Exploring energy minimization to model strain localization as a strong discontinuity using Physics Informed Neural Networks

TL;DR

This work demonstrates that strain localization in elastoplastic solids can be modeled as a free discontinuity problem solved via energy minimization with regularized strong discontinuities, using Physics-Informed Neural Networks to represent both the bulk displacement and the localization jump. By embedding the variational formulation directly into the loss function and encoding the jump with a trainable, band-regularized activation, the method simultaneously identifies the localization band and the displacement jump without resorting to phase-field regularization. The authors validate the approach in 1D and 2D shearing tests, showing accurate recovery of band position, jump magnitude, and cohesive energy consistent with the underlying energy functional, while performing sensitivity analyses on discretization and optimization choices. This variational PINN paradigm offers a promising avenue for handling sharp discontinuities in solid mechanics problems, with potential extensions to fracture, frictional/plastic damage, and propagating discontinuities. The practical impact lies in providing a flexible, mesh-based/discontinuous representation within a neural framework, albeit with notable computational cost that motivates further efficiency enhancements.

Abstract

We explore the possibilities of using energy minimization for the numerical modeling of strain localization in solids as a sharp discontinuity in the displacement field. For this purpose, we consider (regularized) strong discontinuity kinematics in elastoplastic solids. The corresponding mathematical model is discretized using Artificial Neural Networks (ANNs), aiming to predict both the magnitude and location of the displacement jump from energy minimization, $\textit{i.e.}$, within a variational setting. The architecture takes care of the kinematics, while the loss function takes care of the variational statement of the boundary value problem. The main idea behind this approach is to solve both the equilibrium problem and the location of the localization band by means of trainable parameters in the ANN. As a proof of concept, we show through both 1D and 2D numerical examples that the computational modeling of strain localization for elastoplastic solids using energy minimization is feasible.

Figures (23)

• Figure 1: Kinematic decomposition for a one-dimensional problem. A strong discontinuity is recovered as $h$ tends to zero.
• Figure 2: Activation function. Left: effect of varying $\beta$ on $\Phi$. Right: effect of varying $h$ on $\Phi$, with $\beta$ fixed at 100.
• Figure 3: Geometry configuration of the 1D problem.
• Figure 4: Evolution of the displacement field for increasing prescribed displacement, showing the NN-predicted solution (solid green lines) and the expected analytical solution (dotted blue lines).
• Figure 5: Induced cohesive energy and cohesive force, showing the NN-predicted solution (solid green lines) and the expected analytical solution (dotted blue lines).