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Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representations

This paper proposes a physics-informed framework that utilizes convex neural networks to discover anisotropic yield functions directly from full-field displacement and reaction force data, bypassing the need for direct stress observations or predefined parametric forms by embedding the yield function within a differentiable elastoplastic stress integration scheme.

Original authors: Hyeonbin Moon, Donghyuk Cho, Jecheon Yu, Jeong Whan Yoon, Seunghwa Ryu

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hyeonbin Moon, Donghyuk Cho, Jecheon Yu, Jeong Whan Yoon, Seunghwa Ryu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out the exact shape of a hidden, flexible balloon inside a box. You can't see the balloon, you can't touch it, and you don't have a ruler to measure it directly. All you have is a camera watching the outside of the box and a scale measuring how hard you have to push on the box to move it.

This paper presents a clever new way to solve that puzzle for materials like metal sheets. Here is the breakdown of how they did it, using simple analogies.

The Problem: The "Black Box" of Metal

When engineers design things like car bodies or airplane wings, they need to know exactly how the metal will bend and stretch. This behavior is controlled by something called a "yield function." Think of the yield function as the metal's "personality rulebook." It tells the metal: "If I push you this hard in this direction, you will start to permanently bend."

Usually, figuring out this rulebook is hard because:

You can't see the stress inside the metal (like seeing the pressure inside the balloon).
You have to test the metal in many different directions (pulling it left, right, diagonally).
Engineers usually have to guess the shape of the rulebook first (e.g., "Is it a circle? A square?") and then try to fit the data to it. If they guess the wrong shape, the whole model fails.

The Solution: A "Smart Guessing" Robot

The authors created a new framework that acts like a detective. Instead of guessing the shape of the rulebook, they let a computer discover the shape from scratch.

Here is how their "detective" works:

1. The "Black Box" Setup
They took a piece of metal with two holes cut out of it (like a cookie with chocolate chips removed). They pulled on this metal in four different ways (stretching it horizontally, vertically, diagonally, and twisting it).

What they measured: They used a camera to watch how the metal surface moved (displacement) and a scale to measure how much force was needed to pull it (reaction force).
What they didn't measure: They didn't measure the stress inside the metal or the exact plastic strain. They only had the "outside" data.

2. The "Shape-Shifting" AI
Instead of using a fixed mathematical formula, they used a Convex Neural Network.

The Analogy: Imagine a piece of clay that can change its shape. The AI is the sculptor.
The Rules: The sculptor isn't allowed to make just any shape. The paper forces the AI to follow strict physics rules:
- Convexity: The shape must be "bowl-like" (no dents or caves). This ensures the metal behaves stably.
- Symmetry: If you pull the metal, it should behave similarly if you push it (tension-compression symmetry).
- Scaling: If you double the force, the effect should double (positive homogeneity).

3. The "Force Balance" Test
This is the magic part. The AI doesn't look at the stress inside the metal because it can't see it. Instead, it plays a game of "Force Balance."

The AI guesses a shape for the yield function (the rulebook).
It runs a simulation: "If the metal follows this rulebook, and I pull it with this force, how much should the surface move?"
It compares its prediction to the actual camera footage.
The Feedback Loop: If the AI's predicted movement doesn't match the real camera footage, it knows its "rulebook" is wrong. It tweaks the shape of the clay (the neural network) and tries again.
It keeps doing this until the AI's simulation of the metal's movement perfectly matches the real-world camera and scale data.

The Results: Did it Work?

The researchers tested this "detective" on three known types of metal behaviors (some simple, some very complex and directional).

