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Identification of material parameters in low-data limit: application to gradient-enhanced continua

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Abstract

Due to the growing trend towards miniaturization, small-scale manufacturing processes have become widely used in various engineering fields to manufacture miniaturized products. These processes generally exhibit complex size effects, making the behavior of materials highly dependent on their geometric dimensions. As a result, accurate understanding and modeling of such effects are crucial for optimizing manufacturing outcomes and achieving high-performance final products. To this end, advanced gradient-enhanced plasticity theories have emerged as powerful tools for capturing these complex phenomena, offering a level of accuracy significantly greater than that provided by classical plasticity approaches. However, these advanced theories often require the identification of a large number of material parameters, which poses a significant challenge due to limited experimental data at small scales and high computation costs. The present paper aims at evaluating and comparing the effectiveness of various optimization techniques, including evolutionary algorithm, response surface methodology and Bayesian optimization, in identifying the material parameter of a recent flexible gradient-enhanced plasticity model developed by the authors. The paper findings represent an attempt to bridge the gap between advanced material behavior theories and their practical industrial applications, by offering insights into efficient and reliable material parameter identification procedures.

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References

  1. Aifantis EC (1984) On the Microstructural Origin of Certain Inelastic Models. J Eng Mater Technol 106:326–330

    Article  Google Scholar 

  2. Gurtin ME (2002) A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J Mech Phys Solids 50:5–32

    Article  MathSciNet  Google Scholar 

  3. Panteghini A, Bardella L, Niordson CF (2019) A potential for higher-order phenomenological strain gradient plasticity to predict reliable response under non-proportional loading. Proc Royal Soc A Math Phys Eng Sci 475:20190258

    MathSciNet  Google Scholar 

  4. Forest S (2020) Continuum thermomechanics of nonlinear micromorphic, strain and stress gradient media. Phil Trans R Soc A Math Phys Eng Sci 378:20190169

    Article  MathSciNet  Google Scholar 

  5. Jebahi M, Cai L, Abed-Meraim F (2020) Strain gradient crystal plasticity model based on generalized non-quadratic defect energy and uncoupled dissipation. Int J Plast 126:102617

    Article  Google Scholar 

  6. Yuan H, Chen J (2001) Identification of the intrinsic material length in gradient plasticity theory from micro-indentation tests. Int J Solids Struct

  7. Abu Al-Rub RK, Voyiadjis GZ (2004) Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. Int J Plast 20:1139–1182

    Article  Google Scholar 

  8. Peerlings RHJ, Geers MGD, de Borst R, Brekelmans WAM (2001) A critical comparison of nonlocal and gradient-enhanced softening continua. Int J Solids Struct 38:7723–7746

    Article  Google Scholar 

  9. Gurtin ME, Anand L, Lele SP (2007) Gradient single-crystal plasticity with free energy dependent on dislocation densities. J Mech Phys Solids 55:1853–1878

    Article  MathSciNet  Google Scholar 

  10. Poh LH, Peerlings RHJ, Geers MGD, Swaddiwudhipong S (2011) An implicit tensorial gradient plasticity model - Formulation and comparison with a scalar gradient model. Int J Solids Struct 48:2595–2604

    Article  Google Scholar 

  11. Pardoen T, Massart TJ (2012) Interface controlled plastic flow modelled by strain gradient plasticity theory. Comptes Rendus - Mecanique 340:247–260

    Article  Google Scholar 

  12. Fleck NA, Hutchinson JW, Willis JR (2015) Guidelines for Constructing Strain Gradient Plasticity Theories. J Appl Mech 82:071002

    Article  Google Scholar 

  13. Bayerschen E, Böhlke T (2016) Power-Law Defect Energy in a Single-Crystal Gradient Plasticity Framework: A Computational Study. Comput Mech 58:13–27

    Article  MathSciNet  Google Scholar 

  14. Petryk H, Stupkiewicz S (2016) A Minimal Gradient-Enhancement of the Classical Continuum Theory of Crystal Plasticity. Part I: The Hardening Law. Arch Mech 68:459–485

    MathSciNet  Google Scholar 

  15. Martínez-Pañeda E, Niordson CF, Bardella L (2016) A Finite Element Framework for Distortion Gradient Plasticity with Applications to Bending of Thin Foils. Int J Solids Struct 96:288–299

  16. Lebensohn RA, Needleman A (2016) Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms. J Mech Phys Solids 97:333–351

    Article  MathSciNet  Google Scholar 

  17. Panteghini A, Bardella L (2020) Modelling the cyclic torsion of polycrystalline micron-sized copper wires by distortion gradient plasticity. Philos Mag 100:2352–2364

    Article  Google Scholar 

  18. Russo R, Girot Mata FA, Forest S, Jacquin D (2020) A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes. J Manuf Mater Process 4:87

