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
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
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
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'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)
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?
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
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 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. :)
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!
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
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!
Hi, when you are at 4:58 and have triangular body, I don't understand how can you multiply with Tyx, Tyy, Tyz, when Y face doesn't exist at all? As well, why tensor has only 3 vectors, when cube has 6 sides. There should be 6 vectors in a tensor?
On the first question. The surface normal vector has a component in the x and y direction, so face does have a y component - so to speak. I know this is a pretty lame explanation, but the geometry is really tricky. I think you might have to work out an example. You could try an example where you take the little square aligned with the axis and cut a diagonal through it and equate all the forces. It is a little hard to explain in the text box of this comment without drawing another example out. Maybe if you look at "Mohr's circle" in solid mechanics, you this will help. On the second question. We considered only 3 of the six faces. This is because the stress tensor has a LOCAL value. It is evaluated at a point in space. Therefore, for equilibrium to be maintained the Txx and Txx (on the opposite face) must balance out and be equal. Thus the forces on the opposite sides of the face cancel. Thus there is a vector with three component acting on the three faces. The stress tensor then has nine components (3x3 matrix). Conservation of angular says the matrix is symmetric and therefore there are 6 unique components to the tensor. Hope this is helpful
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
I think I understand the question, which is why do we count the x direction force acting on the normal face different than one acting on the other faces? This is a common and good question. I am not sure that I can answer the question well here in the comments, other than to think about how material responds differently to normal (stretching forces) versus shear (sliding forces). I think maybe a key is to think about the differential cube of material and remember that even though we take the limit of the cube getting smaller and smaller, we still think about the cube as a real 3D object. Hope this helps. but just keep working on it!
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.
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.
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'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: czcams.com/video/eCMmmEEyOO0/video.html
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.
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.
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 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.
Your input matrix numbers are scalars representing pressure values. You multiply that by a direction vector, and the answer is another vector whose values are also pressure scalars, I think.
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.
Hi i’m reading a textbook which says that the stress vector is equal to the stress tensor dot the normal unit vector. Here you wrote it the other way around. You are the second source i’ve seen to write it this way and i was wondering if the textbook is wrong. Great video btw!
Confusing right? I think I am correct. The only way to be unambiguous is to use the Einstein index notation - but for teaching concepts I find this gets bogged down in subscripts. The fortunate thing is that since the tensor is symmetric you get the same result with n dot T and T dot n.
And... in matrix notation n dot T is like a row vector on the left multiplying a matrix and T dot n is like a matrix on the left multiplying a column vector.
Question: Why is it that when you wright out the stress tensor, you start with Txx?? Why is it defined like that and not arbitrarily with Tyx as the first entry?
I guess one could devise a different system that would work, I am not sure what it would be. There are some properties of the tensor that I don't discuss here that must be maintained (i.e. you want the tensor to give you the same answer if we change coordinate systems, we want the tensor to conserve angular momentum, and many other things). I could imagine it may be possible to devise a system that satisfies all those constraints. However, sometimes we all just have to have an agreement on a standard and a system to move forward - so maybe that is the best reason of all!
The stress tensor T can also be thought of as three column vectors and three row vectors. In vector notation the order is important, so you can visualize the stress tensor with its nine componets having a header row with x, y, and z and a header column with the same x, y, z. In the convention used here, each element in T has subscripts x, y, or z corresponding to the headers of T. As Brian has answered, this is simply one scheme that works and there are symmetrical counterparts that also work.
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?
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!
Since 2D is a particular case, although is simpler than 3D, some teachers decide to start with 2D, and others starts with 3D. Greetings from Argentina.
1:41 my reaction when I saw the mechanics exam
+Daan Janssen
mechanics was so easy bro!
Multivariate Calculus was a MONSTER! LOL
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.
are you fr?
y y y
Thanks for the chuckle.
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 don't have a college education and even I found this highly intuitive. Thanks Brian, your explanation was a really helpful primer.
that was so lucidly explained and drawn. cant thank you enough for color coding the directions. thank you so much Brian
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.
This is the best explanation of the stress tensor I have found. Thank you!
Thank you, sir! Now I finally understood what a tensor is.
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
What a great video to understand not only the stress tensor, but tensors in general. They're rarely taught in application.
Great explanation, thank you. 1 small addition, the normal vectors which you premultiplied are better noted as transposed imo.
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 Job! The concept of the stress tensor is explained in a very simple and intuitional way.
Hello,Your explanation is the BEST I have encountered. I wish the other lecturers had been as good !
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
Could you elaborate a bit?
Thank you. Explained very well. :)
i was struggling to under the concepts of tensor. now I am clear. lots of thanks
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).
Really clear. A concrete approach to explanation usually works best.
Great sir...great...so nicely explained...now it became clear..thank you
Magnificent video. Somehow I got interested into this, but it is really helpful as part of my major.
Hoooly molly, didn't expect that at the end. I can get why tensors are used in mechanics now.
Thank you for posting this video, it was very helpful. Keep up the good work and best wishes!
Fucking smooth to understand. Thank you.
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)
Nice!
(at 3:25 the vector n is a ROW not a column vector).
Great explanation! Clear and interesting! Very glad I found your site!
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?
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
Wow!! Thank you, you make it look so simple.. I'm so grateful!
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?
at 3:25 did you mean a 'row' vector rather than 'column' vector?
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 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. :)
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
Thankyou sir
Good explanation
i am confused as to why we used the normal vector as [ 1/2 , sqrt 3/2, 0] instead of [swrt3/2, 1/2, 0] at 4:48
How to get normal vector
How do you calculate the normal vector of the hypotenuse of the triangle to be as shown
Yes explanation is good.
3:24 I felt that :D
Great video sir, thanks a lot :)
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!
