Integral fractional viscoelastic models in SPH: LAOS simulations versus experimental data

TL;DR

This work tackles the nonlinear viscoelastic response of polymer melts under Large Amplitude Oscillatory Shear (LAOS) by marrying Smoothed Particle Hydrodynamics (SPH) with a fractional integral $K$-BKZ constitutive model. The authors develop a displacement-based, Lagrangian SPH framework that uses a memory kernel $M(t-t')$ and a damping function $h(oldsymbol{b})$, adopting a Fractional Maxwell Model (FMM) to capture both the power-law and nonlinear responses of the melt. They calibrate the model against SAOS measurements to extract fractional parameters $(oldsymbol{}= ext{e.g. } oldsymbol{}=0.85, eta=0.2)$ and damping characteristics $h(oldsymbol{b})$ with $oldsymbol{b}^* o 2.3$ and $m o 1.9$, then validate the nonlinear LAOS response by Fourier analysis of the stress signal, showing onset of nonlinearity around $oldsymbol{b}_0\, ext{near }1$ and intra-cycle features such as $e_3>0$ and $v_3<0$. The SPH results agree with experiments in the evolution of $G'_M$ and $G'_L$ with strain amplitude and Deborah number, confirming that the integral fractional $K$-BKZ model implemented in SPH can predict both linear and nonlinear viscoelastic behavior of polymer melts under realistic processing conditions. This approach enables simulations of complex flows and geometries beyond SAOS/LAOS, with potential applications to polymer processing and composite systems.

Abstract

The rheological behaviour of a polymer was investigated by performing numerical simulations in complex flow and comparing them to experiments. For our simulations, we employed a Smoothed Particle Hydrodynamics scheme, utilizing an integral fractional model based on the K-BKZ framework. The results are compared with experiments performed on melt-state isotactic polypropylene under medium and large amplitude oscillatory shear. The numerical results are in good agreement with the experimental data, and the model is able to capture and predict both the linear and the non-linear viscoelastic behaviours of the polymer melt. Results show that equipping SPH with an integral fractional model is promising approach for the simulation of complex polymeric materials under realistic conditions.

Figures (14)

• Figure 1: Identification of the $\tan{\delta}$ peak for the iPP material to determine the glass transition temperature (Tg).
• Figure 2: Material in granular state (a,b) and molten state (c,d) prior to testing in oscillatory mode.
• Figure 3: Heat flow versus temperature, identifying $Tm$ at the peak at $171$°C
• Figure 4: Storage and loss moduli as a function of $\bar{\omega}=\omega\lambda$. The explored range of frequencies is indicated by a grey shaded area. $G'(\bar{\omega})$ ($\textcolor{blue}{-}$) and $G"(\bar{\omega})$ ($\textcolor{orange}{- -}$) from the fractional K-BKZ model with parameters $\alpha=0.85$, $\beta=0.2$, $\lambda_c=0.2$ s, $G_c=432$ kPa. For comparison, the artificial Newtonian contribution ($\textcolor{black}{:}$) and the combined behaviour of the fractional model and the artificial Newtonian contribution ($\textcolor{rgb(0,76,0)}{-.}$) are shown.
• Figure 5: Storage and loss moduli from SAOS as a function of $\bar{\omega}=\omega\lambda$: experimental data ($\bullet$ and $\hbox{$\blacksquare$}$ symbols), fractional K-BKZ model (respectively and ) with parameters $\alpha=0.85$, $\beta=0.2$, $\lambda_c=0.2$ s, $G_c=432$ kPa. Artificial Newtonian contribution (:) shown for comparison.