The stress tensor

Рет қаралды 413,200

Brian Storey

Күн бұрын

Lectures for Transport Phenomena course at Olin College
This lecture describes what the stress tensor is.

Пікірлер: 222

@MrDaanjanssen 8 жыл бұрын

1:41 my reaction when I saw the mechanics exam

@guitarttimman 8 жыл бұрын

+Daan Janssen mechanics was so easy bro! Multivariate Calculus was a MONSTER! LOL

@guitarttimman 7 жыл бұрын

crni195 All you do is start by examining the upward and downward forces. Then you start on the torques. (The force times the perpendicular distance from the axis of rotation to the line of action of the force.) Then you set all the forces equal to zero for the purpose of making sure their sums are zero so that nothing is moving. That means equilibrium BRO. Tensile forces work on the same principle. Many years ago, I was an expert in vector calculus, but I've been out of it for a long time. I think I can still do it though. I started as a mathematics major. I wasn't a pee wee either. I tested out of many of those courses and earned A's in advanced mathematics. I have a list of some of my grades posted on google. Although math was my first love, I ended up with a degree in Biology Education. It might sound like I am bragging, but that is only because I didn't even learn how to read until I was 19 years old. Anyway, there's another great thing I did. I designed the first program that straightened out the "ghost" parking ticket dilemma in the City of Chicago way back in the 80's, and that was without any prior knowledge of computer programming. I was naive, and the guy who ran the business, Michael Tellerino, ripped me off big time. I think he should come clean about that and clear his conscience.

@devvv4616 7 жыл бұрын

are you fr?

@carnsoaks1 7 жыл бұрын

y y y

@TheJokeKiller 7 жыл бұрын

Thanks for the chuckle.

@antoniobautista6300 8 жыл бұрын

Your explanation is just great !! it is simple, elegant, smooth and flawless. Great job, I have been looking for so long to understand this concepts. Thanks + Regards

@cr7rulz97 6 жыл бұрын

that was so lucidly explained and drawn. cant thank you enough for color coding the directions. thank you so much Brian

@ozzyfromspace 6 жыл бұрын

I don't have a college education and even I found this highly intuitive. Thanks Brian, your explanation was a really helpful primer.

@oceane230 10 ай бұрын

I can't thank you enough - you answered all the questions I had on this topic in the first 3 minutes! My teacher has been trying to explain these concepts for the last 4 lessons.

@mireksoja9063 5 жыл бұрын

Great Job! The concept of the stress tensor is explained in a very simple and intuitional way.

@dastardlyexpressions 7 жыл бұрын

What a great video to understand not only the stress tensor, but tensors in general. They're rarely taught in application.

@1946Dmitri 4 жыл бұрын

Great explanation! Clear and interesting! Very glad I found your site!

@edroberts1943 3 жыл бұрын

This is the best explanation of the stress tensor I have found. Thank you!

@sdh85204 8 жыл бұрын

Hello,Your explanation is the BEST I have encountered. I wish the other lecturers had been as good !

@LucasMartins-el7kn 6 жыл бұрын

Thank you, sir! Now I finally understood what a tensor is.

@TheGranolaForce 9 жыл бұрын

Thank you for posting this video, it was very helpful. Keep up the good work and best wishes!

@mark_fingerhuth 8 жыл бұрын

great video! thanks for explaining this topic in such a simple way!

@MrSaleh101 9 жыл бұрын

Thank you. Explained very well. :)

@saptarshisikder9707 4 жыл бұрын

Great sir...great...so nicely explained...now it became clear..thank you

@sudha4674 7 жыл бұрын

i was struggling to under the concepts of tensor. now I am clear. lots of thanks

@azuleno17 8 жыл бұрын

Magnificent video. Somehow I got interested into this, but it is really helpful as part of my major.

@guneet6065 5 жыл бұрын

Ultimate explanation hats off to you sir :)

@FZIFzi 6 жыл бұрын

Wow!! Thank you, you make it look so simple.. I'm so grateful!

@SheepEditionStream 7 жыл бұрын

he keeps writting Tzz for Tzy lol, he did it again at 8:12. BUt honestly thank you so much for this. Clear, concise, straight to the point, and everything was relevant to what I needed to know for my exam. Your help was much appreciated

@adamfattal9602 5 жыл бұрын

Great video! The information obtained to time ratio in this video is tremendously high. Thanks a lot Prof. Storey. Not bad for an engineer (Just joking. It's based on a joke that's going around the internet).

