Lode-Angle-Dependent Weighting Function

• Lode-angle-dependent weighting functions are explicit interpolation mechanisms that modulate material responses based on the Lode angle.
• They interpolate between tension, shear, and compression, enabling accurate representation of softening, failure, and plastic flow in constitutive laws.
• Integrating these functions into constitutive models ensures thermodynamic consistency and enhances predictive robustness across diverse loading conditions.

A Lode-angle-dependent weighting function is an explicit interpolation mechanism that modulates material constitutive responses—such as softening, failure, plastic flow, and damage—according to the deformation or stress mode, as quantified by the Lode angle. Such weighting functions provide the means to encode physical tension-compression asymmetry and distortion-mode sensitivity directly within constitutive laws, enabling models to interpolate smoothly between the limiting cases of uniaxial tension, pure shear, and uniaxial compression, and to capture experimentally observed behaviors in both soft solids and metallic materials. Introduced in modern frameworks for hyperelasticity, softening, and coupled elastoplastic-damage mechanics, Lode-angle-dependent weights play a central role in ensuring predictive fidelity and thermodynamic consistency across the full spectrum of loading geometries (Chandrashekar et al., 14 Dec 2025, Hami et al., 2021).

1. Definition of the Lode Angle and Invariants

The Lode angle, often referred to as the “third invariant angle,” parametrizes the state of distortion under deviatoric stress or strain. For isotropic finite-strain settings, the Hencky (logarithmic) strain tensor $E = \ln V$ is commonly adopted as the base measure. The associated invariants are

$$J_2^E = \tfrac{1}{2}\,\text{tr}[({\rm dev}\,E)^2], \qquad J_3^E = \tfrac{1}{3}\,\text{tr}[({\rm dev}\,E)^3].$$

The Lode angle $\theta$ is then given by

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

For stress-based elastoplastic/damage models (Hami et al., 2021), the Cauchy stress invariants are defined as

$$p = \tfrac{1}{3} \rm{tr} \sigma, \qquad \sigma_{\rm eq} = \sqrt{3 J_2}, \qquad X = \frac{3\sqrt{3}}{2} \frac{J_3}{\sigma_{\rm eq}^3}, \quad \cos(3\theta_0) = X,$$

where $J_2$ and $J_3$ are the second and third invariants of the deviatoric stress.

2. Formulation of the Lode-Angle-Dependent Weighting Function

Two classes of Lode-angle-dependent weighting functions are prominent:

2.1. Linear Interpolant for Softening Models (Chandrashekar et al., 14 Dec 2025):

In soft-material constitutive frameworks, the weighting function $w(\theta)$ is defined as a linear map: $$w(\theta) = \frac{\theta + \tfrac{\pi}{6}}{\tfrac{\pi}{3}},\quad \theta\in\left[-\tfrac{\pi}{6},\tfrac{\pi}{6}\right],$$ equivalently,

$w(\theta) = \frac{3\theta}{\pi} + \frac{1}{2}.$

This interpolates continuously between pure compression $(\theta=-\pi/6,\, w=0)$, pure shear $(\theta=0,\, w=1/2)$, and pure tension $(\theta=+\pi/6,\, w=1)$.

2.2. Extended Form for Damage-Coupled Plasticity (Hami et al., 2021):

Elasto-plastic and damage models may employ a generalized weight $\Psi(\eta, \theta_0)$ accounting for both stress triaxiality $\eta=p/\sigma_{\rm eq}$ and the Lode angle $\theta_0$: $$\Psi(\eta, \theta_0) = 1 + d_n(\eta - \eta_0) - d_s |\theta_0|^{m+1}.$$ Here, $d_n$, $d_s$ are model parameters, $\eta_0$ is a reference triaxiality (e.g., $\eta_0=1/3$), and $m$ is a smoothing exponent. This form ensures differentiability at $\theta_0=0,\pi/3$ and captures the combined effect of hydrostatic and shear components on damage evolution.

