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Suppose I have a concrete wall in front of me and I push on it, now it seems that I do work and the energy has magically disappeared.

I am trying to figure out how this works, the explanation I thought for it is that the wall compresses just a little (similar to how a spring does) and my energy gets stored in increasing the energy of the structure of molecules of the wall. Now, the molecules want to go back but I simply can not observe them because the magnitude of the oscillations are that low.

For the above analysis to be correct, I need that that all bodies are necessarily compressed by some amount when put under a load. Is this statement true? In sense has there been studies on this?

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  • $\begingroup$ Your energy did not disappear if you push a fixed wall, it stores the energy as a moment, M = F*h. If you push a free-standing wall, the wall will move or tilt (displacement), now your energy has done the "work". I think you are overthinking in trying to tie the phenomenon of solid mechanics to theromodynamics. $\endgroup$
    – r13
    May 26 at 16:49
  • $\begingroup$ Every solid has a modulus of elasticity, but it might not give you back the energy you put into it as efficiently as a spring does. Also let's make a point of "work" vs "energy expended" ... no displacement, no work. Forget "person vs wall", too complicated. Try "brick sitting on a piece of rubber", with gravity pulling down... After the rubber compresses (storing some finite amount of energy which it will give back), the brick is still pushing exactly as hard, forever. But no more energy expended, and no more work done. $\endgroup$
    – Pete W
    May 27 at 2:07
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TL;DR: Yes, any structure deforms if you put a load on it.

Even adding an ant on top of a granite mountain will change (lower) the height of the mountain - imperceptibly so but it will still change it. The problem is that its not possible to measure it.

That is the whole idea behind Young's Modulus (modulus of elasticity).

Essentially, all materials behave as springs (at some level). One might argue that the forces between molecules in the material are acting as springs.

enter image description here

Figure : source www.tf.uni-kiel.d.

The energy when you are pushing against the wall is stored in the potential energy of those "springs", so when you remove the acting force, the wall will "bounce back".

more formal explanation (not quite)

This is my interpretation (apologies if this does not fall under all material cases).

When the atoms of solid are arranged in space (lets assume a lattice structure like the above example), the distance between the atoms is such so that the potential energy is minimal. The energy between the atoms follows a relationship like the following (this is for the molecule of Hydrogen but in principle a similar exists for crystallic lattices):

enter image description here

Figure 2: A Plot of Potential Energy versus Internuclear Distance for the Interaction between Two Gaseous Hydrogen Atoms. (source: libretexts

The atoms want to settle at the lowest energy point. If you compress/pull them apart then you quickly get a response back, which is the force. The more you pull/push, the higher up you go to the potential well and the higher the force.

Additionally, if you remove the external stimulus (the force you are applying), all that potential energy quickly returns the material to the "resting state".

(That doesn't explain plastic deformation, but its beyond the scope of this question).


Where does the extra energy go

Only a small part of your energy you are expending when you are pushing on the wall is stored as elastic energy.

The rest of the energy that is being burned away, is converted to heat from you body which is trying to maintain the force.

I guess an example that can help you understand that, is if you try to hold a bottle of water on the palm of your hand. You can try it in two ways

  1. like the following image (arm extended)

enter image description here

Figure 3: bottle in hand not supported (source: saltysoulsexperience.com

  1. everything is the same apart from the fact that you can support the back of your hand on a table.

You will be able to sustain position 2. a LOT longer, and the only reason is that the muscles will not need to contract as much in the second case.

excellent resource


An excellent book about this is Prof. Gordon's "The New Science of Strong Materials: Or Why You Don't Fall through the Floor", although it has only a handful of equations if you are interested, and its almost bed time reading (or sometimes a bit more than that).

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    $\begingroup$ This seems to me like an obvious statement that needs to be written as an axiom like the second law of thermodynamics which says you can't get a 100% efficient. So, are there any such statements for the question title? $\endgroup$
    – Buraian
    May 26 at 9:41
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    $\begingroup$ @Buraian To me, that just sounds like you are asking for a list of ideal concepts and the logical conundrums that arise from such ideal concepts. $\endgroup$
    – DKNguyen
    May 26 at 16:26
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    $\begingroup$ @Buraian If you are interested in this sort of thing you should probably study continuum mechanics, not structural engineering. Reminds me of a true story: a physicist was hired to do an engineering job, and needed to find how much clearance was needed for thermal expansion of some component. After a week working on the problem and getting nowhere, the rest of the team were not amused to find he was trying to calculate the expansion from first principles using the chemical composition and atomic structure of the steel alloy, instead of just looking up the thermal expansion coefficient. $\endgroup$
    – alephzero
    May 26 at 17:56
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    $\begingroup$ @Buraian Yes, there is, but it was only worked out a few years ago. It turns out that there is a minimum strain rate that is governed by General Relativity considerations. The bound is millions of times stiffer than any known material, but still, the current model of the universe requires a nonzero amount of strain in order to be consistent. $\endgroup$
    – Phil Sweet
    May 26 at 21:26
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    $\begingroup$ I think ti maybe worth to promote your comment into an answer @Phil Sweet $\endgroup$
    – Buraian
    May 26 at 21:45
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It hasn't disappeared. You have made your hand a little warmer because your own tissue has deformed. Your hand will return to its original shape and the generated heat will be moved to the environment as radiation and by convection. If you tried too hard you could see and remember the consequences at least a while.

The concrete generally happens to be much stiffer than human flesh. The mechanics says that your hand and the concrete get equal and opposite force when you act. I'm afraid if the piece of concrete is heavy enough it doesn't move and your hand is the only participant which has a possibility to get visible changes when the force is not beyond the usual human capabilities.

Actually your own analysis was quite right. Sensitive instruments might see that there's some deformation also inside the concrete - even vibrations. Vibrations will die sooner or later due the friction and acoustic radiation to the environment. I guess the measurable effect is temporary and vanishes (=elastic) as soon as you ease your push - except in case you happen to be The Superman.

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When you push the concrete wall the following happens.

  • your body starts to firm up from the major muscle groups around your back and waist core and gets ready for heavy lifting. Some muscles just tighten your waist and push the disks on your lower backbone tight together causing realignment on them.

  • from there a complex hierarchy of neurons shoots orders down to your feet and shoulder and hand to exert force while balancing the entire action.

  • your feet will push the floor material down and your hand will push the concrete wall in. The muscle near your waist are now loaded to the max and the path of force travels from your feet carrying the reaction of the floor all the way up to your fist. Creating a lot of heat.

  • This will cause small deflection in the cement of the wall and the floor and compresses all your joints closer.

  • while you increase the pressure, the wall and floor even your body will vibrate.

  • you can easily verify this vibration by slapping the concrete wall, the vibration of the cement surface will issue a loud crashing sound.

Conclusion

Nothing is absolutely rigid and any material no matter hough hard under any small load will deform.

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