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# 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:**

When was the last time you used Mohr's circle of strain/stress in your professional work? Was it a week ago? a month? a year? five years? ten years? longer? In what kind of an application or research context?

Please note, I do not mean to ask whether you directly or indirectly used the coordinate transformation equations---the basis for constructing Mohr's circle---to find the principal quantities. The question is: whether you spoke of Mohr's circle itself---and not of the transformation equations---in a direct manner, in a professional activity of yours (apart from teaching Mohr's circles). In other words, whether, in the late 20th and early 21st century, there was any occasion to plot the circle (by hand or using a software) in the practice of engineering, did it directly illuminate something/anything in your work.

In case you are curious, my own answer to this question is: No, never. I would like to know yours.

**Question 2:**

The second question just pursues one of the lines indicated in the first.

In a modern FEM postprocessor, visualizations of stress/strain patterns are provided, usually via field plots and contour lines.

For instance, they show field plots of individual stress tensor components, one at a time.

Recently, there also have been some attempts to try to directly show tensor quantities in full directly, via systematically arranged ellipsoids of appropriate sizes and orientations. The view you get is in a way analogous to the arrow plots for visualizing vector fields in those CFD and EM software packages. Other techniques for tensor visualization are not, IMHO, as successful as the ellipsoids. Mostly, all such techniques still are at the research stage and have not yet made to the commercial offerings.

Some convenience can be had by showing some scalar measures of the tensors such as the von Mises measure, in the usual field/contour plots.

The questions here are:

**(2.a)** Would you like to see an ellipsoids kind of visualization in your engineering FEM software? If yes, would this feature be a "killer" one? Would you consider it to be a decisive kind of advantage?

**(2.b)** Would a simpler, colored cross-bars kind of visualization do? That is, two arrows aligned with the principal directions. The colors and the lengths of the arrows help ascertain the strength of the principal quantities.

**(2.c)** Would you like to see Mohr's circles being drawn for visualization or any other purposes in such a context? If yes, please indicate the specific way in which it would help you.

My own answers to question 2 are: (a) Ellipsoids would be "nice to have" but not "killer." I wouldn't be very insistent on them. Having them is not a decisive adavantage. (b) For 2D, this feature should be provided. (c) Not at all.

Please note, the questions are directed rather at experienced professionals, even engineering managers, but not so much at students as such. The reason is that the ability to buy is an important consideration here, apart from the willingness. Of course, experienced or advanced PhD students and post-docs may also feel free to share their experiences, thoughts and expectations.

Thanks in advance for your comments.

PS: Also posted in my other, personal blog here [^]

[E&OE]

- Ajit R. Jadhav's blog
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## Comments

## Mohr's Circle

Hi Ajit,

Regarding your first question, I personally sometimes sketch (not draw) the Mohr's circle to get a rough estimate of the stress in a specific orientation or the principal stress values (also strains). However, for actual calculations never. It is also more suitable for 2D, and its 3D extension is cumbersome.

Regarding your second question, I prefer the "cross-bars" approach that you describe in (2.b) ... Make it three arrows though instead of 2 for 3D problems. This feature is available in most commercial FE post-processors I have used. For fine finite element meshes though these plots can have too much details to be of practical value (with 3 arrows at every integration point).

Nagi

## Reply to Nagi

Hi Nagi,

Thanks for sharing your views. I appreciate it.

Re. Mohr's circle, personally, I find it cumbersome even for 2D. The double angle thing makes it too indirect. Ellipses (and ellipsoids) are way too direct. It's funny that most beginning texts (and professors teaching these) continue to teach Mohr's circle but not Lame's stress ellipsoids.

Re FEM:

I wouldn't care for stresses to be separately shown for each GP. Guess most commercial software don't show them separately anyway.

One feature I personally would like to see (but may not end up implementing!!) is the stress "cross-bars"/ellipses visualization being shown at *regular* intervals rather than within the individual elements. If the stress is anyway "recovered" before plotting fields or showing contours, then it would do no further harm to show values interpolated at *regular* grid points, regardless of the (ir)regularity of the underlying mesh.

BTW, many post-processors these days also allow you to take an imaginary section (usually a straight plane in 3D and a straight line in 2D) cutting through the domain, and provide a graphical depiction of the local variations of stresses across this cut. They also provide the total resultants (forces and moments) acting across the cut. This way, the analyst doesn't even have to do a "back of the envelope" calculation using Mohr's circle.

The reason I raised this whole question was that I wanted to know if the more experienced people have their imagination better developed using Mohr's circle, and therefore could appreciate it as a feature over and above the abovementioned sectional quantities feature. (Also another reason was that I was wondering what the interviewer was thinking back then---but, yes, it really was a minor matter, not the main driver for this question...)

--Ajit

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[E&OE]

## Mohr's Circle in Mathematics

Has anyone ever seen Mohr’s Circle discussed in a mathematics textbook? I have not directly, but a mathematics problem in “The Green Book of Mathematical Problems,” by K.H. Hardy and K.S. Williams, Dover Publications (1997), has what may interpreted as the equivalent of one. See Problem 10, p. 3, and its solution on pp. 52-53.