pith. machine review for the scientific record. sign in

arxiv: 2604.27775 · v1 · submitted 2026-04-30 · ❄️ cond-mat.mtrl-sci · cs.LG

Recognition: unknown

Data-Efficient Indentation Size Effect Correction in Steels Using Machine Learning and Physics-Guided Augmentation

Radmir Karamov , Tagir Karamov
Authors on Pith no claims yet

Pith reviewed 2026-05-07 07:36 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.LG
keywords nanoindentationindentation size effectmachine learningdata augmentationhardnesssteelsOliver-Pharrneural network
0
0 comments X

The pith

Machine learning with physics-guided augmentation enables accurate indentation size effect correction in steels from small experimental datasets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper demonstrates that a small dataset of nanoindentation measurements on steels can be augmented with physically motivated variations to train machine learning models that predict the reference hardness free from indentation size effect artifacts. The key is combining standard Oliver-Pharr analysis outputs with additional mechanics descriptors such as indentation work partitioning and an area-invariant compliance proxy. Nonlinear models, especially a constrained neural network, achieve high accuracy on a held-out steel specimen and remain reliable even at shallow indentation depths. This approach addresses the limitation of classical methods like Nix-Gao, which require a deep linear regime not always available in volume-constrained samples. The results indicate that data-efficient workflows are feasible for mechanical characterization of thin films and individual phases.

Core claim

The authors establish that the relationship between measured indentation parameters and true hardness is nonlinear and can be learned from limited experimental data when augmented with representations of instrumental noise, session drift, and local multiphase blending. By training on features including the area-invariant proxy P_max/S² and energy-based terms, the constrained neural network produces stable hardness estimates across load ranges, including shallow regimes where traditional Nix-Gao analysis becomes unreliable.

What carries the argument

Physics-guided augmentation of nanoindentation data combined with a constrained neural network that uses Oliver-Pharr values and mechanics descriptors to map to reference hardness.

If this is right

  • Ridge regression fails while nonlinear models succeed, indicating the hardness mapping is nonlinear.
  • The neural network yields stable estimates in the shallow indentation regime.
  • SHAP analysis reveals reliance on area-invariant and energy-based descriptors.
  • The workflow demonstrates feasibility for data-efficient characterization of volume-constrained materials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the augmentation strategy to account for tip geometry variations could broaden applicability to different indenters.
  • Treating the indentation size effect as a supervised learning task rather than assuming a fixed functional form like in Nix-Gao could lead to more flexible corrections.
  • Similar physics-guided augmentation approaches might improve machine learning models in other materials characterization techniques that suffer from limited data.
  • If validated across a wider range of materials, this could enable reliable mechanical testing of thin films and small volumes without requiring deep indentations.

Load-bearing premise

The physically motivated augmentations accurately capture the statistical variations present in real experimental sessions on steels.

What would settle it

Applying the trained model to shallow indentations performed on an additional independent steel specimen and finding that the predicted hardness deviates substantially from the reference value obtained from deep indents or other independent methods.

Figures

Figures reproduced from arXiv: 2604.27775 by Radmir Karamov, Tagir Karamov.

Figure 1
Figure 1. Figure 1: Depth-dependent apparent hardness and representative shallow indentation curve. (a) Apparent Oliver–Pharr view at source ↗
Figure 2
Figure 2. Figure 2: Baseline evaluation of the classical Nix-Gao model, demonstrating that forcing a linear extrapolation through view at source ↗
Figure 3
Figure 3. Figure 3: Parity plots comparing the predicted macroscopic bulk hardness against the experimental ground truth for the view at source ↗
Figure 4
Figure 4. Figure 4: Per-load prediction error (∆H) relative to the 350 mN macroscopic reference for the E212 training steel, illustrating the baseline correction stability of the evaluated architectures across the complete indentation load range. The intermediate specimen E426 is a notable exception: all non-linear architectures yield RMSE above 0.16 GPa, slightly higher than for E212 or E841. This elevated error is not attri… view at source ↗
Figure 5
Figure 5. Figure 5: Application of the ML framework to the unseen validation specimen. Depth-dependent scatter plot comparing view at source ↗
Figure 6
Figure 6. Figure 6: Per-load prediction error (∆H) relative to the 350 mN macroscopic reference (9.215 GPa) for the fused silica calibration standard, demonstrating the deliberate algorithmic failure when applied to an amorphous material lacking an indentation size effect. Both failure modes confirm the same operational boundary: the framework is applicable to crystalline systems governed by dislocation-mediated plasticity wi… view at source ↗
Figure 7
Figure 7. Figure 7: SHAP summary plots illustrating top 10 global feature importance for (a) the Random Forest model, view at source ↗
Figure 8
Figure 8. Figure 8: PCA of the neural network latent space, illustrating autonomous clustering by intrinsic hardness and depth view at source ↗
read the original abstract

