transition temperature region. Eng. Fract. Mech. 2013, 104, 184–197. [Google Scholar] [CrossRef]
- Meshii, T.; Yamaguchi, T. Applicability of the modified Ritchie-Knott-Rice failure criterion to transfer fracture toughness Jc of reactor pressure vessel steel using specimens of different thicknesses–possibility of deterministic approach to transfer the minimum Jc for specified specimen thicknesses. Theor. Appl. Fract. Mech. 2016, 85, 328–344. [Google Scholar] [CrossRef]
- Anderson, T.L.; Stienstra, D.; Dodds, R.H. A theoretical framework for addressing fracture in the ductile-brittle transition region. In Fracture Mechanics; Landes, J.D., McCabe, D.E., Boulet, J.A.M., Eds.; American Society for Testing and Materials: Philadelphia, PA, USA, 1994; Volume 24, pp. 186–214. [Google Scholar]
- Yang, S.; Chao, Y.J.; Sutton, M.A. Higher order asymptotic crack tip fields in a power-law hardening material. Eng. Fract. Mech. 1993, 45, 1–20. [Google Scholar] [CrossRef]
- Meshii, T.; Tanaka, T. Experimental T33-stress formulation of test specimen thickness effect on fracture toughness in the transition temperature region. Eng. Fract. Mech. 2010, 77, 867–877. [Google Scholar] [CrossRef]
- Matvienko, Y.G. The effect of out-of-plane constraint in terms of the T-stress in connection with specimen thickness. Theor. Appl. Fract. Mech. 2015, 80, 49–56. [Google Scholar] [CrossRef]
- Matvienko, Y.G. The effect of crack-tip constraint in some problems of fracture mechanics. Eng. Fail. Anal. 2020, 110, 104413. [Google Scholar] [CrossRef]
- Beremin, F.M.; Pineau, A.; Mudry, F.; Devaux, J.C.; D’Escatha, Y.; Ledermann, P. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metall. Mater. Trans. A 1983, 14, 2277–2287. [Google Scholar] [CrossRef]
- Khalili, A.; Kromp, K. Statistical properties of Weibull estimators. J. Mater. Sci. 1991, 26, 6741–6752. [Google Scholar] [CrossRef]
- Zerilli, F.J.; Armstrong, R.W. Dislocation-mechanics-based constitutive relations for material dynamics calculations. J. Appl. Phys. 1987, 61, 1816–1825. [Google Scholar] [CrossRef] [Green Version]
- Meshii, T. Failure of the ASTM E 1921 master curve to characterize the fracture toughness temperature dependence of ferritic steel and successful application of the stress distribution T-scaling method. Theor. Appl. Fract. Mech. 2019, 100, 354–361. [Google Scholar] [CrossRef]
- Meshii, T. Spreadsheet-based method for predicting temperature dependence of fracture toughness in ductile-to-brittle temperature region. Adv. Mech. Eng. 2019, 11, 1–17. [Google Scholar] [CrossRef]
- Zerbst, U.; Heerens, J.; Pfuff, M.; Wittkowsky, B.U.; Schwalbe, K.H. Engineering estimation of the lower bound toughness in the transition regime of ferritic steels. Fatigue Fract. Eng. Mater. Struct. 1998, 21, 1273–1278. [Google Scholar] [CrossRef]
- Heerens, J.; Pfuff, M.; Hellmann, D.; Zerbst, U. The lower bound toughness procedure applied to the Euro fracture toughness dataset. Eng. Fract. Mech. 2002, 69, 483–495. [Google Scholar] [CrossRef] [Green Version]
- Lu, K.; Meshii, T. Application of T33-stress to predict the lower bound fracture toughness for increasing the test specimen thickness in the transition temperature region. Adv. Mater. Sci. Eng. 2014, 1–8. [Google Scholar] [CrossRef] [Green Version]
- Meshii, T. Characterization of fracture toughness based on yield stress and successful application to construct a lower-bound fracture toughness master curve. Eng. Fail. Anal. 2020, 116, 104713. [Google Scholar] [CrossRef]
- Akbarzadeh, P.; Hadidi-Moud, S.; Goudarzi, A.M. Global equations for Weibull parameters in a ductile-to-brittle transition regime. Nucl. Eng. Des. 2009, 239, 1186–1192. [Google Scholar] [CrossRef]
- Wenman, M.R. Fitting small data sets in the lower ductile-to-brittle transition region and lower shelf of ferritic steels. Eng. Fract. Mech. 2013, 98, 350–364. [Google Scholar] [CrossRef]
- Meshii, T.; Yakushi, G.; Takagishi, Y.; Fujimoto, Y.; Ishihara, K. Quantitative comparison of the predictions of fracture toughness temperature dependence using ASTM E1921 master curve and stress distribution T-scaling methods. Eng. Fail. Anal. 2020, 111, 104458. [Google Scholar] [CrossRef]
- Scikit-Learn. Available online: https://scikit-learn.org/stable/index.html (accessed on 13 October 2021).
