A microstructural rheological model for transient creep in polycrystalline ice

Abstract

The slow creep of glacial ice plays a key role in sea-level rise, yet its transient deformation remains poorly understood. Glen's flow law, where strain rate is simply a function of stress, cannot predict the time-dependent creep behavior observed in experiments. Here we present a physics-based rheological model that captures all three regimes of transient creep in polycrystalline ice. The key components of the model are a series of Kelvin-Voigt mechanical elements that produce a power-law (Andrade) creep, and a single viscous element with microstructure and stress dependence that represents reorientation in the polycrystalline grains. The interplay between these components produces a minimum in the strain rate at approximately 1% strain, which is a universal but unexplained feature reported in experiments. Due to its transient nature, the model exhibits fractional power-law exponents in the stress dependence of the strain rate minimum, which has been conventionally interpreted as independent physical processes. Taken together, we provide a compact, mechanistic framework for transient ice rheology that generalizes to other polycrystalline materials and can be integrated into constitutive laws for ice-sheet models.

Figures (4)

• Figure 1: Rheological model of polycrystalline ice. (a) Microscopic origin of ice deformation. Within individual grains, deformation occurs by the motion of dislocations. Simultaneously, this motion gradually aligns the grains’ $c$-axes, shown as increasingly uniform grain colors. (b) The mechanical analog of the model consists of three components: an elastic spring (green) with modulus $E_0$, a series of Kelvin–Voigt elements (blue) producing power-law (Andrade) creep, and a structural dashpot (red) with viscosity $\eta_s$ dependent on microstructure $\xi$ and stress $\sigma$. (c) Schematics of strains, $\varepsilon$, and strain rates, $\dot\varepsilon$, as functions of time under a step change in stress, for each component in the mechanical analog.
• Figure 2: Mechanical response of the model under constant stress conditions. (a) Total strain rate (black) decomposed into Andrade (blue) and structural dashpot (red) contributions. Primary creep is Andrade-dominated, secondary reflects crossover, and tertiary is controlled by structural evolution. (b–c) Stress dependence of effective viscosity and strain rate, demonstraing softening in the tertiary regime and a minimium consistently at $\approx1$%, respectively. (d) Effect of the initial microstructural parameter, $\xi_0$, which captures anisotropic versus isotropic initial conditions observed in experiments. For all curves, the stress exponents were $p=3$, $n_0=3$, and $n_1=3$.
• Figure 3: Model fits to experimental data using Eqs. \ref{['eq:strain']} and \ref{['eq:strainrate']}. Markers are experimental data; solid curves are model fits. (a) Data from Treverrow et al.Treverrow2012tertiary showing differences in creep behavior between anisotropic and isotropic ice samples. The stress exponent used for the fits is $p=3$. (b) Data from references jacka1984laboratoryjackaMacc1984icegao1987approach showing that $2\lesssim p\lesssim 5$, depending on experimental conditions. All fitting parameters are listed in Table S1-S4.
• Figure 4: Different flow regimes can emerge with apparent power-law exponents in transient dynamics. (a) Minimum strain rate vs. stress (blue squares) using $p=5$, $n_0=4$, and $n_1=2$, demonstrating that different power-law exponents can be observed from a single model with transient dynamics. The inset indicates the apparent exponent, which is the derivative in $\log$ space. The red circles show $\dot\varepsilon_\text{min}$ at a fixed time, $t_c$, which emulates limited experimental observation times. The black diamonds are data from jacka1984time. (b) The time, $t_{\min}$, to reach $\dot\varepsilon_\text{min}$ decreases with stress, yet plateaus at $t_c$. (c) The strain at the minimum, $\varepsilon_\text{min}$, is nearly constant at $\approx1$%, but can deviate at low stresses when $p\neq n_0$, or with finite $t_c$.