Love your videos! I can see i am a bit late to the party but from 10:57 to 11:13 you say that if delta(Uo) is expanded it becomes a product of the partial derivative with respect to epsilon and the variation of epsilon (the strain). Why is this the case? I have seen your video on the delta operator but i can't quite make the connection from that video to the fact that the product mentioned is equal to the variation of Uo. If possible do you have any recommendations on material that explaines this kind of math? Thanks for the amazing effort!
This is a little confusing when proceeding in the direction that I have. It is much more obvious when doing this in reverse. So, take the result δU and take it variation. Since Uo is a function of ε, i.e. Uo = Uo(ε) then it follows that its variation is δUo = ∂Uo/∂ε δε I have simply proceeded in the reverse order in the video. For additional reading material, try Dym & Shames, "Solid Mechanics: A Variational Approach".
Back in 1970 my high school science teacher told us that a structure (system) is always in equilibrium, and in fact it's the maintenance of equilibrium that causes it to collapse if necessary. I've remembered that to this day because it seemed so intuitive. This stuff sounds like the same thing, although the maths is way beyond me. I love maths even though I'm utterly hopeless at it.
I'm a PhD student and your videos helped me a lot. currently, I'm working on phase-field modeling of fracture of brittle materials, that would be very interesting for students if you make a video about that. Thanks again. Navid
Fracture mechanics is somewhat outside the scope of the material I'm trying to present here. It might take me a little while to get there, but I'll add it to the list.
Sorry for the delayed response, but I somehow missed your comment until now. This expression is the strain-displacement relation and comes from the geometric definition of strain: εij = ½(ui,j + uj,i) . The material is presented in most introductory strength of materials or solid mechanics classes. It is written here in index form which is why it might look a little different. It is stating that the strain is the derivation of the displacement field - it measures how much a body deforms from its original shape. For small deformations, it's derived by looking at how nearby points move relative to each other. The variation δεij = ½(δui,j + δuj,i) follows directly by taking the variation of this expression, just like taking a derivative. Since δ is a linear operator, it follows the same rules as differentiation.
Can't recommend any books on history. Typically I get this information by browsing articles online (Wikipedia is always a good start). However, if you are in need of a book that covers this material (ie variational calculus as it applied to structural modeling, then I would recommend Dym & Shames, "Solid Mechanics: A Variational Approach".
Yikes. This is very professional and detailed. Thanks for sharing. Can you do more examples of partial differential equations like heat and wave equations. Maybe even more classical mechanics examples.
at 10:30, V in Eq. 8 and V in Eq. 9 is not the same. In addition, how is the equation Delta (Pi) interpreted as minimum potential energy? I mean why not maximum?
You're right, this is unclear, I used V both for the potential and also to describe the volume integral - probably should have uses a scripted 'V' for the volume to distinguish the two. With regards to whether the stationary point found by setting δπ=0 represents a minimum or maximum, this should really be shown rigorously by taking the 2nd variation and showing it is positive (which can be done). However, simply from the physical nature of the problem at hand, since we are dealing with a combination of the internal strain (elastic) energy and the potential of the external loads, it should be understood that the minimum possible potential energy is 0 while the maximum potential energy is, theoretically, infinite since we've imposed no limitation on this. Therefore any stationary point of the potential energy function will correspond to a minimum value rather than a maximum value.
To denote a virtual displacement, we use the delta operator. We use the same operator to denote the variation of a path. The reason for this is that virtual displacements and path variations are really the same concept - ie a virtual displacement is just a variation of the displacement field..
It would be better to introduce constant forces before the potential. There are forces that didn't admit a potential. But this video is great. You are very good.
First of all, thank you for these fantastic series of videos. I have a question about the definition of the strain tensor \epsilon_ij. In my knowledge, the definition that you put is only valid for small displacements (since there is also a non-linear term in the general equation of the strain tensor). So based on this definition, the principle of virtual work that you obtained, is it still valid for all materials even if they are non-linear (and thus inelastic)?
The strain-displacements were linearized due to the displacements being small, so the model is based on small displacements (ie no geometric nonlinearities). However, we never made any restrictions on the material properties being linear. So this is valid for all materials whether or not the material is linearly elastic. Also, nonlinear stiffness properties does not imply that the material is inelastic - just that the stiffness changes as it strains.
I dont understand why work done by surface forces can be replaced by the Cauchy's formula. Work done by external surface forces are directly equal to work done by internal stresses? (I am confused about the location of the surface in which traction vector acts. Is the point P is on the surface or within the body? ) Why does this surface forces do not appear in the general equilibrium equation (1)?