    Google Scholar 

  19. Cai L, Jebahi M, Abed-Meraim F (2021) Strain Localization Modes within Single Crystals Using Finite Deformation Strain Gradient Crystal Plasticity. Crystals 11:1235

    Article  Google Scholar 

  20. Jebahi M, Forest S (2021) Scalar-based strain gradient plasticity theory to model size-dependent kinematic hardening effects. Continuum Mech Thermodyn

  21. Jebahi M, Forest S (2023) An alternative way to describe thermodynamically-consistent higher-order dissipation within strain gradient plasticity. J Mech Phys Solids 170:105103

    Article  MathSciNet  Google Scholar 

  22. Liu K, Melkote SN (2005) Material Strengthening Mechanisms and Their Contribution to Size Effect in Micro-Cutting. J Manuf Sci Eng 128:730–738

    Article  Google Scholar 

  23. Guha S, Sangal S, Basu S (2014) Numerical investigations of flat punch molding using a higher order strain gradient plasticity theory. Int J Mater Form 7:459–467

    Article  Google Scholar 

  24. Nielsen K, Niordson C, Hutchinson J (2014) Strain gradient effects in periodic flat punch indenting at small scales. Int J Solids Struct 51:3549–3556

    Article  Google Scholar 

  25. Nielsen KL, Niordson CF, Hutchinson JW (2015) Rolling at Small Scales. J Manuf Sci Eng 138

  26. Zhang X, Aifantis K (2015) Interpreting the internal length scale in strain gradient plasticity. Rev Adv Mater Sci 41:72–83

    Google Scholar 

  27. Liu D, Dunstan D (2017) Material length scale of strain gradient plasticity: A physical interpretation. Int J Plast 98:156–174

    Article  Google Scholar 

  28. Begley MR, Hutchinson JW (1998) The mechanics of size-dependent indentation. J Mech Phys Solids 46:2049–2068

    Article  Google Scholar 

  29. Stölken J, Evans A (1998) A microbend test method for measuring the plasticity length scale. Acta Materialia 46:5109–5115

    Article  Google Scholar 

  30. Voyiadjis GZ, Song Y (2019) Strain gradient continuum plasticity theories: Theoretical, numerical and experimental investigations. Int J Plast 121:21–75

    Article  Google Scholar 

  31. Fra̧ś T, Nowak Z, Perzyna P, Pȩcherski R (2011) Identification of the model describing viscoplastic behaviour of high strength metals. Inverse Probl Sci Eng 19:17–30

  32. Gelin J, Ghouati O (1994) An inverse method for determining viscoplastic properties of aluminium alloys. J Mater Process Technol 45:435–440

    Article  Google Scholar 

  33. Herrera-Solaz V, LLorca J, Dogan E, Karaman I, Segurado J (2014) An inverse optimization strategy to determine single crystal mechanical behavior from polycrystal tests: Application to AZ31 Mg alloy. Int J Plast 57:1–15

  34. Meraghni F et al (2014) Parameter identification of a thermodynamic model for superelastic shape memory alloys using analytical calculation of the sensitivity matrix. Eur J Mech A Solids 45:226–237

    Article  Google Scholar 

  35. Saleeb A, Arnold S, Castelli M, Wilt T, Graf W (2001) A general hereditary multimechanism-based deformation model with application to the viscoelastoplastic response of titanium alloys. Int J Plast 17:1305–1350

    Article  Google Scholar 

  36. Andrade-Campos A, Thuillier S, Pilvin P, Teixeira-Dias F (2007) On the determination of material parameters for internal variable thermoelastic–viscoplastic constitutive models. Int J Plast 23:1349–1379

  37. Lewis RM, Torczon V, Trosset MW (2000) Direct search methods: Then and now. J Comput Appl Math 124:191–207

    Article  MathSciNet  Google Scholar 

  38. Kolda TG, Lewis RM, Torczon V (2003) Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods. SIAM Review 45:385–482

    Article  MathSciNet  Google Scholar 

  39. Slowik A, Kwasnicka H (2020) Evolutionary algorithms and their applications to engineering problems. Neural Comput & Applic 32:12363–12379

    Article  Google Scholar 

  40. Nelder JA, Mead R (1965) A Simplex Method for Function Minimization. Comput J 7:308–313

    Article  MathSciNet  Google Scholar 

  41. Lewis RM, Torczon V (1999) Pattern Search Algorithms for Bound Constrained Minimization. SIAM J Optim 9:1082–1099

    Article  MathSciNet  Google Scholar 

  42. Chakraborty A, Eisenlohr P (2017) Evaluation of an inverse methodology for estimating constitutive parameters in face-centered cubic materials from single crystal indentations. Eur J Mech A Solids 66:114–124

