{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T11:13:01Z","timestamp":1762254781522,"version":"3.43.0"},"reference-count":84,"publisher":"ASME International","issue":"11","license":[{"start":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T00:00:00Z","timestamp":1706745600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.asme.org\/publications-submissions\/publishing-information\/legal-policies"}],"funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-22-1-0075"],"award-info":[{"award-number":["FA9550-22-1-0075"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006168","name":"National Nuclear Security Administration","doi-asserted-by":"publisher","award":["DE-NA-0003525"],"award-info":[{"award-number":["DE-NA-0003525"]}],"id":[{"id":"10.13039\/100006168","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["asmedigitalcollection.asme.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2024,11,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Data-driven constitutive modeling frameworks based on neural networks and classical representation theorems have recently gained considerable attention due to their ability to easily incorporate constitutive constraints and their excellent generalization performance. In these models, the stress prediction follows from a linear combination of invariant-dependent coefficient functions and known tensor basis generators. However, thus far the formulations have been limited to stress representations based on the classical Finger\u2013Rivlin\u2013Ericksen form, while the performance of alternative representations has yet to be investigated. In this work, we survey a variety of tensor basis neural network models for modeling hyperelastic materials in a finite deformation context, including a number of so far unexplored formulations which use theoretically equivalent invariants and generators to Finger\u2013Rivlin\u2013Ericksen. Furthermore, we compare potential-based and coefficient-based approaches, as well as different calibration techniques. Nine variants are tested against both noisy and noiseless datasets for three different materials. Theoretical and practical insights into the performance of each formulation are given.<\/jats:p>","DOI":"10.1115\/1.4064650","type":"journal-article","created":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T06:37:27Z","timestamp":1706769447000},"update-policy":"https:\/\/doi.org\/10.1115\/crossmarkpolicy-asme","source":"Crossref","is-referenced-by-count":8,"title":["Stress Representations for Tensor Basis Neural Networks: Alternative Formulations to Finger\u2013Rivlin\u2013Ericksen"],"prefix":"10.1115","volume":"24","author":[{"given":"Jan N.","family":"Fuhg","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University Department of Mechanical Engineering, , Ithaca, NY 14853"},{"name":"Cornell University Department of Mechanical Engineering, , Ithaca, NY 14853"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikolaos","family":"Bouklas","sequence":"additional","affiliation":[{"name":"Cornell University Department of Mechanical Engineering, , Ithaca, NY 14853"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Reese E.","family":"Jones","sequence":"additional","affiliation":[{"name":"Sandia National Laboratories Mechanics of Materials Department, , 7011 East Avenue, Livermore, CA 94550"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"33","published-online":{"date-parts":[[2024,8,6]]},"reference":[{"key":"2025080520020522600_CIT0001","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jcp.2016.05.003","article-title":"Machine Learning Strategies for Systems With Invariance Properties","volume":"318","author":"Ling","year":"2016","journal-title":"J. Comput. Phys."},{"issue":"9\u201310","key":"2025080520020522600_CIT0002","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1080\/14685248.2019.1706742","article-title":"Neural Network Models for the Anisotropic Reynolds Stress Tensor in Turbulent Channel Flow","volume":"21","author":"Fang","year":"2020","journal-title":"J. Turbul."},{"key":"2025080520020522600_CIT0003","doi-asserted-by":"publisher","first-page":"104497","DOI":"10.1016\/j.compfluid.2020.104497","article-title":"Data-Driven Modelling of the Reynolds Stress Tensor Using Random Forests With Invariance","volume":"202","author":"Kaandorp","year":"2020","journal-title":"Comput. Fluids"},{"key":"2025080520020522600_CIT0004","doi-asserted-by":"publisher","first-page":"104532","DOI":"10.1016\/j.jmps.2021.104532","article-title":"Metamodeling of Constitutive Model Using Gaussian Process Machine Learning","volume":"154","author":"Wang","year":"2021","journal-title":"J. Mech. Phys. Solids"},{"issue":"7","key":"2025080520020522600_CIT0005","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1557\/mrs.2019.156","article-title":"Data-Driven Discovery of Formulas by Symbolic Regression","volume":"44","author":"Sun","year":"2019","journal-title":"MRS Bull."