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On geometric discretization of elasticity
This paper presents a geometric discretization of elasticity when
the ambient space is Euclidean. This theory is built on ideas from
algebraic topology, exterior calculus and the recent developments
of discrete exterior calculus. We first review some geometric
ideas in continuum mechanics and show how constitutive equations
of linearized elasticity, similar to those of electromagnetism,
can be written in terms of a material Hodge star operator. In the
discrete theory presented in this paper, instead of referring to
continuum quantities, we postulate the existence of some discrete
scalar-valued and vector-valued primal and dual differential forms
on a discretized solid, which is assumed to be a triangulated
domain. We find the discrete governing equations by requiring
energy balance invariance under time-dependent rigid translations
and rotations of the ambient space. There are several subtle
differences between the discrete and continuous theories. For
example, power of tractions in the discrete theory is written on a
layer of cells with a nonzero volume. We obtain the compatibility
equations of this discrete theory using tools from algebraic
topology. We study a discrete Cosserat medium and obtain its
governing equations. Finally, we study the geometric structure of
linearized elasticity and write its governing equations in a
matrix form. We show that, in addition to constitutive equations,
balance of angular momentum is also metric dependent; all the
other governing equations are topological.
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Comments
Re: Geometry and metamaterials
Arash,
Though I don't understand much of the details of the differential geometry involved, this paper seems to be interesting on a number of fronts. Dr. S. Guenneau is trying to apply similar ideas to metamaterials, c.f. http://gow.epsrc.ac.uk/ViewGrant.aspx?GrantRef=EP/F027125/1. I have two questions:
-- Biswajit
differential geometry
Dear Biswajit :
Thanks for your interest. For learning exterior calculus, I would say
"Differential Forms with Applications to the Physical Sciences" by H.
Flanders would be a good book to start with as he does not assume much
background. For learning differential geometry, one should have a
minimum background in topology and analysis. I can't tell you what the
best book would be but can give you the names of a few I've found
useful. The following three are excellent books that emphasize on
applications too:
- Geometrical methods of mathematical physics by B. Schutz
- Geometry, Topology, and Physics by M. Nakahara
- The Geometry of Physics, An Introduction by T. Frankel
The other good book is "Manifolds, Tensor Analysis, and Applications"
by R. Abraham, J.E. Marsden, and T. Ratiu. John. M. Lee is an
excellent writer and I've found the following three books by him
excellent.
- Introduction to Topological Manifolds
- Riemannian Manifolds
-Introduction to Smooth Manifolds
I've found the following two books by Cartan interesting and fairly
easy to follow but if I were you I would start with some other books.
- Riemannian Geometry in an Orthogonal Frame
- Differential Forms
I believe you're referring to M. P. Do Carmo's "Riemannian Geometry".
He has another book "Differential Geometry of Curves and Surfaces",
which is more concrete and a very good book to start with.
I don't know what a metamaterial means exactly. But I'm assuming you're
referring to what Milton and Willis have been doing recently in the
context of "elastic cloaking". For electromagnetic cloaking, the main
tool is the transformation properties of Maxwell's equations. The
well-known example is the singular map that maps a disk to an annulus.
In the case of elasticity, there are two types of mappings: 1) spatial
maps, and 2) referential maps. When you write linear elasticity in its
classical form, it is not clear which one you're using. I believe
differential geometry can be useful in this direction and in giving a
clear picture of what a given transformation does to different
elasticity quantities, for example.
Let me end this by quoting Kondo (1954), who was one of the first who
realized non-Riemannian manifolds can be used in plasticity.
“Although the remarkable success of the general relativity theory
impressed the importance of tensor calculus and Riemannian geometry on
public opinion, it was unfortunate that it gave a metaphorical
appearance to Riemannian expression, banishing it for a time from the
attention of engineers.” “…it is strange that the first practical field
of application was not the theory of elasticity, especially of residual
strains.”
Re: Differential geometry
Arash,
Thanks for the detailed reply. To your list of references I would like to add the Appendices from Rakotomanana's "A geometric approach to thermomechanics of dissipating continua" for a concise introduction to the basic definitions of p-forms, connections, torsion, and the exterior derivative.
I'm unsure of what you meant by spatial and referential maps in your comment. Could you elaborate?
-- Biswajit
spatial and material changes of frame
Dear Biswajit:
I have read most of that book and to be honest with you am not impressed with it. Not just that book, but many other works related to continua with distributed defects. I don't want to give any specific examples, but to make the long story short, in my opinion, Kondo and Bilby made some seminal contributions (they realized that torsion of a Riemann-Cartan manifold is related to the dislocation density tensor) and after that many people have been playing with some complicated-looking equations for continua with distributed defects without much success. I can give you specific examples of incorrect interpretations like integrating a vector field, which is intrinsically meaningless, etc. My point here is the following. Differential geometry may look complicated but is a useful mathematical tool if used properly. Unfortunately, it has not been used properly in mechanics with the exception of the works of some distinguished people like Jerry Marsden, Tom Hughes, Juan Simo, etc.
