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.
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
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.
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
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
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).
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?
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!
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
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
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!!!
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.
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.
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
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?
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?
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?
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)
Good job! But I wonder, why did you represent some forces with opposite directions? I mean, you placed Tyx and Tyx+(dTyx/dy)dy with opposite directions as if you already knew these forces had that direction. Could you please give me a convincing explanation of why that is? Thank you, Brian Storey.
If I shrink the width of the differential element, dx, to zero - then the sum of the forces must be zero. The forces must be equal in magnitude and opposite in sign. The forces have to balance as a remove the distance between them (there is no mass x acceleration to balance an imbalance in forces). As I type this I realize this is a short explanation for something that may seem confusing and is not as simple as I am claiming. As usual, it is often hard to answer questions in this forum - so I hope this makes sense.
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?
Equilibrium is from conservation of momentum. If the momentum is not changing, then the sum of the fores should be zero. Is this the equilibrium equation you are referring to?
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. :)
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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-eCMmmEEyOO0.html
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?
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.
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
I think it is pretty common to write it without the explicit vector notation and that somehow the vector nature is implied. A quick flip through some of my favorite texts all write it without - so I am at least in distinguished company by neglecting it! From a student perspective, I kind of like the idea of being explicit with the vector notation as it may help with some of the usual confusion around being only able to take the divergence of a vector and not a scalar.
Got a little doubt while studying the momentum conservation equation. I've noticed that in some books the divergence of the shear stress tensor matrix is used with a negative sign. How could it be?
+Agustin Piussi I am not sure without seeing the book, but my guess is that it just depends on what side of the equation you like things. For example for Newton's Law I could write F = ma or F-ma=0 or ma-F = 0. All are equally valid and which way you write it is just a matter of taste.
imgur.com/vLpPWxv, those are the equations I took from Bird's book, as it can be seen, the frist three equations have a minus sign on the tensor divergence. However, once they consider the fluid as newtonian, the equation is exactly the same as the one you derived.
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.
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.
Hello Mr Storey, three questions: 1. Why do we only talk about 3 faces of the cube to define the stress tensor? Is it because the Cauchy's Theorem? 2. When you apply the divergence theorem you leave the normal vector, why? 3. I though the divergence was a scalar field, not a vectorial field. The divergence of a tensors results in a vector? Thank you very much, great explanation.
1) The "three faces" question is a common question and confusing point. The cube is mainly used as a pictorial tool. Perhaps a better way to think is that there are three perpendicular planes that intersect at a point. That is the point that the stress tensor is defined. For each plane that intersects that point, there is a stress vector. These two aspects (the plane and the vector) give the 9 components of the tensor. The cube is easier to draw as the vectors don't overlay each other. 2) I am not sure I understand question 2. 3) Divergence of a vector field is a scalar field. Divergence of a tensor field is a vector field!
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!
