iMechanica - lecture notes - Comments

excellent

Farid Touaiti — Fri, 29 Aug 2008 06:12:24 -0400

thank you very much.

Rui:your initial guesses are correct,but people are NOT careful!

Mike Ciavarella — Sun, 20 Jul 2008 03:51:08 -0400

Rui,

to complete Jim Barber's replies to your question (he actually raised also some more questions) you probably know that you can formulate many problems (for example contact and crack problems) by using displacement DERIVATIVES, and this avoids the problem of the rigid body motion. You can find this also in Barber's book.

To comment on your more recent question, of course strictly speaking ALWAYS at the surface there is a state of plane stress, by definition, as well as ALWAYS there is a TENDENCY to have some stress inside. The stress increases towards the limit plane strain one, that as you know depends on Poisson's ratio s3= nu * (s1 + s2) so this is EXACTLY zero generally only for Poisson=0, or else if the state is of pure shear.

You are rigth that in standard fracture mechanics testing this is more or less known, but I can promise you that in many situation, this is disregarded! For example, I have submitted a paper about Paris' law which shows how the size of the specimen is OFTEN disregarded in what affects the Paris "constant" C and m ---- so people are NOT careful and only GI Barenblatt has strongly commented on this before

The paper is not accepted yet, but you can find a little however in the Paris seminar I did recently A seminar at Paris VI inst. d'alembert - One, no one, and one hundred thousand crack propagation equations: thursday June 5th. 1

Regards, Mike

Re: Flamant Solution - a fine math

Ajit R. Jadhav — Sun, 20 Jul 2008 03:22:25 -0400

OOPS... I realize only now that in my yesterday's post above, there should have been a linefeed or two just before the preface: "(Please correct me if I am wrong here.)" so that that preface would have become separate and got applied to the entire matter following it. Sorry!
Similarly, I did not *actually* mean to imply that Rui was groping... Actually, the groping and grappling is all mine.. I have been thinking of this conjecture to the effect that the approaches like the random walk ought to work out for tensor fields like those of elasticity too... In the final editing of my draft, the placement of that statement went wrong too.

(These days, I first write in a Notepad file my comment, and then, just copy-paste the contents in the TinyMCE or so editor here, because the Internet, the editor (in IE) and the electricity can all crash any moment. So, I first write and then do copy-paste. But still, after such a copy-paste, something more strikes me, and so, I want to revise the initial draft. Sometimes, I end up introducing silly mistakes in this process...)

Anyway, let me take that conjecture off this thread. So, if anyone is interested in discussing it, drop me a line or feel free to start a new thread on that one. Thanks!

-----

Just one more point: Yes, Prof. Barber's comment was remarkable in the sense that despite his background, he spoke of the "a finite system --> an appropriate limiting process --> a higher abstraction involving an infinity" kind of approach. That view is so very unusual to find with mechanicians or applied mathematicians (let alone "pure" mathematicians). I really enjoyed it, too...

-----

...I must wind up or else I can go on and on and on...

Flamant Solution - a fine math

Rui Huang — Sat, 19 Jul 2008 16:10:35 -0400

Ajit's comment seems to bring this discussion to a higher level in terms of mathematics. However, my original question was a humble one. It is not about math. I was trying to understand the physical reality (e.g, the infinite displacement at the remote boundary). It is now understood (thanks to all the comments above) that this is impossible with the physically unrealistic model (e.g., 2D, infinite domain). Jim Barber's comment pretty much resolved my concern.

Re: Flamant Solution -- Is it about 2D vs 3D a la Polya?

Ajit R. Jadhav — Sat, 19 Jul 2008 13:39:41 -0400

Thanks to Biswajit's comment above (no. 7930 dtd. 27 June, 2008) for bringing this fine thread to notice again.

With the benefit of some hindsight, let me add the following:

From his writing, I think that Rui isn't here completely concerned only with the unboundedness of displacements even in the case that the stress drops as 1/r with distance.

Thus, the issue of specifically marking out the difference of the strain field (namely, that it is finite) from the displacement field (namely, that it is infinite) does not appear to be his sole concern here.

