{ "id": "epgb529446", "slug": "hyperelastic-model", "title": "Hyperelastic model", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.engineering", "label": "Engineering and manufacturing", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.engineering" }, { "id": "technology.engineering.engineering.mechanical", "label": "Mechanical engineering", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.engineering.engineering.mechanical" } ], "geo": [ { "id": "geo.nongeo.t1800.technology.engineering", "label": "Non-geographic · 1800 to 1945: Engineering and manufacturing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.technology.engineering", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.technology" }, { "id": "geo.nongeo.t1800.technology.engineering", "label": "Engineering and manufacturing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.technology.engineering" } ] }, { "id": "geo.nongeo.t1946.technology.engineering.engineering.mechanical", "label": "Non-geographic · 1946 to 2000: Mechanical engineering", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.engineering.engineering.mechanical", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.engineering", "label": "Engineering and manufacturing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.engineering" }, { "id": "geo.nongeo.t1946.technology.engineering.engineering.mechanical", "label": "Mechanical engineering", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.engineering.engineering.mechanical" } ] } ], "excerpt": "A hyperelastic model is a constitutive law for rubber-like materials in which stress derives from a strain energy density function, making work path independent; named forms include Mooney-Rivlin, Ogden, and Gent.", "snippet": "A hyperelastic model is a constitutive law for rubber-like materials in which stress derives from a strain energy density function, making work path independent; named forms include Mooney-Rivlin, Ogden, and Gent.", "node": "technology.engineering.engineering.mechanical", "markdown": "# Hyperelastic model\n\nA hyperelastic model is a constitutive law for rubber-like materials in which the stress at any deformation is derived from a scalar strain energy density function W, rather than from a stress-strain law stated directly. Because all work comes from a potential, the mechanical work between two deformation states is path independent.[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) The energy function is a [Helmholtz free energy](https://www.edgepedia.org/helmholtz-free-energy) per unit reference volume, also called the strain energy function or elastic potential.[2](https://onlinelibrary.wiley.com/doi/abs/10.1002/msd2.12013)\n\n| Key fact | Detail |\n|---|---|\n| Defining structure | Stress derived from a strain energy density function W; work between states is path independent[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) |\n| Energy split | W is written as an isochoric part in the invariants Ī1, Ī2 plus a volumetric part W_vol in the volume ratio J[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) |\n| Standard idealizations | Homogeneous, isotropic, incompressible or nearly incompressible, geometrically and physically nonlinear[3](https://www.degruyterbrill.com/document/doi/10.1515/jmbm-2012-0007/html?lang=en) |\n| Calibration tests | Uniaxial tension, planar tension (pure shear), equibiaxial tension, simple shear, and volumetric compression[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) |\n| Forms in Abaqus | Arruda-Boyce, Marlow, Mooney-Rivlin, neo-Hookean, Ogden, polynomial, reduced polynomial, Yeoh, Van der Waals[4](https://abaqus-docs.mit.edu/2017/English/SIMACAEMATRefMap/simamat-c-hyperelastic.htm) |\n| Fitting accuracy | The Marlow form reached \\( R^{2} = 0.999 \\) against uniaxial test data in a nine-model comparison[5](https://www.mdpi.com/2411-9660/7/6/135) |\n| Known pitfall | Curve-fitting the Ogden form is a nonlinear optimization whose solution is not unique for \\( n \\geq 2 \\)[6](https://maths.nuigalway.ie/~destrade/Publis/destrade_154.pdf) |\n\n## How it works\n\nThe model output is a stress response generated by differentiation of W. For isotropic materials, W can be expressed equivalently in terms of the principal invariants \\( I_{1}, I_{2}, I_{3} \\) of the deformation or in terms of the principal stretches \\( \\lambda_{1}, \\lambda_{2}, \\lambda_{3} \\); these two representation routes define the two main lineages of hyperelastic modeling.[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC5719638/) Published surveys identify Rivlin's principal-invariant formulation and the separable principal-stretch form as the most frequently adopted representations for incompressible rubber.