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# Plane elasticity problems

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## Comments

## A line force acting on the surface of a half space

Zhigang:

Once again I try to learn something from you and I did from your note. I taught a similar class last year. After all the hard work, we reached the solution for a line force acting on a half space (2D). Unfortunately I had to tell the students that this solution is problematic because the remote displacement boundary condition is not satisfied. The solution looks even stranger if one considers a line of tangential force. I believe this has been a well-known problem in 2D elasticity. However, I am not aware of any resolution. I wonder if it is possible to integrate the 3D solutions to Boussinesq and Cerruti's problems along one line. Since the 3D solutions have no problem at the remote boundary, would the integration give a more realistic solution for line forces?

Another reason I am interested in this problem is that the solution can be (and has been) used to analyse edge effects and interface fracture in thin films (a class I am teaching this semester). The problematic 2D solutions make me very uncomfortable.

I hope to learn more about this problem from discussions.

RH

## An example of edge effect

Rui:

I'll have to think about your more general comments. For the edge effect, I described toward the end of the notes an experiment with a ceramic laminate. The observation was intriguing, but we were able to explain it. I just talked this example in today's class. I don't know if students enjoyed it, but I did.

Do you have similar things in mind?

## axisymmetric problem with line forces

Zhigang:

I just came up with an interesting thought about this problem. As I mentioned above, the solution to the 2D problem with a line force looks problematic because of the remote boundary condition at infinity. I think the reason might lie in the fact that the line force itself extends to infinity under the plane strain assumption. So, if one integrates the 3D solutions, the result would be the same. On the other hand, your example with a cylinder was intriguing to me. It is a nice application by approximating the cylintrical surface as a planar surface, which is all right to develop the asymptotic behavior near the surface. Turning the problem around, can we solve an axisymmetric problem (also 2D) with a line force acting on the surface of a cylinder with a finite radius R? If such a solution exists, it should be a function of the radius R. At the limiting case, when R approaches infinity, it should recover the solution for the plane strain problem. For a finite radius, the solution should not have any problem in satisfying the remote boundary condition, since now the extend of the line force (around the cylinder) is finite. Does this make sense? Are you aware of such a solution?

Thanks.

RH

## Symmetrical deformation of a circular cylinder

This is the title of Article 143 of Theory of Elasticity by Timoshenko and Goodier. The article does not contain an explicit solution that you want, but the method might apply.

Now, come back to your more general concern about the remote displacement field. Both line force and dislocation result in unbounded displacement field remotely. This feature seems to be inevitable. Here is a dimensional consideration. For the case of line force, let P = force/length and r = distance from the place where the force acts. Stress is linear in P. Thus,

stress (r) = P/r

so that displacement will scale as log(r).

However uncomfortable you are, this unbounded displacement will be with us. So far I haven't encountered any physical situation that it cannot handle.

## Flamant solution

Hi Rui:

I hope all is well. Can you be more specific about what you mean by the Flamant solution being "problematic"? I would be very happy to participate and contribute to this discussion.

## unbounded displacement

As mentioned in the above comment by Zhigang, the displacement in 2D Flamant solution scales as log(r), thus unbounded when r approaches infinity.

RH

## Thanks for the

Thanks for the clarification. As discussed in other replies, the displacement is logarithmic regardless of how you develop the solution (even if you start with 3d and integrate to obtain plane strain 2d).

The physical interpretation that is generally accepted (and that you thought of also) is that there is an infinite total applied force in the problem; the force P in the solution Plogr has units of force per thickness, and in the plane strain problem the thickness is infinite. The physically acceptable result is then that the displacement becomes infinite.

Elasticity if quite beautiful and instructive. A related issue is the following. Students (and also elasticity-challenged individuals :) ) often ask why we accept a 1/r singularity in stress (and thus infinite strain energy) in the Flamant solution, but not in the crack problem (we limit the sincularity to inverse square root). The reason is that there is no source of infinite energy at the tip of the crack, but there is in the point force solution. The displacement at r=0 is infinite, and thus the force does infinite work.

## Unbounded displacement

Hi Rui,

I'm interested in this discussion. If you don't mind, I'd like to join here.

- If you're dealing with 2-D(say, XY plane) problem, it seems natural that the displacement scales as log(r). I think that if you integrate the 3D point force solution along an infinite line in the direction of Z, you will get the same result. This unboundedness of displacement seems inevitable because of 2-D nature and an infinite domain. I'm not sure why you're worry about unboundedness of the displacement. When you think of an infinite solid under a uniform stress field, you'll get unbounded displacement field which scales as r. Are you worry about the multi-valuedness of the displacement?

Jae-Hyun

## displacement in an infinite domain

Jae-Hyun:

Thanks for your comments.

I agree with you that the unbounded displacement may be inevitable for the 2D problem. In 3D, the displacement in an infinite domain remains bounded under a concentrated force. Consequently, the effect of the load stays local, decaying from the point of action. In 2D, however, the effect of a concentrated force (line force for plane strain) extends to infinity, which seems unphysical to me. As I mentioned in a previous comment, this may be due to the fact that the line force itself extends to infinity by plane strain assumption. In some sense, infinity (as a mathematic idealization) behaves like one point, no matter how you approach it. So, if the force extends to the infinity, the effect is there too. Another infinite quantity here is the total force implied by the 2D plane strain problem, while in 3D the total force is finite.

