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Description
This work introduces a computational framework for the finite-element simulation of finite-strain nonlinear viscoelasticity using the Deep Rheological Element (DRE). The DRE represents a neural-network-augmented dashpot whose viscosity function is learned from data while remaining embedded in a thermodynamically consistent architecture. We integrate this element into a finite-strain Generalized Maxwell Model (GMM) based on the multiplicative decomposition of the deformation gradient, enabling the prediction of complex phenomena, such as the Payne effect in filled elastomers.
The implementation is developed within the FEniCSx environment, where the entire finite-strain formulation, including the neural-network-driven viscosity, is expressed directly in the Unified Form Language (UFL). For incompressible isotropic materials, the deviatoric viscosity is expressed as an isotropic scalar function of the invariants of the total and viscous unimodular left Cauchy–Green deformation tensors. In this demonstrative implementation, the general invariant set is reduced to a dependence on the single invariant $J_2 = \| \mathbf{T}_e^D \|$, allowing the model to recover a power-law behavior.
The robustness of the variational implementation is assessed through two main numerical experiments: (i) a shear-block benchmark under amplitude-sweep oscillatory loading to verify storage and loss moduli against analytical DRE predictions, and (ii) a 3D cylinder subjected to large-angle torsion. In the latter, the DRE accurately reproduces the torque and normal force responses, demonstrating numerical stability in realistic boundary-value problems with heterogeneous deformations.