Shallow nanoindentation enables mechanical characterization of thin films, individual phases and other volume-constrained materials, but measured hardness is often inflated by the indentation size effect (ISE), contact-area errors and tip-geometry artifacts. Classical ISE corrections such as the Nix-Gao require a deep linear regime and are unreliable when only shallow measurements are used. This study investigates how a small experimental dataset can be used to predict a reference hardness with physics-guided feature engineering and augmentation. Approximately 700 experimental indentations were collected from three steel reference specimens covering a hardness range of 2-6.5 GPa and augmented using physically motivated variations representing instrumental noise, session-level drift, and local multiphase boundary blending. The input space combined Oliver-Pharr values with mechanics descriptors, including indentation work partitioning, ($H\text{/}E_{r}$), and the area-invariant compliance proxy ($P_{\max}\text{/}S^{2}$). Ridge Regression (RR), Random Forest, XGBoost, and Neural Networks (NN) were evaluated using a quarantined fourth steel specimen tested at staggered loads. The hardness mapping was nonlinear: RR failed, whereas nonlinear models achieved ($R^2 > 0.98$) internally. A constrained (64-8-64) NN gave the best results, reaching RMSE = 0.470 GPa, MAPE = 5.4% on the quarantined steel. Unlike Nix-Gao analysis, the NN produced stable estimates in the shallow regime. SHAP and latent-space analysis showed reliance on area-invariant and energy-based descriptors. The results demonstrate the feasibility of a this workflow for ISE correction in steels using small datasets and suggest a pathway toward data-efficient characterization of any volume constrained materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a machine learning workflow for correcting the indentation size effect (ISE) in nanoindentation of steels using a small experimental dataset augmented with physics-motivated variations. Approximately 700 indents from three specimens are augmented for instrumental noise, session-level drift, and local multiphase boundary blending. Models (ridge regression, random forest, XGBoost, and neural networks) are trained to predict reference hardness from Oliver-Pharr values combined with mechanics descriptors such as indentation work partitioning, H/E_r, and P_max/S^2. A constrained (64-8-64) neural network achieves the best held-out performance on a quarantined fourth specimen (RMSE = 0.470 GPa, MAPE = 5.4%), with R^2 > 0.98 internally, and produces stable estimates in the shallow regime unlike Nix-Gao analysis. SHAP and latent-space analysis highlight reliance on area-invariant and energy-based features.

Significance. If the augmentations accurately capture real session-to-session variability, the work demonstrates a data-efficient pathway for reliable ISE correction in volume-constrained materials without requiring a deep linear regime. Strengths include the use of a quarantined specimen for external validation, physics-guided rather than purely statistical augmentation, and interpretability via SHAP. This could be particularly useful for thin films and multiphase systems where classical Nix-Gao methods fail at shallow depths.

major comments (3)
  1. [Methods] Methods (augmentation procedure): No direct statistical comparison (e.g., distribution matching, Kolmogorov-Smirnov tests, or variance decomposition) is reported between the augmented training features and the actual feature distributions in the quarantined hold-out specimen. This is load-bearing for the generalization claim, as the reported RMSE/MAPE on the fourth specimen assumes the synthetic variations (noise, drift, blending) match real experimental session variability.
  2. [Results] Results (reference hardness targets): The procedure for obtaining the reference hardness values used as training targets is not specified in sufficient detail. It is unclear whether these are derived from deep indents on the same specimens, literature values, or other means; without this, the mapping learned by the NN cannot be fully assessed for physical consistency or potential bias.
  3. [Results] Results (Nix-Gao comparison): The claim of superior stability in the shallow regime is central but lacks quantitative detail on the specific load ranges or indentation depths where the NN remains stable while Nix-Gao diverges, including error bars or per-depth RMSE values on the hold-out data.
minor comments (2)
  1. [Abstract] Abstract: The notation (H/E_r) and (P_max/S^2) should be defined at first use, and E_r should be explicitly identified as reduced modulus.
  2. [Methods] The manuscript would benefit from a sensitivity analysis on the augmentation magnitudes (noise, drift, blending parameters) to show robustness of the reported performance.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and detailed review of our manuscript. The comments have identified areas where additional clarity and quantitative support will strengthen our claims. We provide point-by-point responses to the major comments and commit to the indicated revisions in the updated version.

read point-by-point responses

  1. Referee: [Methods] Methods (augmentation procedure): No direct statistical comparison (e.g., distribution matching, Kolmogorov-Smirnov tests, or variance decomposition) is reported between the augmented training features and the actual feature distributions in the quarantined hold-out specimen. This is load-bearing for the generalization claim, as the reported RMSE/MAPE on the fourth specimen assumes the synthetic variations (noise, drift, blending) match real experimental session variability.

