you are such a great teacher. thank you very much for your lecture

Wonderful video. Very informative and crystal clear explanation!

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.

Very nice material! Thank you for sharing, It concatenates a lot of concepts! A big fan here, hello from Brazil.

Thanks a lot, for the amazing lectures.

Outstanding video lecture.

A very good explaining. Thanks

Amazing Video, thank you!!

Amazing content, please keep it up

Your lectures are morning breeze in summer.

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.

I guess i will fall in love with the math with your lectures😊😊

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.

@Freeball99

2 жыл бұрын

What you have stated is pretty much a description of D'Alembert's Principal.

That's great. Could you please upload such an explanatory videos for LU Decomposition and Newton raphsons method ?

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.

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

@Freeball99

Жыл бұрын

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.

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.

@navsquid32

4 ай бұрын

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.

@Freeball99

4 ай бұрын

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.

Hey, truly insightful videos, loved it. Can you suggest any book to read and know about the history of scientists

@Freeball99

9 ай бұрын

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".

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)?

@Freeball99

2 жыл бұрын

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.

@shahramkhazaie

2 жыл бұрын

@@Freeball99 Thank you very much for your answer.

second equation of 2 is it sigma 12 or sigma 21? awesome work neatly explained. big fan❤

@simeon7450

3 жыл бұрын

the Cauchy matrix is symmetric, therefore sigma 12 = sigma 21

Great content! There is a typo at 2:29, second equation first term index should be 2,1.

@Freeball99

Жыл бұрын

Yep, you're correct. Although due to the symmetric nature of the stress tensor, σ12 = σ21. Thanks for catching that.

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!

@Freeball99

5 ай бұрын

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".

Good morning, please do you have notes on derivation of thin plate equation using total potential energy?

@Freeball99

2 жыл бұрын

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.

@ 3:47 I don't remember you saying anything about virtual displacement. How can it be used as variational?

@Freeball99

3 жыл бұрын

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..

Bro which is the name of the font appearing on previews of your videos?

@Freeball99

3 жыл бұрын

It’s called Trocchi - a Google font.

At 2:03 I believe that should be del (sigma_21) / del (x_1)

@Freeball99

3 ай бұрын

σ_12 is equal to σ_21 due to the symmetry of the stress tensor.

Hi, I've a doubt concerning the strain energy density there should be a 1/2 in front of it??

@Freeball99

3 жыл бұрын

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

@francescoindolfo

3 жыл бұрын

@@Freeball99 Ok 👍 Thanks for your quick reply!!

@5:48 it is the divergence of a Vector not the gradient no?

@Freeball99

2 жыл бұрын

Yes, I misspoke; it's the divergence (which is why its called the Divergence Theorem).

Is the principle of minimum potential energy the same as Castiglianos first theorem?

@Freeball99

Жыл бұрын

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.

Are expressions 13 and 14 valid in a non linear case?

@Freeball99

Жыл бұрын

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.

I fail to understand how differentiation of Uo wrt to epsilon ij . Delta Epsilon ij equal to delta Uo

@Freeball99

2 жыл бұрын

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

@Freeball99

6 ай бұрын

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 ...

@Freeball99

8 ай бұрын

These derivations do get very mathematical. Perhaps I will solve a problem using the principle and is will be easier to follow.