A Minimal Nonlocal Theory of Thixotropic Flow

Abstract

Dense amorphous materials exhibit both nonlocal flow cooperativity and pronounced history dependence, yet existing continuum models capture only one of these features at a time. Nonlocal rheologies are intrinsically memoryless, while thixotropic models remain local. Here we introduce a coupling between structural memory and nonlocal fluidity to include aging and rejuvenation in nonlocal granular fluidity. The resulting model reproduces hysteresis in shear-rate sweeps and delayed yielding in creep, while preserving nonlocal flow profiles. By introducing memory augmented non local granular fluidity, MNGF, we show that nonlocality alone cannot encode history, and memory alone cannot encode spatial cooperativity, but their coupling is essential and minimal. These results demonstrate that memory and nonlocality must be treated jointly to describe history dependent flows, and provide a unified framework for modeling time-dependent rheology in dense amorphous materials.

Figures (3)

• Figure 1: Kymographs of the flow velocity (left column) and structure parameter $\lambda$ (right column) during planar shear flow. Top row shows an initially fully destructured state ($\lambda \approx 0$), and the bottom row shows a fully structured initial condition ($\lambda \approx 1$). Simulations are performed with parameters $t_0 = 0.01\,\text{s}$, $A = 0.84$, $d = 8.0\times 10^{-4}\,\text{m}$, $\mu_s = 0.29$, $\rho = 2450\,\text{kg}/\text{m}^3$, $P = 10\,\text{kPa}$, $\beta = 0.1$, $a_{\text{age}} = 5.0 \,\text{s}^{-1}$, $\tau = 20.0\,\text{s}$, $\alpha = 1.0$, and $b = 1.5$.
• Figure 2: Top: Ramp down/up flows of the NGF and MNGF models. The blue curve corresponds to the NGF limit without structural evolution, while the black and red curves show the down- and up-sweep MNGF flows, respectively. Bottom: Velocity–profile comparison for a representative shear–rate sweep in the MNGF model with a dwell time of $\delta t = 1.0$ s. From left to right: (d) Down-sweep segment, (e) Up-sweep segment mirrored [in time] for visual comparison with the down-sweep flow, and (f) Flow difference between the down- and mirrored up-sweep flows, $\lvert \Delta (u/u_{\mathrm{top}}) \rvert$. Vertical dashed lines mark the time at which the highest shear stress difference in down- and up-sweep flows is observed, also marked in the flow curves of the top row. Simulation parameters are $\rho=2450~\mathrm{kg\,m^{-3}}$, $A=0.84$, $d=8\times10^{-4}~\mathrm{m}$, $\mu_s=0.29$, $\beta=0.10$, $t_0=0.01~\mathrm{s}$, $a_{\mathrm{age}}=5.0$, $b=1.5$, $\alpha=1.0$, $\tau=20.0~\mathrm{s}$, with $\dot{\gamma}_{\min}=10^{-6}~\mathrm{s^{-1}}$, $\dot{\gamma}_{\max}=10^{3}~\mathrm{s^{-1}}$, and dwell times $\delta t=\{1,10,100\}~\mathrm{s}$.
• Figure 3: Creep response of the MNGF model under constant applied stress $\sigma$ (color-coded). Time evolution of the shear rate $\dot{\gamma}(t)$ (top) without rest and (bottom) after a $20~\mathrm{s}$ rest period, during which the structural parameter $\lambda$ rebuilds and increases the effective yield threshold $\sigma_y$.