    Authors: We agree that a direct statistical comparison would strengthen the evidence supporting the augmentation procedure. In the revised manuscript we will add Kolmogorov-Smirnov tests, quantile-quantile plots, and a variance decomposition for the principal input features (Oliver-Pharr hardness and modulus, work partitioning ratios, H/E_r, and P_max/S^2) comparing the augmented training distribution to the quarantined specimen. These analyses will be placed in the Methods section immediately following the description of the augmentation pipeline. revision: yes

  2. Referee: [Results] Results (reference hardness targets): The procedure for obtaining the reference hardness values used as training targets is not specified in sufficient detail. It is unclear whether these are derived from deep indents on the same specimens, literature values, or other means; without this, the mapping learned by the NN cannot be fully assessed for physical consistency or potential bias.

    Authors: The reference hardness targets for the three training specimens were obtained from deep indentation measurements (depths > 2 μm) performed on the same specimens, where the indentation size effect is negligible; values were averaged over at least 20 indents per specimen across two independent sessions. We will expand the Methods and Results sections to state the exact depth threshold, the number of indents and sessions used for averaging, and any consistency checks against published hardness values for the steel grades. This added detail will permit readers to evaluate the physical consistency of the learned mapping. revision: yes

  3. Referee: [Results] Results (Nix-Gao comparison): The claim of superior stability in the shallow regime is central but lacks quantitative detail on the specific load ranges or indentation depths where the NN remains stable while Nix-Gao diverges, including error bars or per-depth RMSE values on the hold-out data.

    Authors: We agree that quantitative, depth-resolved metrics would make the stability comparison more rigorous. In the revision we will add a table (or supplementary figure) reporting per-depth RMSE and MAPE for both the neural network and the Nix-Gao model on the quarantined specimen, binned in 50 nm increments from 50 nm to 1000 nm, together with error bars representing the standard deviation across indents at each depth. The table will also indicate the load ranges corresponding to each depth bin. This material will be placed in the Results section alongside the existing Nix-Gao comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity: hold-out evaluation on independent specimen prevents reduction to fitted inputs.

full rationale

The paper's workflow collects ~700 experimental indentations from three steel specimens, applies physics-motivated augmentations (instrumental noise, session drift, multiphase blending) that are not fitted to the target hardness, trains ML models (RR, RF, XGBoost, NN) on the augmented feature space (Oliver-Pharr plus H/Er and Pmax/S² descriptors), and reports RMSE/MAPE on a quarantined fourth specimen tested at staggered loads. The central performance numbers are computed on unseen experimental data rather than on training or self-generated targets. No equations or self-citations are invoked that would make the reported hardness predictions equivalent to the model's own inputs by construction; the augmentations are externally motivated and the test set is drawn from a distinct experimental session. This satisfies the self-contained criterion against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The paper rests on experimental nanoindentation data plus standard ML techniques augmented by domain-motivated synthetic variations. No new physical entities are introduced. The main assumptions concern the fidelity of the augmentation process and the representativeness of the held-out specimen.

free parameters (2)
  • Neural network architecture and hyperparameters
    The (64-8-64) constrained architecture and training details were selected to achieve best performance on the validation set.
  • Augmentation magnitudes for noise, drift, and blending
    Specific variation levels were chosen to represent instrumental and session-level effects but are not stated as derived from first principles.
axioms (2)
  • domain assumption Physically motivated augmentations accurately represent real experimental variations in noise, drift, and multiphase blending.
    Invoked to expand the small experimental dataset while preserving physical plausibility.
  • domain assumption The target hardness values used for training represent the true reference hardness free of ISE.
    Required to define the supervised learning target.

pith-pipeline@v0.9.0 · 5628 in / 1862 out tokens · 64754 ms · 2026-05-07T07:36:25.274557+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 53 canonical work pages

  1. [1]

    Determination of mechanical properties by instrumented indentation.Meccanica, 42(1):19–29, February 2007

    Jaroslav Menˇcík. Determination of mechanical properties by instrumented indentation.Meccanica, 42(1):19–29, February 2007. ISSN 0025-6455, 1572-9648. doi:10.1007/s11012-006-9018-6

    work page doi:10.1007/s11012-006-9018-6 2007

  2. [2]

