Decoupling deformation modes: a physically interpretable neural network for hyperelastic constitutive modeling

Abstract Hyperelastic constitutive models traditionally rely on a single global strain energy density function defined by deformation invariants. While this approach ensures thermodynamic consistency and mathematical simplicity, it implicitly forces a uniform energy landscape across all deformation modes. As demonstrated by the classical Treloar rubber dataset, this assumption is physically restrictive: uniaxial, equibiaxial, and planar deformations exhibit distinct stiffening behaviors that a single invariant-based formulation depending only on the first invariant I 1 , such as the Gent model, struggles to capture simultaneously. We systematically examine whether this deficiency is intrinsic to invariant-based modeling in general, or specific to I 1 -only formulations, by comparing six constitutive representations calibrated on the same multi-mode Treloar data: the classical Gent model ( I 1 only), the classical Mooney-Rivlin model (fixed-slope I 1 and I 2 ), a three-term Ogden model (principal stretches), a three-term Yeoh model ( I 1 only, higher order), a fully coupled neural energy network $$W_\theta(I_1,I_2)$$ , and a Split Energy Neural Network (SENN) that represents the energy as an additive, decoupled sum $$W_1(I_1)+W_2(I_2)$$ . We find that simply including I 2 at all, even through the two fixed constants of Mooney-Rivlin, closes most of the gap Gent leaves on equibiaxial loading (normalized mean squared error, NMSE). while Yeoh, despite having one more parameter than Gent, recovers almost none of this gap because it remains I 1 -only; Ogden, built from principal stretches rather than an additive invariant split, reaches comparable biaxial accuracy to Mooney-Rivlin by a different route. Both neural formulations fit the training data to near machine precision. However, evaluating each model’s tangent stiffness on a dense stretch grid, rather than only at the sparse experimental points, reveals that both neural energy surfaces oscillate substantially between training points, including physically inadmissible regions of negative stiffness; the coupled network is, if anything, less prone to this than the decoupled one. A held-out generalization test further shows the decoupled model generalizing markedly worse than the coupled model on the equibiaxial mode, the same mode central to this paper’s original motivation. We report these findings as a rigorous negative result for the specific hypothesis that architecturally decoupling I 1 and I 2 improves on a coupled representation, while confirming that invariant coverage itself, not decoupling, is what resolves the Gent model’s structural deficiency. We further identify that sparse-point tangent-stiffness evaluation, common practice in this literature, can mask substantial unphysical oscillation that only becomes visible under dense-grid evaluation, a methodological point we believe generalizes beyond this dataset. The contribution of this work is therefore a matched-condition diagnostic framework, five independent tests applied identically to a coupled and a decoupled neural energy architecture, that separates the effect of invariant coverage from the effect of architectural decoupling. To our knowledge this separation has not been made explicit elsewhere in the constitutive neural network literature, and it changes the answer: what resolves the deficiency documented across three decades of invariant-based hyperelastic modeling is access to I 2 , not any particular way of combining it with I 1 .