[Math] Physical Meaning of Constant Velocity Gradient

Question

I'm interested in representing homogeneous elastic deformations using Lie groups/algebras. Homogeneous deformations are those with a deformation gradient F which depends only on time (not position). If the velocity gradient L = (df/dt)F^-1 also is constant (independent of time or position), F = e^(Lt) where F is Lie (sub)group of GL(3,R) & L is Lie (sub)algebra of gl(3,R). This obviously is what I want but I'm unsure what a constant L means physically in terms of the deformation. Thanks, John.

Answer 1

I'm not sure this is what you're after, but are you familiar with pseudo-rigid bodies? These are elastic media where the deformation tensor $F$ is constant in time and hence is an element of $GL(3)$. I'm a little vague on the details, but different continuum models are specified in terms of different subgroups of $GL(3)$. For instance, rigid bodies are obviously described in terms of $SO(3)$ while homogeneous fluids can be described in terms of $SL(3)$.

I have never checked the literature, but apparently Chandrasekhar used these descriptions for the study of relative equilibria of rotating, self-gravitating blobs of gas. Casey has a number of introductory papers on the continuum aspects, and this seems to be a good survey. I found this paper, Pseudo-rigid bodies: a geometric lagrangian approach, a good introduction to the geometric aspects, which IMHO are more interesting anyway.