Transient Elasticity -- A Unifying Framework for Thixotropy, Polymers, and Granular Media

TL;DR

The paper addresses the complexity of thixotropic yield-stress fluids and granular media by proposing Transient Elasticity (TE), a hydrodynamic framework that keeps elastic structure alive under shear via a finite relaxation time $\tau$ and introduces a second, mesoscopic temperature $T_m$. By coupling $\varepsilon^e$ and $T_m$ within a simple, thermodynamically consistent set of equations, TE reproduces a wide range of non-Newtonian phenomena, including static/dynamic yield, viscosity bifurcation, rate-jump hysteresis, aging, shear bands, and elastic shear waves. It clarifies the connections to classical viscoelastic models (Maxwell/Jeffrey) and argues for the essential role of the second temperature in TYF and granular systems, while presenting clear predictions and testable scenarios. Overall, TE offers a minimal yet broad unifying framework that can describe complex TYF-like behavior across polymers, granular media, and soils, with potential extensions to improve realism where needed.

Abstract

Having the elastic strain relax while observing energy conservation and entropy production yields a model called hydrodynamics of transient elasticity (HoTE). It interpolates between solid- and fluid-dynamics and provides a broad framework for many systems such as polymers, granular media, thixotropic and yield-stress fluids. Focusing on the last two systems, this paper shows that, with little efforts at model sculpting, HoTE easily accounts for many effects, including: static and dynamic yield stress, over- and under-shoot of stresses at rate jumps, hysteresis at rate ramps and rate inversion, "viscosity bifurcation", acceleration on a tilted plane, shear band, elastic shear waves, aging and rejuvenation.

Figures (9)

• Figure 1: Modifying solid-dynamics by introducing a plastic rate, TE describes an elastic structure that persists under shear rates -- like wind-bent trees -- yet imitates a viscous behavior if the rate is stationary. Part of the structure disentangles and slips, while others reconnect, maintaining the stress-load.
• Figure 2: A log-log plot of the critical stress $\sigma=K\varepsilon^c+\eta{\dot\varepsilon}$ vs. rate ${\dot\varepsilon}$, with $\varepsilon^c$ of Eq.(\ref{['eq5']}), $\tau$ of Eq.(\ref{['eq11']}), and the parameters of $^{\ref{['fn4']}}$. It is essentially the same plot as Fig. 6 of the "everything flows"-review by Barnes thix3. There are two (Newtonian) viscosity plateaus, for ${\dot\varepsilon}>10^3$ and ${\dot\varepsilon}<10^{-5}$, which sandwich a long stretch of constant stress $K\varepsilon^c$, for 6 orders of magnitude of ${\dot\varepsilon}$.
• Figure 3: Elastic strain $\varepsilon^e(t)$ vs. time $t$, after a rate jump at $t=0$, from ${\dot\varepsilon}_i$ to ${\dot\varepsilon}$, drawn employing Eqs.(\ref{['eq14a']},\ref{['eq14']}), with $\tau=1$ s, ${\dot\varepsilon}_i\tau=0.1$, starting formerly at $\varepsilon^e=0$ to show the change after the jump. The monotonic curve in straight line has ${\dot\varepsilon} \tau=0.2$ (implying ${\dot\varepsilon} \tau\ll1$, cf. Eq.(\ref{['eq13']})), and is zoomed up by a factor of 3 for a clearer view. The dash line has ${\dot\varepsilon} \tau=2\gtrsim1$. With $\tau=$ const, both display polymeric behavior, with the latter showing the rotational contribution.
• Figure 4: Elastic strain $\varepsilon^e(t)$ vs. time $t$ after a rate jump (${\dot\varepsilon}_i\to{\dot\varepsilon}$ at $t=0$) in a thixotropic fluid (ie, $1/\tau=\lambda_1T_m$), showing over- and undershoot. These are obtained by inserting Eq.(\ref{['eq9']}) and $1/\tau=\lambda_1T_m$ into ${\frac{\partial}{\partial t}}\varepsilon^e={\dot\varepsilon}-\lambda_1T_m\varepsilon^e$, Eqs.(\ref{['eq3a']}), taking: ${\dot\varepsilon} =2$/s, $r_m=3$/s, $\lambda_1=10$. The initial conditions are $\varepsilon^e_i=1/\lambda_1$, for the upper curve $T_m={\dot\varepsilon}_i=1$/s, for the lower one $T_m{\dot\varepsilon}_i=4$/s. (Overshoot depends on $\tau=1/(\lambda{\dot\varepsilon})=1/20\lesssim1/r_m=1/3$ s.)
• Figure 5: The first figure is a simple plot of the evolution of the elastic strain after three consecutive jumps, dotted line: $0\to{\dot\varepsilon}$, dash line: ${\dot\varepsilon}\to-{\dot\varepsilon}$, and solid line: $-{\dot\varepsilon}\to{\dot\varepsilon}$, with $\lambda_1=10, {\dot\varepsilon}=0.5$/s. The second figure (with the time $t$ stretched by $\sim$2x) turns the time backward for the dash line, suggesting a dependence of the elastic strain $\varepsilon^e$ (or stress) on the total strain $\varepsilon=|{\dot\varepsilon}| t$, thus causing hysteresis.