The mechanics and physics of tofu: Understanding hydrated soft solids through feature networks

Abstract

Tofu remains one of the worlds most important plant-based foods, famous for its cultural legacy, nutritional benefits, and environmental sustainability. Made up of only two ingredient-soy beans and water-it has nourished people for centuries; yet, its rheology remains incompletely understood. Here, we use tofu as a minimal model system to study how composition governs the mechanics and physics of hydrated soft solids. We perform over a hundred controlled compression tests on silken, firm, and extrafirm tofu, and observe a strong nonlinearity, a pronounced viscoelasticy, and a more than ten-fold increase in stress with only six percent decrease in water content. We demonstrate that automated model discovery can identify inelastic constitutive laws that accurately capture these unique characteristics across multiple loading magnitudes and rates. The discovered models autonomously separate elastic and inelastic mechanisms and reveal that tofu elasticity depends primarily on the second isochoric invariant, while tofu inelasticity depends on a combination of the second and third deviatoric invariants. This functional form is universal across all three tofu types, but scales in magnitude with water content. Our new water-content-based feature network reveals that this correlation in not linear, as assumed by traditional bi-phasic theories, but rather highly non-linear and sensitive to the individual model terms. These results position tofu as a quantitative benchmark for nonlinear poroviscoelastic solids and demonstrate how physics-informed machine learning can uncover the constitutive structure of hydrated soft materials. Our source code and examples are available at https://doi.org/10.5281/zenodo.16993236.