    Oliver and G.M

    W.C. Oliver and G.M. Pharr. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments.Journal of Materials Research, 7(6):1564–1583, June 1992. ISSN 0884-2914, 2044-5326. doi:10.1557/JMR.1992.1564

    work page doi:10.1557/jmr.1992.1564 1992

  3. [3]

    Characterization of mechanical properties of thin films us- ing nanoindentation test.Mechanics of Materials, 38(12):1182–1198, December 2006

    Hervé Pelletier, Joël Krier, and Pierre Mille. Characterization of mechanical properties of thin films us- ing nanoindentation test.Mechanics of Materials, 38(12):1182–1198, December 2006. ISSN 01676636. doi:10.1016/j.mechmat.2006.02.011

    work page doi:10.1016/j.mechmat.2006.02.011 2006

  4. [4]

    Hossein Besharatloo and Jeffrey M. Wheeler. Influence of indentation size and spacing on statistical phase analysis via high-speed nanoindentation mapping of metal alloys.Journal of Materials Research, 36(11):2198–2212, June

  5. [5]

    doi:10.1557/s43578-021-00214-5

    ISSN 0884-2914, 2044-5326. doi:10.1557/s43578-021-00214-5

    work page doi:10.1557/s43578-021-00214-5 2044

  6. [6]

    Characterization of Mechanical Property Degradation of Ion-Irradiated Materials.Frontiers in Materials, 9:849209, April 2022

    Luyao Mei, Xun Guo, and Ke Jin. Characterization of Mechanical Property Degradation of Ion-Irradiated Materials.Frontiers in Materials, 9:849209, April 2022. ISSN 2296-8016. doi:10.3389/fmats.2022.849209

    work page doi:10.3389/fmats.2022.849209 2022

  7. [7]

    A comparative study of indentation size effect models for different materials.Scientific Reports, 14(1):20010, August 2024

    Peina Wang, Yu Gao, and Peihuan Wang. A comparative study of indentation size effect models for different materials.Scientific Reports, 14(1):20010, August 2024. ISSN 2045-2322. doi:10.1038/s41598-024-71136-5

    work page doi:10.1038/s41598-024-71136-5 2024

  8. [8]

    Nix and Huajian Gao

    William D. Nix and Huajian Gao. Indentation size effects in crystalline materials: A law for strain gradient plasticity.Journal of the Mechanics and Physics of Solids, 46(3):411–425, March 1998. ISSN 00225096. doi:10.1016/S0022-5096(97)00086-0

    work page doi:10.1016/s0022-5096(97)00086-0 1998

  9. [9]

    Review of Nanoindentation Size Effect: Experiments and Atomistic Simulation.Crystals, 7(10):321, October 2017

    George V oyiadjis and Mohammadreza Yaghoobi. Review of Nanoindentation Size Effect: Experiments and Atomistic Simulation.Crystals, 7(10):321, October 2017. ISSN 2073-4352. doi:10.3390/cryst7100321

    work page doi:10.3390/cryst7100321 2017

  10. [10]

    Elmustafa and D.S

    A.A. Elmustafa and D.S. Stone. Nanoindentation and the indentation size effect: Kinetics of deformation and strain gradient plasticity.Journal of the Mechanics and Physics of Solids, 51(2):357–381, February 2003. ISSN 00225096. doi:10.1016/S0022-5096(02)00033-9

    work page doi:10.1016/s0022-5096(02)00033-9 2003

  11. [11]

    Aifantis and A.H.W

    Katerina E. Aifantis and A.H.W. Ngan. Modeling dislocation—grain boundary interactions through gradient plasticity and nanoindentation.Materials Science and Engineering: A, 459(1-2):251–261, June 2007. ISSN 09215093. doi:10.1016/j.msea.2007.01.028. 16 Data-Efficient ISE Correction in Steels Using ML and Physics-Guided AugmentationA PREPRINT

    work page doi:10.1016/j.msea.2007.01.028 2007

  12. [12]

    Wu, J.S.C

    D. Wu, J.S.C. Jang, and T.G. Nieh. Elastic and plastic deformations in a high entropy alloy inves- tigated using a nanoindentation method.Intermetallics, 68:118–127, January 2016. ISSN 09669795. doi:10.1016/j.intermet.2015.10.002

    work page doi:10.1016/j.intermet.2015.10.002 2016

  13. [13]

    Lewis, C.H

    Osman Sahin, Orhan Uzun, Malgorzata Sopicka-Lizer, Hasan Gocmez, and U ˘gur Kölemen. Analysis of load- penetration depth data using Oliver–Pharr and Cheng–Cheng methods of SiAlON–ZrO2 ceramics.Journal of Physics D: Applied Physics, 41(3):035305, February 2008. ISSN 0022-3727, 1361-6463. doi:10.1088/0022- 3727/41/3/035305

    work page doi:10.1088/0022- 2008

  14. [14]