There are several things that need to be addressed here. Let me try to tease it apart for you... 1. In general, tractions can act anywhere on the surface of the body. It depends on where loads are applied on the surface. But we want to convert/relate these to internal stresses at any point within the body in order to simplify calculations. 2. Cauchy's Formula allow us to do exactly this, but in order to understand why this is would require its own video. Note, we are just converting the tractions - not the work. 3. We then use Gauss' Theorem to convert the surface integral into a volume integral which makes the math easier (eqn 6). 4. For a body in equilibrium, the Principle of Virtual Work, leads us to the conclusion that work done by external surface forces equals work done by internal stresses. This is shown in the video (eqn. 8). 5. The point P is an arbitrary point within the body. It could be anywhere, but probably easier to think about it being internal rather than on the surface. 6. The surface tractions do not appear in the equilibrium equation 1 because this equation come from solid mechanics and represents a force balance within the volume. We need to convert the surface tractions into internal stresses first so that we can apply this equation. This is what we use Cauchy's Formula for.
@@Freeball99 Thank you very much for answer and for the lecture. Indeed my main question is related with 1. and 2. how can we relate surface loads with internal stresses through Cauchy's formula? Where can i read about these derivation? A video about these would be great! Thank you again
@@jorgeluismedina1548 If you search TH-cam for "Cauchy's Stress Formula" you'll find several videos on the subject. I haven't watched them all, but I can vouch for this one by Clay Petit. He has some great content. th-cam.com/video/CGDziWoEEgo/w-d-xo.html
Traction in solid mechanics is not limited to a pulling force. It includes any kind of force exerted over an area, whether it's pulling, pushing, twisting, or shearing.
So, the force exerted by a gas on the walls of its container is traction? I don’t think so. Traction is normally used for tangential forces, but can also be used to refer to tensile forces exerted by dry friction.
In the context of solid or continuum mechanics, you are correct that tractions can refer to any kind of surface force. In mechanical engineering in general, however, traction is commonly used to describe pulling forces.
In general, no, but it depends on the type nonlinearity. It would handle geometric and material nonlinearities, but would not handle a non-conservative force.
No these are not the same thing. Fundamentally the difference is that Castigliano's Theorem is based upon minimizing work while The Principle of Minimum Potential Energy is based upon minimizing the strain energy. Castigliano's Theorem (also know as the theorem of minimum work) allows one to find the forces from the potential/strain energy (First Theorem) or the displacements from the strain energy (Second Theorem). This is a necessary step in deriving the Principle of Minimum Potential Energy (a "sub-component" if you will), but they are not the same thing. So I would describe Castigliano's Theorem as a direct consequence or result of the Principle of Minimum Potential Energy.
The strain energy density definition is correct. The 1/2 appears in the strain energy comes from integrating the strain energy density. If you substitute σ = Eε and the integrate with respect to ε, you will get U = ½Eε²
I don't have many examples of the Principle of Minimum Potential Energy. However, I extend this theory in the next video to Hamilton's Principle by incorporating the dynamic case and the several videos that follow that contain examples.
![Why less energy means more stability? [Pure Logic]](http://i.ytimg.com/vi/76n7BmqAeOs/mqdefault.jpg)
Absolutely loving your videos. I graduated from my Maths degree in 2000. I don't recall any of this being explained so clearly before. Good work.
you are such a great teacher. thank you very much for your lecture
Wonderful video. Very informative and crystal clear explanation!
Your lectures are morning breeze in summer.
Very nice material! Thank you for sharing, It concatenates a lot of concepts! A big fan here, hello from Brazil.
Love your videos!
I can see i am a bit late to the party but from 10:57 to 11:13 you say that if delta(Uo) is expanded it becomes a product of the partial derivative with respect to epsilon and the variation of epsilon (the strain). Why is this the case? I have seen your video on the delta operator but i can't quite make the connection from that video to the fact that the product mentioned is equal to the variation of Uo.
If possible do you have any recommendations on material that explaines this kind of math?
Thanks for the amazing effort!
This is a little confusing when proceeding in the direction that I have. It is much more obvious when doing this in reverse. So, take the result δU and take it variation.
Since Uo is a function of ε, i.e. Uo = Uo(ε) then it follows that its variation is δUo = ∂Uo/∂ε δε
I have simply proceeded in the reverse order in the video.
For additional reading material, try Dym & Shames, "Solid Mechanics: A Variational Approach".
