A constitutive framework for distortional-mode-dependent failure in soft materials: Tension-compression asymmetry and beyond

Soft materials exhibit pronounced tension-compression asymmetry (TCA) in their softening and failure, a feature that conventional hyperelastic and continuum-damage formulations fail to capture in a unified framework. We present a Lode-invariant-based hyperelastic softening model for distortional-mode-dependent failure in soft materials, where mode dependence is introduced through a bi-failure construction with distinct tensile and compressive energy limiters. The proposed model extends Volokh's classical energy-limiting approach by embedding a Lode-angle-dependent weighting function, ensuring a smooth and physically consistent transition in failure across distortion modes within the constitutive description of the bulk response, without introducing internal damage variables. Agarose hydrogels (1, 2, and 3 % w/v) serve as the validation system. The framework reproduces experimental stress-stretch responses in uniaxial tension and compression, capturing concentration-dependent stiffness and failure energetics. Using parameters calibrated solely from combined uniaxial data, the model predicts pure shear behavior, including softening and failure, thereby demonstrating strong cross-mode predictive capability. To further assess thermodynamic consistency and distortion-mode sensitivity, the model's free-energy landscape is analyzed across the full Lode-invariant space, confirming a smooth and physically consistent response under diverse loading conditions. Parameter evolution with concentration follows power-law scaling, enabling interpolation and predictive validation at intermediate concentrations (evaluated at 2.5 % w/v). Overall, the proposed formulation provides a physically interpretable constitutive framework for tension-compression-asymmetric softening and distortional-mode-dependent failure, and establishes a foundation for three-dimensional failure mapping in soft materials.