Perfect Conditions: When the data was clean (no noise), the AI successfully "rediscovered" the hidden rulebooks. It drew the correct shapes, even for the complex ones, without ever being told what the answer was.
Noisy Conditions: In the real world, cameras aren't perfect; they have a little bit of "static" or noise. The team tested this by adding fake "static" to the data. The AI got a little less accurate as the noise got louder, but it didn't break. It still found a shape that was very close to the truth.
The "Blind Spots": The AI was very confident about the parts of the rulebook that were tested by the four pulling directions. However, in areas where no one pulled the metal (like pulling in a direction that was never tested), the AI was less sure. This is logical: if you never test a direction, you can't be 100% sure how the metal behaves there.

The Final Step: Making it Usable

The AI's "clay" shape is mathematically complex and a bit jagged (it's a neural network). Real-world engineering software (used to design cars) needs smooth, simple formulas to work.

The Analogy: The AI found the perfect, complex shape of the balloon. Now, the researchers took that shape and traced it onto a piece of paper using a smooth, simple polynomial curve.
The Result: This smooth curve was almost identical to the AI's complex shape and could be easily plugged into standard engineering software to simulate the metal's behavior.

Summary

This paper shows that you don't need to guess the mathematical formula for how metal bends. Instead, you can use a physics-constrained AI to learn the shape of the bending rule just by watching how the metal moves and how hard you have to pull it. It's like figuring out the shape of a hidden object by only watching how the shadows it casts move, but with a robot that knows the laws of physics so it never makes impossible guesses.

Technical Summary: Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representations

1. Problem Statement

Identifying anisotropic yield functions remains a significant challenge in solid mechanics. Traditional approaches rely on selecting a specific analytical form (e.g., von Mises, Hill 1948, Barlat–Yoon) and calibrating parameters against experimental data. However, this process faces several hurdles:

Data Limitations: Yielding is not directly observed in full-field mechanical measurements (such as Digital Image Correlation, DIC). Experimental data typically provide displacement fields and reaction forces, but not direct stress fields, plastic strain measurements, or yield surface data.
Calibration Complexity: Determining anisotropic behavior often requires loading in many directions, and selecting an appropriate analytical form that is sufficiently expressive yet compact is nontrivial.
Model Structure: Existing data-driven methods often approximate stress responses directly or require stress-space supervision, which fails to reveal the underlying constitutive structure (the yield function) required for conventional elastoplastic stress integration.

The core problem addressed is how to infer an anisotropic yield function directly from full-field displacement and reaction force data, without stress observations, plastic strain measurements, or a prescribed parametric form, while ensuring the result remains compatible with the mechanical structure of elastoplastic stress integration.

2. Methodology

The authors propose a physics-informed framework that treats the yield function as a mechanically constrained constitutive component within an elastoplastic stress integration loop, rather than as a direct supervised mapping in stress space.

A. Neural Representation of the Yield Function
The unknown yield function, $\bar{\sigma}(\sigma)$ , is represented by a Convex Neural Network (CNN) rather than a candidate library of analytical equations. To satisfy the mechanical requirements of rate-independent, pressure-insensitive plasticity with an associated flow rule (AFR), the network enforces:

Convexity: Ensured via an input convex neural network architecture where weights multiplying previous-layer activations are constrained to be non-negative.
Positive Homogeneity of Degree One: Enforced by using ReLU activations and removing bias terms, ensuring $\bar{\sigma}(\rho\sigma) = \rho\bar{\sigma}(\sigma)$ .
Tension-Compression Symmetry: Imposed by symmetrizing the network output over sign transformations of the stress components.
Smoothness: The architecture relaxes global $C^1$ smoothness to a $C^0$ , piecewise differentiable representation to satisfy the convexity and homogeneity constraints simultaneously.

B. Physics-Informed Training Strategy
The framework utilizes a differentiable elastoplastic stress update algorithm to train the neural network using only displacement and force data:

Data Input: Time-resolved full-field displacement data (converted to strain increments) and reaction force data from multiple loading cases (Uniaxial Tension in x/y, Biaxial Tension, Simple Shear).
Differentiable Stress Integration: An explicit forward Euler scheme is used to update stress and plastic strain. To maintain differentiability with respect to the neural network parameters, the sharp plasticity indicator (which switches between elastic and plastic states) is replaced by a smooth sigmoid approximation.
Loss Function: The training objective is the force equilibrium residual. The internal force increments, predicted by assembling the stress updates over the finite element mesh, are compared against the external force increments derived from measured reaction forces. The neural yield function is optimized to minimize this mismatch.
Optimization: The process uses a gradient-based optimizer (Adam) to update the network parameters across all loading cases simultaneously.

