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.

翻译：高可压缩固体（如泡沫）表现出复杂的力学响应，包括显著的拉伸-压缩非对称性。在统一超弹性框架内捕捉此类行为仍具挑战性。基于不变量的超弹性模型通常通过标准试验（如均匀单轴拉伸/压缩和简单剪切）进行识别，并隐含假设能量表示的唯一性。本文证明该假设存在根本性违背，且通常选择哪一项主导仅取决于偏好。通过将基于样条的应变能密度函数作为数据自适应工具，并结合弹性体泡沫的应力-应变实验数据，我们揭示了这种往往被低参数公式所掩盖的非唯一性。利用拉伸、压缩和剪切的均匀实验数据，我们的框架成功捕捉了用于竞速鞋的超轻泡沫的体变形特性。我们在（$\bar{I}_1$，$\bar{I}_2$，$J$）空间中构建了过完备的可分离与不可分离能量Ansatz，其形式兼具Mooney-Rivlin特征。这些由乘法分解定义的构造不仅类似经典不变量模型，更将其推广为数据驱动的样条表示，具有双重目的：（i）捕捉复杂体变形模式下的响应特征；（ii）使识别问题中的非唯一性自然呈现。研究发现，等容变形与体变形之间的耦合项（如$Ψ(\bar{I}_1,J)$或$Ψ(\bar{I}_2,J)$）具有必要性，而额外耦合项虽有助于建模但并非完全必需——它们反而凸显了非唯一性。因此，不同模型在现有数据上可能难以区分。至关重要的是，这些挑战并非样条方法所特有，而是普遍存在于传统模型与神经网络模型中。