Strain-rate enhancements

Given an anisotropic rheology \({\bf D}({\bf S})\), where \({\bf D}\) and \({\bf S}\) are the strain-rate and deviatoric stress tensors, respectively, the directional strain-rate enhancement factors \(E_{ij}\) are defined as the \(({\bf e}_i, {\bf e}_j)\)-components of \({\bf D}\) relative to that of the rheology in the isotropic limit (isotropic CPO):

for a stress state aligned with \(({\bf e}_i, {\bf e}_j)\):

In this way:

• \({E_{11}}\) is the longitudinal strain-rate enhancement along \({\bf e}_{1}\) when subject to compression along \({\bf e}_{1}\)

• \({E_{12}}\) is the \({\bf e}_{1}\)—\({\bf e}_{2}\) shear strain-rate enhancement when subject to shear in the \({\bf e}_{1}\)—\({\bf e}_{2}\) plane

and so on.

Eigenenhancements

Eigenenhancements are defined as the enhancement factors w.r.t. the CPO symmetry axes (\({\bf m}_i\)):

These are the enhancements factors needed to specify the viscous anisotropy in bulk rheologies:

Transversely isotropic	Orthotropic

Grain homogenization

Calculating \(E_{ij}\) using (1) for a given CPO requires an effective rheology that takes the microstructure into account.

In the simplest case, polycrystals may be regarded as an ensemble of interactionless grains (monocrystals), subject to either a homogeneous stress field over the polycrystal scale:

or a homogeneous stain-rate field:

where \({\bf S}'\) and \({\bf D}'\) are the microscopic (grain-scale) stress and strain-rate tensors, respectively.

The effective rheology can then be approximated as the ensemble-averaged monocrystal rheology for either case:

where \(\langle \cdot \rangle^{-1}\) inverts the tensorial relationship.

If a linear combination of the two homogenizations is considered, equation (1) can be written as

or simply

where \(\alpha\) is a free parameter.

Transversely isotropic grains

Monocrystal	Polycrystal

If grains are approximately transversely isotropic, the grain rheology can be modelled using the transversely isotropic power-law rheology. This requires specifying the grain eigenenhancements \(E_{mm}'\) and \(E_{mt}'\), the power-law exponent \(n'\), and the Taylor—Sachs weight \(\alpha\).

Example for glacier ice

For glacier ice, we follow the literature and rename

The below code example shows how to calculate \(E_{ij}\) given a4 (or nlm) assuming the grain parameters proposed by Rathmann and Lilien (2021):

import numpy as np
from specfabpy import specfab as sf
lm, nlm_len = sf.init(8) 

### Synthetic unidirectional CPO (all c-axes aligned in z-direction)
m = np.array([0,0,1]) 
a4 = np.einsum('i,j,k,l', m,m,m,m) # 4-times repeated outer product of m
nlm = np.zeros((nlm_len), dtype=np.complex64)
nlm[:sf.L4len] = sf.a4_to_nlm(a4) # derive corresponding expansion coefficients

### Basis for enhancement factor calculations
(e1,e2,e3, eigvals) = sf.frame(nlm, 'e') # enhancement factors are w.r.t. a^(2) basis (i.e. eigenenhancements)
#(e1,e2,e3) = np.eye(3) # enhancement factors are w.r.t. Cartesian basis (x,y,z)

### Transversely isotropic monocrystal parameters for ice (Rathmann & Lilien, 2021)
n_grain   = 1        # power-law exponent: n=1 => linear grain rheology, nonlinear (n>1) is unsupported.
Eij_grain = (1, 1e3) # grain eigenenhancements (Ecc,Eca) for compression along c-axis (Ecc) and for shear parallel to basal plane (Eca)
alpha     = 0.0125   # Taylor--Sachs weight

### Calculate enhancement factors w.r.t. (e1,e2,e3)
Eij = sf.Eij_tranisotropic(nlm, e1,e2,e3, Eij_grain,alpha,n_grain) # Eij=(E11,E22,E33,E23,E13,E12)

Orthotropic grains

Monocrystal	Polycrystal

If grains are approximately orthotropic, the grain rheology can be modelled using the orthotropic power-law rheology. This requires specifying the grain eigenenhancements \(E_{ij}'\), the power-law exponent \(n'\), and the Taylor—Sachs weight \(\alpha\).

Example for olivine

Not yet available.