A filler-conditioned neural quasi-linear viscoelastic (QLV) model for carbon-black-filled rubber

We propose a filler-conditioned quasi-linear viscoelastic (QLV) model for carbon-black filled rubber: a finite-strain constitutive law built from two neural networks – an equilibrium energy and a continuous relaxation spectrum – both conditioned on the filler volume fraction φ . Both are thermodynamically admissible by construction : the equilibrium energy ψ polyconvex, the relaxation spectrum non-negative – so the relaxation decays monotonically in time because of fading memory. Trained directly on quasi-static pure-shear data of five filler levels, the equilibrium energy matches the curves to a mean relative RMS error of ≈3 %, outperforming a per-level Yeoh fit at every level. For the time dependence, the shear relaxation function is built from a continuous relaxation spectrum – a non-negative neural network, smooth in the relaxation time τ and monotone in φ by construction. We fit this spectrum directly to the storage-modulus master curves G′(ω;φ) measured in small-amplitude oscillatory simple shear, across five filler levels and about eighteen decades of frequency. Both parts are deployed as an ensemble of independently trained networks: its mean preserves every by-construction guarantee, while its spread provides an epistemic (inter-member) uncertainty band. Leave-one-filler-out tests generalization to unseen filler levels, where averaging largely removes the strong scatter that a single network shows across random initializations.