Physics-informed data-driven constitutive modeling of compressible, nonlinear soft materials under multiaxial cyclic loading

Modeling nonlinear, history-dependent viscoelastic behavior under multiaxial cyclic loading remains challenging because existing approaches address only parts of the problem. Purely data-driven models can learn time-dependent responses but often require large datasets, lack interpretability, and may violate physical constraints. Conversely, regression-based surrogates, including GPR and polynomial models, efficiently capture smooth equilibrium behavior but cannot represent path dependence or long-term memory effects. Existing hybrid models typically combine neural networks with physics-based constraints, yet they rarely integrate separate surrogates for equilibrium and nonequilibrium responses within a unified constitutive framework, nor do they enforce objectivity, material symmetry, and thermodynamic consistency at all stages. To address this gap, we propose a physics-informed hybrid framework in which GPR learns the smooth volumetric and isochoric equilibrium response, while RNN–LSTM networks capture the history-dependent viscoelastic contribution. The stress is decomposed into volumetric, isochoric hyperelastic, and isochoric viscoelastic components, each treated with its own surrogate and trained under explicit physical constraints. Training data are generated using the nonlinear Holzapfel differential viscoelastic model, and both short-term and long-term datasets are constructed to span distinct memory behaviors. The framework is evaluated under diverse multiaxial loading paths, including variable stretch ratios, multiple strain rates, and both tension and compression, with additional tests conducted beyond the training domain. The results show that the hybrid surrogate preserves objectivity, material symmetry, and thermodynamic admissibility while accurately reproducing nonlinear, rate-dependent, and history-sensitive behavior.