Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
I don't quite understand what the formulas mean, but I think you want $L$ to be a function from (time,position) configuration space to the Lie algebra, rather than having $L$ be an actual Lie algebra. In your special case, I think it should a constant function. –  S. Carnahan Oct 1 '10 at 3:00
add comment

2 Answers

up vote 2 down vote accepted

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.

share|improve this answer
    
Thanks for your answer. I read the Casey papers on pseudo-rigid bodies which were quite useful. My interest is inhomogeneous deformation but in the finite element context (so finite number of nodes &, hence, dofs). Thanks again, John –  John Craighead Apr 3 '11 at 20:37
    
Hi John, this sounds like an interesting problem. I'm sure there would be a lot of nice geometry involved if you modelled a continuum as something which is locally a pseudo-rigid body. Moreover, this could lead to some novel ways of doing FEM. –  jvkersch Apr 4 '11 at 3:22
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.