    Article  MathSciNet  Google Scholar 

  43. Vaz M, Luersen MA, Muñoz-Rojas PA, Trentin RG (2016) Identification of inelastic parameters based on deep drawing forming operations using a global–local hybrid Particle Swarm approach. Comptes Rendus Mécanique 344:319–334

  44. Agius D et al (2017) Sensitivity and optimisation of the Chaboche plasticity model parameters in strain-life fatigue predictions. Mater Des 118:107–121

    Article  Google Scholar 

  45. Kapoor K et al (2021) Modeling Ti–6Al–4V using crystal plasticity, calibrated with multi-scale experiments, to understand the effect of the orientation and morphology of the \(\alpha \) and \(\beta \) phases on time dependent cyclic loading. J Mech Phys Solids 146:104192

  46. Qu J, Jin Q, Xu B (2005) Parameter identification for improved viscoplastic model considering dynamic recrystallization. Int J Plast 21:1267–1302

    Article  Google Scholar 

  47. Chaparro B, Thuillier S, Menezes L, Manach P, Fernandes J (2008) Material parameters identification: Gradient-based, genetic and hybrid optimization algorithms. Comput Mater Sci 44:339–346

    Article  Google Scholar 

  48. Furukawa T, Sugata T, Yoshimura S, Hoffman M (2002) An automated system for simulation and parameter identification of inelastic constitutive models. Comput Methods Appl Mech Eng 191:2235–2260

    Article  Google Scholar 

  49. Lundstedt T, Seifert E, Abramo L, Thelin B (1998) Experimental design and optimization

  50. Stander N, Craig K, Müllerschön H, Reichert R (2005) Material identification in structural optimization using response surfaces. Struct Multidiscip Optim 29:93–102

    Article  Google Scholar 

  51. Sedighiani K et al (2020) An efficient and robust approach to determine material parameters of crystal plasticity constitutive laws from macro-scale stress–strain curves. Int J Plast 134:102779

  52. Kakaletsis S, Lejeune E, Rausch MK (2023) Can machine learning accelerate soft material parameter identification from complex mechanical test data? Biomech Model Mechanobiol 22:57–70

    Article  Google Scholar 

  53. Greenhill S, Rana S, Gupta S, Vellanki P, Venkatesh S (2020) Bayesian Optimization for Adaptive Experimental Design: A Review. IEEE Access 8:13937–13948

    Article  Google Scholar 

  54. Kuhn J, Spitz J, Sonnweber-Ribic P, Schneider M, Böhlke T (2022) Identifying material parameters in crystal plasticity by Bayesian optimization. Optim Eng 23:1489–1523

    Article  MathSciNet  Google Scholar 

  55. Veasna K, Feng Z, Zhang Q, Knezevic M (2023) Machine learning-based multi-objective optimization for efficient identification of crystal plasticity model parameters. Comput Methods Appl Mech Eng 403:115740

    Article  MathSciNet  Google Scholar 

  56. Shahriari B, Swersky K, Wang Z, Adams RP, de Freitas N (2016) Taking the Human Out of the Loop: A Review of Bayesian Optimization. Proc IEEE 104:148–175

    Article  Google Scholar 

  57. Vrajitoru D (2000) in Large Population or Many Generations for Genetic Algorithms? Implications in Information Retrieval (eds Crestani F, Pasi G) Soft Computing in Information Retrieval: Techniques and Applications Studies in Fuzziness and Soft Computing, 199–222 (Physica-Verlag HD, Heidelberg)

  58. Hansen N, Ostermeier A (2001) Completely Derandomized Self-Adaptation in Evolution Strategies. Evol Comput 9:159–195

  59. Hansen N (2006) in The CMA Evolution Strategy: A Comparing Review (eds Lozano JA, Larrañaga P, Inza I, Bengoetxea E) Towards a New Evolutionary Computation: Advances in the Estimation of Distribution Algorithms Studies in Fuzziness and Soft Computing, 75–102 (Springer, Berlin, Heidelberg)

  60. Hansen N (2023) The CMA Evolution Strategy: A Tutorial. arXiv

  61. Cauvin L, Raghavan B, Bouvier S, Wang X, Meraghni F (2018) Multi-scale investigation of highly anisotropic zinc alloys using crystal plasticity and inverse analysis. Mater Sci Eng A 729:106–118

    Article  Google Scholar 

  62. Sobol’ I (1967) On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys 7:86–112

    Article  MathSciNet  Google Scholar 

  63. Owen AB (1998) Scrambling Sobol’ and Niederreiter–Xing Points. J Complex 14:466–489

  64. Matoušek J (1998) On theL2-Discrepancy for Anchored Boxes. J Complex 14:527–556

    Article  Google Scholar 

  65. Hussain MF, Barton RR, Joshi SB (2002) Metamodeling: Radial basis functions, versus polynomials. Eur J Oper Res 138:142–154

