The constitutive relation is developed as a
generalized Newtonian fluid, where the deviatoric stress and strain tensors are related by a viscosity scalar: where \mu is the viscosity (units of Pa s), \boldsymbol\tau is the deviatoric stress tensor, and \boldsymbol \dot{\epsilon} is the strain rate tensor. In some derivations, \lambda=(2\mu)^{-1} (units of Pa−1 s−1) is substituted. •
Isotropy, as the single proportionality scalar is the same for all tensor components. •
Incompressibility, as volumetric stress is ignored and only the deviatoric stress can do work. • That corresponding components of the two tensors are directly proportional to one another, i.e. \tau_{ij} \propto \dot{\epsilon}_{ij}. Theoretically, this assumption results from ignoring the third principle invariant of the tensors; physically, this means that the strain rate can only change along the same axes as the
principal stresses. While incompressibility is an accurate assumption for glacial ice, glacial ice can be anisotropic and in general the strain rate may respond perpendicularly to the principal stress.
With these assumptions, the stress and strain rate tensors here are symmetric and have a trace of zero, properties that allow their invariants and squares to be simplified from the general definitions. The deviatoric stress tensor is related to an
effective stress by its second principal invariant: :\tau_e^2 = II_{\boldsymbol{\tau}} = \frac{1}{2}\tau_{ij} \tau_{ij} where
Einstein notation implies summation over repeated indices. The same is defined for an effective strain rate: :\dot{\epsilon}_e^2 = II_{\boldsymbol{\dot{\epsilon}}} = \frac{1}{2}\dot{\epsilon}_{ij} \dot{\epsilon}_{ij} From this form, we can recognize that: : \boldsymbol\tau^2 = \tau_{ij} \tau_{ij} =2\tau_e^2 and : \boldsymbol{\dot{\epsilon}}^2 = \dot{\epsilon}_{ij} \dot{\epsilon}_{ij} =2\dot{\epsilon}_e^2 The viscosity is scalar and cannot be negative (a fluid cannot gain energy as it flows), so \mu can be expressed in terms of the invariant effective stress and effective strain rate. \mu= \frac{\boldsymbol{\tau}}{2\dot{\boldsymbol{\epsilon}}}=\frac{1}{2} \tau_e \dot{\epsilon}_e^{-1} Here, the Glen–Nye flow law allows us to substitute for either \tau_e or \dot{\epsilon}_e, and \mu can be defined in terms of either the effective strain rate or effective stress alone: where B=A^{-1/n} (units of Pa s^{1/n}) is sometimes substituted. ==Parameter values==