- Raschka, S.; Mrjalili, V. Python Machine Learning, 2nd ed.; Packt Publishing: Birmingham, UK, 2017. [Google Scholar]
- Kingma, D.P.; Ba, J.L. ADAM: A METHOD FOR STOCHASTIC OPTIMIZATION. In Proceedings of the 3rd International Conference for Learning Representations, San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
- Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning representations by back-propagating errors. Nature 1986, 323, 533–536. [Google Scholar] [CrossRef]
- Miura, N. Study on Fracture Toughness Evaluation Method for Reactor Pressure Vessel Steels by Master Curve Method Using Statistical Method. Ph.D. Thesis, The University of Tokyo, Tokyo, Japan, 2014. (In Japanese). [Google Scholar]
- Gopalan, A.; Samal, M.K.; Chakravartty, J.K. Fracture toughness evaluation of 20MnMoNi55 pressure vessel steel in the ductile to brittle transition regime: Experiment & numerical simulations. J. Nucl. Mater. 2015, 465, 424–432. [Google Scholar] [CrossRef]
- Rathbun, H.J.; Odette, G.R.; Yamamoto, T.; Lucas, G.E. Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition—A single variable experiment and database. Eng. Fract. Mech. 2006, 73, 134–158. [Google Scholar] [CrossRef]
- García, T.; Cicero, S. Application of the master curve to ferritic steels in notched conditions. Eng. Fail. Anal. 2015, 58, 149–164. [Google Scholar] [CrossRef]
- Cicero, S.; García, T.; Madrazo, V. Application and validation of the notch master curve in medium and high strength structural steels. J. Mech. Sci. Technol. 2015, 29, 4129–4142. [Google Scholar] [CrossRef] [Green Version]
Heat No. | Material | C | Si | Mn | P | S | Ni | Cr | Mo | V | Cu | Nb | Ti | Al |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | MiuraSFVQ1A [38] | 0.18 | 0.18 | 1.46 | 0.002 | <0.001 | 0.90 | 0.12 | 0.52 | <0.01 | - | - | - | - |
0.17 | 0.17 | 1.39 | 0.002 | <0.001 | 0.87 | 0.11 | 0.50 | <0.01 | - | - | - | - | ||
2 | Gopalan20MnMoNi55 [39] | 0.20 | 0.24 | 1.38 | 0.011 | 0.005 | 0.52 | 0.06 | 0.30 | - | - | 0.032 | - | 0.068 |
3 | ShorehamA533B [40] | 0.21 | 0.24 | 1.23 | 0.004 | 0.008 | 0.63 | 0.09 | 0.53 | - | 0.08 | - | - | 0.04 |
4 | MiuraSQV2Ah1 [38] | 0.22 | 0.25 | 1.44 | 0.021 | 0.028 | 0.54 | 0.08 | 0.48 | - | 0.10 | - | - | - |
5 | MiuraSQV2Ah2 [38] | 0.22 | 0.25 | 1.46 | 0.002 | 0.002 | 0.69 | 0.11 | 0.57 | - | - | - | - | - |
6 | GarciaS275JR [41] | 0.18 | 0.26 | 1.18 | 0.012 | 0.009 | <0.085 | <0.018 | <0.12 | <0.02 | 0.06 | - | 0.022 | 0.034 |