C. Finite Element Deployment
Since the discovered neural yield function is piecewise differentiable ( $C^0$ ) and not directly suitable for standard implicit finite element solvers (which require smooth analytical derivatives for return mapping), a polynomial surrogate is fitted post-discovery. This surrogate is a smooth, positively homogeneous polynomial function fitted to the level sets of the identified neural yield function, enabling deployment in conventional implicit FE simulations.

3. Key Contributions

Direct Identification from Observable Data: The framework identifies yield functions from full-field displacement and reaction force data without requiring stress observations, plastic strain measurements, direct yield surface data, or a prescribed analytical form.
Mechanically Constrained Representation: It introduces a convex neural yield function representation and a differentiable elastoplastic training algorithm that enforces key mechanical constraints (convexity and positive homogeneity) while optimizing based on force equilibrium.
Comprehensive Validation: The study validates the approach using finite element benchmarks (von Mises, Hill 1948, and Yld2000-2d), assessing:
- Yield contour agreement under noise-free conditions.
- Sensitivity to displacement measurement noise (simulating DIC uncertainty).
- Identifiability and epistemic uncertainty via deep ensembles.
- Deployment feasibility via a smooth polynomial surrogate.

4. Results and Validation

Noise-Free Discovery: The framework successfully recovered yield contours for isotropic (von Mises) and anisotropic (Hill 1948, Yld2000-2d) benchmarks. The inferred contours matched the ground truth trends, including directional anisotropy, despite the absence of direct stress supervision.
Noise Sensitivity: The method demonstrated robustness to sub-micrometer displacement noise (0.1–0.3 µm). While contour distortion increased with noise levels, the inferred yield surfaces remained consistent with ground truth trends, showing gradual degradation rather than sudden failure.
Identifiability and Uncertainty: Analysis of plastically active stress states revealed that regions of the yield surface visited by the specific loading cases were well-constrained. Conversely, regions with sparse stress state coverage (e.g., opposite-sign normal stresses) exhibited higher epistemic uncertainty, as confirmed by deep ensemble standard deviations.
Deployment: A polynomial surrogate fitted to the discovered neural function successfully replicated the yield contours and produced finite element field results (stress and plastic strain) comparable to the ground truth for the Yld2000-2d benchmark.

5. Significance and Claims

The paper claims that the yield function can be identified as an unknown constitutive component inside elastoplastic stress integration, rather than as a supervised contour fit in stress space. By embedding the neural representation within the stress update and constraining it via force equilibrium, the framework bridges the gap between data-driven discovery and the structural requirements of classical plasticity theory.

The authors emphasize that this approach provides a mechanics-constrained pathway for discovering anisotropic yield functions. It isolates the yield function identification problem from the need for direct stress measurements or pre-selected analytical forms, provided that the hardening law and flow rule are known or prescribed.

Limitations Acknowledged by the Authors:

The validation relies on synthetic data generated by FE simulations, not experimental DIC measurements.
The hardening law, flow rule (AFR), and elasticity are prescribed; the study does not perform simultaneous identification of hardening or non-associated flow rules.
The current implementation assumes plane-stress conditions and small deformations.
The neural yield function is piecewise differentiable, necessitating the post-processing step of fitting a polynomial surrogate for standard implicit FE deployment.

The authors conclude that while the framework successfully demonstrates the feasibility of this approach on benchmark data, future work is required to extend it to experimental settings, simultaneous identification of yield and hardening, and three-dimensional stress states.