Thermodynamically Consistent Hybrid and Permutation-Invariant Neural Yield Functions for Anisotropic Plasticity

TL;DR

This paper introduces two neural network frameworks that incorporate thermodynamic constraints to model anisotropic plasticity in metals, demonstrating improved generalization over traditional models with minimal data.

Contribution

The authors develop hybrid and permutation-invariant neural network models that enforce thermodynamic consistency and effectively capture anisotropic yield behavior.

Findings

PI-ICNN models outperform traditional yield criteria on validation data.

Hybrid models fit training data well but overfit, showing less robustness.

PI-ICNN models generalize better, accurately predicting yield loci and ratios.

Abstract

Plastic anisotropy in metals remains challenging to model. This is partly because conventional phenomenological yield criteria struggle to combine a highly descriptive, flexible representation with constraints, such as convexity, dictated by thermodynamic consistency. To address this gap, we employ architecturally-constrained neural networks and develop two data-driven frameworks: (i) a hybrid model that augments the Hill yield criterion with an Input Convex Neural Network (ICNN) to get an anisotropic yield function representation in the six-dimensional stress space and (ii) a permutation-invariant input convex neural network (PI-ICNN) that learns an isotropic yield function representation in the principal stress space and embeds anisotropy through linear stress transformations. We calibrate the proposed frameworks on a sparse Al-7079 extrusion experimental dataset comprising 12 uniaxial samples with measured yield stresses and Lankford ratios. To test the robustness of each framework, nine datasets were generated using k-fold cross-validation. These datasets were then used to quantitatively compare Hill-48, Yld2004-18p, pure ICNNs, the hybrid approach, and the PI-ICNN frameworks. While ICNNs and hybrid approaches can almost perfectly fit the training data, they exhibit significant over-fitting, resulting in high validation and test losses. In contrast, both PI-ICNN frameworks demonstrate better generalization capabilities, even outperforming Yld2004-18p on the validation and test data. These results demonstrate that PI-ICNNs unify physics-based constraints with the flexibility of neural networks, enabling the accurate prediction of both yield loci and Lankford ratios from minimal data. The approach opens a path toward rapid, thermodynamically consistent constitutive models for advanced forming simulations and future exploration of coupled hardening or microstructure-informed design.