Lode-Invariant Hyperelastic Softening Model

• The paper introduces a unified constitutive framework that embeds Lode-angle sensitive invariants into hyperelastic softening models without using internal damage variables.
• It captures tension-compression asymmetry and mode-selective energy limits through smooth transitions across distortion modes using analytic and machine-learning approaches.
• The model's efficacy is demonstrated via parameter calibration and cross-mode generalizability, yielding robust predictions for soft materials such as hydrogels.

A Lode-invariant-based hyperelastic softening model is a constitutive framework for soft materials in which the distortion-mode dependence of softening and failure is embedded directly into the bulk free-energy description using Lode-angle–sensitive invariants. This approach enables the unified modeling of tension-compression asymmetry, mode-selective energy limits, and smooth transitions across the space of distortion modes, without recourse to internal damage variables. Two archetypes for this modeling paradigm are the recent generalizations of energy-limiting hyperelasticity to Lode-invariant spaces for softening with cross-mode predictions and the machine-learning-augmented, invariant-based surrogate approaches with physics-based constraints (Upadhyay et al., 2023, Chandrashekar et al., 14 Dec 2025).

1. Fundamental Invariants and Lode Angle

The kinematic basis for Lode-invariant-based softening models lies in the use of both principal and deviatoric invariants of strain or strain-rate measures. The right Cauchy–Green tensor, $\mathbf C = \mathbf F^T \mathbf F$, supplies the principal invariants:

$$I_1 = \mathrm{tr}\,\mathbf C, \qquad I_2 = \tfrac12\bigl[(\mathrm{tr}\,\mathbf C)^2 - \mathrm{tr}(\mathbf C^2)\bigr], \qquad I_3 = \det \mathbf C.$$

For incompressible or nearly incompressible applications, distortional (isochoric) behavior is captured by the deviatoric invariants:

$$J_2 = \tfrac12 \operatorname{dev} \mathbf A : \operatorname{dev} \mathbf A, \qquad J_3 = \det [\operatorname{dev} \mathbf A],$$

where $$\operatorname{dev} \mathbf A = \mathbf A - \tfrac13(\mathrm{tr}\,\mathbf A)\mathbf I$$ for a symmetric tensor $\mathbf A$. The Lode angle, $\theta$, is a third-deviatoric invariant that parameterizes the distortion mode:

$$\theta = \frac{1}{3}\arccos\left(\frac{3\sqrt{3}\,J_3}{2(J_2)^{3/2}}\right), \qquad \theta \in \left[-\frac{\pi}{6}, \frac{\pi}{6}\right].$$

These invariants furnish a kinematic description suitable for constructing mode-sensitive constitutive potentials and softening (damage) mechanisms (Chandrashekar et al., 14 Dec 2025).

2. Hyperelastic Potentials with Lode-invariant Softening

The foundation of the intact (undamaged) material response is a hyperelastic energy density expressed in terms of Lode-invariant measures, such as the Prasad–Kannan (PK) potential in Hencky-invariant space:

$$W_{\text{PK}}(K_2, K_3) = \frac{\mu}{2} K_2^2 + \frac{a}{\mathcal{G}(K_3)} \left( e^{K_2 \mathcal{G}(K_3)} - 1 \right) - \frac{a}{2} K_2^2 \mathcal{G}(K_3) - a K_2,$$

with $K_2 = \|\operatorname{dev}(\ln\mathbf V)\|$, $$K_3 = \frac{1}{3}\sin^{-1}\left(\frac{\sqrt{6}\operatorname{tr}[(\operatorname{dev}\ln\mathbf V)^3]}{\|\operatorname{dev}(\ln\mathbf V)\|^3}\right)$$, $\mu>0$ the small-strain shear modulus, $a>0$ the stiffening parameter, and $\mathcal{G}(K_3)$ a smooth mode-dependent function parameterized by $b_0, b_1$:

$$\mathcal{G}(K_3) = b_0 \left[\frac{e^{b_1/2 - b_1 \cos(K_3+\pi/6)}}{b_1} + \cos(K_3 + \pi/6) + \frac{\sqrt{7}-2}{6}\right].$$

This construction guarantees adherence to Baker–Ericksen inequalities and smoothly interpolates between compression and tension (Chandrashekar et al., 14 Dec 2025).

Softening is implemented via limiting energies, $\psi_f^\pm$, and reduction functions, with explicit dependence on tension (positive branch) and compression (negative branch):

$$\psi_f^{\pm} = \frac{\Phi^{\pm}}{m^{\pm}} \,\Gamma\left(\frac{1}{m^{\pm}},0\right), \qquad \psi_e^{\pm}(W_{\text{PK}}) = \frac{\Phi^{\pm}}{m^{\pm}} \,\Gamma\left(\frac{1}{m^{\pm}},[W_\text{PK}/\Phi^{\pm}]^{m^{\pm}}\right).$$

The “stress-reduction” functions (softening factors) are

$$\Phi_t(W) = \exp\left[-(W/\Phi^+)^{m^+}\right],\quad \Phi_c(W) = \exp\left[-(W/\Phi^-)^{m^-}\right].$$

Here, $\Phi^{\pm}$ are pseudo-failure energies and $m^{\pm}$ control “sharpness”.