7 | GarciaS355J2 [41] | 0.2 | 0.31 | 1.39 | <0.012 | 0.008 | 0.09 | 0.05 | <0.12 | 0.02 | 0.06 | - | 0.022 | 0.014 |
8 | CiceroS460M [42] | 0.12 | 0.45 | 1.49 | 0.012 | 0.001 | 0.016 | 0.062 | - | 0.066 | 0.011 | 0.036 | 0.003 | 0.048 |
9 | CiceroS690Q [42] | 0.15 | 0.40 | 1.42 | 0.006 | 0.001 | 0.160 | 0.020 | - | 0.058 | 0.010 | 0.029 | 0.003 | 0.056 |
10 | MeshiiFY2017SCM440 [25] | 0.39 | 0.17 | 0.62 | 0.011 | 0.002 | 0.07 | 1.02 | 0.17 | - | 0.10 | - | - | - |
11 | MeshiiFY2012S55C [6] | 0.55 | 0.17 | 0.61 | 0.015 | 0.004 | 0.07 | 0.08 | - | - | 0.13 | - | - | - |
12 | MeshiiFY2016S55C [26] | 0.54 | 0.17 | 0.61 | 0.014 | 0.003 | 0.06 | 0.12 | - | - | - | - | - | - |
Heat No. | Material | Specimen | Temps. | Num. of Temps. | σYS | σYSRT | σBRT | Num. of Specimens | T0 |
---|---|---|---|---|---|---|---|---|---|
Type | (°C) | (MPa) | (MPa) | (MPa) | (°C) | ||||
1 | MiuraSFVQ1A [38] | 1TC(T) | −120~−60 | 4 | 530~640 | 454 | 594 | 32 | −98 |
2TC(T) | −120~−60 | 4 | 530~640 | 454 | 594 | 16 | −98 | ||
4TC(T) | −100~−80 | 2 | 560~607 | 454 | 594 | 12 | −98 | ||
0.4TC(T) | −140~−80 | 4 | 560~695 | 454 | 594 | 34 | −98 | ||
0.4TSE(B) | −140~−80 | 4 | 560~695 | 454 | 594 | 29 | −98 | ||
2 | Gopalan20MnMoNi55 [39] | 1TC(T) | −140~−80 | 3 | 560~667 | 479 | 616 | 18 | −133 |
0.5TC(T) | −140~−80 | 3 | 560~667 | 479 | 616 | 12 | −133 |
Heat No. | Material | Specimen | Temps. | Num. of Temps. | σYS | σYSRT | σBRT | Num. of Specimens | T0 |
---|---|---|---|---|---|---|---|---|---|
Type | (°C) | (MPa) | (MPa) | (MPa) | (°C) | ||||
3 | ShorehamA533B [40] | 1TC(T) * | −100~−64 | 3 | 551~586 | 488 | 644 | 18 | −91 |
4 | MiuraSQV2Ah1 [38] | 1TC(T) | −100~−60 | 3 | 544~600 | 473 | 625 | 14 | -93 |
2TC(T) | −100~−60 | 3 | 544~600 | 473 | 625 | 14 | −93 | ||
4TC(T) | −80~−60 | 2 | 544~566 | 473 | 625 | 12 | −93 | ||
0.4TC(T) | −120~−60 | 4 | 544~658 | 473 | 625 | 32 | −93 | ||
0.4TSE(B) | −120~−60 | 4 | 544~658 | 473 | 625 | 29 | −93 | ||
5 | MiuraSQV2Ah2 [38] | 1TC(T) | −140~−80 | 4 | 542~709 | 461 | 602 | 23 | −121 |
2TC(T) | −100~−80 | 2 | 542~607 | 461 | 602 | 12 | −121 | ||
4TC(T) | −100~−80 | 2 | 542~607 | 461 | 602 | 12 | −121 | ||
0.4TC(T) | −140~−80 | 4 | 542~709 | 461 | 602 | 33 | −121 | ||
0.4TSE(B) | −140~−80 | 4 | 542~709 | 461 | 602 | 32 | −121 |
Heat No. | Material | Specimen | Temps. | Num. of Temps. | σYS | σYSRT | σBRT | Num. of Specimens | T0 |
