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Ajit R. Jadhav's blog

Stress is defined as the quantity equal to ... what?

In introducing the very concept of the stress tensor to the beginning student, text-books always present only indirect relations involving the concept. Thus, you have the relations like "traction = (stress-transposed)(unit normal)" (i.e. Cauchy's formula, for uniform stress), or the relations for the coordinate transformations of the stress tensor, or the divergence theorem (for non-uniform stress). These are immediately followed or interspersed with alternative notations, and the rules for using them.

But what you never ever get to see, in text-books or references, is this: a *direct* definition of the stress tensor, i.e. an equation in which there is only the stress tensor on the left hand-side, and some expression involving some *other* quantities on right hand-side. Why? What possibly could be the conceptual and pedagogical advantages of giving a direct definition of this kind, and its physical meaning? I would like to ponder on these matters here, giving my answers to these and similar questions in the process.

Also remember Alcoa

Also remember Alcoa.

Yes I know about the [essentials of] QM!

Check out here [at my personal blog] [^] and the post before that.


Have a happy holiday season!





A ``small'' but interesting riddle from the theory of vibrations

A ``small'' but interesting riddle from the basic theory of vibrations. Haven't run into it in any physics/classical mechanics text/reference.

Explicit vs. implicit FDM: Could you please suggest a reference?

The context is the finite difference modeling (FDM) of the transient diffusion equation (the linear one: $\dfrac{\partial T}{\partial t} = \alpha \dfrac{\partial^2 T}{\partial x^2}$).

Two approaches are available for modeling the evolution of $T$ in time: (i) explicit and (ii) implicit (e.g., the Crank-Nicolson method).

It was obvious to me that the explicit approach has a local (or compact) support whereas the implicit approach has a global support.

Expansion of a function into a basis set

Consider a ``neat'' function such as what an engineer is most likely to use in his typical theory/work. Such a function would typically be: (i) defined over a single finite interval, (ii) continuous throughout, and (iii) smooth enough. In other words, the sort of a function they used when they introduced the very idea of a graph of a function to you, back in high-school. ... Feel free to add any other qualifications you like, but note them explicitly, e.g., (iv) bounded throughout, and (v) periodic.

I am [still] confused about gradients, vectors, deformation gradient, etc.

I am creating this blog entry to have my confusions about gradients, vectors, and deformation gradient, etc., straightened out once (and hopefully for all time!) ... My confusions got exposed (even to me) while commenting on a thread started by Prof. Suo here [^]. In particular, I realized my confusions after writing this comment [^] there.

MWR for the first- and third-order differential equations

Hi all,

In engineering sciences, we usually end up using either the second- or the fourth-order differential equations, and the MWR (the method of weighted residuals) works pretty well for them.

The question is: how about the first- and the third-order differential equations? Why don't we see any applications of MWR for these odd-ordered differential equations? What gives?

Those were not waves: A bit historical re. Huygens' principle

A few points that might be of general interest:

1. The dates: The date of Huygens' first written down material, which was orally presented to the French Academy of Sciences, is 1678---in contrast to the oft-quoted date of 1690. 1690 was the year of the first, French, publication of these notes (plus other material) in the form of a book.

Journals in Physics and Engineering, and Preprint Servers Like arXiv

Hi all,


1. In the past, we have had quite some discussion regarding both open-access and open-access journals. However the slant in this blog post is different. I am not concerned here much about open-access journals per say.

What would you choose as the Top 5 Equations? Top 10?

Equations are of central importance in all of science and engineering, but especially so in mechanics.

Even leaving aside algebraic equations, handbooks on PDEs alone list hundreds of equations. However, a few of these do stand out, either because they encapsulate some fundamental aspect of physics/science/engg., or because they serve as simpler prototypes for more complex situtations, or simply because they are so complex as to be fascinating by themselves. There might be other considerations too... But the fact is, some equations really do stand out as compared to others.

An interesting arXiv paper: "Precession optomechanics"

Hi all,

Just thought that the following paper archived at the arXiv yesterday could be of general interest to any mechanician:

Xingyu Zhang, Matthew Tomes, Tal Carmon (2011) "Precession optomechanics," arXiv:1104.4839 [^]

The fig. 1 in it makes the matter conceptually so simple that the paper can be recommended to any mechanician for his general reading, and not only to a specialist in the field.



Any tips/comments regarding the latest version of the C++ library: Eigen (v. 3.0)?