},{"key":"2025080520020522600_CIT0006","doi-asserted-by":"publisher","first-page":"100052","DOI":"10.1016\/j.apples.2021.100052","article-title":"Application of Symbolic Regression for Constitutive Modeling of Plastic Deformation","volume":"6","author":"Kabliman","year":"2021","journal-title":"Appl. Eng. Sci."},{"key":"2025080520020522600_CIT0007","doi-asserted-by":"publisher","first-page":"106557","DOI":"10.1016\/j.compstruc.2021.106557","article-title":"Development of Interpretable, Data-Driven Plasticity Models With Symbolic Regression","volume":"252","author":"Bomarito","year":"2021","journal-title":"Comput. Struct."},{"key":"2025080520020522600_CIT0008","doi-asserted-by":"publisher","first-page":"104742","DOI":"10.1016\/j.jmps.2021.104742","article-title":"Establish Algebraic Data-Driven Constitutive Models for Elastic Solids With a Tensorial Sparse Symbolic Regression Method and a Hybrid Feature Selection Technique","volume":"159","author":"Wang","year":"2022","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0009","doi-asserted-by":"publisher","first-page":"111967","DOI":"10.1016\/j.commatsci.2022.111967","article-title":"Establishing a Data-Driven Strength Model for \u03b2-Tin by Performing Symbolic Regression Using Genetic Programming","volume":"218","author":"de Oca Zapiain","year":"2023","journal-title":"Comput. Mater. Sci."},{"issue":"9","key":"2025080520020522600_CIT0010","doi-asserted-by":"publisher","first-page":"2093","DOI":"10.1002\/nme.7203","article-title":"Automatic Generation of Interpretable Hyperelastic Material Models by Symbolic Regression","volume":"124","author":"Abdusalamov","year":"2023","journal-title":"Int. J. Numer. Methods Eng."},{"key":"2025080520020522600_CIT0011","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1007\/s00466-019-01723-1","article-title":"A Cooperative Game for Automated Learning of Elasto-plasticity Knowledge Graphs and Models With AI-Guided Experimentation","volume":"64","author":"Wang","year":"2019","journal-title":"Comput. Mech."},{"issue":"5923","key":"2025080520020522600_CIT0012","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1126\/science.1165893","article-title":"Distilling Free-Form Natural Laws From Experimental Data","volume":"324","author":"Schmidt","year":"2009","journal-title":"Science"},{"key":"2025080520020522600_CIT0013","doi-asserted-by":"publisher","first-page":"105076","DOI":"10.1016\/j.jmps.2022.105076","article-title":"NN-EUCLID Deep-Learning Hyperelasticity Without Stress Data","volume":"169","author":"Thakolkaran","year":"2022","journal-title":"J. Mech. Phys. Solids"},{"issue":"1","key":"2025080520020522600_CIT0014","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-022-00752-4","article-title":"Discovering Plasticity Models Without Stress Data","volume":"8","author":"Flaschel","year":"2022","journal-title":"npj Comput. Mater."},{"key":"2025080520020522600_CIT0015","doi-asserted-by":"publisher","first-page":"115225","DOI":"10.1016\/j.cma.2022.115225","article-title":"Bayesian-EUCLID, Discovering Hyperelastic Material Laws With Uncertainties","volume":"398","author":"Joshi","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0016","first-page":"229","article-title":"Representation of Material Behavior: Neural Network-Based Models","author":"Wu","year":"1990"},{"key":"2025080520020522600_CIT0017","first-page":"701","article-title":"Material Modeling With Neural Networks","author":"Ghaboussi","year":"1990"},{"issue":"1","key":"2025080520020522600_CIT0018","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1061\/(ASCE)0733-9399(1991)117:1(132)","article-title":"Knowledge-Based Modeling of Material Behavior With Neural Networks","volume":"117","author":"Ghaboussi","year":"1991","journal-title":"J. Eng. Mech."