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.
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$\begingroup$ 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. $\endgroup$– S. Carnahan ♦Commented Oct 1, 2010 at 3:00
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2 Answers
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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.
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$\begingroup$ 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 $\endgroup$ Commented Apr 3, 2011 at 20:37
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$\begingroup$ 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. $\endgroup$– jvkerschCommented Apr 4, 2011 at 3:22
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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