    Pharr, Erik G

    George M. Pharr, Erik G. Herbert, and Yanfei Gao. The Indentation Size Effect: A Critical Examination of Experimental Observations and Mechanistic Interpretations.Annual Review of Materials Research, 40(1):271–292, June 2010. ISSN 1531-7331, 1545-4118. doi:10.1146/annurev-matsci-070909-104456

    work page doi:10.1146/annurev-matsci-070909-104456 2010

  15. [15]

    S. Qu, Y . Huang, W.D. Nix, H. Jiang, F. Zhang, and K.C. Hwang. Indenter tip radius effect on the Nix–Gao relation in micro- and nanoindentation hardness experiments.Journal of Materials Research, 19(11):3423–3434, November 2004. ISSN 0884-2914, 2044-5326. doi:10.1557/JMR.2004.0441

    work page doi:10.1557/jmr.2004.0441 2004

  16. [16]

    Abu Al-Rub and George Z

    Rashid K. Abu Al-Rub and George Z. V oyiadjis. Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments.International Journal of Plasticity, 20(6):1139–1182, June 2004. ISSN 07496419. doi:10.1016/j.ijplas.2003.10.007

    work page doi:10.1016/j.ijplas.2003.10.007 2004

  17. [17]

    M. Dao, N. Chollacoop, K.J. Van Vliet, T.A. Venkatesh, and S. Suresh. Computational modeling of the forward and reverse problems in instrumented sharp indentation.Acta Materialia, 49(19):3899–3918, November 2001. ISSN 13596454. doi:10.1016/S1359-6454(01)00295-6

    work page doi:10.1016/s1359-6454(01)00295-6 2001

  18. [18]

    Kang, A.A

    J.J. Kang, A.A. Becker, and W. Sun. Determining elastic–plastic properties from indentation data obtained from finite element simulations and experimental results.International Journal of Mechanical Sciences, 62(1):34–46, September 2012. ISSN 00207403. doi:10.1016/j.ijmecsci.2012.05.011

    work page doi:10.1016/j.ijmecsci.2012.05.011 2012

  19. [19]

    K.K. Tho, S. Swaddiwudhipong, Z.S. Liu, K. Zeng, and J. Hua. Uniqueness of reverse analysis from conical indentation tests.Journal of Materials Research, 19(8):2498–2502, August 2004. ISSN 0884-2914, 2044-5326. doi:10.1557/JMR.2004.0306

    work page doi:10.1557/jmr.2004.0306 2004

  20. [20]

    Xi Chen, Nagahisa Ogasawara, Manhong Zhao, and Norimasa Chiba. On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials.Journal of the Mechanics and Physics of Solids, 55(8):1618–1660, August 2007. ISSN 00225096. doi:10.1016/j.jmps.2007.01.010

    work page doi:10.1016/j.jmps.2007.01.010 2007

  21. [21]

    Bock, Roland C

    Frederic E. Bock, Roland C. Aydin, Christian J. Cyron, Norbert Huber, Surya R. Kalidindi, and Benjamin Klusemann. A Review of the Application of Machine Learning and Data Mining Approaches in Continuum Materials Mechanics.Frontiers in Materials, 6:110, May 2019. ISSN 2296-8016. doi:10.3389/fmats.2019.00110

    work page doi:10.3389/fmats.2019.00110 2019

  22. [22]

    Huber and Ch

    N. Huber and Ch. Tsakmakis. Determination of constitutive properties fromspherical indentation data using neural networks. Part i:the case of pure kinematic hardening in plasticity laws.Journal of the Mechanics and Physics of Solids, 47(7):1569–1588, June 1999. ISSN 00225096. doi:10.1016/S0022-5096(98)00109-4

    work page doi:10.1016/s0022-5096(98)00109-4 1999

  23. [23]

    K Genel. Application of artificial neural network for predicting strain-life fatigue properties of steels on the basis of tensile tests.International Journal of Fatigue, 26(10):1027–1035, October 2004. ISSN 01421123. doi:10.1016/j.ijfatigue.2004.03.009

    work page doi:10.1016/j.ijfatigue.2004.03.009 2004

  24. [24]

    Haijie Wang, Bo Li, Jianguo Gong, and Fu-Zhen Xuan. Machine learning-based fatigue life prediction of metal materials: Perspectives of physics-informed and data-driven hybrid methods.Engineering Fracture Mechanics, 284:109242, May 2023. ISSN 00137944. doi:10.1016/j.engfracmech.2023.109242

    work page doi:10.1016/j.engfracmech.2023.109242 2023

  25. [25]