[8](https://link.springer.com/article/10.1007/s10659-022-09982-5)\n\nFor compressible treatment, the energy is split as an isochoric part \\( W_{\\mathrm{iso}} \\) in the reduced invariants \\( \\bar{I}_{1}, \\bar{I}_{2} \\) and a volumetric part \\( W_{\\mathrm{vol}} \\) depending on the volume change J.[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) Full incompressibility remains the standard idealization, supported by experiment for most rubbers.[3](https://www.degruyterbrill.com/document/doi/10.1515/jmbm-2012-0007/html?lang=en) Notably, the second law of thermodynamics imposes no restrictions on constructing W, so admissibility rests on objectivity and physical plausibility arguments instead.[1](https://link.springer.com/article/10.1007/s11831-022-09865-x)\n\n## How it is done\n\nCalibration follows a fixed workflow. First, the material is characterized with up to five experiments: uniaxial tension, pure shear (planar tension), equibiaxial tension, simple shear, and confined (volumetric) compression.[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) Abaqus accepts four of these as test datasets: uniaxial, equibiaxial, planar, and volumetric compression, entered as nominal strain (change in length per unit original length) and nominal stress (force per unit original cross-sectional area).[4](https://abaqus-docs.mit.edu/2017/English/SIMACAEMATRefMap/simamat-c-hyperelastic.htm)\n\nSecond, the coefficients are found by least squares. Abaqus minimizes the relative error in stress over the n nominal stress-strain data pairs, where \\( T_{i}^{\\mathrm{test}} \\) is the test stress and \\( T_{i}^{\\mathrm{h}} \\) comes from the model's nominal stress expression; relative rather than absolute error is used because it fits better at low strains.[9](https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-fithyperconst.htm) In general terms, fitting is an optimization that searches for the parameter set minimizing the difference between the model response and the experimental data.[1](https://link.springer.com/article/10.1007/s11831-022-09865-x) Third, the stress measure matters: optimizing against Cauchy stress gives the smallest deviations at large stretches (\\( \\lambda > 7.5 \\)), the second Piola-Kirchhoff stress performs best for \\( 1.75 < \\lambda < 6.5 \\), and the first Piola-Kirchhoff stress is a tradeoff between them.[1](https://link.springer.com/article/10.1007/s11831-022-09865-x)\n\n## Origin\n\nThe 1940 paper \"A Theory of Large Elastic Deformation\" by M. Mooney, in the [Journal of Applied Physics](https://www.edgepedia.org/journal-of-applied-physics), is the landmark starting point of rubber-like elasticity theory.[10](https://doi.org/10.1063/1.1712836) Mooney's construction rested on three assumptions: the material is isotropic, the deformation is isochoric, and the traction in simple shear in any isotropic plane is proportional to the amount of shear.[6](https://maths.nuigalway.ie/~destrade/Publis/destrade_154.pdf) The resulting strain energy form,\n\n\\[ W = C_{1}(I_{1} - 3) + C_{2}(I_{2} - 3), \\]\n\nwith \\( C_{1} \\) and \\( C_{2} \\) physical constants, was taken up in rubber mechanics in the late 1940s.[11](https://royalsocietypublishing.org/rspa/article-pdf/195/1043/463/39706/rspa.1949.0004.pdf) A series of papers in the [Philosophical Transactions of the Royal Society](https://www.edgepedia.org/philosophical-transactions-of-the-royal-society) then developed the theory of large elastic deformations of isotropic materials systematically, connecting large-strain measures back to definitions going back to Cauchy.[12](https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1948.0024) Later work extended the two-constant form by replacing \\( C_{2} \\) with an unknown function of \\( I_{2} - 3 \\), and the principal-stretch-based Ogden form introduced material parameters \\( \\mu_{i} \\) and \\( \\alpha_{i} \\), with \\( \\mu_{r} \\alpha_{r}/2 \\) giving the initial shear modulus and non-integer \\( \\alpha_{r} \\) allowed.[6](https://maths.nuigalway.ie/~destrade/Publis/destrade_154.pdf)\n\n## Variants\n\nThe named forms differ mainly in which deformation variables they use and how many parameters they carry. The neo-Hookean model is the simplest physically based model.[13](https://hal.science/hal-01004686v1/file/2003-Comparaison%20of%20hyperelastic%20models%20for%20rubber-like%20materials-GM%20EV.pdf) In invariant terms, neo-Hookean depends solely on \\( I_{1} \\), Blatz-Ko on \\( I_{2} \\), Mooney-Rivlin combines \\( I_{1} \\) and \\( I_{2} \\), and Yeoh uses only \\( I_{1} \\) in a higher-order polynomial expansion.