},{"issue":"28\u201330","key":"2025080520020522600_CIT0019","doi-asserted-by":"publisher","first-page":"3265","DOI":"10.1016\/S0045-7825(03)00350-5","article-title":"Artificial Neural Network as an Incremental Non-linear Constitutive Model for a Finite Element Code","volume":"192","author":"Lefik","year":"2003","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1\u20133","key":"2025080520020522600_CIT0020","doi-asserted-by":"publisher","first-page":"608","DOI":"10.1016\/j.cma.2006.06.006","article-title":"Characterizing Rate-Dependent Material Behaviors in Self-learning Simulation","volume":"196","author":"Jung","year":"2006","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0021","doi-asserted-by":"publisher","first-page":"113008","DOI":"10.1016\/j.cma.2020.113008","article-title":"A Machine Learning Based Plasticity Model Using Proper Orthogonal Decomposition","volume":"365","author":"Huang","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0022","doi-asserted-by":"publisher","first-page":"114217","DOI":"10.1016\/j.cma.2021.114217","article-title":"Local Approximate Gaussian Process Regression for Data-Driven Constitutive Models: Development and Comparison With Neural Networks","volume":"388","author":"Fuhg","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0023","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.jmps.2019.03.004","article-title":"Exploring the 3D Architectures of Deep Material Network in Data-Driven Multiscale Mechanics","volume":"127","author":"Liu","year":"2019","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0024","doi-asserted-by":"publisher","first-page":"112875","DOI":"10.1016\/j.cma.2020.112875","article-title":"SO (3)-Invariance of Informed-Graph-Based Deep Neural Network for Anisotropic Elastoplastic Materials","volume":"363","author":"Heider","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0025","doi-asserted-by":"publisher","first-page":"110072","DOI":"10.1016\/j.jcp.2020.110072","article-title":"Learning Constitutive Relations Using Symmetric Positive Definite Neural Networks","volume":"428","author":"Xu","year":"2021","journal-title":"J. Comput. Phys."},{"key":"2025080520020522600_CIT0026","doi-asserted-by":"publisher","first-page":"114124","DOI":"10.1016\/j.cma.2021.114124","article-title":"Learning Viscoelasticity Models From Indirect Data Using Deep Neural Networks","volume":"387","author":"Xu","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0027","doi-asserted-by":"publisher","first-page":"115930","DOI":"10.1016\/j.cma.2023.115930","article-title":"Modular Machine Learning-Based Elastoplasticity: Generalization in the Context of Limited Data","volume":"407","author":"Fuhg","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0028","doi-asserted-by":"publisher","first-page":"104925","DOI":"10.1016\/j.euromechsol.2023.104925","article-title":"Enhancing Phenomenological Yield Functions With Data: Challenges and Opportunities","author":"Fuhg","year":"2023","journal-title":"Euro. J. Mech.-A\/Solids"},{"issue":"3","key":"2025080520020522600_CIT0029","doi-asserted-by":"publisher","DOI":"10.31614\/cmes.2018.04285","article-title":"Machine Learning Models of Plastic Flow Based on Representation Theory","volume":"117","author":"Jones","year":"2018","journal-title":"CMES-Comput. Model. Eng. Sci."},{"issue":"3","key":"2025080520020522600_CIT0030","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1615\/JMachLearnModelComput.2022042917","article-title":"A Neural Ordinary Differential Equation Framework for Modeling Inelastic Stress Response Via Internal State Variables","volume":"3","author":"Jones","year":"2022","journal-title":"J. Mach. Learn. Model. Comput."},{"key":"2025080520020522600_CIT0031","doi-asserted-by":"publisher","first-page":"104703","DOI":"10.1016\/j.jmps.2021.104703","article-title":"Polyconvex Anisotropic Hyperelasticity With Neural Networks","volume":"159","author":"Klein","year":"2022","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0032","doi-asserted-by":"publisher","first-page":"105022","DOI":"10.1016\/j.jmps.2022.105022","article-title":"Learning Hyperelastic Anisotropy From Data Via a Tensor Basis Neural Network","volume":"168","author":"Fuhg","year":"2022","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0033","doi-asserted-by":"publisher","first-page":"104277","DOI":"10.1016\/j.jmps.2020.104277","article-title":"Thermodynamics-Based Artificial Neural Networks for Constitutive Modeling","volume":"147","author":"Masi","year":"2021","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0034","doi-asserted-by":"publisher","first-page":"110010","DOI":"10.1016\/j.jcp.2020.110010","article-title":"Constitutive Artificial Neural Networks: A Fast and General Approach to Predictive Data-Driven Constitutive Modeling by Deep Learning","volume":"429","author":"Linka","year":"2021","journal-title":"J. Comput. Phys."