In theoretical glaciology and continuum mechanics, the Glen–Nye flow law, also referred to as Glen's flow law, is an empirically derived constitutive relation widely used as a model for the rheology of glacial ice. The Glen–Nye flow law treats ice as a purely viscous, incompressible, isotropic, non-Newtonian fluid, with a viscosity determined by a power law relation between strain rate and stress:
ϵ ˙ e = A τ e n {\displaystyle {\dot {\epsilon }}_{e}=A\tau _{e}^{n}}
The effective strain rate ϵ ˙ e {\displaystyle {\dot {\epsilon }}_{e}} (units of s−1) and effective stress τ e {\displaystyle \tau _{e}} (units of Pa) are related to the second principle invariants of their respective tensors. The parameters A {\displaystyle A} and n {\displaystyle n} are scalar constants which have been estimated through a combination of theory and measurements. The exponent n {\displaystyle n} is dimensionless, and the rate factor A {\displaystyle A} takes on the units Pa− n {\displaystyle n} s−1. The Glen–Nye flow law simplifies the viscous stress tensor to a single scalar value μ {\displaystyle \mu } , the dynamic viscosity, which is determined by tensor invariants of the deviatoric stress tensor τ {\displaystyle {\boldsymbol {\tau }}} and the strain rate tensor ϵ ˙ {\displaystyle {\boldsymbol {\dot {\epsilon }}}} . Under the application of sustained force ice will flow as a fluid, and changes to the force applied will result in non-linear changes to the resulting flow. This fluid behavior of ice, which the Glen–Nye flow law is intended to represent, is accommodated within the solid ice by creep, and is a dominant mode of glacial ice flow.
Viscosity definition The constitutive relation is developed as a generalized Newtonian fluid, where the deviatoric stress and strain tensors are related by a viscosity scalar:
where μ {\displaystyle \mu } is the viscosity (units of Pa s), τ {\displaystyle {\boldsymbol {\tau }}} is the deviatoric stress tensor, and ϵ ˙ {\displaystyle {\boldsymbol {\dot {\epsilon }}}} is the strain rate tensor. In some derivations, λ = ( 2 μ ) − 1 {\displaystyle \lambda =(2\mu )^{-1}} (units of Pa−1 s−1) is substituted. This construction makes several assumptions:
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. τ i j ∝ ϵ ˙ i j {\displaystyle \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:
τ e 2 = I I τ = 1 2 τ i j τ i j {\displaystyle \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:
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