---|---|---|---|---|---|---|---|---|---|
Type | (°C) | (MPa) | (MPa) | (MPa) | (°C) | ||||
6 | GarciaS275JR [41] | 1TC(T) | −50~−10 | 3 | 338~349 | 328 | 519 | 14 | −26 |
7 | GarciaS355J2 [41] | 1TC(T) | −150~−100 | 3 | 426~528 | 375 | 558 | 13 | −134 |
8 | CiceroS460M [42] | 0.6TSE(B) | −140~−100 | 3 | 597~686 | 473 | 595 | 14 | −92 |
9 | CiceroS690Q [42] | 0.6TSE(B) | −140~−100 | 3 | 899~988 | 775 | 832 | 13 | −111 |
10 | MeshiiFY2017SCM440 [25,30] | 0.9TSE(B) | −55~100 | 4 | 410~524 | 459 | 796 | 18 | 17 |
0.5TSE(B) | −55~100 | 4 | 410~524 | 459 | 796 | 22 | 17 | ||
11 | MeshiiFY2012S55C [6] | 0.5TSE(B) | −25~20 | 3 | 394~444 | 394 | 707 | 17 | 27 |
12 | MeshiiFY2016S55C [26,30] | 0.9TSE(B) | −45~35 | 3 | 375~475 | 382 | 685 | 17 | 15 |
0.5TSE(B) | −85~20 | 3 | 382~562 | 382 | 685 | 19 | 15 |
Parameters | Value |
---|---|
Number of hidden layers | 4 |
Number of hidden layer nodes | 100, 50, 25, 10 |
Activation function | ReLU |
Solver | Adam |
α | 0.01 |
η | 0.001 |
β1 | 0.9 |
β2 | 0.999 |
1.0 × 10−8 |
Heat No. | Material | W (mm) | T (°C) | σYSRT (MPa) | σYS (MPa) | σBRT (MPa) |
---|---|---|---|---|---|---|
1 | Miura SFVQ1A | 20 | −140, −120, −100, −80 | 454 | 695, 640, 607, 560 | 594 |
50.8 | −120, −100, −80, −60 | 454 | 640, 607, 560, 530 | 594 | ||
101.6 | −120, −100, −80, −60 | 454 | 640, 607, 560, 530 | 594 | ||
203.2 | −80, −100 | 454 | 607, 560 | 594 | ||
10 | MeshiiFY2017SCM440 | 25 | −55, 20, 60 100 | 459 | 524, 459, 435, 410 | 796 |
46 | −55, 20, 60, 100 | 459 | 524, 459, 435, 410 | 796 |
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Ishihara, K.; Kitagawa, H.; Takagishi, Y.; Meshii, T. Application of an Artificial Neural Network to Develop Fracture Toughness Predictor of Ferritic Steels Based on Tensile Test Results. Metals 2021, 11, 1740. https://doi.org/10.3390/met11111740
Ishihara K, Kitagawa H, Takagishi Y, Meshii T. Application of an Artificial Neural Network to Develop Fracture Toughness Predictor of Ferritic Steels Based on Tensile Test Results. Metals. 2021; 11(11):1740. https://doi.org/10.3390/met11111740
Chicago/Turabian StyleIshihara, Kenichi, Hayato Kitagawa, Yoichi Takagishi, and Toshiyuki Meshii. 2021. "Application of an Artificial Neural Network to Develop Fracture Toughness Predictor of Ferritic Steels Based on Tensile Test Results" Metals 11, no. 11: 1740. https://doi.org/10.3390/met11111740