},{"key":"2025080520020522600_CIT0035","doi-asserted-by":"publisher","first-page":"113299","DOI":"10.1016\/j.cma.2020.113299","article-title":"Geometric Deep Learning for Computational Mechanics Part I: Anisotropic Hyperelasticity","volume":"371","author":"Vlassis","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"2025080520020522600_CIT0036","doi-asserted-by":"publisher","first-page":"035005","DOI":"10.1088\/2632-2153\/ab9299","article-title":"Prediction of the Evolution of the Stress Field of Polycrystals Undergoing Elastic-Plastic Deformation With a Hybrid Neural Network Model","volume":"1","author":"Frankel","year":"2020","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"2025080520020522600_CIT0037","doi-asserted-by":"publisher","first-page":"114915","DOI":"10.1016\/j.cma.2022.114915","article-title":"On Physics-Informed Data-Driven Isotropic and Anisotropic Constitutive Models Through Probabilistic Machine Learning and Space-filling Sampling","volume":"394","author":"Fuhg","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0038","doi-asserted-by":"publisher","first-page":"105363","DOI":"10.1016\/j.jmps.2023.105363","article-title":"Neural Networks Meet Hyperelasticity: A Guide to Enforcing Physics","author":"Linden","year":"2023","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0039","doi-asserted-by":"publisher","first-page":"109099","DOI":"10.1016\/j.commatsci.2019.109099","article-title":"Predicting the Mechanical Response of Oligocrystals With Deep Learning","volume":"169","author":"Frankel","year":"2019","journal-title":"Comput. Mater. Sci."},{"key":"2025080520020522600_CIT0040","first-page":"1","article-title":"The Non-linear Field Theories of Mechanics","volume-title":"The Non-linear Field Theories of Mechanics","author":"Truesdell","year":"1965"},{"issue":"5","key":"2025080520020522600_CIT0041","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1007\/s00466-022-02260-0","article-title":"FE ANN: An Efficient Data-Driven Multiscale Approach Based on Physics-Constrained Neural Networks and Automated Data Mining","volume":"71","author":"Kalina","year":"2023","journal-title":"Comput. Mech."},{"key":"2025080520020522600_CIT0042","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/BF00279992","article-title":"Convexity Conditions and Existence Theorems in Nonlinear Elasticity","volume":"63","author":"Ball","year":"1976","journal-title":"Archive Rat. Mech. Anal."},{"key":"2025080520020522600_CIT0043","doi-asserted-by":"publisher","first-page":"115731","DOI":"10.1016\/j.cma.2022.115731","article-title":"A New Family of Constitutive Artificial Neural Networks Towards Automated Model Discovery","volume":"403","author":"Linka","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2025080520020522600_CIT0044","doi-asserted-by":"publisher","first-page":"115248","DOI":"10.1016\/j.cma.2022.115248","article-title":"Data-Driven Tissue Mechanics With Polyconvex Neural Ordinary Differential Equations","volume":"398","author":"Tac","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"2025080520020522600_CIT0045","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s00466-019-01731-1","article-title":"Model-Free Data-Driven Methods in Mechanics: Material Data Identification and Solvers","volume":"64","author":"Stainier","year":"2019","journal-title":"Comput. Mech."},{"key":"2025080520020522600_CIT0046","doi-asserted-by":"publisher","first-page":"1281","DOI":"10.1007\/s00466-013-0876-1","article-title":"Lie-Group Interpolation and Variational Recovery for Internal Variables","volume":"52","author":"Mota","year":"2013","journal-title":"Comput. Mech."},{"year":"1894","author":"Finger","key":"2025080520020522600_CIT0047"},{"key":"2025080520020522600_CIT0048","first-page":"323","article-title":"Stress-Deformation Relations for Isotropic Materials","volume":"4","author":"Rivlin","year":"1955","journal-title":"J. Ration. Mech. Anal."},{"volume-title":"An Introduction to Continuum Mechanics","year":"1982","author":"Gurtin","key":"2025080520020522600_CIT0049"},{"issue":"1","key":"2025080520020522600_CIT0050","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1615\/JMachLearnModelComput.2020033325","article-title":"Tensor Basis Gaussian Process Models of Hyperelastic Materials","volume":"1","author":"Frankel","year":"2020","journal-title":"J. Mach. Learn. Model. Comput."