    Sergeichev

    Radmir Karamov, Iskander Akhatov, and Ivan V . Sergeichev. Prediction of Fracture Toughness of Pultruded Composites Based on Supervised Machine Learning.Polymers, 14(17):3619, September 2022. ISSN 2073-4360. doi:10.3390/polym14173619

    work page doi:10.3390/polym14173619 2022

  26. [26]

    Shiyu He, Yanming Wang, Zhengyang Zhang, Fei Xiao, Shungui Zuo, Ying Zhou, Xiaorong Cai, and Xuejun Jin. Interpretable machine learning workflow for evaluation of the transformation temperatures of TiZrHfNiC- oCu high entropy shape memory alloys.Materials & Design, 225:111513, January 2023. ISSN 02641275. doi:10.1016/j.matdes.2022.111513

    work page doi:10.1016/j.matdes.2022.111513 2023

  27. [27]

    Machine learning framework for determination of elastic modulus without contact model fitting.International Journal of Solids and Structures, 256:111976, December

    Linh Thi Phuong Nguyen and Bernard Haochih Liu. Machine learning framework for determination of elastic modulus without contact model fitting.International Journal of Solids and Structures, 256:111976, December

  28. [28]

    doi:10.1016/j.ijsolstr.2022.111976

    ISSN 00207683. doi:10.1016/j.ijsolstr.2022.111976

    work page doi:10.1016/j.ijsolstr.2022.111976 2022

  29. [29]

    Artificial neural network model for material characterization by indentation.Modelling and Simulation in Materials Science and Engineering, 12(5):1055–1062, September

    K K Tho, S Swaddiwudhipong, Z S Liu, and J Hua. Artificial neural network model for material characterization by indentation.Modelling and Simulation in Materials Science and Engineering, 12(5):1055–1062, September

  30. [30]

    doi:10.1088/0965-0393/12/5/019

    ISSN 0965-0393, 1361-651X. doi:10.1088/0965-0393/12/5/019. 17 Data-Efficient ISE Correction in Steels Using ML and Physics-Guided AugmentationA PREPRINT

    work page doi:10.1088/0965-0393/12/5/019

  31. [31]

    Quan Jiao, Yongchao Chen, Jong-hyoung Kim, Chang-Fu Han, Chia-Hua Chang, and Joost J. Vlassak. A machine learning perspective on the inverse indentation problem: Uniqueness, surrogate modeling, and learning elasto-plastic properties from pile-up.Journal of the Mechanics and Physics of Solids, 185:105557, April 2024. ISSN 00225096. doi:10.1016/j.jmps.2024.105557

    work page doi:10.1016/j.jmps.2024.105557 2024

  32. [32]

    Lu Lu, Ming Dao, Punit Kumar, Upadrasta Ramamurty, George Em Karniadakis, and Subra Suresh. Extrac- tion of mechanical properties of materials through deep learning from instrumented indentation.Proceed- ings of the National Academy of Sciences, 117(13):7052–7062, March 2020. ISSN 0027-8424, 1091-6490. doi:10.1073/pnas.1922210117

    work page doi:10.1073/pnas.1922210117 2020

  33. [33]

    Small data machine learning in materials science

    Pengcheng Xu, Xiaobo Ji, Minjie Li, and Wencong Lu. Small data machine learning in materials science.npj Computational Materials, 9(1):42, March 2023. ISSN 2057-3960. doi:10.1038/s41524-023-01000-z

    work page doi:10.1038/s41524-023-01000-z 2023

  34. [34]

    Machine learning for material characterization with an application for predicting mechanical properties.GAMM-Mitteilungen, 44(1):e202100003, March 2021

    Anke Stoll and Peter Benner. Machine learning for material characterization with an application for predicting mechanical properties.GAMM-Mitteilungen, 44(1):e202100003, March 2021. ISSN 0936-7195, 1522-2608. doi:10.1002/gamm.202100003

    work page doi:10.1002/gamm.202100003 2021

  35. [35]

    Physics-informed few-shot deep learning for elastoplastic constitutive relationships.Engineering Applications of Artificial Intelligence, 126: 106907, November 2023

    Chen Wang, You-quan He, Hong-ming Lu, Jian-guo Nie, and Jian-sheng Fan. Physics-informed few-shot deep learning for elastoplastic constitutive relationships.Engineering Applications of Artificial Intelligence, 126: 106907, November 2023. ISSN 09521976. doi:10.1016/j.engappai.2023.106907

    work page doi:10.1016/j.engappai.2023.106907 2023

  36. [36]