  66. Roy A, Chakraborty S (2020) Support vector regression based metamodel by sequential adaptive sampling for reliability analysis of structures. Reliab Eng Syst Saf 200:106948

    Article  Google Scholar 

  67. Dasari SK, Cheddad A, Andersson P (2019) MacIntyre J, Maglogiannis I, Iliadis L, Pimenidis E (eds) Random Forest Surrogate Models to Support Design Space Exploration in Aerospace Use-Case. (eds MacIntyre J, Maglogiannis I, Iliadis L, Pimenidis E) Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, 532–544 (Springer International Publishing, Cham)

  68. Rasmussen CE, Williams CKI (2006) Gaussian Processes for Machine Learning Adaptive Computation and Machine Learning. MIT Press, Cambridge, Mass

    Google Scholar 

  69. Gardner M, Dorling S (1998) Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences. Atmos Environ 32:2627–2636

  70. Torregrosa S, Champaney V, Ammar A, Herbert V, Chinesta F (2022) Surrogate parametric metamodel based on Optimal Transport. Math Comput Simul 194:36–63

    Article  MathSciNet  Google Scholar 

  71. Sancarlos A, Champaney V, Cueto E, Chinesta F (2023) Regularized regressions for parametric models based on separated representations. Adv Model Simul Eng Sci 10:4

    Article  Google Scholar 

  72. Soares do Amaral JV, Montevechi JAB, Miranda RdC, Junior WTdS (2022) Metamodel-based simulation optimization: A systematic literature review. Simul Model Pract Theory 114:102403

  73. Yang K, Emmerich M, Deutz A, Bäck T (2019) Efficient computation of expected hypervolume improvement using box decomposition algorithms. J Glob Optim 75:3–34

    Article  MathSciNet  Google Scholar 

  74. Auger A, Bader J, Brockhoff D, Zitzler E (2012) Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications. Theor Comput Sci 425:75–103

    Article  MathSciNet  Google Scholar 

  75. Blank J, Deb K (2020) Pymoo: Multi-Objective Optimization in Python. IEEE Access 8:89497–89509

    Article  Google Scholar 

  76. Picheny V et al (2023) Trieste: Efficiently Exploring The Depths of Black-box Functions with TensorFlow. arXiv

  77. Pedregosa F et al (2011) Scikit-learn: Machine Learning in Python. J Mach Learn Res 12:2825–2830

    MathSciNet  Google Scholar 

  78. Chen Z, Liu Y (2022) Individuals redistribution based on differential evolution for covariance matrix adaptation evolution strategy. Sci Rep 12:986

    Article  Google Scholar 

  79. Daulton S, Balandat M, Bakshy E (2020) Differentiable expected hypervolume improvement for parallel multi-objective Bayesian optimization. Proc 34th Int Conf Neural Inf Process Syst 9851–9864

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Acknowledgements

M. Jebahi acknowledges the financial support of the French National Research Agency (ANR) under reference ANR-20-CE08-0010 (SGP-GAPS project https://www.sgpgaps.fr/).

Funding

This work was supported by the French National Research Agency (ANR) under reference ANR-20-CE08-0010 (SGP-GAPS project https://www.sgpgaps.fr/)

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Authors and Affiliations

  1. Arts et Metiers Institute of Technology, CNRS, Université de Lorraine, LEM3, F-57000, Metz, France

    Duc-Vinh Nguyen & Mohamed Jebahi

  2. ESI Group Chair @ PIMM Lab, Arts et Métiers Institute of Technology, 151 Boulevard de l’Hôpital, Paris, F-75013, France

    Victor Champaney & Francisco Chinesta

  3. ESI Group, 3 bis Saarinen, Parc Icade, Immeuble le Seville, Rungis Cedex, 94528, France

    Francisco Chinesta

  4. CNRS@CREATE LTD, 1 Create Way, #08-01 CREATE Tower, Singapore, 138602, Singapore

    Francisco Chinesta

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  1. Duc-Vinh Nguyen
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  2. Mohamed Jebahi
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  3. Victor Champaney
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  4. Francisco Chinesta
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Contributions

Conceptualization: D.V.N., M.J., V.C., F.C.; Methodology: D.V.N., M.J., V.C., F.C.; Formal analysis and investigation: D.V.N., M.J., V.C.; Writing - original draft preparation: D.V.N., M.J.; Writing - review and editing: D.V.N., M.J., V.C., F.C.; All authors read and approved the final manuscript.

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Correspondence to Mohamed Jebahi.

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Nguyen, DV., Jebahi, M., Champaney, V. et al. Identification of material parameters in low-data limit: application to gradient-enhanced continua. Int J Mater Form 17, 10 (2024). https://doi.org/10.1007/s12289-023-01807-7

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