Moreno Mateos MA, Wiesheier S, Steinmann P, Kuhl E (2026)
Publication Type: Journal article
Publication year: 2026
Original Authors: Miguel Angel Moreno-Mateos, Simon Wiesheier, Paul Steinmann, Ellen Kuhl
Book Volume: 218
Article Number: 106827
DOI: 10.1016/j.jmps.2026.106827
Highly compressible solids, such as foams, exhibit complex responses that often include pronounced tension-compression asymmetry. Capturing such responses 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 can be fundamentally violated and that 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 supervised stress-strain experimental data for elastomeric foams, we expose this non-uniqueness, often hidden in low-parameter representations. 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 (I¯1, 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 deformation modes and (ii) to allow non-uniqueness in the identification problem to emerge naturally. Importantly, we verify that, independently of the calibrated constitutive representation, constitutive descriptors such as the infinitesimal shear and bulk moduli do not inherit this non-uniqueness (ambiguity) but consistently reflect the same material behavior. We find that a coupling term between isochoric and volumetric invariants, such as Ψ(I¯1,J) or Ψ(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, individual energy terms might lose physical interpretability and 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.
APA:
Moreno Mateos, M.A., Wiesheier, S., Steinmann, P., & Kuhl, E. (2026). Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations. Journal of the Mechanics and Physics of Solids, 218. https://doi.org/10.1016/j.jmps.2026.106827
MLA:
Moreno Mateos, Miguel Angel, et al. "Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations." Journal of the Mechanics and Physics of Solids 218 (2026).
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