Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations

TL;DR

This paper introduces a spline-based, data-driven hyperelastic model for ultra-compressible solids like foams, revealing fundamental non-uniqueness in energy representations and emphasizing the importance of coupling terms.

Contribution

It demonstrates the non-uniqueness of hyperelastic energy models using spline functions and stress-strain data, highlighting the role of coupling terms in capturing complex behaviors.

Findings

Spline-based models expose non-uniqueness in energy representations.

Coupling terms between volumetric and isochoric deformations are essential.

Non-uniqueness challenges extend beyond splines to other modeling approaches.

Abstract

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 \`a 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 $\Psi(\bar{I}_1,J)$ or $\Psi(\bar{I}_2,J)$, 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.