The stress–strain constitutive relationship of materials is a critical indicator for assessing their mechanical properties and serves as a vital bridge connecting microstructures with macroscopic mechanical behaviors. This study focuses on 106 metal alloy materials, employing the finite element method to establish uniaxial tensile numerical models, thereby simulating and obtaining a comprehensive dataset of stress–strain relationship curves. Innovatively, this research proposes a MA-LSTM-DLnet prediction model that integrates multi-head attention mechanisms with long short-term memory networks, based on deep learning methodologies. The model utilizes conventional key parameters such as material strain, yield strength, tensile strength, elastic modulus, density, strength coefficient, and stress as input features to construct a multi-dimensional training dataset. It particularly explores the learning capabilities and predictive performance of the MA-LSTM-DLnet model under both complete and limited feature datasets. The results demonstrate that the trained MA-LSTM-DLnet model can accurately predict the stress–strain constitutive relationship curves of new materials by merely acquiring conventional physical parameters like yield strength, tensile strength, and elastic modulus, with a similarity exceeding 95% compared to test data. Compared to traditional empirical formulas and numerical simulation methods, the main innovation of this study lies in eliminating key parameters that are difficult to determine in mathematical models (such as the strength coefficient in the Ludwik isotropic hardening model), significantly reducing repetitive work dependent on experiments to obtain constitutive relationships, and achieving efficient and precise material performance evaluation.
La relation constitutive contrainte-déformation des matériaux est un indicateur essentiel pour évaluer leurs propriétés mécaniques, servant de lien crucial entre les microstructures et les comportements mécaniques macroscopiques. Cette étude porte sur 106 alliages métalliques, utilisant la méthode des éléments finis pour établir des modèles numériques de traction uniaxiale, permettant ainsi de simuler et d’obtenir un ensemble de données complet de courbes contrainte-déformation. De manière innovante, cette recherche propose un modèle de prédiction MA-LSTM-DLnet intégrant des mécanismes d’attention multi-tête avec des réseaux de mémoire courte et longue (LSTM), basé sur des méthodologies d’apprentissage profond. Le modèle utilise des paramètres clés conventionnels tels que la déformation du matériau, la limite d’élasticité, la résistance à la traction, le module d’élasticité, la densité, le coefficient de résistance et la contrainte comme caractéristiques d’entrée pour construire un ensemble de données d’entraînement multidimensionnel. L’étude explore particulièrement les capacités d’apprentissage et les performances prédictives du modèle MA-LSTM-DLnet, aussi bien avec des ensembles de données complets qu’avec des données limitées. Les résultats démontrent que le modèle entraîné peut prédire avec précision les courbes de relations constitutives contrainte-déformation de nouveaux matériaux, en se basant uniquement sur des paramètres physiques conventionnels comme la limite d’élasticité, la résistance à la traction et le module d’élasticité, avec une similarité dépassant 95 % par rapport aux données de test. Comparé aux formules empiriques traditionnelles et aux méthodes de simulation numérique, la principale innovation de cette étude réside dans l’élimination de paramètres clés difficiles à déterminer dans les modèles mathématiques (comme le coefficient de résistance dans le modèle d’écrouissage isotrope de Ludwik), réduisant ainsi considérablement le travail expérimental répétitif nécessaire pour obtenir les relations constitutives, et permettant une évaluation efficace et précise des performances des matériaux.