    Waghmare

    Narendra Kumar, Padmini Rajagopalan, Praveen Pankajakshan, Arnab Bhattacharyya, Suchismita Sanyal, Janaki- raman Balachandran, and Umesh V . Waghmare. Machine Learning Constrained with Dimensional Analysis and Scaling Laws: Simple, Transferable, and Interpretable Models of Materials from Small Datasets.Chemistry of Materials, 31(2):314–321, January 2019. ...

    work page doi:10.1021/acs.chemmater.8b02837 2019

  37. [37]

    Oliver, G.M

    W.C. Oliver and G.M. Pharr. Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology.Journal of Materials Research, 19(1):3–20, January 2004. ISSN 0884-2914, 2044-5326. doi:10.1557/jmr.2004.19.1.3

    work page doi:10.1557/jmr.2004.19.1.3 2004

  38. [38]

    Investigation on morphology and chemistry of the Beilby layer on polished fused silica.Ceramics International, 49(11):17116–17122, June 2023

    Zehua Wu, Gang Li, Yong Jia, Qipeng Lv, Songwen Deng, and Yuqi Jin. Investigation on morphology and chemistry of the Beilby layer on polished fused silica.Ceramics International, 49(11):17116–17122, June 2023. ISSN 02728842. doi:10.1016/j.ceramint.2023.02.074

    work page doi:10.1016/j.ceramint.2023.02.074 2023

  39. [39]

    Feng and A

    G. Feng and A. H. W. Ngan. Effects of Creep and Thermal Drift on Modulus Measurement Using Depth- sensing Indentation.Journal of Materials Research, 17(3):660–668, March 2002. ISSN 0884-2914, 2044-5326. doi:10.1557/JMR.2002.0094

    work page doi:10.1557/jmr.2002.0094 2002

  40. [40]

    Relationships between hardness, elastic modulus, and the work of indenta- tion.Applied Physics Letters, 73(5):614–616, August 1998

    Yang-Tse Cheng and Che-Min Cheng. Relationships between hardness, elastic modulus, and the work of indenta- tion.Applied Physics Letters, 73(5):614–616, August 1998. ISSN 0003-6951, 1077-3118. doi:10.1063/1.121873

    work page doi:10.1063/1.121873 1998

  41. [41]

    Bolshakov and G

    A. Bolshakov and G. M. Pharr. Influences of pileup on the measurement of mechanical properties by load and depth sensing indentation techniques.Journal of Materials Research, 13(4):1049–1058, April 1998. ISSN 0884-2914, 2044-5326. doi:10.1557/JMR.1998.0146

    work page doi:10.1557/jmr.1998.0146 1998

  42. [42]

    D. L. Joslin and W. C. Oliver. A new method for analyzing data from continuous depth-sensing microin- dentation tests.Journal of Materials Research, 5(1):123–126, January 1990. ISSN 0884-2914, 2044-5326. doi:10.1557/JMR.1990.0123

    work page doi:10.1557/jmr.1990.0123 1990

  43. [43]

    J., & Wistrich, A

    Dane Morgan and Ryan Jacobs. Opportunities and Challenges for Machine Learning in Materials Science.Annual Review of Materials Research, 50(1):71–103, July 2020. ISSN 1531-7331, 1545-4118. doi:10.1146/annurev- matsci-070218-010015

    work page doi:10.1146/annurev- 2020

  44. [44]

    Data augmentation for deep-learning-based electroen- cephalography.Journal of Neuroscience Methods, 346:108885, December 2020

    Elnaz Lashgari, Dehua Liang, and Uri Maoz. Data augmentation for deep-learning-based electroen- cephalography.Journal of Neuroscience Methods, 346:108885, December 2020. ISSN 01650270. doi:10.1016/j.jneumeth.2020.108885

    work page doi:10.1016/j.jneumeth.2020.108885 2020

  45. [45]

    Progress in determination of the area function of indenters used for nanoindentation.Thin Solid Films, 377–378:394–400, December 2000

    K Herrmann, N.M Jennett, W Wegener, J Meneve, K Hasche, and R Seemann. Progress in determination of the area function of indenters used for nanoindentation.Thin Solid Films, 377–378:394–400, December 2000. ISSN 00406090. doi:10.1016/S0040-6090(00)01367-5

    work page doi:10.1016/s0040-6090(00)01367-5 2000

  46. [46]

    Dauphin, and David Lopez-Paz

    Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. Mixup: Beyond Empirical Risk Minimization, April 2018

    2018

  47. [47]

