A Euler-lagrange Model of dynamic internal friction

A Euler-Lagrange model of dynamic internal friction is proposed and is shown to match the frequency and decay of oscillations in both simple extension (pull) and cantilever beam experiments. The proposed dynamic internal frictional stress, ${\tau }_{ij}$, is proportional to the rate of change of the engineering stress, ${\dot{\sigma }}_{ij}$. i.e.${\tau }_{ij}={\mu }_m{\dot{\sigma }}_{ij},$with ${\mu }_m$ the dynamic internal friction coefficient. A single value of the dynamic internal friction coefficient is shown to match the results of the experiments for a number of different geometries of the silicon rubber, Dragon Skin^TM. Dragon Skin^TM is used in skin effects for movies and in prosthetics and cushioning applications. It is chosen here because of its ease of preparation and relatively simple non-linear stress-strain response. Because of these characteristics, it provides a simple starting place for simulating more complicated synthetic rubber and biological materials, which are used in a myriad of commercial applications.