The viscoelastic rheology of transient diffusion creep

TL;DR

This work develops a grain-scale viscoelastic model for diffusionally-accommodated/-assisted grain-boundary sliding (transient diffusion creep) using periodic tessellations of hexagonal grains in 2D and tetrakaidecahedra in 3D to derive upscaled, effective rheology. The macroscale response is captured by an extended Burgers model, exhibiting Maxwell-like behavior at low frequencies and Andrade-like behavior at high frequencies, with parameters tied to grain geometry and diffusion physics through $\tau_M$, $\tau_A$, $\alpha$, and $\Delta$. Key findings include $\alpha$≈0.367 for hexagons (from triple-junction angles) and $\alpha$=1/3 for tetrakaidecahedra, with $\tau_A/\tau_M$ ~ 1.3–1.7 and $\Delta$ around 0.55–0.6, and excellent fits to finite-element results via the extended Burgers model. Comparison with laboratory data shows good qualitative and quantitative alignment but indicates additional dissipative mechanisms are needed to fully explain attenuation, especially at high homologous temperatures and seismic frequencies. The model offers a principled, geometry-informed framework for extrapolating grain-scale diffusion creep to planetary-scale rock rheology, with extensions to interior diffusion, melts, and microstructural anisotropy as future directions.

Abstract

Polycrystalline materials have a viscoelastic rheology where the strains produced by stresses depend on the timescale of deformation. Energy can be stored elastically within grain interiors and dissipated by a variety of different mechanisms. One such dissipation mechanism is diffusionally-accommodated/-assisted grain boundary sliding, also known as transient diffusion creep. Here we detail a simple reference model of transient diffusion creep, based on finite element calculations with simple grain shapes: a regular hexagon in 2D and a tetrakaidecadedron in 3D. The linear viscoelastic behaviour of the finite element models can be well described by a parameterised extended Burgers model, which behaves as a Maxwell model at low frequencies and as an Andrade model at high frequencies. The parametrisation has a specific relaxation strength, Andrade exponent and Andrade time. The Andrade exponent depends only on the angles at which grains meet at triple junctions, and can be related to the exponents of stress singularities that occur at triple junctions in purely elastic models without diffusion. A comparison with laboratory experiments shows that the simple models here provide a lower bound on the observed attenuation. However, there are also clearly additional dissipative processes occurring in laboratory experiments that are not described by these simple models.

Figures (10)

• Figure 1: Properties of the model for hexagonal grains as a function of Poisson's ratio $\nu$. a) Scaled unrelaxed shear modulus $G_0/\mu$ and scaled steady-state shear modulus $G_{ss}/\mu$. b) Relaxation strength $\Delta$ associated with grain boundary diffusion, $\Delta = (G_0/G_{ss}) - 1$. c) Ratio of the Andrade time $\tau_A$ to the Maxwell time $\tau_M$. The data in these plots can be well fit by simple rational functions of $\nu$ (see \ref{['sec:rational']}). The plots show the behaviour for the full allowable range of Poisson's ratio for linear elasticity, $-1 \leq \nu \leq \tfrac{1}{2}$; a typical value for rock is $\nu \sim 0.3$.
• Figure 2: A summary of the viscoelastic response for hexagonal grains with a Poisson's ratio $\nu = 0.3$. a) $|G^*|/G_0$, the ratio of the absolute shear modulus to the unrelaxed modulus as a function of dimensionless frequency $\omega \tau_M$. b) $Q^{-1}$, inverse quality factor (attenuation) as a function of dimensionless frequency. c) Scaled compliances $(G_0 J_1 - 1)/\Delta$ and $(G_0 J_2 - 1/\omega \tau_M)/\Delta$ as a function of dimensionless frequency. d) Cole-Cole plot of the scaled compliances against each other. In all plots solid lines show the finite element solutions, dashed lines an extended Burgers model fit, dotted lines an Andrade model fit, dash-dotted lines a Maxwell model fit.
• Figure 3: Images of boundary normal stress $\sigma_{nn}$ for time-harmonic deformation of frequency $\omega$ for tetrakaidecahedral grains. The imposed macroscale strain is proportional to $\cos \omega t$ where $t$ is time, and the principal axes of the deformation are aligned with the square faces which are at right angles to each other. Orange colours are positive, blue colours are negative, and off-white is zero. Left column shows images at low dimensionless frequency $\omega \tau_M=0.1$; right column shows images at high dimensionless frequency $\omega \tau_M=500$. The top row is at time $t=0$ in the cycle, the bottom row at a quarter of a period later. The bottom left image is similar to that shown for steady-state creep in Figure 9 (networked) in Rudge2018. At high dimensionless frequency diffusion only influences a narrow boundary layer in the neighbourhood of the triple line (the lengthscale $l$ in \ref{['eq:ldef']}); at low dimensionless frequency diffusion influences the whole grain boundary.
• Figure 4: The same properties as \ref{['fig:hex_moduli']}, but for a tessellation of tetrakaidecahedrons in 3D. The properties of such a tiling are anisotropic. Shaded bands are used to show how the parameters can vary with different orientations. The solid lines show results from Voigt-averaging the corresponding complex moduli.
• Figure 5: The same viscoelastic response functions as \ref{['fig:hex_all']} but for the case of tetrakaidecahedral grains. The Poisson's ratio $\nu=0.3$ and the curves are shown for Voigt-averaged values.