@deniz.7200 4 жыл бұрын

Great explanation, thank you. 1 small addition, the normal vectors which you premultiplied are better noted as transposed imo.

@ingGS 4 жыл бұрын

Beautiful explanation. Thank you!

@fernandb.6162 4 жыл бұрын

3:24 I felt that :D Great video sir, thanks a lot :)

@isabeln.93 6 жыл бұрын

this is so good. thank you so much! precious help

@fanruihu330 4 жыл бұрын

This is so great!!

@bhangrafan4480 4 жыл бұрын

Really clear. A concrete approach to explanation usually works best.

@tonyzahtila9217 4 жыл бұрын

Great explanation Brian!

@RahulSharmaSingularity 2 жыл бұрын

Fantastic !!!

@eamonnsiocain6454 6 жыл бұрын

Excellent! Very clear.

@kl-nc5rc 6 жыл бұрын

Superb Video!!!!

@dimitarstoyanov9932 5 жыл бұрын

How do you calculate the normal vector of the hypotenuse of the triangle to be as shown

@manaoharsam4211 3 жыл бұрын

Yes explanation is good.

@sabamalik1798 8 жыл бұрын

Excellent video thanks !

@amanravan9795 Жыл бұрын

Thankyou sir Good explanation

@alikarimi-langroodi5402 2 жыл бұрын

Excellant. Thank you

@johnspivack6520 10 ай бұрын

Good and clear. Thanks.

@luk45ful 2 жыл бұрын

Really good explanation!

@diemitdenententanzt 8 жыл бұрын

Thank you! Your explanation is great! I just wondered if the origin of the Txy force should be on the edge of the cube since you placed the origin of the coordinate system in the lower left corner or doesn't it matter? Sorry, I am quite new to this topic

@Ulas_Aldag 3 жыл бұрын

Wow that was amazing

@IceyJunior 5 жыл бұрын

I'm cool with the governing equations for CFD, which can be written in integral (conservation of mass, linear momentum, angular momentum, and energy) or differential (conservation of mass, linear momentum, and energy) form. But I'm not quite sure about the governing equation(s) for CSM. Is this stress tensor the governing equations for CSM? Is it the only one used in CSM?

@user-fh4cd5up8d 4 жыл бұрын

That’s amazing! Thank you!!

@verygood6625 5 жыл бұрын

nice job mate... thanks

@Madmetroid99 7 жыл бұрын

Thanls a lot, great explanation

@odijiechrisobhione 2 жыл бұрын

Brian, thanks a lot.

@prashikbhagat 8 жыл бұрын

Great explanation

@bsp6496 6 ай бұрын

Hoooly molly, didn't expect that at the end. I can get why tensors are used in mechanics now.

@rahulsharma-wi7xn 5 жыл бұрын

great explanation

@Tuba672 9 жыл бұрын

very good ... thank you!!

@jackal5096 4 жыл бұрын

At 03:25, you said "the normal vector is a column vector", but wrote it on your whiteboard as a row vector (horizontally). I was watching more of what you wrote, and less of what you said, and became totally confused. Went through 3 of my old textbooks, looking for dot product of vector and tensor, which all showed writing the vector as a standard vector, i.e., a column vector. Finally, I went back and listened to the video. Very, very frustrating. But otherwise, a great tutorial. I saw Bruce's comment below while I was writing this

@dannyboy12357 7 жыл бұрын

Can you do a video on the stress-energy tensor that has 16 components ie. the space-time components.

@harleyspeedthrust4013 4 жыл бұрын

This has applications in machine learning. The backpropagation algorithm can be vectorized and tensors can be used to represent the weight gradients between two layers

@rares60 4 жыл бұрын

Could you elaborate a bit?

@suryakarla8628 2 жыл бұрын

Thank you. Very helpful!

@RTD553 6 жыл бұрын

Excellent.

@Guthans09 9 жыл бұрын

Question about the normal vector in the triangle example. Wouldn't the components of the normal vector, i.e, 1/2 and sqrt(3)/2 be switched since the cosine is in the x-direction (thus making it first) and the sine is in y-direction (making it second)? Assuming we are defining a vector as v = [x , y , z] ? EDIT: I SCREWED UP Ayyy lmao, nevermind. I just did the geometry. Carry on. Thanks for this video!

@rawanalharbi6267 2 жыл бұрын

I have a question! Why used the partial derivatives ?

