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.

Answer 2

I have received two answers from chatting with colleagues:

1) Constant L assumes constant strain throughout the body. For example, a 3-noded 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