A constitutive framework for tension-compression failure asymmetry in soft materials

Abstract

Soft materials often exhibit pronounced tension-compression asymmetry (TCA) in their softening and failure behavior, a feature that conventional hyperelastic and continuum-damage formulations fail to capture within a unified framework. This work presents a Lode-invariant-based hyperelastic softening model that explicitly incorporates deformation-mode dependence 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, which ensures a smooth and thermodynamically consistent transition of failure behavior across distortion modes, achieved directly within the constitutive description of the bulk response, without introducing internal damage variables. Agarose hydrogels (1, 2, and 3% w/v) serve as the model system for validation. The framework accurately 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, demonstrating strong cross-mode generalizability. To further assess thermodynamic stability and deformation-mode sensitivity, the model's energy landscape was analyzed across the Lode-invariant space, confirming stable behavior under diverse loading conditions. Parameter evolution with concentration follows power-law scaling, enabling interpolation and predictive validation at intermediate concentrations (2.5% w/v). By establishing a physically interpretable damage framework over the Lode invariant space, this work provides a unified framework for tension-compression-asymmetric softening and lays the foundation for distortional-mode-sensitive, three-dimensional failure mapping of soft materials.

Figures (12)

• Figure 1: Unit cube in the (a) reference configuration and (b) under uniaxial deformation. Tension ($\lambda>1$) is shown in red; compression ($0<\lambda<1$) in magenta.
• Figure 3: Typical uniaxial engineering stress–stretch responses of the Volokh damage model, considering the one-term Ogden strain energy density to represent intact material behavior. The dashed blue curve shows the undamaged response, while the solid curves demonstrate the softening response for different parameter pairs $(\Phi, m)$, as indicated in the legend. All curves use the following Ogden parameters: $\mu = 300~\mathrm{kPa}$ and $\alpha = 7$.
• Figure 4: A schematic of the proposed Lode-invariants-based bi-failure framework, with $\psi_{\mathrm{prop}}(W(K_2, K_3))$ as the mode-dependent strain energy density. The colored boundary curves correspond to uniaxial compression $(K_3=-\pi/6)$ (magenta), pure shear $(K_3=0)$, and uniaxial tension $(K_3=\pi/6)$ (red), with distinct compressive and tensile failure energies $\psi_f^{-}$ and $\psi_f^{+}$ indicated. The inset shows the Lode-weighting function $\beta(K_3)$ that provides a smooth linear interpolation between the compressive $(\beta=0)$ and tensile $(\beta=1)$ branches, enabling a continuous, mode-dependent description of softening and failure across the $(K_2,K_3)$ space and capturing TCA.
• Figure 5: Experimental setup for mechanical testing of agarose gels: (a) uniaxial compression between parallel platens; (b) uniaxial tension of a dog-bone specimen (inset shows the assembled fixture with mounting plates); and (c) pure shear of a rectangular strip clamped between rigid jaws. Prior to loading, a high-contrast speckle pattern was applied on all specimens using white acrylic spray paint to facilitate DIC (excluding compression testing).
• Figure 6: Combined experimental uniaxial engineering stress–stretch data (black) and the corresponding simultaneous fits of the proposed asymmetric-failure model for agarose gels at (a) 1% w/v (orange), (b) 2% w/v (red), and (c) 3% w/v (magenta), shown for each replicate (I--IV). Note: Y-axis scales differ between subplots for clarity.