A physics-augmented neural network framework for modeling and detecting thermo-visco-plastic behavior

Although considerable attention has been devoted to the development of models for isothermal, rate-independent plasticity, many high-consequence performance assessments involve viscoplastic processes that generate substantial heat. In addition, materials may transit from a nearly isothermal, rate-independent regime to a viscous, temperature-dependent regime during these processes, which makes modeling more challenging. In this work, we develop a physics-augmented neural network (PANN) framework for modeling general temperature-dependent, rate-dependent inelastic processes firmly based on physical principles, including the second law of thermodynamics and coordinate equivariance. These embedded properties are enabled by a number of architectural innovations in the structure and training of an input convex and potential-based neural ordinary differential equation framework. The resulting neural network models are capable of representing a wide spectrum of rate- and temperature-dependence ranging from isothermal, rate-independent elastic-plastic phenomenology to rate-dependent fully viscous inelastic behavior, as we demonstrate. We also show that the framework is capable of modeling complex microstructural inelasticity and predicting the conversion of plastic work to heating when calibrated to stress-temperature observations.

💡 Research Summary

The paper introduces a Physics‑Augmented Neural Network (PANN) framework designed to model thermo‑visco‑plastic behavior across the full spectrum of temperature‑ and rate‑dependent inelasticity while rigorously respecting thermodynamic principles. Traditional constitutive models such as Johnson‑Cook or Cowper‑Symonds treat temperature and strain‑rate effects as empirical modifiers, often using a fixed Taylor‑Quinney coefficient to convert plastic work into heat. These approaches lack a unified, physics‑based description and can fail when a material transitions from an isothermal, rate‑independent regime to a viscous, temperature‑sensitive regime.

To overcome these limitations, the authors embed two fundamental potentials—Helmholtz free energy (ψ) and dissipation potential (ϕ)—directly into the neural network architecture. ψ is modeled as an input‑convex neural network (ICNN) of strain and temperature; its gradient with respect to strain yields the second‑Piola‑Kirchhoff stress. ϕ, also constructed as an ICNN, is a convex, positively homogeneous function of the internal state forces; its gradient defines the evolution of internal variables (e.g., plastic strain, hardening variables). By construction, the dissipation inequality Γ = K·ṡκ ≥ 0 is automatically satisfied, ensuring compliance with the second law of thermodynamics.

Key architectural innovations include: (1) a partially input‑convex network that guarantees the dissipation inequality; (2) a joint observable‑hidden state representation that allows internal variables to influence stress in a tensorial, coordinate‑equivariant manner; (3) a rectified dissipation potential together with a custom loss term that penalizes violations of the duality relationship between ψ and ϕ, thereby enforcing rate‑independent or rate‑dependent flow as appropriate; and (4) regularization terms that inject prior knowledge such as a finite elastic domain and yield‑flow duality, which help the network learn physically realistic elastic limits and plastic yield surfaces.

The theoretical development is grounded in classical continuum thermodynamics, specifically the Coleman‑Gurtin internal‑state‑variable (ISV) framework and the Generalized Standard Material (GSM) theory. The authors show how the traditional yield function, flow rule, and maximum dissipation principle emerge naturally from the gradients of ψ and ϕ. They also discuss the role of positive homogeneity (ϕ(αK)=α^pϕ(K)) in embedding rate dependence, and they demonstrate how the network can recover both rate‑independent plasticity (p=1) and fully viscous behavior (p>1) within a single formulation.

Three numerical experiments validate the approach. First, the PANN is trained on synthetic data spanning isothermal, rate‑independent plasticity, rate‑dependent viscoplasticity, and fully viscous regimes. The model accurately reproduces stress–strain curves, captures thermal softening, and outperforms conventional phenomenological models in predictive accuracy. Second, the framework is applied to homogenize the response of stochastic crystal‑plasticity volume elements that lack closed‑form constitutive equations. Despite the high dimensionality of the microstructural state space, the PANN learns an effective macroscopic law that matches direct numerical simulations. Third, the network is calibrated to stress‑temperature histories from a tension test, enabling it to infer a state‑dependent Taylor‑Quinney coefficient rather than a fixed constant. The learned coefficient varies with temperature, strain rate, and accumulated plastic work, providing a more realistic description of heat generation during deformation.

Overall, the paper demonstrates that embedding thermodynamic constraints directly into neural network architecture yields models that are both highly expressive and physically trustworthy. The input‑convex, potential‑based design ensures training stability, respects energy conservation, and maintains interpretability through the explicit representation of free energy and dissipation potentials. The authors argue that this approach bridges the gap between data‑driven flexibility and the rigorous requirements of continuum mechanics, making it a promising tool for multiscale simulations, complex material systems, and real‑time predictive modeling. Future work is suggested in extending the framework to damage and fracture, model compression for high‑performance computing, and active experimental design to maximize information gain from limited data.