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

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.

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.

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 KZfaq 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. kzfaq.info/get/bejne/ea10rcyI1Kupl6M.html

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

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".

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".

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

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.

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

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

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.

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..

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

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

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.

It’s called Trocchi - a Google font.

Which equation or what point in the video are you referring to?

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.

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