Back in 1970 my high school science teacher told us that a structure (system) is always in equilibrium, and in fact it's the maintenance of equilibrium that causes it to collapse if necessary. I've remembered that to this day because it seemed so intuitive.
This stuff sounds like the same thing, although the maths is way beyond me.
I love maths even though I'm utterly hopeless at it.
What you have stated is pretty much a description of D'Alembert's Principal.
Great food for hungry minds. Thank you, Sir.
Great content! There is a typo at 2:29, second equation first term index should be 2,1.
Yep, you're correct. Although due to the symmetric nature of the stress tensor, σ12 = σ21. Thanks for catching that.
I'm a PhD student and your videos helped me a lot. currently, I'm working on phase-field modeling of fracture of brittle materials, that would be very interesting for students if you make a video about that. Thanks again. Navid
Fracture mechanics is somewhat outside the scope of the material I'm trying to present here. It might take me a little while to get there, but I'll add it to the list.
Thank you! But I do not understand why the assumption at 3:00-3:12 holds / where it comes from.
Sorry for the delayed response, but I somehow missed your comment until now.
This expression is the strain-displacement relation and comes from the geometric definition of strain: εij = ½(ui,j + uj,i) . The material is presented in most introductory strength of materials or solid mechanics classes. It is written here in index form which is why it might look a little different. It is stating that the strain is the derivation of the displacement field - it measures how much a body deforms from its original shape. For small deformations, it's derived by looking at how nearby points move relative to each other.
The variation δεij = ½(δui,j + δuj,i) follows directly by taking the variation of this expression, just like taking a derivative. Since δ is a linear operator, it follows the same rules as differentiation.
That's great.
Could you please upload such an explanatory videos for LU Decomposition and Newton raphsons method ?
I guess i will fall in love with the math with your lectures😊😊
@5:48 it is the divergence of a Vector not the gradient no?
Yes, I misspoke; it's the divergence (which is why its called the Divergence Theorem).
Thanks a lot, for the amazing lectures.
Hey, truly insightful videos, loved it. Can you suggest any book to read and know about the history of scientists
Can't recommend any books on history. Typically I get this information by browsing articles online (Wikipedia is always a good start). However, if you are in need of a book that covers this material (ie variational calculus as it applied to structural modeling, then I would recommend Dym & Shames, "Solid Mechanics: A Variational Approach".
Amazing Video, thank you!!
A very good explaining. Thanks
Outstanding video lecture.
Yikes. This is very professional and detailed. Thanks for sharing. Can you do more examples of partial differential equations like heat and wave equations. Maybe even more classical mechanics examples.
at 10:30, V in Eq. 8 and V in Eq. 9 is not the same. In addition, how is the equation Delta (Pi) interpreted as minimum potential energy? I mean why not maximum?
You're right, this is unclear, I used V both for the potential and also to describe the volume integral - probably should have uses a scripted 'V' for the volume to distinguish the two.
With regards to whether the stationary point found by setting δπ=0 represents a minimum or maximum, this should really be shown rigorously by taking the 2nd variation and showing it is positive (which can be done). However, simply from the physical nature of the problem at hand, since we are dealing with a combination of the internal strain (elastic) energy and the potential of the external loads, it should be understood that the minimum possible potential energy is 0 while the maximum potential energy is, theoretically, infinite since we've imposed no limitation on this. Therefore any stationary point of the potential energy function will correspond to a minimum value rather than a maximum value.
@ 3:47 I don't remember you saying anything about virtual displacement. How can it be used as variational?
To denote a virtual displacement, we use the delta operator. We use the same operator to denote the variation of a path. The reason for this is that virtual displacements and path variations are really the same concept - ie a virtual displacement is just a variation of the displacement field..
perfect!!!
At 2:03 I believe that should be del (sigma_21) / del (x_1)
σ_12 is equal to σ_21 due to the symmetry of the stress tensor.
It would be better to introduce constant forces before the potential. There are forces that didn't admit a potential. But this video is great. You are very good.
First of all, thank you for these fantastic series of videos. I have a question about the definition of the strain tensor \epsilon_ij. In my knowledge, the definition that you put is only valid for small displacements (since there is also a non-linear term in the general equation of the strain tensor). So based on this definition, the principle of virtual work that you obtained, is it still valid for all materials even if they are non-linear (and thus inelastic)?
The strain-displacements were linearized due to the displacements being small, so the model is based on small displacements (ie no geometric nonlinearities). However, we never made any restrictions on the material properties being linear. So this is valid for all materials whether or not the material is linearly elastic. Also, nonlinear stiffness properties does not imply that the material is inelastic - just that the stiffness changes as it strains.