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
Beautiful explanation. Thank you!
Excellant. Thank you
great video! thanks for explaining this topic in such a simple way!
Good and clear. Thanks.
why is he multiplying the vector and matrix in that order? I learned in math that we cant multiply vector with matrix but matrix x vector is possible.
Fantastic !!!
How is the value for normal vector obtained at 4.48?
Tell me tooo
this is a really good video (although requires some self calculation to figure out how divergance of tensor has meaning)
sir you are a masterpiece .
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?
I have a question! Why used the partial derivatives ?
Thank you.
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!
nice job mate... thanks
Ultimate explanation hats off to you sir :)
Brian, thanks a lot.
Can you do a video on the stress-energy tensor that has 16 components ie. the space-time components.
Hi, when you are at 4:58 and have triangular body, I don't understand how can you multiply with Tyx, Tyy, Tyz, when Y face doesn't exist at all?
As well, why tensor has only 3 vectors, when cube has 6 sides. There should be 6 vectors in a tensor?
On the first question. The surface normal vector has a component in the x and y direction, so face does have a y component - so to speak. I know this is a pretty lame explanation, but the geometry is really tricky. I think you might have to work out an example. You could try an example where you take the little square aligned with the axis and cut a diagonal through it and equate all the forces. It is a little hard to explain in the text box of this comment without drawing another example out. Maybe if you look at "Mohr's circle" in solid mechanics, you this will help.
On the second question. We considered only 3 of the six faces. This is because the stress tensor has a LOCAL value. It is evaluated at a point in space. Therefore, for equilibrium to be maintained the Txx and Txx (on the opposite face) must balance out and be equal. Thus the forces on the opposite sides of the face cancel. Thus there is a vector with three component acting on the three faces. The stress tensor then has nine components (3x3 matrix). Conservation of angular says the matrix is symmetric and therefore there are 6 unique components to the tensor.
Hope this is helpful
this is so good. thank you so much! precious help
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
Could someone explain why forces acting in the same direction arent just the same force? I mean they’re coming from the same point right?
I think I understand the question, which is why do we count the x direction force acting on the normal face different than one acting on the other faces? This is a common and good question. I am not sure that I can answer the question well here in the comments, other than to think about how material responds differently to normal (stretching forces) versus shear (sliding forces). I think maybe a key is to think about the differential cube of material and remember that even though we take the limit of the cube getting smaller and smaller, we still think about the cube as a real 3D object. Hope this helps. but just keep working on it!
why don't you consider the other 3 faces of the cube?
Excellent
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.
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.
very nice and simple explanation. very good sir. can u make a video on "elastic constants ( C11, C12 etc.)"?
Thank you. Very helpful!
How u supposed these values in normal vector n please
Really good explanation!
why there are only forces on the 3 faces ?
Great explanation Brian!
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.
Does a stress energy event update spacetime at the speed of light?
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: czcams.com/video/eCMmmEEyOO0/video.html
Wow that was amazing
Excellent video thanks !
Superb Video!!!!
That’s amazing! Thank you!!
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.
Excellent! Very clear.
please answer me for 0:46 is the 2d tensor both forces or just one?
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.
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?
For consistency of notation, the del operator should carry an overbar arrow to denote it as a vector.
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.
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.
Just transpose the vector And matrix and change the order of multiplication.
Thank
This is so great!!
Thanls a lot, great explanation
So is the answer to a stress tensor problem a simple vector?
Your input matrix numbers are scalars representing pressure values. You multiply that by a direction vector, and the answer is another vector whose values are also pressure scalars, I think.
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.
Базар жоқ. Мықты мықты.
hello guy. i am studying maxwell stress tensor in rectangular shape. Can you help me?
Great stuff ... but where'd you learn to write so fast?
Hi i’m reading a textbook which says that the stress vector is equal to the stress tensor dot the normal unit vector. Here you wrote it the other way around. You are the second source i’ve seen to write it this way and i was wondering if the textbook is wrong. Great video btw!
Confusing right? I think I am correct. The only way to be unambiguous is to use the Einstein index notation - but for teaching concepts I find this gets bogged down in subscripts. The fortunate thing is that since the tensor is symmetric you get the same result with n dot T and T dot n.
And... in matrix notation n dot T is like a row vector on the left multiplying a matrix and T dot n is like a matrix on the left multiplying a column vector.
Thank you sir
Question: Why is it that when you wright out the stress tensor, you start with Txx?? Why is it defined like that and not arbitrarily with Tyx as the first entry?
I guess one could devise a different system that would work, I am not sure what it would be. There are some properties of the tensor that I don't discuss here that must be maintained (i.e. you want the tensor to give you the same answer if we change coordinate systems, we want the tensor to conserve angular momentum, and many other things). I could imagine it may be possible to devise a system that satisfies all those constraints.
However, sometimes we all just have to have an agreement on a standard and a system to move forward - so maybe that is the best reason of all!
Oh i see, fair enough ! Thanks :)
The stress tensor T can also be thought of as three column vectors and three row vectors. In vector notation the order is important, so you can visualize the stress tensor with its nine componets having a header row with x, y, and z and a header column with the same x, y, z. In the convention used here, each element in T has subscripts x, y, or z corresponding to the headers of T.
As Brian has answered, this is simply one scheme that works and there are symmetrical counterparts that also work.
great explanation
very good ... thank you!!
hi, can you give me the exercise solution of Alexander Mendelson's books "Plasticity: Theory and application"
Give me too
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?
Tell me too
Thank you sir😊....
how do u know when the n vector matrix is a column or row
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!
wouldn't it be easier to use a 2d shape, before applying it to a 3d shape?
Since 2D is a particular case, although is simpler than 3D, some teachers decide to start with 2D, and others starts with 3D.
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