Strain rate enhancement

Given a bulk anisotropic rheology \({\bf D}({\bf S})\), where \({\bf D}\) and \({\bf S}\) are the bulk 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 limit of an isotropic CPO:

\[ E_{ij} = \frac{ {\bf e}_i \cdot {\bf D}({\bf S}) \cdot {\bf e}_j }{ {\bf e}_i \cdot {\bf D}_{\mathrm{iso}}({\bf S}) \cdot {\bf e}_j } , \qquad(1) \]

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

\[ {\bf S}({\bf e}_i, {\bf e}_j) = \tau_0 \begin{cases} {\bf I}/3 - {\bf e}_i \otimes {\bf e}_i \;\;\quad\quad\text{if}\quad i=j \\ {\bf e}_i \otimes {\bf e}_j + {\bf e}_j \otimes {\bf e}_i \quad\text{if}\quad i\neq j \\ \end{cases} . \]

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.

Hard or soft

\(E_{ij}>1\) implies the material response is softened due to fabric (compared to an isotropic CPO), whereas \(E_{ij}<1\) implies hardening.

Eigenenhancements

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

\[{\bf e}_i = {\bf m}_i .\]

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

Transversely isotropic Orthotropic

Grain homogenization

Taylor—Sachs

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:

\[ {\bf S}' = {\bf S} , \qquad\qquad \text{(Sachs's hypothesis)} \]

or a homogeneous stain-rate field:

\[ {\bf D}' = {\bf D} , \qquad\qquad \text{(Taylor's hypothesis)} \]

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:

\[ {\bf D}^{\mathrm{Sachs}} = \langle {\bf D}'({\bf S}') \rangle = \langle {\bf D}'({\bf S}) \rangle , \qquad\qquad \text{(Sachs homogenization)} \]
\[ \qquad {\bf D}^{\mathrm{Taylor}} = \langle {\bf S}'({\bf D}') \rangle^{-1} = \langle {\bf S}'({\bf D}) \rangle^{-1} , \qquad \text{(Taylor homogenization)} \]

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

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

\[ E_{ij} = (1-\alpha) \frac{{\bf e}_i \cdot {\bf D}^{\mathrm{Sachs}}({\bf S}) \cdot {\bf e}_j} {{\bf e}_i \cdot {\bf D}^{\mathrm{Sachs}}_{\mathrm{iso}}({\bf S}) \cdot {\bf e}_j } + {\alpha} \frac{ {\bf e}_i \cdot {\bf D}^{\mathrm{Taylor}}({\bf S}) \cdot {\bf e}_j } { {\bf e}_i \cdot {\bf D}^{\mathrm{Taylor}}_{\mathrm{iso}}({\bf S}) \cdot {\bf e}_j } , \]

or simply

\[ E_{ij} = (1-\alpha)E_{ij}^{\mathrm{Sachs}} + {\alpha}E_{ij}^{\mathrm{Taylor}} , \]

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

Grain parameters

The grain viscous parameters used for homogenization should be understood as the effective values needed to reproduce deformation experiments on polycrystals; they are not the values derived from experiments on single crystals.

Azuma—Placidi

🚧 Not yet documented.