I personally believe that if a theory makes sense one should be able to explain it in simple words. An excellent example of a book with many deep concepts explained in very simple words is the little book by Richard Feynmann "The Character of Physical Law", which I recommend to everybody. One can clearly see in this very easy to follow book that the author had a very clear and deep understanding of fundamentals of physics and was able to explain his knowledge in simple and honest words.
By spatial and referential maps I meant the following. In continuum mechanics, it is usually assumed that there is a given reference configuration. For solids this can be any of the possible natural configurations, i.e. stress-free states of the body. One would then define motions with respect to this reference configuration. The configuration space of elasticity (in some sense where the theory is defined) is the space of maps between a reference manifold to an embedding space manifold. Embedding space manifold is in general a Riemannian manifold with a given background metric (metric is not dynamic unlike general relativity where metric is dynamic and is governed by Einstein's equations). The ambient space for most applications is the Euclidean space. A spatial map (or change of frame) is a map from the ambient space to itself. For example, when one talks about material-frame-indifference, internal energy density is assumed to be invariant under isometries (i.e. length preserving) of the ambient Euclidean space, i.e. translations and rotations in the current configuration.
Reference manifold is not as well understood in the mechanics community, in my opinion. This can be best described in terms of evolution of defects (and this of course goes back to the seminal works of Eshelby on defect forces). For example, if a cracked elastic solid is under external forces and if the crack starts to propagate, there is a time-dependent reference manifold. Obviously, crack propagation does not follow the standard deformation mapping and the extra kinematical process is represented by the time-dependence of the material manifold. One important difference between ambient space and material manifolds is the lack of homogeneity in the material manifold, in general. A material mapping (or change of frame) is a map that acts in the reference configuration and maps it to a new reference manifold. Now in the setting of nonlinear elasticity, "cloaking", in my opinion, should be studied using material maps because a spatial map is roughly speaking looking at the same material through a warped lens. Again, if one starts from classical linear elasticity, there is no way to distinguish between spatial and material mappings.
In summary, there are two manifolds in any elasticity problem and different elasticity quantities are defined with respect to these two manifolds. For example, deformation gradient is a two-point tensor with one leg in the reference manifold and one leg in the current configuration manifold. And because of this it is meaningless to talk about symmetry of deformation gradient. When there are two completely different manifolds involved, one would expect to have, in general, different maps acting on them. I hope this is clear.
Arash
Re: Geometry, Kondo
Arash,
Thanks for your reply. I will now have to look at Rakotomanana with a jaundiced eye though I still feel that the Appendices provide a nice starting point :)
Those of us who are interested in Kondo will find the following article intriguing: http://arxiv.org/ftp/arxiv/papers/0712/0712.0641.pdf. It's called "The Natural Philosophy of Kazuo Kondo" by Grenville Croll.
Also, you say that "... integrating a vector field, which is intrinsically meaningless ...". I find that hard to grasp. Most of classical mechanics is based on summing forces. If these forces can be expressed as a function of position why is summing them meaningless?
Biswajit
Integrating vector fields
Dear Biswajit:
Thanks for the article, which I will read with great interest.
In a linear space (or vector space) you can always add two vectors by
definition of a vector space. A vector field on a manifold is by
definition to associate a vector to each point of the manifold (of
course each vector lies in the tangent space at that point). Tangent
spaces at different points are different linear spaces and adding two
vectors at two different tangent spaces is meaningless. This is why one
would need more "structure" if such an operation is necessary. The
extra structure is called "connection". In Riemannian geometry the most
natural connection is the Levi-Civita connection, which is the unique
linear connection that preserves inner products and is torsion free.
Now having a connection one can define parallel transport of a vector
along a given curve.
If you want to integrate a vector field, for example
on some sub-manifold, you need to pick a point and then parallel
transport all the other vectors to this point. The problems is that
parallel transport of a vector from a point p_1 to another point p_2
explicitly depends on the curve connecting the two points unless the
manifold does not have curvature. As a matter of fact, one usually
defines the curvature tensor in terms of parallel transport on
infinitesimal closed curves. In the Euclidean space (where classical
mechanics is usually formulated), there is no curvature and thus
parallel transport is independent of the path you choose. And this is
why you can integrate vector fields. In a general manifold, a vector
field cannot be integrated and for this reason the integral form of
balance of linear momentum, for example, is geometrically meaningless.
In a geometric setting, one can start with balance
of energy (a scalar) because balance of energy can always be written on
any manifold. Balance laws then follow from symmetries of energy
balance. There are, of course, other ways of obtaining balance laws,
e.g. Hamilton's action principle, etc.
Arash
a question
Dear prof.Yavari,
I am a postdoc at university of Minnesota doing Discrete Exterior Calculus in Elasticity, I have a problem about your paper " On Geometry Discretization of elasticity", what is the explicit form of material hodge star? Since I am doing coding to realize the discretization in computer, do you think the material hodge star is a diagnoal matrix since for angular moment conversation the traction must be parellel to the discrete strain ,right? and the material hodge star relates the discrete strain to traction(stress), so through the mathematical analysis I found that the material hodge star must be a diagonal matrix . If you can give me some help about this I will be very appreciated! Thank you so much!