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Accepted:
Published online:
Mots-clés : Apprentissage profond, Contrainte-déformation, Mécanisme d’attention, Réseau de mémoire courte et longue (LSTM), Prédiction des performances des matériaux
Lei Han 1; Chunmei Duan 2, 3; Taochuan Zhang 2, 3
CC-BY 4.0
@article{CRMECA_2025__353_G1_953_0,
author = {Lei Han and Chunmei Duan and Taochuan Zhang},
title = {Predicting stress{\textendash}strain constitutive relationships: a deep learning approach based on multi-head attention mechanism and long short-term memory networks},
journal = {Comptes Rendus. M\'ecanique},
pages = {953--988},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {353},
doi = {10.5802/crmeca.315},
language = {en},
}
TY - JOUR AU - Lei Han AU - Chunmei Duan AU - Taochuan Zhang TI - Predicting stress–strain constitutive relationships: a deep learning approach based on multi-head attention mechanism and long short-term memory networks JO - Comptes Rendus. Mécanique PY - 2025 SP - 953 EP - 988 VL - 353 PB - Académie des sciences, Paris DO - 10.5802/crmeca.315 LA - en ID - CRMECA_2025__353_G1_953_0 ER -
%0 Journal Article %A Lei Han %A Chunmei Duan %A Taochuan Zhang %T Predicting stress–strain constitutive relationships: a deep learning approach based on multi-head attention mechanism and long short-term memory networks %J Comptes Rendus. Mécanique %D 2025 %P 953-988 %V 353 %I Académie des sciences, Paris %R 10.5802/crmeca.315 %G en %F CRMECA_2025__353_G1_953_0
Lei Han; Chunmei Duan; Taochuan Zhang. Predicting stress–strain constitutive relationships: a deep learning approach based on multi-head attention mechanism and long short-term memory networks. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 953-988. doi: 10.5802/crmeca.315
[1] Thermo-elastic material parameters identification using modified error in constitutive equation approach, Inverse Probl. Sci. Eng., Volume 26 (2018), pp. 1356-1382 | DOI | Zbl
[2] Constitutive equations for thermo-elasto-plastic metallic materials undergoing large temperature variations, Mech. Mater., Volume 181 (2023), 104637 | DOI
[3] Determination of a material constitutive equation that includes the effects of strain rate and temperature using an optimization method, Ocean Eng., Volume 287 (2023), 115719 | DOI
[4] High-temperature flow behaviour and constitutive equations for a TC17 titanium alloy, High Temp. Mater. Processes, Volume 38 (2019), pp. 168-177 | DOI
[5] Critical state elastoplastic constitutive model of angular-shaped and fragile granular materials, Mar. Georesour. Geotechnol., Volume 39 (2020), pp. 1-14
[6] Extracting true stresses and strains from nominal stresses and strains in tensile testing, Strain, Volume 57 (2021), e12396 | DOI
[7] Use of hardness, PIP and tensile testing to obtain stress–strain relationships for metals, Mech. Mater., Volume 187 (2023), 104846 | DOI
[8] Comparison analysis of dynamic modulus of asphalt mixture: indirect tension and uniaxial compression test, Transportmetrica A: Transp. Sci., Volume 15 (2019), pp. 165-178 | DOI
[9] Calculation method for cold flow stress of Al6082 based on tensile test and compression test results, Int. J Precis. Eng. Man., Volume 22 (2021), pp. 1337-1344 | DOI
[10] Measurement of true stress-true strain curve up to large strain extent at elevated temperature in Ti-6Al-4V alloy with image analysis tensile test method, J. Jpn. Soc. Technol. Plast., Volume 62 (2021), pp. 177-182 | DOI
[11] Description of stress–strain curves by three parameters (1943) no. 93R10904 (Technical Note)
[12] Determination of Johnson-Cook constitutive equation for Inconel 601, J. Mech. Sci. Technol., Volume 32 (2018), pp. 1569-1574 | DOI
[13] Determination of mechanical properties from sharp dynamic indentation, J. Strain Anal. Eng. Des., Volume 57 (2021), pp. 607-613 | DOI
[14] A general constitutive model of soft elastomers, J. Mech. Phys. Solids, Volume 117 (2018), pp. 110-122 | DOI
[15] Temperature and strain rate dependent constitutive model of TRIP steels for low-temperature applications, Comput. Mater Sci., Volume 50 (2011), pp. 2014-2027 | DOI
[16] Constitutive modeling for intercalated PMMA/clay nanocomposite foams, Polym. Eng. Sci., Volume 46 (2010), pp. 1787-1796 | DOI
[17] Calibration of constitutive models using genetic algorithms, Mech. Mater., Volume 189 (2024), p. 1.1-1.10 | DOI
[18] The static constitutive model of fiber reinforced cellular materials, Constr. Build Mater., Volume 187 (2018), pp. 903-906 | DOI
[19] A review of physically based and thermodynamically based constitutive models for soft materials, J. Appl. Mech.: Transac. ASME, Volume 87 (2020), 110801 | DOI
[20] -GM: a purely micromechanical constitutive model for granular materials, Mech. Mater., Volume 126 (2018), pp. 57-74 | DOI
