A Note on Hypoelastic Models: A Benchmark-Oriented FEM Lagrangian Formulation in FeniCSx

Abstract

Hypoelastic formulations are commonly used for modeling materials undergoing large elastic deformations. Yet their lack of an underlying energy potential raises concerns about integrability and energy consistency. This work addresses the longstanding gap between the theoretical notion of integrability and its numerical realization in finite element implementations. We develop a total Lagrangian framework for hypoelasticity and implement it within the open-source FEniCSx platform, enabling direct comparison of various objective stress rates. Two stress-integration algorithms, namely a forward Euler and an implicit midpoint, are formulated and compared, together with a sub-iterative variant to enhance accuracy. The framework is tested through benchmark problems involving cyclic and non-proportional shear loading. We systematically assess the interplay between constitutive rate choice and numerical integration accuracy. The results confirm that classical Jaumann and Green–Naghdi stress rates are non-integrable for grade-zero hypoelasticity, producing residual stresses and spurious energy accumulation. Conversely, the logarithmic rate preserves path independence for grade-zero hypoelasticity. Moreover, we show that even theoretically integrable models can exhibit numerical non-integrability when discretized inconsistently. Finally, the framework is applied to a non-grade-zero hypoelastic model featuring a non-constant stiffness matrix derived from the Exponentiated Hencky energy, assessing its numerical integrability and performance. The proposed framework provides a transparent, extensible basis for evaluating hypoelastic models in boundary value problems, bridging theoretical consistency and computational practice. This supports the development of integrable constitutive formulations for large-deformation mechanics.