Hi all,

1. A new version of Eigen (v 3.0 now) is out (on March 23, 2011), and it seems promising. First, a few links:

The main page for the project is here: [^]. The page for v.3.0 is here: [^]. It seems to be very fast: [^].

Open House: Can you define FEM in one line?

Can you define FEM in one line?

If yes, what would it be? And, in that case, permit me a second question: How?


...Really interested in knowing what the members of this community think (of this matter), if they do...





What would you like for an undergraduate book on QM to explain to you?


1. Background:

A couple of things concerning books happened recently, in the last week or two.

(i) Dr. Biswajit Banerjee announced last week that a new book on metamaterials and waves in composites authored by him is coming out in print within a few months.

How about having a special Mechanics Gallery here?

Here is an idea I submit for consideration by all iMechanicians, but esp. so by the admins and moderators. Discussion is welcome.


Idea: Why not have a Mechanics Gallery section here?

Origin: Recently, I was browsing for some OpenGL-encapsulating C++ class libraries, e.g. OpenSceneGraph, VTK, the game development libraries, etc. The Web sites of all such libraries always carry a "Gallery" page which is designed to attract the potential users. The Gallery page shows the capabilities and advantages of that library/framework.

How to supply a visualization for the displacement gradient tensor

Hi all, 

[Warning: The writing is long, as is usually the case with my posts :)]

It all began with a paper that I proposed for an upcoming conference in India. The extended abstract got accepted, of course, but my work is still in progress, and today I am not sure if I can meet the deadline. So, I may perhaps withdraw it, and then submit a longer version of it to a journal, later.

Mohr's Circle---When Was the Last Time You Used It in Your Professional Engineering Work?

As a consultant in computational mechanics, I currently help write some FEM-related code, and while doing this job, an episode from a recent past came to my mind. Let me go right on to the technical issue, keeping aside the (not so good) particulars of that episode. (In case you are curious: it happened outside of my current job, during a job interview.)

If you are a design engineer, FE analyst, researcher, or any professional dealing with stress analysis in your work, I seek answers to a couple of questions from you:

Question 1:

Use Only the Angular Quantities in Analysis? Three Sample Problems to Consider...

A recent discussion at iMechanica following my last post here [^] leads to this post. The context of that discussion is assumed here.

I present here three sample problems, thought of almost at random, just to see how the suggestions made by Jaydeep in the above post work out.

Are Linear and Angular Momenta Interconvertible?

To the best of my knowledge, the two momentum conservation principles, namely, the conservation of linear- and angular-momentum, operate completely independent of each other. For an isolated object, there is no possibility of conversion of one form of momentum to the other.

Wanted: Fast FEA Solvers...


I am thinking of informally conducting a specific case-study concerning the FEA solvers. The reference problem is a very simple but typical problem from stress analysis, leading of course to the linear systems: Ax = b and Ax = Lx.

I seek advice as to what software libraries currently available in the public domain would be best to use---the ones that would be fastest in terms of execution time for the reference problem.

I have a personal and longer-term research interest with certain issues related to the solvers technologies.

An Urgent Appeal for Your Support of My Job Application at COEP's Mechanical Engineering Department

Dear iMechanicians,

I have applied for the job of "Associate Professor" in the Department of Mechanical Engineering at COEP, Pune, India [^]---the same place from where I did my PhD (Mech.) research.

I most earnestly make an appeal to you to provide me with an informal support for my job application by way of a brief email recommendation. My resume may be found here [^].

Food for Thought: A Few Recent arXiv Papers

Since my research touches on the basics of QM, I have developed this habit of visiting every now and then. Last week or so, at, I found a couple of interesting articles on physics in general. I would like to share these with you.

My Ph.D. Defence

I am pleased to inform you that I will be defending my Ph.D. thesis, formally in mechanical engineering, at COEP, University of Pune, India, on the next Sunday (i.e. 20th September, 2009).

The title of my thesis is: "A New Approach to Computer Modeling and Analysis of Certain Fundamental Field Problems from Engineering Sciences." 

A Different Kind of a Book Involving Electromagnetism and Potential Theory

Unlike other blog-posts of mine, I am not going "own" this particular thread. By that, I mean to say: I am going to only begin this thread and immediately turn it over to you completely. I am not going to watch over whether the discussion here continues to stick to its main theme or not, whether it slides into some minor side issues, whether it deserts the main theme altogether, etc., the way I usually do.

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