The above are my freeform thinking about this problem. One way to prove this could be solving the axisymmetric cylinder problem, also mentioned above.

RH

## Flamant solution

The difficulty really comes from the concept of an 'infinite body'. I prefer to take the finitist point of view here. There are no infinite bodies. The solution for a problem involving an infinite body should be thought of as the limit of the corresponding solution for a finite body when the dimensions of the body go to infinity. There is no reason to assume that all physical quantities in a finite problem will remain finite if the body is allowed to become infinite. Some quantities will converge and others will not. No meaningful statements can be made about quantities which do not converge.

Suppose we could solve the problem of a point force acting on the surface of a rectangle, the lower surface of the rectangle being supported (say) on a rigid plane frictionless surface. Then increase the size of the rectangle. If the origin is taken at the point of application of the force, you will find that the displacement of all points will increase as the size of the body increases. However, the displacements in some region near to the force (compared with the rectangle dimensions) will change only by the addition of a rigid-body displacement with increasing rectangle size.

An alternative problem that can be solved in closed form is that of a semi-circle whose flat side rests on a frictionless rigid surface and which is loaded by a concentrated force at the mid-point of the curved boundary. This can be solved by taking the solution for the complete disk in Timoshenko and Goodier Article 41 and cutting the disk alomg the symmetry axis. If we now let the radius of the disk increase without limit, the curvature will tend to zero and we shall recover the Flamant solution. However, displacements will increase without limit in the same way as in the rectangle example.

The logarithmically infinite displacement causes problems in the formulation of two-dimensional contact problems, since among other things it prevents us being able to define a convergent measure of the stiffness of a contact.

As suggested by others, you can also get a bounded solution by starting with a three-dimensional half space and applying a line load, provided the line load does not extend all the way to infinity on either side. J.J.Kalker developed some interesting solutions to 'almost' two-dimensional contact problems by exploiting this feature. He considered cases where the contact area was long and narrow, such as an ellipse with ellipticity close to unity. In this case, a two-dimensional cross-section through the stress field looks very like a two dimensional state, but there is slow variation along the axis. Kalker exploited the concept of matched asymptotic expansions to solve this problem, essentially defining a deformed coordinate system involving the small ratio between representative contact dimensions in the two directions. See J.J.Kalker, On elastic line contact, ASME J.Appl.Mech., Vol. 39 (1972), 1125-1132, J.J.Kalker, The surface displacement of an elastic half-space loaded in a slender bounded curved surface region with application to the calculation of the contact pressure under a roller, Journal of the Institute of Mathematics and its Applications, Vol. 19 (1977), 127--144. I have also used these methods in J.R. Barber, The rolling contact of misaligned elastic cylinders, J. Mech. Eng. Sci., 22 (1980), 125-128, Jose Castillo and J.R.Barber, Contact problems involving beams, Proc.Roy.Soc. (London), Vol.A453 (1997), pp. 2397--2412.

One final point: This difficulty arises in many other mechanics applications. One example close to my own interests regards the thermoelastic displacement in a three-dimensional body subjected to localized surface heating. The temperature in such cases IS bounded if the body is infinite, but the temperature and hence the strains decay with the reciprocal of distance from the heated region. It follows that when we integrate the strains to get displacement, the latter is logarithmically unbounded. This was missed by no less a mathematician than Ian Sneddon (who literally wrote the book on Fourier Transforms) (see D.L.George and I.N.Sneddon, The axisymmetric Boussinesq problem for a heated punch, J.Math.Mech., 11 (1962), 665-689, Z.Olesiak and I.N.Sneddon, The distribution of thermal stress in an infinite elastic solid containing a penny shaped crack, Arch.Rat.Mech. Anal., 4 (1960), 238-254). Because of the unbounded displacements, Sneddon's solutions to some of these thermoelastic contact problems are not well-posed and they will be found to lead to non-convergent inverse transforms.

## Re: Flamant solution

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

## another question about plane elasticity

For a circular hole in an infinite sheet under remote shear or tension, should the solution depend on the sheet thickness?

Here is my thinking. The solution for the in-plane stresses are identical for plane stress and plane strain, while the strain and displacement are different. Whether it is plane stress or plane strain depends on the thickness of the sheet in comparison to the hole size (two relevant length scales). If the thickness is small compared to the hole radius, the plane stress condition applies. In the other limit, when the thickness is much larger than the hole radius, the plane strain condition applies. However, for a sheet of finite thickness, the stress state is always plane stress at the surface, no matter how thick it is. Apparently, for finite thickness, the stress/strain near the hole cannot be assumed to be plane stress or plane strain; rather, it is a three-dimensional problem. How would this affect the stress concentration as well as the deformation near the hole?

A similar situation coming to mind is the thickness effect in the standard fracture toughness test using the compact tension specimen that we taught in an undergraduate materials lab class.

RH

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

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

## sorry

I am sorry to post thie comment ''a student ...'' though I have not been aware of it until now! Without knowing the function enough, I did this mistake at the first time I registered. I will do everything more carefully.

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

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).

## Flamant Solution - a fine math

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.

RH

## Re: Flamant Solution - a fine math

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!

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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...

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