},{"issue":"12","key":"2025080520020522600_CIT0051","doi-asserted-by":"publisher","first-page":"2445","DOI":"10.1016\/S0022-5096(00)00023-5","article-title":"An Invariant Basis for Natural Strain Which Yields Orthogonal Stress Response Terms in Isotropic Hyperelasticity","volume":"48","author":"Criscione","year":"2000","journal-title":"J. Mech. Phys. Solids"},{"key":"2025080520020522600_CIT0052","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970340","volume-title":"Topics in Finite Elasticity","author":"Gurtin","year":"1981"},{"key":"2025080520020522600_CIT0053","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198567783.001.0001","volume-title":"Finite Elasticity Theory","author":"Steigmann","year":"2017"},{"key":"2025080520020522600_CIT0054","first-page":"146","article-title":"Input Convex Neural Networks","author":"Amos","year":"2017"},{"key":"2025080520020522600_CIT0055","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1512\/iumj.1959.8.58033","article-title":"The Derivation of Stress-Deformation Relations for a Stokesian Fluid","author":"Serrin","year":"1959","journal-title":"J. Math. Mech."},{"key":"2025080520020522600_CIT0056","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/BF00042459","article-title":"Smoothness of the Scalar Coefficients in the Representation","volume":"40","author":"Man","year":"1995","journal-title":"J. Elasticity"},{"issue":"1","key":"2025080520020522600_CIT0057","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1177\/108128659600100106","article-title":"Smoothness of the Scalar Coefficients in Representations of Isotropic Tensor-Valued Functions","volume":"1","author":"Scheidler","year":"1996","journal-title":"Math. Mech. Solids"},{"key":"2025080520020522600_CIT0058","first-page":"1","article-title":"Basic Issues Concerning Finite Strain Measures and Isotropic Stress-Deformation Relations","volume":"67","author":"Xiao","year":"2002","journal-title":"J. Elasticity Phys. Sci. Solids"},{"issue":"1","key":"2025080520020522600_CIT0059","first-page":"107","article-title":"The Mechanics of Rubber Elasticity","volume":"48","author":"Treloar","year":"1974","journal-title":"J. Polym. Sci.: Polym. Sym."},{"issue":"11","key":"2025080520020522600_CIT0060","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1088\/0022-3727\/8\/11\/007","article-title":"The Properties of Rubber in Pure Homogeneous Strain","volume":"8","author":"Jones","year":"1975","journal-title":"J. Phys. D: Appl. Phys."},{"issue":"3","key":"2025080520020522600_CIT0061","doi-asserted-by":"publisher","first-page":"361","DOI":"10.5254\/1.3538206","article-title":"Recent Advances in the Phenomenological Theory of Rubber Elasticity","volume":"59","author":"Ogden","year":"1986","journal-title":"Rubber Chem. Technol."},{"issue":"9","key":"2025080520020522600_CIT0062","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1063\/1.1712836","article-title":"A Theory of Large Elastic Deformation","volume":"11","author":"Mooney","year":"1940","journal-title":"J. Appl. Phys."},{"issue":"835","key":"2025080520020522600_CIT0063","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1098\/rsta.1948.0024","article-title":"Large Elastic Deformations of Isotropic Materials IV. Further Developments of the General Theory","volume":"241","author":"Rivlin","year":"1948","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"2025080520020522600_CIT0064","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/s10659-010-9279-0","article-title":"A Strain Energy Function for Vulcanized Rubbers","volume":"103","author":"Carroll","year":"2011","journal-title":"J. Elast."},{"key":"2025080520020522600_CIT0065","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1007\/s10409-021-01064-4","article-title":"Improved Carroll\u2019s Hyperelastic Model Considering Compressibility and Its Finite Element Implementation","volume":"37","author":"Melly","year":"2021","journal-title":"Acta Mech. Sin."},{"issue":"1","key":"2025080520020522600_CIT0066","doi-asserted-by":"publisher","first-page":"59","DOI":"10.5254\/1.3538357","article-title":"A New Constitutive Relation for Rubber","volume":"69","author":"Gent","year":"1996","journal-title":"Rubber Chem. Technol."},{"issue":"5","key":"2025080520020522600_CIT0067","doi-asserted-by":"publisher","first-page":"839","DOI":"10.5254\/1.3547687","article-title":"A Note on the Gent Model for Rubber-Like Materials","volume":"75","author":"Pucci","year":"2002","journal-title":"Rubber Chem. Technol."},{"issue":"9","key":"2025080520020522600_CIT0068","doi-asserted-by":"publisher","first-page":"2287","DOI":"10.1002\/pen.25757","article-title":"A Consistently Compressible Mooney\u2013Rivlin Model for the Vulcanized Rubber Based on the Penn\u2019s Experimental Data","volume":"61","author":"Peng","year":"2021","journal-title":"Polym. Eng. Sci."