[14](https://arxiv.org/html/2508.12063) Ignoring the second invariant in the general polynomial gives the reduced polynomial form.[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC10421088/) The Ogden form, written directly in \\( \\lambda_{1}, \\lambda_{2}, \\lambda_{3} \\), recovers both the neo-Hookean and Mooney-Rivlin models for specific parameter choices.[16](https://par.nsf.gov/servlets/purl/10376314)\n\nLimiting-chain models add a maximum stretch. The Gent model is a two-parameter empirical form with an infinitesimal shear modulus and a maximum allowable strain parameter; it reduces to neo-Hookean at small strains, reflects severe strain-stiffening at large strains, and exhibits a \"locking stretch\" in simple extension at which the tensile stress becomes unbounded.[17](https://www.sciencedirect.com/science/article/abs/pii/S0020746214001127) Arruda-Boyce is a micromechanical (network) model, while van der Waals is a hybrid; the remaining common forms are phenomenological.[5](https://www.mdpi.com/2411-9660/7/6/135)\n\nQuantitatively, in a nine-model comparison against uniaxial test data the Marlow form matched the data over the entire strain range with \\( R^{2} = 0.999 \\).[5](https://www.mdpi.com/2411-9660/7/6/135) With multiple test datasets, the Ogden and Van der Waals forms fit experimental results more accurately; with only one dataset, the Marlow form is recommended.[4](https://abaqus-docs.mit.edu/2017/English/SIMACAEMATRefMap/simamat-c-hyperelastic.htm)\n\nMachine-learned constitutive models have entered hyperelasticity. GI-CANN, a generalized-invariant-based constitutive artificial neural network by Martonová, Goriely, and Kuhl (2025), published on arXiv, learns the optimal input invariants and the strain energy function simultaneously, removing the need to predefine a strain energy form or run a sequential two-step identification.[14](https://arxiv.org/html/2508.12063) A complementary line enforces mathematical guarantees in hyperelastic physics-augmented neural networks (PANNs) by concurrent enforcement of polyconvexity and true-stress-true-strain monotonicity.[18](https://arxiv.org/html/2605.20031)\n\n## Applications\n\nElastomer components analyzed with hyperelastic models include tires, engine mounts, seals, conveyor belts, and base isolations that protect buildings and bridges from earthquakes.[3](https://www.degruyterbrill.com/document/doi/10.1515/jmbm-2012-0007/html?lang=en) The Ogden model is widely used in biomechanics, where soft tissues are treated as (pseudo-)elastic.[16](https://par.nsf.gov/servlets/purl/10376314) The Gent model has likewise been applied beyond rubber elasticity to the biomechanics of soft biomaterials.[17](https://www.sciencedirect.com/science/article/abs/pii/S0020746214001127) On the software side, Abaqus offers the nine strain energy potential forms listed above.[4](https://abaqus-docs.mit.edu/2017/English/SIMACAEMATRefMap/simamat-c-hyperelastic.htm)\n\n## Limitations and alternatives\n\nA single experiment cannot fully characterize a rubber-like material even under the elasticity assumption; a series of biaxial tests has been shown sufficient for comprehensive characterization.[5](https://www.mdpi.com/2411-9660/7/6/135) Fitting has uniqueness constraints: for the general polynomial and Ogden models, planar test data must be accompanied by uniaxial or biaxial data, or the least-squares solution will not be unique.[4](https://abaqus-docs.mit.edu/2017/English/SIMACAEMATRefMap/simamat-c-hyperelastic.htm) The Ogden curve-fit is non-unique for \\( n \\geq 2 \\), so several parameter combinations give the same optimal fit.[6](https://maths.nuigalway.ie/~destrade/Publis/destrade_154.pdf) Stability is a further failure mode: in one nine-model study only the two-term polynomial and Yeoh models experienced Drucker instability, defined as a non-positive gradient of the nominal strain-nominal stress function.[5](https://www.mdpi.com/2411-9660/7/6/135) [Extrapolation](https://www.edgepedia.org/extrapolation) occurs in two distinct ways, from observed deformation modes to unknown ones and across an observed strain regime, and the second type has received considerably less attention.[18](https://arxiv.org/html/2605.20031) One structural difference remains between learned and classical forms: limiting-chain models such as Arruda-Boyce and Gent always ensure that stress diverges at a particular level of deformation, whereas a hyperelastic PANN generally does not include such information even when monotonicity is prescribed.