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.

I'm not sure this is what you're after, but are you familiar with pseudorigid 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, selfgravitating 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, Pseudorigid bodies: a geometric lagrangian approach, a good introduction to the geometric aspects, which IMHO are more interesting anyway. 


I have received two answers from chatting with colleagues: 1) Constant L assumes constant strain throughout the body. For example, a 3noded triangle in finite element analysis (fea). 2) Constant L assumes constant acceleration which is a common fea assumption & reasonable with small time steps. I'm comfortable with these explanations so wanted to share. Thanks, John. PS: Re Scott's comment: Scott Thanks for your thoughts. Sorry that my thoughts often are a bit vague. But, I'm using the def'n of a matrix Lie algebra: "Lie algebra g of a matrix Lie group G is the set of all matrices X such that e^(Xt) is in G for all real numbers t". That's why I'm calling matrices L a Lie algebra. L has to be constant for df/dt = LF = Le^(Lt) (above equ'n rearranged) to be true. But, I'm still puzzled as to what a constant L means physically in terms of deformation. It certainly means that velocity varies linearly with position but unsure beyond that. The assumption of L being constant is made very often in the literature as a simplifying assumption but never explained beyond that. Thanks, John 