@science_10523 3 жыл бұрын

very nice and simple explanation. very good sir. can u make a video on "elastic constants ( C11, C12 etc.)"?

@hawraaraheem2449 Жыл бұрын

How u supposed these values in normal vector n please

@user-fn9go5xb9l 3 жыл бұрын

Базар жоқ. Мықты мықты.

@fatkraken3140 4 жыл бұрын

why there are only forces on the 3 faces ?

@zizili7917 7 жыл бұрын

super nice!

@anomalyanomaly 9 жыл бұрын

Fucking smooth to understand. Thank you.

@alwysrite 7 жыл бұрын

at 3:25 did you mean a 'row' vector rather than 'column' vector?

@marquez2390 5 жыл бұрын

No. He means column vector. As dot product of A and B is defined as (A^T)(B) so what you thought was row vector was just the transpose of the column vector he was referring to.

@ejijojo 4 жыл бұрын

Thank you.

@-ul7lh 10 ай бұрын

Excellent

@bijoybasumatary4651 4 жыл бұрын

You are great

@reup6943 Жыл бұрын

I've seen in other documents: S = -T .n (S: surface stress, n: normal) with 'T' the 1st Piola Kirchhoff stress. Where does the sign difference and multiplication inversion stems from? (in the video we have S = n.T)

@danpoles2864 5 жыл бұрын

how do u know when the n vector matrix is a column or row

@brianstorey7830 5 жыл бұрын

Yeah, this is always one of the most confusing things. It is always a row vector, but since the tensor is symmetric - it is OK if you mix it up. If you work through an example or two yourself with the sketch of what the components are with simple normal vectors (like [1 0 0]) you'll see how symmetry saves you!

@jeanpaul4294 3 жыл бұрын

please answer me for 0:46 is the 2d tensor both forces or just one?

@brianstorey7830 3 жыл бұрын

The 2D pictures are just easier to draw. Everything is conceptually the same for 2 and 3D. For 2D, we are just working in the plane of the paper you are drawing on. Here at the beginning I was just trying to explain that for stress the direction of the force and the direction of the face upon which it acts are both important.

@lopezb 4 жыл бұрын

Nice! (at 3:25 the vector n is a ROW not a column vector).

@teymourtb 9 ай бұрын

Hello, thank you for this video. One question: why did you call the back face Txx in the second drawing when it was on the opposite face for the previous drawing? 9:09

@twinaibots5549 7 жыл бұрын

Great video sir..

@AfirSraftGarrier 7 жыл бұрын

Great, thx.

@BoZhaoengineering 4 жыл бұрын

can we use column vector form to describe both tensor matrix and normal vector? that will be consistent with vector form/notation in linear algebra.

@AntoninKrovina 4 жыл бұрын

Just transpose the vector And matrix and change the order of multiplication.

@manikhorajina2662 2 жыл бұрын

How is the value for normal vector obtained at 4.48?

@mediwise2474 Жыл бұрын

Tell me tooo

@akamaor 4 жыл бұрын

good one!

@gaiuspliniussecundus1455 Жыл бұрын

What if your object under deformation is a parametric function of two variables, u and v, producing a vector in x,y,z? So f(u,v):R^2->R^3. Doesn't the tensor needs to be symmetric? What to do, and how to compute the magnutude of the deformation between a undeformed and deformed object in this case?

@darthnegativehunter8659 3 жыл бұрын

this is a really good video (although requires some self calculation to figure out how divergance of tensor has meaning)

@00PedroM 7 жыл бұрын

Great video! I'd like to start recording lessons like you do, but I'm stuck with some technical problems... I don't know how I can support the device I'm going to use for recording (camera or cell phone) at a good distance while I write... Can you tell me how you did this and what tools did you use? Thanks!!!

@brianstorey7830 7 жыл бұрын

I just used one of these document cameras - really no different than a standard web cam but has a stand for writing under. www.ipevo.com/prods/point-2-view-usb-camera An external mic is usually needed to get better sound quality (rather than the built in laptop mic I had anyway) A desk lamp and play around with the lighting. That's about it. Pretty minimal.

@robertbrandywine 5 ай бұрын

So is the answer to a stress tensor problem a simple vector?

@chandrahasam836 6 жыл бұрын

nice helped lot

@Farzan1World 6 жыл бұрын

Very well explained. Thank you. Can you refer me somewhere on the web that makes practical use of this with numbers generated, say in fluid dynamics or stress analysis?

