A Viscoelastic Constitutive Relation Describing Primary and Secondary Creep and Solid Elastic Behaviour of Ice

The flow of glacier ice is widely treated by a stress-strain relation commonly referred to as Glen's flow law. The Glen flow law is a pure viscous relation, hence a certain amount of stress immediately corresponds to a rate of deformation, independent of time and possible relaxation. This behaviour is appropriate for stationary (secondary) creep of ice. However, recent detailed ice flow measurements on Gornergletscher, Switzerland, performed during the drainage of a glacier dammed lake, have identified particular unexplained flow changes where significant variations occur within a few hours to several days. It was suggested that elastic effects may play a role in such a rapid response of glacier ice. From an engineering point of view, the loads required to produce displacements of the ice similar to those observed is on the order of 1 to 10 bar. In this range, linear viscoelastic behaviour of ice may be inappropriate and one must consider non-linear viscoelastic response of the ice. However, many experiments and creep tests conducted in the past have shown non-stationary (primary) creep to be effective for approximate durations of a few hours to a few days. We therefore conjecture that primary creep plays a role in short-time glacier flow variations. To test our hypothesis and elucidate the influence of possible viscoelastic effects, we constructed a constitutive relation which is able to reproduce primary and secondary creep as well as elastic effects. A modified Rivlin-Eriksen fluid model, where the stress is related to both strain rate and strain accelerations is used to describe primary and secondary creep of the ice. We couple the viscous fluid model with a non-linear elastic Kelvin-type stress-strain relation to enable the material to exhibit elastic behaviour of a solid. The constitutive relation obeys thermodynamic requirements. The transient response in ice flow is thus assigned to the effects of viscoelastic relaxation and primary creep. The decoupled formulation allows to study those effects separately or in combination. Some first numerical results using the finite element method have been obtained for two benchmark problems of (i) flow in a channel and (ii) flow over an inclined slab.