Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908967567884288 |
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| author | Moreno-Mateos, Miguel Angel Wiesheier, Simon Steinmann, Paul Kuhl, Ellen |
| author_facet | Moreno-Mateos, Miguel Angel Wiesheier, Simon Steinmann, Paul Kuhl, Ellen |
| contents | Highly compressible solids, such as foams, exhibit complex responses, including pronounced tension-compression asymmetry. Capturing such behaviors within unified hyperelastic frameworks remains challenging. Invariant-based hyperelastic models are commonly identified from standard tests such as homogeneous uniaxial tension/compression and simple shear, implicitly assuming a unique energy representation. Here we show that this assumption is fundamentally violated and that, oftentimes, the choice of which term should prevail is just a matter of taste. Using spline-based strain-energy density functions as a data-adaptive tool and stress-strain experimental data for elastomeric foams, we expose this non-uniqueness, often hidden in low-parameter formulations. Our framework captures the volumetric deformation of ultra-light foams used in racing shoes using homogeneous experimental data from tension, compression, and shear. We formulate an overly rich ansatz of separable and non-separable energies in the ($\bar{I}_1$, $\bar{I}_2$, $J$) space à la Money-Rivlin. These constructs, defined by multiplicative decompositions, resemble classical invariant-based models while generalizing them to a data-driven spline representation. This serves two purposes: (i) to capture the response under complex volumetric deformation modes and (ii) to allow non-uniqueness in the identification problem to emerge naturally. We find that a coupling term between isochoric and volumetric deformation, such as $Ψ(\bar{I}_1,J)$ or $Ψ(\bar{I}_2,J)$, is essential and that additional coupling terms help but are not fully necessary; rather, they pronounce the non-uniqueness. As a consequence, different models may be indistinguishable on available data. Importantly, these challenges are not specific to splines but extend to traditional and neural network-based models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14264 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations Moreno-Mateos, Miguel Angel Wiesheier, Simon Steinmann, Paul Kuhl, Ellen Computational Engineering, Finance, and Science Highly compressible solids, such as foams, exhibit complex responses, including pronounced tension-compression asymmetry. Capturing such behaviors within unified hyperelastic frameworks remains challenging. Invariant-based hyperelastic models are commonly identified from standard tests such as homogeneous uniaxial tension/compression and simple shear, implicitly assuming a unique energy representation. Here we show that this assumption is fundamentally violated and that, oftentimes, the choice of which term should prevail is just a matter of taste. Using spline-based strain-energy density functions as a data-adaptive tool and stress-strain experimental data for elastomeric foams, we expose this non-uniqueness, often hidden in low-parameter formulations. Our framework captures the volumetric deformation of ultra-light foams used in racing shoes using homogeneous experimental data from tension, compression, and shear. We formulate an overly rich ansatz of separable and non-separable energies in the ($\bar{I}_1$, $\bar{I}_2$, $J$) space à la Money-Rivlin. These constructs, defined by multiplicative decompositions, resemble classical invariant-based models while generalizing them to a data-driven spline representation. This serves two purposes: (i) to capture the response under complex volumetric deformation modes and (ii) to allow non-uniqueness in the identification problem to emerge naturally. We find that a coupling term between isochoric and volumetric deformation, such as $Ψ(\bar{I}_1,J)$ or $Ψ(\bar{I}_2,J)$, is essential and that additional coupling terms help but are not fully necessary; rather, they pronounce the non-uniqueness. As a consequence, different models may be indistinguishable on available data. Importantly, these challenges are not specific to splines but extend to traditional and neural network-based models. |
| title | Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations |
| topic | Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2604.14264 |