@mediwise2474 Жыл бұрын

Tell me too

@wulphstein 5 жыл бұрын

Does a stress energy event update spacetime at the speed of light?

@dieselguitar1440 5 жыл бұрын

I'm not sure if I know exactly what that is, but I would guess that it would be at the speed of sound of the material in question. If I'm assuming correctly, sound would actually be a stress energy event in constant oscillation. Here's a link where if I remember right they talk about tension in a slinky released into free fall moving at the speed of sound, or if it wasn't the speed of sound, it definitely wasn't light speed. The comment section is also filled with people's own theories, but I'm pretty sure the contents of the video are known facts: kzfaq.info/get/bejne/m6l9oNB2qt-zf2Q.html

@armins1752 9 жыл бұрын

Thank you very much

@bens4446 6 жыл бұрын

Great video. Net force per unit volume--so, basically the net force density? But then there are three (x, y, z) components. How to think intuitively about the ith component of density? Density in the ith direction? What's that?

@brianstorey7830 6 жыл бұрын

Ah, I think maybe this is just confusion over the word "density". Usually when we use the word density, we mean "mass density" - mass per unit volume. That is a scalar and thus has no direction. By force density, we just mean the force (vector) divided by the volume over which that force acts. So ho g is the force density due to gravity. It has a component only in the direction of the g vector. Does this actaully answer your question?

@khaledplus1021 4 жыл бұрын

Thanks 💯

@aniken184 5 жыл бұрын

i think at 8:00 minute see divergence of stress tensor gives components in terms of (del T ij / del xi )j so it might be right only in case of stress symmetry. But if stress tensor represented as column vector combination of stress on each plane then first column will give stress on plane perpendicula to x and so divergence of it gives del Tij / del xj ) i in general . is it correct or not?

@brianstorey7830 5 жыл бұрын

So the order of things is always easy to confuse and something I tend to screw up a lot. Is it Tij or Tji? It is a common mistake, and one I have trouble with. The good thing is that in the case of stress, T is ALWAYS symmetric. So it doesn't matter.... As you note, using index notation is a better way to be clear about which components you are talking about, but that was not something I wanted to introduce here.

@user-fz2ir8kc6z 3 жыл бұрын

Thank you sir

@slowsatsuma3214 4 жыл бұрын

Beautiful

@Harsh.Parekh 6 жыл бұрын

scalar also has 3 component in x,y and z component?

@brianstorey7830 6 жыл бұрын

Scalar just has a single value. Temperature is an example of a scalar. It has no directional components. A vector, like velocity, has x, y, and z components.

@kindintiudaykiran2426 3 жыл бұрын

Thanks 🙂🙂

@VikiJoker1924 Жыл бұрын

sorry, @ 4:43 how is normal vector equal to what is shown? I understand sin30 =1/2 and cos30= sqrt3/2, but where's the 0 from?

@jeffreychavey4161 7 жыл бұрын

Great stuff ... but where'd you learn to write so fast?

@quocanhnguyen7275 8 жыл бұрын

Thanks a lot

@aminegc9353 4 жыл бұрын

hello sir , i just would like to tell you that i speak and understand french cours more better that english , but your cours is too much well explained than in french , i understood more better what you explain for us, i would like too to give us more cours about elasticity and FEM to beguinner untel to the advenced level, thank you sir another time. :)

@DiceMaster740 4 жыл бұрын

Maybe you could use the knowledge that you gained from this video, and your ability to speak french, and make a better French video to explain it

@Amr-hb2wh 8 жыл бұрын

sir you are a masterpiece .

@MrCooldude4172 5 жыл бұрын

Hi there, I am confused about one thing: Does it matter if you do n . T or T . n, i.e. the order of the dot product of the tensor with the normal vector? I get 2 different results. I know with a vector, it does not matter.

@brianstorey7830 5 жыл бұрын

So it is different if you think of the 3x3 tensor multiplying a column vector, n or a row vector n multiplying the 3x3 tensor. However, the stress tensor is always symmetric (from angular momentum considerations) therefore for the symmetric tensor you get the same result! If you do much more with tensors, it is usually better to work in index notation, but that opens up more complexity than I wanted here.

@pandas896 3 жыл бұрын

Thank

@tehlolzfactor 6 жыл бұрын

I know this video is old but I just wanted to point out that at 8:00, the y component of the vector shouldnt be partial of Tzz with respect to z it should be Tzy with respect to z