Rheology of Earth Materials :

Closing the gap between timescales in the laboratory and in the mantle

21 December 2020 - Why wadsleyite is hard to deform in the lab ?

Bridging atomic and polycrystal scales: Why wadsleyite is hard to deform in the lab?

            Solid-state convection is the main process by which the rocky mantle evacuates heat from the Earth’s interior. The rheology of (Mg,Fe)2SiO4 olivine, the most abundant mineral of the upper mantle, controls plastic deformation, and hence mantle flow, until a depth of ~410 km. Between ~410 and ~660 km depth, olivine transforms into its high-pressure polymorphs, the most abundant constituents of the mantle transition zone. The transition zone is a major boundary layer between the Earth’s upper and lower mantle and is expected to influence the extent of whole mantle convection depending on its rheological properties by controlling mass transfer between the upper and lower mantle via subducting slabs and upwelling plumes. Entering the mantle transition zone beyond ~410 km depth with increasing pressure (P) and temperature (T), olivine transforms first into its high-P polymorph wadsleyite. Knowledge of the mechanical behavior of wadsleyite is primarily derived from deformation experiments. These experiments are technically challenging since they are performed at P,T conditions of ~15 GPa and ~1600 K. Moreover, the resulting constitutive equations of viscoplastic behavior have to be extrapolated by more than 10 orders of magnitude to the appropriate strain rate conditions prevailing in Earth’s mantle, not accounting for the intrinsic effect of strain rate on the dominant deformation mechanism. A non-empirical computational approach represents an alternative way to quantify the viscoplastic behavior of minerals as wadsleyite and offers a way to handle the extremely low strain rate conditions. However, such theoretical predictions at low mantle strain rates cannot be empirically verified yet. The present work aims to provide a theoretical framework which enables to explain the high stress values observed during deformation experiments of polycrystalline wadsleyite at in-situ laboratory conditions. The viscoplastic behavior of polycrystalline wadsleyite is obtained by bridging several characteristic length scales, from the sub-nanometer scale at the intracrystalline level to the sub-meter scale of the polycrystalline level.

            Intracrystalline plasticity of wadsleyite has been modeled based on stress-assisted and thermally activated glide of dislocations belonging to the easiest slip systems for a wide range of temperatures and at in-situ laboratory strain rate conditions (Ritterbex et al. 2016). The model relies on the computation of the critical energies of kink-pair formation on the rate controlling ½<111> and [100] screw dislocations which have been modeled by a combination of ab initio calculations of generalized stacking fault surfaces and the continuum-mechanical Peierls-Nabarro-Galerkin method (Fig. 1). The obtained energy barriers for dislocation glide are combined with Boltzmann statistics to provide a constitutive equation for each slip system at the grain level.

Figure 1. Dislocation core modeling in Mg2SiO4 wadsleyite at 15 GPa. (a) Generalized stacking fault surface of the {101} plane with the easiest rigid-body shear path along <111>. (b) ½<111>{101} dislocation core structure computed with the Peierls-Nabarro-Galerkin method.

 

To address the rheology of polycrystalline wadsleyite, a second upscaling has been carried out from the grain to the polycrystal level. Our results obtained at the grain level are implemented in two grain-polycrystal scale transition models: (1) A newly developed mean field method (Fully-Optimized Second-Order Viscoplastic Self-consistent scheme) (FOSO-SC) allowing rapid evolution of the effective viscosity of a polycrystalline aggregate, only applied yet to porous sea ice (Shuvrangsu et al. 2019), and the (2) full field (FFT) method allowing to investigate many inter- and intragranular features such as stress and strain localization during the development of a typical polycrystalline microstructure and the heterogeneous activation of slip systems in the interior of grains. For the application to wadsleyite, we considered a periodical three-dimensional unit cell microstructure containing 1000 grains randomly generated by Voronoi tessellation (Fig. 2).

 

Figure 2. Representative periodic microstructure considered for FFT computations.

 

The effective rheology has been obtained by averaging the model output for 10 random realizations of such synthetic polycrystal aggregates, leading to an error of only 0.1% on the effective flow stress. Micromechanical modeling predicts no activation of the second easiest slip system [100](010) and shows that wadsleyite is able to deform with only four independent slip systems of the ½<111>{101} family, in contrast to the von Mises criterion. The FFT results further show that large stress fluctuations occur between different grains and also in the interior of individual grains (Fig. 3). The stress heterogeneities inside grains are due to the viscoplastic anisotropy of the grain of interest, and is related to its crystal orientation, the mechanical interaction of grains depending on the behavior of neighboring grains, the grain shape and the overall polycrystal behavior. Results demonstrate that deformation of individual grains in polycrystalline wadsleyite are significantly influenced by the behavior of neighboring grains.

 

Figure 3. (left) Distribution of normalized equivalent stress and (right) normalized equivalent strain rate in a 2D section of a 3D microstructure of polycrystalline wadsleyite at 1700 K using the FFT method.

 

 

Theoretical predictions of the effective viscoplastic behavior of polycrystalline wadsleyite at 15 GPa, transition zone temperatures and in-situ laboratory strain rates using the FOSO-SC approach have been successfully compared with the FFT computational homogenization method demonstrating the superior estimations provided by the FOSO-SC scheme. The results (Fig. 4) show that the obtained flow stresses of polycrystalline wadsleyite matches very well the available results from previous deformation experiments and lie within experimental uncertainties at least for temperatures ranging between 1500-2100 K. This agreement shows that the sole contribution of dislocation glide is able to reproduce the high stress levels required to deform polycrystalline wadsleyite at P,T conditions of the transition zone at in-situ laboratory strain rates. Modeling of the thermally-activated processes responsible for glide has shown that dislocations experience substantial lattice friction in wadsleyite at transition zone pressure and temperatures (Ritterbex et al. 2016), partly due to the strong increase of the elastic constants under pressure. Linking the atomic to the polycrystal scale now shows that high lattice friction opposed to ½<111>{101} glide is the main cause of a strong non-linear viscoplastic behavior of polycrystalline wadsleyite controlling its mechanical behavior at laboratory conditions.

Figure 4. Flow stress of polycrystalline wadsleyite at 15 GPa as a function of temperature. Predictions of FOSO-SC and FFT polycrystal methods are included. The response of individual plastic slip on ½<111>{101} as well as the static lower bound (LB) are shown for comparison.

 

References:

  • S. Ritterbex, Ph. Carrez, P. Cordier (2016). Modeling dislocation glide and lattice friction in Mg2SiO4 wadsleyite in conditions of the Earth’s transition zone. Am. Mineral. 101(9), 2085-2094. (Selected by the editors as “Notable paper”).
  • D. Shuvrangsu, P. Ponte Castañeda (2019). A multiphase homogenization model for the viscoplastic response of intact sea ice: the effect of porosity and crystallographic texture. J. Multiscale Comput. Engin. 17, 121-150.

 

To learn more:

O. Castelnau, K. Derrien, S. Ritterbex, Ph. Carrez, P. Cordier & H. Moulinec (2020) Multiscale modeling of the effective viscoplastic behavior of Mg2SiO4 wadsleyite: bridging atomic and polycrystal scales. Comptes Rendus Mécanique, 348(10-11), 827-846. https://doi.org/10.5802/crmeca.61. [https://hal.archives-ouvertes.fr/hal-03012651].