[21] Deep alloys: metal materials empowered by deep learning, Mat. Sci. Semicon. Proc., Volume 179 (2024), 108514 | DOI
[22] Cross-modal learning for material perception using deep extreme learning machine, Int. J. Mach. Learn. Cyb., Volume 11 (2020), pp. 813-823 | DOI
[23] Improving direct physical properties prediction of heterogeneous materials from imaging data via convolutional neural network and a morphology-aware generative model, Comput. Mater. Sci., Volume 150 (2017), pp. 212-221 | DOI
[24] Accelerating materials discovery with bayesian optimization and graph deep learning, Mater. Today, Volume 51 (2021), pp. 126-135 | DOI
[25] Deep learning based automated fracture identification in material characterization experiments, Adv. Eng. Inform., Volume 60 (2024), 102402 | DOI
[26] et al. Deep learning in two-dimensional materials: characterization, prediction, and design, Front. Phys.-Beijing, Volume 19 (2024), 53601
[27] Deep learning unlocks X-ray microtomography segmentation of multiclass microdamage in heterogeneous materials, Adv. Mater., Volume 34 (2022), 2107817 | DOI
[28] Deep learning method for predicting the strengths of microcracked brittle materials, Eng. Fract. Mech., Volume 271 (2022), 108600
[29] Deep learning for time series-based acoustic emission damage classification in composite materials, Russ. J. Nondestruct.+, Volume 59 (2023), pp. 665-676
[30] Deep learning model for early prediction of material fracture in tensile testing, Neural Comput. Appl., Volume 36 (2024), pp. 10461-10474 | DOI
[31] A combined machine learning and numerical approach for evaluating the uncertainty of 3D angle-interlock woven composites, Compos. Struct., Volume 294 (2022), 115726 | DOI
[32] Adaptive deep homogenization theory for periodic heterogeneous materials, Compos. Struct., Volume 340 (2024), 118171
[33] Deep learning, reinforcement learning, and world models, Neural Netw.: Official J. Int. Neural Netw. Soc., Volume 152 (2022), pp. 267-275 | DOI
[34] Deep learning of temperature–dependent stress–strain hardening curves, C. R. Mec., Volume 351 (2023), pp. 151-170 | DOI
[35] Estimation of texture-dependent stress–strain curve and r-value of aluminum alloy sheet using deep learning+1, Mater. Trans., Volume 61 (2020), pp. 2276-2283 | DOI
[36] Application of LSTM approach for modelling stress–strain behaviour of soil, Appl. Soft Comput., Volume 100 (2021), 106959 | DOI
[37] Prediction of local elasto-plastic stress and strain fields in a two-phase composite microstructure using a deep convolutional neural network, Comput. Method Appl. M, Volume 421 (2024), 116816 | DOI | Zbl
[38] Data-driven bio-mimetic composite design: direct prediction of stress–strain curves from structures using cGANs, J. Mech Phys. Solids, Volume 193 (2024), 105857 | DOI
[39] Advanced meta-modeling framework combining machine learning and model order reduction towards real-time virtual testing of woven composite laminates in nonlinear regime, Compos. Sci. Technol., Volume 262 (2025), 111055 | DOI
[40] Multiscale thermodynamics-informed neural networks (MuTINN) towards fast and frugal inelastic computation of woven composite structures, J. Mech. Phys. Solids, Volume 186 (2024), 105604 | DOI | MR
[41] A survey on LSTM memristive neural network architectures and applications, Eur. Phys. J. Spec. Top., Volume 228 (2019), pp. 2313-2324 | DOI
[42] A review on the long short-term memory model, Artif. Intell. Rev.: An Int. Sci. Eng. J., Volume 53 (2020), pp. 5929-5955 | DOI
[43] et al. Transformers in computational visual media: a survey, Comput. Vis. Media, Volume 8 (2022), pp. 33-62 | DOI
[44] Introduction to Elasticity and Plasticity, Tsinghua University Press, Beijing, 2013
[45] Experimental study and parametric analysis on the stability behavior of 7A04 high-strength aluminum alloy angle columns under axial compression, Thin Wall. Struct., Volume 108 (2016), pp. 305-320 | DOI
[46] Lattice structures of Cu–Cr–Zr copper alloy by selective laser melting: microstructures, mechanical properties and energy absorption, Mater. Design, Volume 187 (2020), 108406
[47] Full-range stress–strain curves for aluminum alloys, J. Struct. Eng., Volume 147 (2021), 04021060
[48] Strain hardening exponents and strength coefficients for aeroengine isotropic metallic materials—a reverse engineering approach, J. Mech. Behav. Mater., Volume 23 (2014), pp. 101-106 | DOI
[49] Estimation of the strength coefficient and strain hardening exponent from monotonic tensile properties of steels, Int. J. Steel Struct., Volume 19 (2019), pp. 1951-1968 | DOI
[50] Attention is all you need, 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA (2017)
[51] Long short-term memory, Neural Comput., Volume 9 (1997), pp. 1735-1780 | DOI
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