That issue---of strains vs. displacements---would be too simple a matter... By way of a simple analogy (and please permit me one), it would be like saying that even if the net yearly population growth rate of a nation (i.e. the analog of strain) goes on reducing monotonically, its actual total population (i.e. the analog of displacement) would continue to grow in an *unbounded* manner (or tend to infinity in infinite time) provided that net growth rate were to be a +ve number. I mean, all this is too simple a mathematics, and, judging by his writing, I guess, someone like Rui wouldn't be concerned with it. (Please correct me if I am wrong here.)

I think Rui here is trying to make a truly fine observation about the essential difference of behaviour in between the 2D and the 3D situation.

In particular, I think that Rui's query (and call it groping/grappling if you wish), when put in a wider context, is about a continuum (tensor) elasticity version of that which is known as Polya's recurrence theorem in the discrete electric network theory. (For a discussion of the latter theory, for example, see: Doyle and Snell, "Random Walks and Electric Networks," arXiv:math.PR/0001057). Such a thing may not be possible in complete generality. (In any case, I haven't thought separately about it---whether it would be possible or not.) But at least, that kind of a definitive and essential difference between the 2D and 3D situations is what Rui seems to be hitting at.

Could any mathematician please shed some light on this matter? (I would have loved to answer this question definitively, but, mathematics, as such, has never been my main interest in life---not even a side interest, for that matter.) Thanks in advance for any clarification (which is also accessible to an engineer like me).

Thank you zhangzhuo

hitzhangzhuo — Mon, 30 Jun 2008 07:59:13 -0400

Thank you

zhangzhuo hitzhangzhuo@gamil.com

Re: Flamant solution

Biswajit Banerjee — Thu, 26 Jun 2008 21:40:36 -0400

Recently I revisited this excellent discussion on the Flamant solution in linear elasticity (we need more such discussions on iMechanica). However, one of the problems that I noticed in the discusion was that the Flamant solution was not really explained to the uninitiated reader. Given that many of our readers may be new to linear elasticity, I went ahead and added a page on the Flamant solution to Wikpedia (based on Prof. Barber's book). You can find it at http://en.wikipedia.org/wiki/Flamant_solution.

Please add some insights and correct any mistakes in the article.

-- Biswajit

Thanks for sharing. Yi

Yi Han — Thu, 12 Jun 2008 13:00:10 -0400

Thanks for sharing.

Yi Han hyxjtu@gmail.com

thanks Zhigang Suo

Muller Abera — Thu, 08 May 2008 10:33:24 -0400

share what we have

Comprehensive Distance Education

Noel M. Dioyan — Fri, 11 Apr 2008 09:56:43 -0400

That is very interesting idea sir,if imechanica can provide a distance education from all field of engineering mechanics.The whole world is watching all the comments and suggestion in this site. Maybe, We here from asia can be benefited in your great proposal " Education without boundary" reagardless of distance,races,ethnics and culture..

thanks its very useful

ashwath.nr — Tue, 12 Feb 2008 11:19:25 -0500

Ashwath NR
Solid Mechanics Division
Department of Applied Mechanics
Indian Institute of Technology Madras,
chennai-600036
India

METAL FORMING

prabukarthi — Sat, 09 Feb 2008 08:42:17 -0500

material for warm deep drawing

Primary unknowns of FEM formulation

surajitdas — Tue, 29 Jan 2008 04:14:16 -0500

In the FE formulation of beam element, generally, nodal displacement (v) and slope (θ) are taken as primary unknowns. What will be the obstacles if the nutral axis curvature (κ) is used instead of slope as primary unknown, along with nodal displacement?

Thanks

kisnaforme — Tue, 25 Dec 2007 07:02:47 -0500

Thank you

Its very useful...

Muni

Thank you Prof. Nix

leila ladani — Thu, 06 Dec 2007 13:31:26 -0500

Thank you for your useful notes. I am teaching micromechanics of materials next semester and I am sure that I could use some of your notes.