    Constantinides, K.S

    G. Constantinides, K.S. Ravi Chandran, F.-J. Ulm, and K.J. Van Vliet. Grid indentation analysis of composite microstructure and mechanics: Principles and validation.Materials Science and Engineering: A, 430(1-2): 189–202, August 2006. ISSN 09215093. doi:10.1016/j.msea.2006.05.125

    work page doi:10.1016/j.msea.2006.05.125 2006

  48. [48]

    Hoerl and Robert W

    Arthur E. Hoerl and Robert W. Kennard. Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12(1):55–67, February 1970. ISSN 0040-1706, 1537-2723. doi:10.1080/00401706.1970.10488634. 18 Data-Efficient ISE Correction in Steels Using ML and Physics-Guided AugmentationA PREPRINT

    work page doi:10.1080/00401706.1970.10488634 1970

  49. [49]

    Machine Learning , author =

    Leo Breiman. Random Forests.Machine Learning, 45(1):5–32, October 2001. ISSN 0885-6125, 1573-0565. doi:10.1023/A:1010933404324

    work page doi:10.1023/a:1010933404324 2001

  50. [50]

    XGBoost: A Scalable Tree Boosting System

    Tianqi Chen and Carlos Guestrin. XGBoost: A Scalable Tree Boosting System. InProceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 785–794, San Francisco California USA, August 2016. ACM. ISBN 978-1-4503-4232-2. doi:10.1145/2939672.2939785

    work page doi:10.1145/2939672.2939785 2016

  51. [51]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization, January 2017

    2017

  52. [52]

    G. E. Hinton and R. R. Salakhutdinov. Reducing the Dimensionality of Data with Neural Networks.Science, 313 (5786):504–507, July 2006. ISSN 0036-8075, 1095-9203. doi:10.1126/science.1127647

    work page doi:10.1126/science.1127647 2006

  53. [53]

    A Unified Approach to Interpreting Model Predictions, 2017

    Scott Lundberg and Su-In Lee. A Unified Approach to Interpreting Model Predictions, 2017

    2017

  54. [54]

    Andrea Cristina McGlinchey and Peter J

    Scott M. Lundberg, Gabriel Erion, Hugh Chen, Alex DeGrave, Jordan M. Prutkin, Bala Nair, Ronit Katz, Jonathan Himmelfarb, Nisha Bansal, and Su-In Lee. From local explanations to global understanding with explainable AI for trees.Nature Machine Intelligence, 2(1):56–67, January 2020. ISSN 2522-5839. doi:10.1038/s42256-019-0138-9

    work page doi:10.1038/s42256-019-0138-9 2020

  55. [55]

    Williams

    Hervé Abdi and Lynne J. Williams. Principal component analysis.WIREs Computational Statistics, 2(4):433–459, July 2010. ISSN 1939-5108, 1939-0068. doi:10.1002/wics.101

    work page doi:10.1002/wics.101 2010

  56. [56]

    Newbury* and Nicholas W

    Dale E. Newbury* and Nicholas W. M. Ritchie. Is Scanning Electron Microscopy/Energy Dispersive X-ray Spectrometry (SEM/EDS) Quantitative?Scanning, 35(3):141–168, May 2013. ISSN 0161-0457, 1932-8745. doi:10.1002/sca.21041

    work page doi:10.1002/sca.21041 2013

  57. [57]

    ASM International, Materials Park, 2nd ed edition, 2015

    George Krauss, editor.Steels: Processing, Structure, and Performance. ASM International, Materials Park, 2nd ed edition, 2015. ISBN 978-1-62708-084-2

    2015

  58. [58]

    Hardie, Steve G

    Christopher D. Hardie, Steve G. Roberts, and Andy J. Bushby. Understanding the effects of ion irradiation using nanoindentation techniques.Journal of Nuclear Materials, 462:391–401, July 2015. ISSN 00223115. doi:10.1016/j.jnucmat.2014.11.066

    work page doi:10.1016/j.jnucmat.2014.11.066 2015

  59. [59]

    Yang and H

    B. Yang and H. Vehoff. Dependence of nanohardness upon indentation size and grain size – A local examination of the interaction between dislocations and grain boundaries.Acta Materialia, 55(3):849–856, February 2007. ISSN 13596454. doi:10.1016/j.actamat.2006.09.004

    work page doi:10.1016/j.actamat.2006.09.004 2007

  60. [60]

    Trelewicz, C.A

    Jason R. Trelewicz and Christopher A. Schuh. The Hall–Petch breakdown in nanocrystalline metals: A crossover to glass-like deformation.Acta Materialia, 55(17):5948–5958, October 2007. ISSN 13596454. doi:10.1016/j.actamat.2007.07.020. 19

    work page doi:10.1016/j.actamat.2007.07.020 2007