Constitutive laws#

Equation (1) describes buoyancy-driven flow in an isotropic fluid where strain rate is related to stress by a scalar (possibly spatially variable) multiplier, $\eta$. For some material models it is useful to generalize this relationship to anisotropic materials, or other exotic constitutive laws. For these cases ASPECT can optionally include a generalized, fourth-order tensor field as a material model state variable which changes equation (1) to

(16)#\[ -\nabla \cdot \left[2\eta \left(C \varepsilon(\mathbf u) - \frac{1}{3}(tr(C \varepsilon(\mathbf u)))\mathbf 1\right) \right] + \nabla p = \rho \mathbf g \qquad \textrm{in $\Omega$}\]

and the shear heating term in equation (3) to

(17)#\[\begin{split}\begin{aligned} \dots \notag \\ + 2 \eta \left(C \varepsilon(\mathbf u) - \frac{1}{3}(tr(C \varepsilon(\mathbf u)))\mathbf 1\right) : \left(\varepsilon(\mathbf u) - \frac{1}{3}(\nabla \cdot \mathbf u)\mathbf 1\right) \\ \dots \notag \end{aligned}\end{split}\]

where $C = C_{ijkl}$ is defined by the material model. For physical reasons, $C$ needs to be a symmetric rank-4 tensor: i.e., when multiplied by a symmetric (strain rate) tensor of rank 2 it needs to return another symmetric tensor of rank 2. In mathematical terms, this means that $C_{ijkl}=C_{jikl}=C_{ijlk}=C_{jilk}$. Energy considerations also require that $C$ is positive definite: i.e., for any $\varepsilon \neq 0$, the scalar $\varepsilon : (C \varepsilon)$ must be positive.

This functionality can be optionally invoked by any material model that chooses to define a $C$ field, and falls back to the default case ($C=\mathbb I$) if no such field is defined. It should be noted that $\eta$ still appears in equations (16) and (17). $C$ is therefore intended to be thought of as a “director” tensor rather than a replacement for the viscosity field, although in practice either interpretation is okay.