3. Incorporation into Constitutive Models

The Lode-angle-dependent weighting function enters constitutive formulations at multiple levels:

• Energy Softening: In hyperelastic models for soft TCA materials, $w(\theta)$ blends tensile and compressive energy limiters in the mode-sensitive softening potential,

$$\Psi(F) = (1-w(\theta))\, \Psi_f^- + w(\theta)\, \Psi_f^+ - \left[(1-w(\theta))\, \psi_e^-(W) + w(\theta)\, \psi_e^+(W)\right],$$

where $\Psi_f^{\pm}$ are limiting energies and $\psi_e^{\pm}$ are recoverable elastic portions (Chandrashekar et al., 14 Dec 2025).

• Flow Stress and Plasticity: In metal plasticity, the weighting function modifies the flow stress according to both triaxiality and Lode angle,

$$\sigma_y(\epsilon^p, \eta, \theta_0) = \sigma(\epsilon^p)\; [1 - C_n(\eta - \eta_0)]\; [c_s + (c_{ar} - c_s) |\theta_0|^{m+1}],$$

with $C_n,\, c_s,\, c_{ar},\, m$ determined empirically (Hami et al., 2021).

• Damage Evolution: The damage rate is weighted by $\Psi(\eta, \theta_0)$,

$$\dot{D} = (1-\Psi D)^{-\alpha} \langle Y - Y_0 \rangle^\beta$$

controlling enhancement or suppression of damage according to loading state (Hami et al., 2021).

4. Physical Interpretation and Parameterization

The endpoints of the weighting function correspond to distinct physical modes: $\theta=+\pi/6$ (pure tension, tensile limiter), $\theta=-\pi/6$ (pure compression, compressive limiter), and $\theta=0$ (pure shear, arithmetic mean of limits). The linear slope ($d w/d \theta = 3/\pi$) governs the sharpness of the interpolated transition.

In metallic damage models, $d_n$ and $d_s$ control sensitivity to triaxiality and Lode angle respectively, while $m$ sets the smoothness of the transition between pure tension, pure shear, and pure compression. These parameters are calibrated against a suite of uniaxial, shear, and compression experiments to ensure fidelity across the Lode-invariant space (Hami et al., 2021).

5. Thermodynamic Consistency and Smoothness

Ensuring the interpolated constitutive response is thermodynamically admissible and free from spurious artifacts is central. For linear $w(\theta)$, both the weight and its derivative are continuous and bounded, yielding a strain energy $\Psi(F)$ that is smooth in all arguments. This design avoids loss-of-ellipticity, nonconvexity, or unphysical kinks in the stress response.

Analogous continuity is maintained in elastoplastic-damage models by selection of exponents and prefactors to regularize the weighting function at transitions and endpoints (Chandrashekar et al., 14 Dec 2025, Hami et al., 2021).

6. Predictive Robustness and Cross-Mode Validation

Lode-angle-dependent weighting enables predictive generalization to untested loading modes. In soft gels, models calibrated solely to uniaxial tension and compression can correctly anticipate pure shear failure and softening, as the Lode-weighted interpolation specifies the energetic limit in such intermediate modes. Similarly, in metallic alloys, the weighting function ensures the evolution and localization of damage follows observed “damage locus” trends across a gamut of stress states.

This capacity is rooted in the explicit, physically interpretable nature of the weight, and its embedding directly into the constitutive law, as opposed to ad hoc damage variables or auxiliary phenomenological adjustments (Chandrashekar et al., 14 Dec 2025, Hami et al., 2021).

7. Extensions and Future Directions

While current frameworks employ a linear or power-law form for the Lode-angle-dependent weighting, the interpolation scheme can be generalized. Higher-order (e.g., sinusoidal or polynomial) interpolants introduce additional “shape” parameters, enabling sharper or more gradual cross-mode transitions if demanded by mixed-mode fracture experiments, but at the cost of additional complexity. The explicit separation of tensile and compressive energetics is preserved in such generalizations, maintaining the physical transparency and predictive capabilities of the approach (Chandrashekar et al., 14 Dec 2025).

Model	Weighting Function	Calibration Material
Soft hydrogels	$w(\theta) = \frac{3\theta}{\pi}+\frac12$	Agarose gels (1–3% w/v)
Al alloys (damage)	$\Psi(\eta, \theta_0)$ as above	Al 2024-T351

Present methodologies lay the groundwork for extension to three-dimensional, distortional-mode-sensitive failure mapping across a broad range of materials exhibiting tension-compression asymmetry.