@@Freeball99 Thank you very much for your answer.
I dont understand why work done by surface forces can be replaced by the Cauchy's formula. Work done by external surface forces are directly equal to work done by internal stresses? (I am confused about the location of the surface in which traction vector acts. Is the point P is on the surface or within the body? ) Why does this surface forces do not appear in the general equilibrium equation (1)?
There are several things that need to be addressed here. Let me try to tease it apart for you...
1. In general, tractions can act anywhere on the surface of the body. It depends on where loads are applied on the surface. But we want to convert/relate these to internal stresses at any point within the body in order to simplify calculations.
2. Cauchy's Formula allow us to do exactly this, but in order to understand why this is would require its own video. Note, we are just converting the tractions - not the work.
3. We then use Gauss' Theorem to convert the surface integral into a volume integral which makes the math easier (eqn 6).
4. For a body in equilibrium, the Principle of Virtual Work, leads us to the conclusion that work done by external surface forces equals work done by internal stresses. This is shown in the video (eqn. 8).
5. The point P is an arbitrary point within the body. It could be anywhere, but probably easier to think about it being internal rather than on the surface.
6. The surface tractions do not appear in the equilibrium equation 1 because this equation come from solid mechanics and represents a force balance within the volume. We need to convert the surface tractions into internal stresses first so that we can apply this equation. This is what we use Cauchy's Formula for.
@@Freeball99 Thank you very much for answer and for the lecture. Indeed my main question is related with 1. and 2. how can we relate surface loads with internal stresses through Cauchy's formula? Where can i read about these derivation? A video about these would be great! Thank you again
@@jorgeluismedina1548 If you search TH-cam for "Cauchy's Stress Formula" you'll find several videos on the subject. I haven't watched them all, but I can vouch for this one by Clay Petit. He has some great content. th-cam.com/video/CGDziWoEEgo/w-d-xo.html
Traction in solid mechanics is not limited to a pulling force. It includes any kind of force exerted over an area, whether it's pulling, pushing, twisting, or shearing.
So, the force exerted by a gas on the walls of its container is traction? I don’t think so.
Traction is normally used for tangential forces, but can also be used to refer to tensile forces exerted by dry friction.
In the context of solid or continuum mechanics, you are correct that tractions can refer to any kind of surface force. In mechanical engineering in general, however, traction is commonly used to describe pulling forces.
second equation of 2 is it sigma 12 or sigma 21?
awesome work neatly explained. big fan❤
the Cauchy matrix is symmetric, therefore sigma 12 = sigma 21
Are expressions 13 and 14 valid in a non linear case?
In general, no, but it depends on the type nonlinearity. It would handle geometric and material nonlinearities, but would not handle a non-conservative force.
Is the principle of minimum potential energy the same as Castiglianos first theorem?
No these are not the same thing. Fundamentally the difference is that Castigliano's Theorem is based upon minimizing work while The Principle of Minimum Potential Energy is based upon minimizing the strain energy.
Castigliano's Theorem (also know as the theorem of minimum work) allows one to find the forces from the potential/strain energy (First Theorem) or the displacements from the strain energy (Second Theorem). This is a necessary step in deriving the Principle of Minimum Potential Energy (a "sub-component" if you will), but they are not the same thing.
So I would describe Castigliano's Theorem as a direct consequence or result of the Principle of Minimum Potential Energy.
Good morning, please do you have notes on derivation of thin plate equation using total potential energy?
Unfortunately, I do not have any written notes on thin plates. However, I will be making a video on this in the future. That one is on my to-do list.
Bro which is the name of the font appearing on previews of your videos?
It’s called Trocchi - a Google font.
Hi, I've a doubt concerning the strain energy density there should be a 1/2 in front of it??
The strain energy density definition is correct. The 1/2 appears in the strain energy comes from integrating the strain energy density. If you substitute σ = Eε and the integrate with respect to ε, you will get U = ½Eε²
@@Freeball99 Ok 👍 Thanks for your quick reply!!
I fail to understand how differentiation of Uo wrt to epsilon ij . Delta Epsilon ij equal to delta Uo
Which equation or what point in the video are you referring to?
you should give a example this is not a good way 2 explain this toppic
I don't have many examples of the Principle of Minimum Potential Energy. However, I extend this theory in the next video to Hamilton's Principle by incorporating the dynamic case and the several videos that follow that contain examples.
There is to much math not enough graphs and images ...
These derivations do get very mathematical. Perhaps I will solve a problem using the principle and is will be easier to follow.