The Strain Tensor and its Weird Formula

This video is part of a series of videos on continuum mechanics (see playlist: • Continuum Mechanics .
The strain tensor is a mathematical construct to quantify the deformation of matter in continuum mechanics. But the formula for the strain tensor looks unintuitive at a first glance. In this video, the (small or infinitesimal) strain tensor is introduced, and its formula is explained.
Keywords: continuum mechanics, solid mechanics, fluid mechanics, partial differential equations, boundary value problems, linear elasticity, small strain elasticity, infinitesimal strain elasticity, kinematics
Music:
Prelude 2 - VGM Mark H / prelude-2
Abandond Station - VGM Mark H / abandoned-station

[@DrSimulate]

Hi, in this video I use the gradient of the displacement without much elaboration. Would you be interested in a video on gradient, divergence and rotation of fields?

[@Cookstein2]

Oh go on then

[@erayyildiz9562]

Absolutely.

[@sagsolyukariasagi]

Keep going on! Perfect explanations with great visualizations. I stuck at large deformations (differential geometry, tangent spaces), I hope you touch them in future, too.

of course!

[@stitaswain7349]

Yes very much. Plz keep explaining topics in such beautiful manner.

[@vasiliisivovol7943]

Thank you for your work! This invaluable information was explained clearly.

[@Cookstein2]

Great! looking like the start of a valuable series of videos

[@DrSimulate]

Thanks :D

[@meer911]

High quality stuff you putting out here. Nailing the animations and the concept flow. Would love to see more from you.

[@DrSimulate]

Thanks a lot! Appreciate it 🚀✨

[@jimpal5119]

Excellent explanation and visuals! Would love a video covering any nonlinear topic!

[@DrSimulate]

Thanks a lot! :) More videos on nonlinear continuum mechanics are planned. I find nonlinear CM very interesting but it was hard to understand when I saw it the first time ...

[@prateekskaushik]

Really well explained, thank you for providing such qualitative content!

[@DrSimulate]

Thanks! :)

[@vegetablebake]

Another great intuition explained visually.

[@Alfurwan]

i love how your videos always leave me smarter than before!

[@DrSimulate]

[@yunusemresurmeli9980]

Great video, I liked the detailed philosophy of this topic. I'm waiting for other videos.

[@DrSimulate]

Thank you !! ♥

[@amarug]

Well done, love it!

[@josuecorleto569]

Great explanation! I like the use of animations to illustrate these concepts.

[@PK24_Singh]

Well thanks for the mathematical interlude that helps the understanding! How was that arrived at..must be some story behind it?

[@thescientist1839]

Keep making videos man. Slowly your channel will get 1 M subscriber within a year.

[@DrSimulate]

Thank you so much! ❤️

[@apoorvvyas52]

Good explanation. Good topic. Good animations. Overall an excellent video. Please also cover computational fluid dynamics topics also in some of your videos.

[@DrSimulate]

Thank you!!!

[@interhaker]

A beauty to behold. Keep us posted the algorithim will definitely make you blow up

[@joaopedrorocha5693]

This was amazing! Thanks!

[@CAEProfHan]

great video! Thank you.

[@edhead76]

I'm learning about tensors in AI, and tensors have been making my mind mush for a few days. Visual aids really help, even when the physics is a bit over my head. Nice video!

[@DrSimulate]

Thanks!

[@elshons1576]

Can you do a video about finite strain and why is defined as the difference of squares of two infinitesimal line segments?

[@DrSimulate]

I want to do a video about finite strain and the different strain measures. Stay tuned :)

[@Prashanth-yn9zd]

The gradient of displacement tensor is zero for the translation of a body coz, translation is defined as u(x) = u_0, where u_0 is constant

[@DrSimulate]

That's right. Do I say something contradicting in the video? If yes, can you tell me were? :)

[@Prashanth-yn9zd]

@@DrSimulate sorry, you did not say anything contradicting. I added this info so that if someone wants to know why the displacement tensor is zero for the translation, they can find it in the comment section.

[@DrSimulate]

@@Prashanth-yn9zd Ahh, I see. Thanks, appreciate it! :D

[@shivamrajput6368]

Thankyou for visualising this concept using animation…. I am currently working on large deformation, can you please suggest me some related resources.

[@DrSimulate]

I worked a lot with "Nonlinear Solid Mechanics" by Gerhard Holzapfel, but I find it a bit too detailed for a beginner. Let me know if you come across a more didactic explanation. I would be very interested! :)

[@rezaafshar3832]

Superb videos! thanks a lot. Keep it up!

[@Bledy49]

Great video, what's your background?

[@DrSimulate]

Thanks! I studied computational engineering and did my PhD in computational mechanics :)

[@GranHerrmanno]

Klasse Video! Wann geht's hier mit Kontinuumsmechanik weiter?😅

[@DrSimulate]

@@GranHerrmanno ich hoffe bald :) ich will auch unbedingt nichtlineare Sachen zeigen...

[@lorenzoferrara8652]

Very good explanation! A question: how can we link the skew-simmetric part of the displacement gradient to the rotation? Thanks

[@imaginarynumber416]

Hi could you make a video about the material and spatial coordinates with intuitive explanation?

[@DrSimulate]

Hey, it's definitely planned, but unfortunately not in the immediate future, because I want to finish other videos first. Stay tuned and thanks for your patience :)

[@TheChoosenBoi]

Very Nice channel

[@DrSimulate]

Thanks! :)

[@plutothetutor1660]

It's notable that (A + A^T)/2 is always diagonalisible.

[@DrSimulate]

May be addressed in a future video :)

[@RahulSuresh-r2y]

I have a doubt. When the rotation is done, I find that the cube side dimension will increase..i.e. the point at origin is same, but the point on the X3 axis shifts right, so length increased. Then there is strain no? I took x = X + U for transformation. Can you please explain?

[@DrSimulate]

@@RahulSuresh-r2y You are right. This is all under the small strain assumption. For large strains, this displacement field would describe a rotation and deformation of the infinitesimal element.

[@hamzazaheer3783]

Thanks for your Videos. I am fresh graduate in mechanical engineer (BS). I want to get into computational simulation especially multiscale modelling of composite. Right now i am learning continuum mechanics and FEA (Basic Concept). Do you have any advice for me ?

[@DrSimulate]

Sounds great! 💪If you want to learn multiscale modeling, check out these unsurpassable lecture notes by Dennis Kochmann: ethz.ch/content/dam/ethz/special-interest/mavt/mechanical-systems/mm-dam/documents/Notes/CompMultMod_Notes.pdf

[@hamzazaheer3783]

@@DrSimulate Thanks for your help . Can't wait to see your upcoming videos.

[@aj-uo3uh]

"The gradient of the displacement field" Displacement is a function from R^3 to R^3 while the gradient (looking at wikipedia) works on functions from R^3 to R. So for me its not clear what this means.

[@aj-uo3uh]

Ah I see a little further in the video you mean with gradient what I would call the total derivative. Never mind :)

[@VahidNegahdari]

thanks

[@VahidNegahdari]

The examples provided were not rotations at all. Note that a rotation matrix cannot have a row of zeros.

[@DrSimulate]

@@VahidNegahdari Here you are mixing two different concepts. You are talking about rotation matrices that if multiplied by a vector rotate that vector. I'm not talking about those in this video. :)