A multiplicative finite viscoelastic model for ice using an exponential update formulation

Understanding the long-term deformation behavior of glacier ice at the calving front is a crucial point for the derivation of calving criteria regarding ice mass loss. To some extent, linear models are sufficient to capture the elastic as well as viscous response of the motion of ice. However, finite deformation models are inevitable, especially for simulations taking into account deformations over decades. Following this, a finite viscoelastic Maxwell model is employed to capture the elastic and short-term response, which is denoted by e.g. changing loading conditions, as well as the long-term and viscous response by the creep behavior of the ice flow. A further extension of the standard Maxwell model is given by including Glen’s flow law, resulting in nonlinear stress-dependent viscosities and thus, nonlinear viscous flow. The model here presented utilizes a multiplicative split of the deformation gradient into an elastic and viscous part, followed by the derivation of the evolution equation based on a Lie derivative. An essential part is the application of an exponential map as the integrator for the update of the internal viscous variables in time, ensuring an isochoric viscous flow throughout the whole simulation. The performance of the model is demonstrated on two numerical examples, illustrating the enforcement of the incompressible viscous flow and numerical stability as well as an ice shelf benchmark that serves as a comparative study for the monitoring of occurring stresses and displacements assuming a constant as well as stress-dependent viscosity using Glen’s flow law.