[18](https://arxiv.org/html/2605.20031)\n\nHyperelastic models are purely elastic. Coupled viscoelasticity and Mullins stress softening in filled rubber require formulations beyond a purely hyperelastic treatment, for example composite treatments with rigid filler particles.[19](https://www.sciencedirect.com/science/article/abs/pii/S0022509624001169) A widely cited criterion for model choice, due to Marckmann and Verron, is that the best model describes the complete elastomer behavior with a minimal number of parameters determinable from experiments and without instability.[3](https://www.degruyterbrill.com/document/doi/10.1515/jmbm-2012-0007/html?lang=en)\n\n## References\n\n1. [Systematic Fitting and Comparison of Hyperelastic Continuum Models for Elastomers](https://link.springer.com/article/10.1007/s11831-022-09865-x)\n2. [A review on material models for isotropic hyperelasticity (Melly, 2021)](https://onlinelibrary.wiley.com/doi/abs/10.1002/msd2.12013)\n3. [More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study](https://www.degruyterbrill.com/document/doi/10.1515/jmbm-2012-0007/html?lang=en)\n4. [Hyperelastic behavior of rubberlike materials (Abaqus 2017 documentation)](https://abaqus-docs.mit.edu/2017/English/SIMACAEMATRefMap/simamat-c-hyperelastic.htm)\n5. [Comparative Analysis of Various Hyperelastic Models and Element Types for Finite Element Analysis](https://www.mdpi.com/2411-9660/7/6/135)\n6. [The Ogden model of rubber mechanics: 50 years of impact on nonlinear elasticity](https://maths.nuigalway.ie/~destrade/Publis/destrade_154.pdf)\n7. [How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity](https://pmc.ncbi.nlm.nih.gov/articles/PMC5719638/)\n8. [Large Isotropic Elastic Deformations: On a Comprehensive Model to Correlate the Theory and Experiments for Incompressible Rubber-Like Materials](https://link.springer.com/article/10.1007/s10659-022-09982-5)\n9. [Fitting of hyperelastic and hyperfoam constants (Abaqus 2025 documentation)](https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-fithyperconst.htm)\n10. [M. Mooney (1940). A Theory of Large Elastic Deformation. Journal of Applied Physics.](https://doi.org/10.1063/1.1712836)\n11. [Large elastic deformations of isotropic materials. V. The problem of flexure](https://royalsocietypublishing.org/rspa/article-pdf/195/1043/463/39706/rspa.1949.0004.pdf)\n12. [Large elastic deformations of isotropic materials IV. Further developments of the general theory](https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1948.0024)\n13. [Comparison of hyperelastic models for rubber-like materials (2003)](https://hal.science/hal-01004686v1/file/2003-Comparaison%20of%20hyperelastic%20models%20for%20rubber-like%20materials-GM%20EV.pdf)\n14. [Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials (GI-CANN)](https://arxiv.org/html/2508.12063)\n15. [A Modified Constitutive Model for Isotropic Hyperelastic Polymeric Materials and Its Parameter Identification](https://pmc.ncbi.nlm.nih.gov/articles/PMC10421088/)\n16. [An introduction to the Ogden model in biomechanics](https://par.nsf.gov/servlets/purl/10376314)\n17. [The remarkable Gent constitutive model for hyperelastic materials](https://www.sciencedirect.com/science/article/abs/pii/S0020746214001127)\n18. [Concurrent enforcement of polyconvexity and true-stress-true-strain monotonicity in incompressible isotropic hyperelasticity: application to neural network constitutive models](https://arxiv.org/html/2605.20031)\n19. [Experiments and modeling of the coupled viscoelasticity and Mullins effect in filled rubber materials](https://www.sciencedirect.com/science/article/abs/pii/S0022509624001169)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Engineering and manufacturing › Mechanical engineering*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/hyperelastic-model", "markdown_url": "https://www.edgepedia.org/hyperelastic-model.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Hyperelastic model\", Edgepedia (EdgeChat), https://www.edgepedia.org/hyperelastic-model. Edgepedia Community License 1.0.", "credit_md": "\"[Hyperelastic model](https://www.edgepedia.org/hyperelastic-model)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/hyperelastic-model](https://www.edgepedia.org/hyperelastic-model). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Hyperelastic model\", Edgepedia (EdgeChat), https://www.edgepedia.org/hyperelastic-model. Edgepedia Community License 1.0.", "speakable": "A hyperelastic model is a constitutive law for rubber-like materials in which stress derives from a strain energy density function, making work path independent; named forms include Mooney-Rivlin, Ogden, and Gent." }