Learning Physics-Consistent Material Behavior from Dynamic Displacements

TL;DR

This paper addresses the challenge of inferring physics-consistent constitutive laws from deformation data without access to boundary forces or stress measurements. It introduces uLED, a dynamics-based, unsupervised framework that learns a surrogate $\widehat{\mathcal{P}}(\mathbf{F}; \theta)$ via an ICNN to ensure convexity of the energy in Green-Lagrange strain, enforcing momentum balance within a subdomain using only displacement data. The method demonstrates accurate recovery of energy density and stress across multiple hyperelastic models, shows transferability to unseen geometries, and remains robust under partial observations, coarse data, and moderate noise, with extensions to dissipative materials. The work emphasizes practical applicability to in-situ measurement scenarios and strain-rate dependent materials, while outlining future directions toward 3D extension and symbolic forms of the learned constitutive relations.

Abstract

Accurately modeling the mechanical behavior of materials is crucial for numerous engineering applications. The quality of these models depends directly on the accuracy of the constitutive law that defines the stress-strain relation. However, discovering these constitutive material laws remains a significant challenge, in particular when only material deformation data is available. To address this challenge, unsupervised machine learning methods have been proposed to learn the constitutive law from deformation data. Nonetheless, existing approaches have several limitations: they either fail to ensure that the learned constitutive relations are consistent with physical principles, or they rely on boundary force data for training which are unavailable in many in-situ scenarios. Here, we introduce a machine learning approach to learn physics-consistent constitutive relations solely from material deformation without boundary force information. This is achieved by considering a dynamic formulation rather than static equilibrium data and applying an input convex neural network (ICNN). We validate the effectiveness of the proposed method on a diverse range of hyperelastic material laws. We demonstrate that it is robust to a significant level of noise and that it converges to the ground truth with increasing data resolution. We also show that the model can be effectively trained using a displacement field from a subdomain of the test specimen and that the learned constitutive relation from one material sample is transferable to other samples with different geometries. The developed methodology provides an effective tool for discovering constitutive relations. It is, due to its design based on dynamics, particularly suited for applications to strain-rate-dependent materials and situations where constitutive laws need to be inferred from in-situ measurements without access to global force data.

Figures (19)

• Figure 1: Illustration of the reference configuration $\Omega$ and the deformation over time. The displacement $\prescript{t}{}{}{\mathbf{u}}$ of an arbitrary material point at each time $t$ is the difference between its position $\prescript{t}{}{}{\mathbf{x}}$ at time $t$ and its original position ${\mathbf{X}}$ in the reference configuration $\Omega$, i.e., $\prescript{t}{}{}{\mathbf{u}} = \prescript{t}{}{}{\mathbf{x}} - {\mathbf{X}}$.
• Figure 2: The concept and architecture of the proposed uLED method. $\widehat{\mathcal{P}}({\mathbf{F}}; \theta)$ is the neural-network based surrogate model designed to approximate the constitutive material law, i.e., $\widehat{\mathcal{P}}({\mathbf{F}}; \theta) \approx \mathcal{P}({\mathbf{F}})$, where $\widehat{\mathcal{P}}({\mathbf{F}};\theta)$ is parameterized by $\theta$. (a) The neural network architecture of the surrogate model $\widehat{\mathcal{P}}$. From the measured displacement field ${\mathbf{u}}$, we compute the deformation gradient ${\mathbf{F}}$, the Green-Lagrangian strain ${\mathbf{E}}$, and strain invariants $I_1$ and $I_2$. The neural network (ICNN amos2017input) takes strain invariants as input and transforms them through a series of latent representations ${\mathbf{z}}_1, {\mathbf{z}}_2, \ldots, {\mathbf{z}}_5$ via multiple layers. The output at the last layer corresponds to the predicted energy density $\widehat{W}$, i.e., $\widehat{W}={\mathbf{z}}_5$. The predicted first Piola-Kirchhoff stress $\widehat{{\mathbf{P}}}$ is computed by taking the gradient of the energy density. The learnable parameters $\theta$ in $\widehat{\mathcal{P}}({\mathbf{F}}; \theta)$ include the non-negative learnable layer-wise parameters $\theta^{\text{(cvx)}}$ and normal learnable parameters $\theta^{\text{(fc)}}$ without the non-negative restriction. (b) Parameter optimization of the surrogate constitutive model $\widehat{\mathcal{P}}({\mathbf{F}}; \theta)$. The learning objective (Eq. \ref{['eq:objective_function']}) is formulated as a constrained optimization task and projected gradient descent is adopted to efficiently optimization parameters under the non-negative constraint.
• Figure 3: Illustration of the experimental configuration. (a) We simulate the transient motion of a 2D plate that is fixed at its left boundary and loaded with a temporally evolving traction on its right boundary. We record the displacements $\prescript{t}{}{}{\mathbf{u}}^{a}$ of sampled material points (indicated by the yellow dots) from the FEM simulation. These sampled displacements $\prescript{t}{}{}{\mathbf{u}}^{a}$ are used as the input for uLED. (b) The proposed uLED method is trained using the sampled displacements $\prescript{t}{}{}{\mathbf{u}}^{a}$. (c) The model predicts the energy density value and the first Piola-Kirchhoff stress tensor for an arbitrary element given the deformation gradient.
• Figure 4: Comparison between the predicted and ground-truth energy density and stress fields for a Neo-Hookean material. Data is shown for two arbitrarily chosen time steps (a) $t=7$, and (b) $t=1407$. On the left side, the predicted energy density ($\widehat{W}$) is compared to the ground-truth value ($W$) via their normalized difference. On the right side, the magnitude of the first Piola-Kirchhoff stress fields ($|\widehat{{\mathbf{P}}}|$) is compared to its ground-truth values ($|{\mathbf{P}}|$). The predicted values shown are the mean of three independent experiments.
• Figure 5: Validation of the predicted energy density and stress of the Neo-Hookean material. We compare curves of the prediction to the ground-truth of the energy density (top) and the stress (bottom) for different deformation states (see Eq. \ref{['eq:F_path']}) at a single material point. The predictions shown are the mean of three independent experiments.