An Open-Source Finite-Element Implementation of the Deep Rheological Element for Large-Strain Nonlinear Viscoelastic Behavior of Elastomers

We present an open-source finite-element implementation of the Deep Rheological Element (DRE), a nonlinear dashpot whose constitutive behaviour is governed by a neural-network (NN) deviatoric viscosity. The DRE is embedded into a finite-strain Generalized Maxwell Model (GMM) based on the multiplicative decomposition F = Fe Fv. The resulting framework captures amplitude-dependent softening and dissipation in filled elastomers, i.e. the Payne effect. The implementation is carried out within FEniCSx, where the entire finite-strain viscoelastic formulation, including the NN viscosity, is expressed directly in Unified Form Language (UFL). For incompressible isotropic materials, the deviatoric viscosity can be expressed as an isotropic scalar function of invariants of the total and viscous unimodular Cauchy–Green deformation tensors. In our demonstrative implementation, the general invariant set is reduced to two invariants, following the Kumar & Lopez–Pamies (KLP) law used to generate training data. Two numerical experiments are conducted: (i) a shear-block benchmark under amplitude-sweep oscillatory loading, from which storage and loss moduli are computed and compared between the KLP and DRE predictions; (ii) a finite-element model of a 3D cylinder in torsion (ramp-and-hold twist), comparing KLP and DRE responses in torque and normal force.