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.