The metric description of viscoelasticity and instabilities in viscoelastic solids

Many manmade and naturally occurring materials form viscoelastic solids. The increasing use of biologically inspired elastomeric material and the extreme mechanical response these elastomers offer drive the need for a better, more intuitive and quantitative understanding of the mechanical response of such continua. This is in particular important in determining the stability of viscoelastic structure over time where in lieu of robust rules we often must resort to simulations. In this work we put forward a metric description of viscoelasticity in which the continua is characterized by temporally evolving reference lengths quantified by a rest reference metric. This rest reference metric serves as a state variable describing the result of the viscoelastic flow in the system, and allows us to provide robust claims regarding stability of incompressible isotropic viscoelastic media. We demonstrate these claims for a simple bistable systems of three standard-linear-solid spring dashpot assemblies where the predicted rules can be verified by explicit calculations, and also show quantitative agreement with recent experiments in viscoelastic silicone rubber shells that display delayed stability loss.