},{"key":"2025080520020522600_CIT0069","doi-asserted-by":"publisher","first-page":"484","DOI":"10.1007\/s00466-004-0593-y","article-title":"Fitting Hyperelastic Models to Experimental Data","volume":"34","author":"Ogden","year":"2004","journal-title":"Comput. Mech."},{"key":"2025080520020522600_CIT0070","first-page":"8026","article-title":"Pytorch: An Imperative Style, High-Performance Deep Learning Library","volume":"32","author":"Paszke","year":"2019","journal-title":"Adv. Neural Inform. Process. Syst."},{"key":"2025080520020522600_CIT0071","first-page":"315","article-title":"Deep Sparse Rectifier Neural Networks","author":"Glorot","year":"2011"},{"year":"2014","author":"Kingma","key":"2025080520020522600_CIT0072"},{"issue":"5","key":"2025080520020522600_CIT0073","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1016\/S0020-7462(03)00059-3","article-title":"The Attainable Region of Strain-Invariant Space for Elastic Materials","volume":"39","author":"Currie","year":"2004","journal-title":"Int. J. Non-Linear Mech."},{"key":"2025080520020522600_CIT0074","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-7091-2810-7","volume-title":"Applications of Tensor Functions in Solid Mechanics","author":"Boehler","year":"1987"},{"volume-title":"Computational Inelasticity","year":"2006","author":"Simo","key":"2025080520020522600_CIT0075"},{"volume-title":"Plasticity Theory","year":"2008","author":"Lubliner","key":"2025080520020522600_CIT0076"},{"key":"2025080520020522600_CIT0077","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511762956","volume-title":"The Mechanics and Thermodynamics of Continua","author":"Gurtin","year":"2010"},{"volume-title":"Mechanics of Solid Materials","year":"1994","author":"Lemaitre","key":"2025080520020522600_CIT0078"},{"issue":"1","key":"2025080520020522600_CIT0079","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1115\/1.3225775","article-title":"A Continuous Damage Mechanics Model for Ductile Fracture","volume":"107","author":"Lemaitre","year":"1985","journal-title":"J. Eng. Mater. Technol."},{"year":"2023","author":"Upadhyay","key":"2025080520020522600_CIT0080"},{"issue":"26\u201327","key":"2025080520020522600_CIT0081","doi-asserted-by":"publisher","first-page":"3455","DOI":"10.1016\/S0020-7683(97)00217-5","article-title":"A Theory of Finite Viscoelasticity and Numerical Aspects","volume":"35","author":"Reese","year":"1998","journal-title":"Int. J. Solids Struct."},{"issue":"2166","key":"2025080520020522600_CIT0082","doi-asserted-by":"publisher","first-page":"20140058","DOI":"10.1098\/rspa.2014.0058","article-title":"On Nonlinear Viscoelastic Deformations: A Reappraisal of Fung\u2019s Quasi-linear Viscoelastic Model","volume":"470","author":"De Pascalis","year":"2014","journal-title":"Proc. R. Soc. A: Math. Phys. Eng. Sci."},{"key":"2025080520020522600_CIT0083","doi-asserted-by":"publisher","first-page":"115867","DOI":"10.1016\/j.cma.2022.115867","article-title":"Automated Discovery of Generalized Standard Material Models With EUCLID","volume":"405","author":"Flaschel","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"2025080520020522600_CIT0084","doi-asserted-by":"publisher","first-page":"676","DOI":"10.1137\/15M1030170","article-title":"How Bad Are Vandermonde Matrices?","volume":"37","author":"Pan","year":"2016","journal-title":"SIAM J. Matrix Anal. Appl."}],"container-title":["Journal of Computing and Information Science in Engineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/asmedigitalcollection.asme.org\/computingengineering\/article-pdf\/24\/11\/111007\/7358730\/jcise_24_11_111007.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/asmedigitalcollection.asme.org\/computingengineering\/article-pdf\/24\/11\/111007\/7358730\/jcise_24_11_111007.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T00:02:50Z","timestamp":1754438570000},"score":1,"resource":{"primary":{"URL":"https:\/\/asmedigitalcollection.asme.org\/computingengineering\/article\/24\/11\/111007\/1195179\/Stress-Representations-for-Tensor-Basis-Neural"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,6]]},"references-count":84,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2024,11,1]]}},"URL":"https:\/\/doi.org\/10.1115\/1.4064650","relation":{},"ISSN":["1530-9827","1944-7078"],"issn-type":[{"type":"print","value":"1530-9827"},{"type":"electronic","value":"1944-7078"}],"subject":[],"published":{"date-parts":[[2024,8,6]]},"article-number":"111007"}}