3. Lode-angle-dependent Blending and Unified Energy Limiters

The key innovation to achieve distortion-mode-sensitive softening is the introduction of a Lode-angle weighting function:

$w(\theta) = \frac{\theta + \pi/6}{\pi/3},$

which satisfies $w(-\pi/6)=0$ (pure compression), $w(\pi/6)=1$ (pure tension), and continuously interpolates across all distortion modes, ensuring a smooth, thermodynamically admissible transition between tensile and compressive failure behaviors. The unified strain-energy is then

$$\psi(F) = \psi_f^{\rm prop}(K_3) - \psi_e^{\rm prop}(W_{\text{PK}})$$

with

$$\begin{aligned} \psi_f^{\rm prop}(K_3) &= [1-w(K_3)]\psi_f^{-} + w(K_3)\psi_f^{+}, \ \psi_e^{\rm prop}(W_{\text{PK}}) &= [1-w(K_3)]\psi_e^{-}(W_{\text{PK}}) + w(K_3)\psi_e^{+}(W_{\text{PK}}). \end{aligned}$$

This construction enables the total energy (and hence, via $\partial\psi/\partial W_{\text{PK}}$, the stress) to exhibit monotonic, mode-dependent softening and capacity, with convexity and smoothness guaranteed by analytic properties of $w(K_3)$ and the PK potential (Chandrashekar et al., 14 Dec 2025).

4. Machine-learning Surrogates for Lode-invariant Softening

An alternative, data-driven protocol employs surrogates trained on integrity-basis decompositions (Upadhyay et al., 2023). Here, the stress is additively split:

$$\mathbf{S} = \mathbf{S}_\text{vol} + \mathbf{S}_{h,\rm iso} + \mathbf{S}_{v,\rm iso}$$

with the isochoric hyperelastic component

$$\mathbf{S}_{h,\rm iso} = J^{-2/3} [\Gamma_1(\bar I_1,\bar I_2)\,\mathrm{Dev}(\mathbf I) + \Gamma_2(\bar I_1,\bar I_2)\,\mathrm{Dev}(\bar{\mathbf{C}})].$$

Lode-invariant softening is incorporated by either (A) multiplying the surrogate outputs by explicit softening functions $f(I_2, I_3, \theta)$, for example,

$$f(\theta) = 1 - B |\sin(3\theta)|^p, \quad B \in [0,1), \; p\ge1,$$

or (B) training the surrogate directly on the augmented feature set $[\bar I_1, \bar I_2, \theta]$, allowing the machine-learnt map to internalize Lode-angle-driven softening. Physics-based constraints—objectivity, isotropy, reference normalization ($f(3,3,0)=1$), monotonicity—are imposed at both data selection and model fitting stages. Thermodynamic consistency (e.g., $\Xi_{\rm int} \ge 0$) is enforced via constrained GPR (Upadhyay et al., 2023).

5. Thermodynamic Consistency and Stability

Thermodynamic admissibility is guaranteed by several properties:

• The softening factors $\partial\psi/\partial W \in (0,1]$, enforcing that $\psi(W)$ is monotonic and saturates at $\psi_f$.
• The intact PK potential, via its $\mathcal{G}(K_3)$ form, ensures satisfaction of the Baker–Ericksen inequalities and absence of loss of ellipticity under small strains.
• The linear interpolation $w(K_3)$ maintains $C^1$-smoothness, confirmed by three-dimensional energy landscape plots showing no loss of convexity or unphysical snap-back, ensuring positive incremental moduli under all loading paths.
• Where desired, unloading irreversibility is imposed by locking the energy at the maximal value achieved ($\psi_f$) using a Heaviside switch (Chandrashekar et al., 14 Dec 2025).

6. Parameter Identification, Cross-mode Generalizability, and Scaling

The Lode-invariant-based softening model can be calibrated by fitting a small set of physically interpretable parameters to experimental tension-compression data. For agarose hydrogels (1%, 2%, 3% w/v), simultaneous fits of all eight model parameters (shear modulus $\mu$, stiffening $a$, mode-smoothness $b_0, b_1$, tensile and compressive energy limiters $\Phi^{\pm}$, and softening sharpness $m^{\pm}$) achieved sub-10% residual errors by nonlinear optimization. Each parameter $Y$ follows a power law in concentration $c$:

$$\begin{aligned} \mu &\sim c^{1.66}, \;\; a \sim c^{1.47}, \;\; b_0 \sim c^{-0.35}, \;\; b_1 \sim c^{0.76}, \ \Phi^+ &\sim c^{2.13}, \;\; \Phi^- \sim c^{3.54}, \;\; m^- \sim c^{0.28}. \end{aligned}$$

Parameters interpolated to untested concentrations yielded accurate predictions for both uniaxial and pure-shear loading paths, demonstrating the model’s cross-mode generalizability. Prediction surfaces $\psi(K_2, K_3)$ show monotonic energy growth up to failure, saturation at mode-dependent maxima, and smooth distortional interpolation (Chandrashekar et al., 14 Dec 2025).

7. Summary and Significance

The Lode-invariant-based hyperelastic softening model provides a unified, distortion-mode-sensitive description of softening and failure in soft materials. By embedding separate tensile and compressive energy limits through direct Lode-angle weighting and forgoing internal damage variables, these models deliver robust cross-mode predictions, thermodynamic rigor, and physically interpretable composition scaling. Both analytic (energy-limiting) and data-driven (integrity-basis surrogate) instantiations are possible, each enforcing objectivity, isotropy, and stability by construction (Upadhyay et al., 2023, Chandrashekar et al., 14 Dec 2025). This framework underpins generalized constitutive descriptions enabling three-dimensional, distortion-sensitive failure mapping of soft matter classes, with immediate applications in hydrogel mechanics, biological tissue modeling, and soft robotics.