A physics-augmented neural network framework for finite strain incompressible viscoelasticity

TL;DR

This work integrates physics-based constraints into neural networks to model finite-strain incompressible viscoelasticity within the GSM framework. By employing a multiplicative decomposition with unimodular inelastic deformation, invariant-based NN potentials for the free energy and dual dissipation potential, and an implicit exponential time integrator, the approach guarantees thermodynamic consistency, objectivity, and material symmetry. A trainable gate along with $\ell_p$ regularization enables automatic reduction of the number of internal variables during training, and the model is calibrated against both synthetic ground-truth data and real experimental measurements, showing excellent interpolation and plausible extrapolation. The framework also recovers linear viscoelastic behavior via linearization and offers a pathway to integration with finite element methods for complex loading scenarios.

Abstract

We propose a physics-augmented neural network (PANN) framework for finite strain incompressible viscoelasticity within the generalized standard materials theory. The formulation is based on the multiplicative decomposition of the deformation gradient and enforces unimodularity of the inelastic deformation part throughout the evolution. Invariant-based representations of the free energy and the dual dissipation potential by monotonic and fully input-convex neural networks ensure thermodynamic consistency, objectivity, and material symmetry by construction. The evolution of the internal variables during training is handled by solving the evolution equations using an implicit exponential time integrator. In addition, a trainable gate layer combined with lp regularization automatically identifies the required number of internal variables during training. The PANN is calibrated with synthetic and experimental data, showing excellent agreement for a wide range of deformation rates and different load paths. We also show that the proposed model achieves excellent interpolation as well as plausible and accurate extrapolation behaviors. In addition, we demonstrate consistency of the PANN with linear viscoelasticity by linearization of the full model.

Figures (13)

• Figure 1: Visualization of fictitious intermediate configurations ${}_\xi\!\mathcal{B}^\text{i}$ implied by the multiplicative decompositions $\boldsymbol {F} := {}_\xi\!\boldsymbol {F}^\text{e} \cdot {}_\xi\!\boldsymbol {F}^\text{i}$ in finite strain viscoelasticity modeling. Figure inspired by Yamakawa2021.
• Figure 2: Rheological model of an incompressible generalized Maxwell model ($\det \boldsymbol {F} = 1$) in finite strain viscoelasticity. The model consists of a spring for the equilibrium part and $N$ Maxwell elements. The tangents of equilibrium $\mathbb C^\text{eq}(\bar{\boldsymbol C})$ and non-equilibrium ${}_\xi\! \mathbb C^\text{neq}(\bar{\boldsymbol C}, {}_\xi\! \bar{\boldsymbol C}^\text{i})$ components may depend nonlinearly on the isochoric parts of deformation $\bar{\boldsymbol C}$ and inelastic deformations ${}_\xi\! \bar{\boldsymbol C}^\text{i}$, respectively. Similarly, the viscosity tensors ${}_\xi\! \mathbb V({}_\xi\! \boldsymbol {A}, {}_\xi\! \boldsymbol {C}^\text{i}, \bar{\boldsymbol C})$ can depend nonlinearly on conjugate thermodynamic forces ${}_\xi\! \boldsymbol {A}$ and ${}_\xi\! \boldsymbol {C}^\text{i}$, $\bar{\boldsymbol C}$. The pressure-like Lagrangian multiplier $\tilde{p}$ enforces incompressibility.
• Figure 3: Neural network-based potential $\psi^\text{eq,PANN}$ for the description of the free energy equilibrium part of the finite strain viscoelastic PANN. A monotonic FICNN with skip connections is used, where the network inputs are the invariants $\boldsymbol{\mathcal{I}}^\text{eq} =(\bar{I}_1, \bar{I}_2)$ of the isochoric right Cauchy-Green deformation $\bar{\boldsymbol C}$. The correction term $\psi^\text{NN}(\boldsymbol{\mathcal{I}}^\text{eq})|_\textit{1}$ enforces zero energy in the undeformed state.
• Figure 4: Neural network-based potential ${}_\xi\! \psi^\text{neq,PANN}$ for the description of the $\xi$th free energy non-equilibrium part of the finite strain viscoelastic PANN. A monotonic FICNN with skip connections is used, where the network inputs are the invariants ${}_\xi\! \boldsymbol{\mathcal{I}}^\text{neq} = ({}_\xi\! \bar{I}_1^\text{e}, {}_\xi\! \bar{I}_2^\text{e})$ of the isochoric part of the $\xi$th elastic right Cauchy-Green deformation ${}_\xi\! \bar{\boldsymbol C}^\text{e}$. A gate layer is placed behind the FICNN, which has the task of switching off unneeded Maxwell elements during training. The correction term ${}_\xi\!\psi^\text{NN}({}_\xi\!\boldsymbol{\mathcal{I}}^\text{neq})|_{(\textit{1},\textit{1})}$ enforces zero energy in the unloaded state.
• Figure 5: Neural network-based potential ${}_\xi\! \phi^{*,\text{PANN}}$ for the description of the $\xi$th dual dissipation potential of the finite strain viscoelastic PANN. A monotonic FICNN with skip connections is used, where the network inputs are mixed isotropic invariants ${}_\xi\!\boldsymbol{\mathcal{I}}^{\phi^*} = ({}_\xi\! I_1^{\phi^*}, {}_\xi\! I_2^{\phi^*},\ldots,{}_\xi\! I_9^{\phi^*})$ of the $\xi$th projected thermodynamic forces ${}_\xi\! \boldsymbol {A}^\text{p}$ and the isochoric right Cauchy-Green deformation $\bar{\boldsymbol C}$. A gate layer is placed behind the FICNN, which has the task of switching off unneeded Maxwell elements during training. The correction terms ${}_\xi\! \phi^{*,\text{corr}}$, defined in Eq. \ref{['eq:PANN_phi']}, enforce ${}_\xi\!\phi^{*,\text{PANN}}({}_\xi\! \boldsymbol{\mathcal{I}}^{\phi^*})|_{({}_\xi\! \boldsymbol {A}^\text{p}(\textit{0},{}_\xi\! \boldsymbol {C}^\text{i}),\bar{\boldsymbol C})}=0$ and $\partial_{{}_\xi\!\boldsymbol {A}}{}_\xi\!\phi^{*,\text{PANN}}|_{({}_\xi\! \boldsymbol {A}^\text{p}(\textit{0},{}_\xi\! \boldsymbol {C}^\text{i}),\bar{\boldsymbol C})} = \textit{0}$.

Theorems & Definitions (33)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Remark 4
• Theorem 2
• proof
• Remark 5
• Remark 6