[Paper Review] Physics-informed neural networks for modeling rate- and temperature-dependent plasticity

Rajat Arora, Pratik Kakkar+2 more|arXiv (Cornell University)|Jan 20, 2022

Force Microscopy Techniques and Applications22 citations

TL;DR

A physics-informed neural network framework models rate- and temperature-dependent elastic-viscoplastic deformation using mixed-variable outputs and a balanced physics/data loss, demonstrated on two test problems with good agreement to FEM results.

ABSTRACT

This work presents a physics-informed neural network (PINN) based framework to model the strain-rate and temperature dependence of the deformation fields in elastic-viscoplastic solids. To avoid unbalanced back-propagated gradients during training, the proposed framework uses a simple strategy with no added computational complexity for selecting scalar weights that balance the interplay between different terms in the physics-based loss function. In addition, we highlight a fundamental challenge involving the selection of appropriate model outputs so that the mechanical problem can be faithfully solved using a PINN-based approach. We demonstrate the effectiveness of this approach by studying two test problems modeling the elastic-viscoplastic deformation in solids at different strain rates and temperatures, respectively. Our results show that the proposed PINN-based approach can accurately predict the spatio-temporal evolution of deformation in elastic-viscoplastic materials.

Motivation & Objective

• Motivate efficient, real-time modeling of elastic-viscoplastic deformation under varying strain rates and temperatures.
• Develop a PINN framework that handles rate and temperature dependencies with minimal gradient-balancing overhead.
• Address output variable selection to faithfully solve mechanical PDEs with PINNs.
• Demonstrate predictive accuracy against finite element method (FEM) references for two test cases.
• Provide guidance on loss function construction and training stability for nonlinear plasticity problems.

Proposed method

• Use two separate physics-informed neural networks to predict displacement, stress, plastic strain, and strength (u, σ, εp, S) with inputs capturing rate and temperature effects.
• Employ a mixed-variable formulation to avoid accuracy issues, enabling constitutive relation satisfaction and stable training.
• Construct a composite physics-based loss including PDE, Dirichlet/Neumann boundary, initial conditions, and constitutive/strength/plastic strain-rate losses.
• Introduce a novel Modified Mean Squared Error (MMSE) loss to mitigate stiffness from power-law plasticity terms.
• Balance loss components with a simple, non-iterative scaling strategy using nondimensionalization and reference scales (no added computational overhead).
• Train PINNs with a 9-layer, 120-neuron-per-layer multilayer perceptron per network, using Adam and ReduceLROnPlateau, on data generated by deal.II solving the governing equations on a 32x32 grid.

Experimental results

Research questions

• RQ1 How can PINNs model spatio-temporal deformation fields in elastic-viscoplastic solids under varying strain rates and temperatures?
• RQ2 What outputs should PINNs predict to faithfully satisfy constitutive laws and equilibrium in elastic-viscoplastic problems?
• RQ3 How can one balance physics-based and data-driven losses without introducing gradient instability or excessive computation?
• RQ4 What is the predictive accuracy of PINN surrogates compared to FEM references across training and out-of-training-range inputs?
• RQ5 Do rate- and temperature-dependent PINN models generalize to inputs beyond the training set?

Key findings

• PINN predictions for stress, plastic strain, and displacement show great agreement with FEM references in the tested rate- and temperature-dependent cases.
• Within the training input range, errors remain small, while extrapolation beyond the training range leads to increasing errors (up to ~10% in some regions).
• Model I (with outputs including stress) converges reliably, whereas Model II (omitting σ as an output) exhibits convergence challenges and higher final loss.
• A simple, constant-weighting strategy for the physics-based loss components effectively mitigates gradient imbalance without extra computational overhead.
• The MMSE loss for plastic-strain-rate residual helps stabilize training by addressing the nonlinearity of the rate-dependent term.
• Two case studies demonstrate the framework’s capability to predict spatio-temporal evolution under different strain-rate and temperature settings.