Design of a specimen to train path-dependent deep learning material models from a single uniaxial test: eliciting strain diversity via automatically differentiable elastoplastic topology optimization

TL;DR

This work tackles the data-bottleneck in training path-dependent neural material models by designing a single uniaxial specimen via automatically differentiable elastoplastic topology optimization to maximize strain-path diversity. An entropy-based objective guides topology optimization, FE-based data generation produces rich stress–strain trajectories, and ADiMU trains a large GRU surrogate from this dataset. The approach yields GRU models with substantially lower prediction errors than those trained on standard dogbone or notched specimens and demonstrates robustness to data redundancy and cross-model generalization (e.g., to Drucker–Prager plasticity). If realized experimentally, this framework could enable Material Testing 2.0, reducing experimental burden and enabling data-efficient constitutive modeling from a single optimized test.

Abstract

Artificial neural networks accurately learn nonlinear, path-dependent material behavior. However, training them typically requires large, diverse datasets, often created via synthetic unit cell simulations. This hinders practical adoption because physical experiments on standardized specimens with simple geometries fail to generate sufficiently diverse stress-strain trajectories. Consequently, an unreasonably large number of experiments or complex multi-axial tests would be needed. This work shows that such networks can be trained from a single specimen subjected to simple uniaxial loading, by designing the specimen using a novel automatically differentiable elastoplastic topology optimization method. Our strategy diversifies the stress-strain states observed in a single test involving plastic deformation. We then employ the automatically differentiable model updating (ADiMU) method to train the neural network surrogates. This work demonstrates that topology-optimized specimens under simple loading can train large neural networks, thereby substantially reducing the experimental burden associated with data-driven material modeling.

Figures (18)

• Figure 1: What is the optimal geometry to train (update or calibrate) a material model? The standard dogbone specimen (left) generates limited stress–strain paths under uniaxial loading, leading to less informative data that can only train a conventional material model with few parameters; the dogbone specimen is inadequate to train a neural network material model with millions of parameters. In contrast, the aim of this article is to find an optimized specimen geometry (right) that generates diverse strain–stress data from a single uniaxial test from which a neural network material model can be trained.
• Figure 2: Schematic workflow of the proposed framework. Step 1: Automatically-differentiable elastoplastic topology optimization to find a specimen under uniaxial monotonic loading that maximizes the strain diversity within that specimen. Step 2: The optimized specimen is subjected to a cyclic tension–compression loading to generate a local stress–strain dataset that spans a wide range of deformation states, as illustrated by the cloud of trajectories in the $\epsilon_{11} - \epsilon_{22}$ plane. Step 3: The effectiveness of the topology-optimized specimen is evaluated by training a GRU-based recurrent neural network model based on the dataset and testing on an unseen randomly generated polynomial dataset.
• Figure 3: One-dimensional schematic of the objective-function definition. (a) The predefined strain interval is partitioned into several equal-width cells. A Gaussian probability-density function (PDF), centered at the midpoint of each cell, is assigned as a weighting kernel. The standard deviation is selected so that the PDF decays to $\approx 0$ outside its own cell; after normalization, a single strain sample therefore contributes $\approx 1$ to its host cell and negligibly elsewhere. (b) An aggregate histogram is built by summing these individual contributions over all samples. Because each sample behaves as a unit impulse within its cell, the resulting bar heights closely approximate the discrete point counts in the corresponding bins, providing a differentiable surrogate for cell occupancy. The dataset achieves maximum diversity when the histogram is a uniform distribution.
• Figure 4: Visualization of the 2D test set: (a) Accumulated plastic strain versus load step, demonstrating that all test paths undergo plastic deformation; (b) Probability density distribution of the accumulated plastic strain across the test set; (c) A representative sample of strain paths; (d) The corresponding stress response for the strain paths shown in (c).
• Figure 5: Representative 2D design evolution: (a) Optimized design obtained directly from the topology optimization process; (b) Binary design after applying a thresholding technique; (c) Final cleaned design after post-processing to remove tiny features. This design is used for subsequent finite element simulations and dataset generation.