Plastic spin

In bulk simple shear, the dominant slip-system orientation of grains tends to align with the bulk shear-plane system. Thus, if the two are perfectly aligned, the slip-system orientation should be unaffected by continued shearing (i.e. be in steady state). Clearly, slip systems do therefore not simply co-rotate with the bulk continuum spin ($\bf W$) like passive material line elements embedded in a flow field, i.e. ${\bf \dot{n}} \neq {\bf W} \cdot {\bf n}$. Rather, slip systems must be subject to an additional contribution — a plastic spin ${\bf W}_\mathrm{p}$ — such that the bulk spin is exactly counteracted to achieve steady state:

\[ {\bf \dot{n}} = ({\bf W} + {\bf W}_{\mathrm{p}}) \cdot {\bf n} = {\bf 0} \quad\text{for ${\bf b}$–${\bf n}$ shear}. \]

Here, the functional form of ${\bf W}_\mathrm{p}$ is breifly discussed following Aravas and Aifantis (1991) and Aravas (1994) (among others).

For a constant rate of shear deformation ($1/T$) aligned with the ${\bf b}$—${\bf n}$ system,

\[ {\bf F}_{\mathrm{S}} = {\bf I} + \frac{t}{T} {\bf b}\otimes{\bf n} \quad \Rightarrow\quad \nabla {\bf u} = \frac{1}{T} {\bf b}\otimes{\bf n} , \]

the bulk strain-rate and spin tensors are, respectively,

\[ {\bf D} = \frac{1}{2T} ({\bf b}\otimes{\bf n} + {\bf n}\otimes{\bf b}), \\ {\bf W} = \frac{1}{2T} ({\bf b}\otimes{\bf n} - {\bf n}\otimes{\bf b}) . \]

Since ${\bf W}_\mathrm{p} = -{\bf W}$ is required in steady state, it follows from eliminating $1/(2T)$ by calculating ${\bf D} \cdot {\bf n}\otimes{\bf n}$, ${\bf n}\otimes{\bf n} \cdot {\bf D}$, ${\bf D} \cdot {\bf b}\otimes{\bf b}$, and ${\bf b}\otimes{\bf b} \cdot {\bf D}$, that

\[ {\bf W}_\mathrm{p}({\bf D},{\bf n}) = +{\bf n}\otimes{\bf n} \cdot {\bf D} - {\bf D} \cdot {\bf n}\otimes{\bf n} , \\ {\bf W}_\mathrm{p}({\bf D},{\bf b}) = -{\bf b}\otimes{\bf b} \cdot {\bf D} + {\bf D} \cdot {\bf b}\otimes{\bf b} , \]

so that

\[ {\bf \dot{n}} = ({\bf W} + {\bf W}_{\mathrm{p}}({\bf D},{\bf n})) \cdot {\bf n}, \\ {\bf \dot{b}} = ({\bf W} + {\bf W}_{\mathrm{p}}({\bf D},{\bf b})) \cdot {\bf b}. \]

Indeed, this result agrees with representation theorems for isotropic functions (Wang, 1969), stating that an anti-symmetric tensor-value function of a symmetric tensor (${\bf D}$) and a vector (${\hat {\bf r}}$) is to lowest order given by

\[ {\bf W}_{\mathrm{p}}({\bf D},{\hat {\bf r}}) = \iota({\hat {\bf r}}\otimes{\hat {\bf r}}\cdot{\bf D} - {\bf D}\cdot{\hat {\bf r}}\otimes{\hat {\bf r}}) . \]

To be consistent with the above, $\iota = +1$ for ${\hat {\bf r}}={\bf n}$ and $\iota = -1$ for ${\hat {\bf r}}={\bf b}$.

${\bf W}_\mathrm{p}({\bf D},{\bf n})$ and ${\bf W}_\mathrm{p}({\bf D},{\bf b})$ are then generally taken to relevant for other modes of deformation, too.