I was wondering if it is possible to drive a nail through, for example, concrete by dropping a light object on the head of the nail over many iterations.
I.e. is there a certain threshold of force that must be reached for the nail to make even the slightest progress into the concrete? Or is it possible for a small force applied thousands of times to drive the nail?
Driving a nail through something, physically speaking, amounts to breaking crystalline, chemical or extra-molecular (e.g. van der Waals) structure: a certain material forms some sort of structure and you destroy it by passing a nail in the middle. Now, in quantum mechanics, we have the interesting phenomenon that not all interactions produce meaningful effects. For example: an electron in an atom will not react after being hit by photons with random energies - only the ones that have just the right ones will result in a quantum jump between levels. Since photons carry momentum, we can interpret a standing-still nail as being constantly hit by a huge amount of very tiny hammer blows, one for each photon that hits it perpendicularly to the surface. These tiny hammer blows are not enough to destroy the bonds in the material you want to nail, showing that there is indeed a [quantum] threshold that must be surpassed in order to drive a nail through some solid material.
My initial answer to that is no. This is because there is some maximum normal force $\vec{F_n}$ that the material can exert back on an object before it is deformed. Unless the object can exceed this $\vec{F_n}$, then it will not deform, and therefore not be driven through the material.
There are of course so further questions to be asked. If the nail has a wedged head, then it is "mathematically plausible" for a nail to have such a high slope that the mechanical advantage is high enough to drive through, but this is clearly unrealistic. There is also the chance that slight forces on the nail might create a form of erosion over time, but this isn't really the nail being "driven" so I'd settle on saying that no it's not possible.
You got theoretical answers, here's an experimental one.
I was wondering if it is possible to drive a nail through, for example, concrete by dropping a light object on the head of the nail over many iterations.
That's how rotary hammer drills work. The motor actuates a piston which pressurizes an air cylinder which propels another piston (the hammer) until it hits a steel rod which whacks the back of the drill bit. The reason why it uses a pneumatic system is to reduce recoil and vibration. The wimpy version (hammer drill) is not pneumatic and uses a pair of toothed wheels rotating against each other to move the chuck back and forth, which is much less efficient and vibrates a heck of a lot more.
The interesting bit is that we know the impact energy of a rotary hammer, usually it will be between about 1 Joule and 30 Joules depending on the quality of the tool and how heavy it is.
1 Joule is pretty small. It isn't the fastest rotary hammer but it will make holes. Hammer drills have much lower impact energy, maybe a tenth of a Joule, and they also make holes (slowly and with lots of noise though).
So it is quite possible to drive things into concrete with repeated applications of a small amount of energy, but there are limits.
Also there are several big differences between a drill bit and a nail.
When the pneumatic hammer hits the back of the drill bit, it creates a pressure wave inside the steel which propagates along the length of the bit and delivers the energy into the tip, which pulverizes a tiny bit of work material with each hit. The bit itself moves very little front to back, as any energy expended in accelerating it back and forth is wasted. The only energy that makes a hole is energy delivered at the tip. This pressure wave propagation thingy is why you can have a one meter long drill and make a one meter deep hole with it. If the machine had to wiggle it back and forth it would take forever.
Once the nail has entered the material however, it is squeezed inside the hole, and in order to go deeper it has to overcome this friction on top of crushing material in front and compressing it in order to go through. So static friction is going to be your problem. Once the nail is in, even a bit, if you hit it with not enough force to overcome static friction, you will compress the metal a bit like a spring, then it will bounce back to its normal length, but it won't go in deeper at all.
You can do this experiment easily with a nail and a bit of wood: drive a nail through until the tip comes out on the other side of the plank. Now the only thing holding the nail is friction, and if you don't hit it hard enough to overcome friction, it won't move at all. You can even measure the static friction experimentally, using the favorite tool of any theoretical physicist:
So the reason why I bothered to explain about drills is that a drill bit evacuates material out of the hole, which solves the friction problem, but a nail does not do that. With enough patience, you could probably make a hole with a drill bit using really tiny impacts, as long as each impact removes at least one atom/molecule of material.
So, for a drill the limit would be how much energy it takes to break either covalent bonds or Van der Waals forces, depending on what holds the material together. For concrete, it will be covalent bonds inside the crystalline aggregates (ie, rocks) so that's pretty hard. I'd say ultrasonic drilling would be a good illustration of "lots of tiny shocks".
And for a nail the limit would be static friction.
That's why a nailgun for concrete uses one big explosive charge (it is quite spectacular) while the optimum strategy for drilling is lots of repeated impacts.
Bonus material:
The drill bit (or the nail) has to be harder than the work material. Drills for concrete use tungsten carbide tips. Concrete nails are made of hardened steel so that does the job for a single-use nail, but hardened steel is brittle, so they will chip, break, and send pointy razor sharp bits flying. Always wear eye and ear protection!
Also, ancient Egyptians machined granite with copper tools. That's not possible because copper is very soft. They did it by mixing in some fine sand (probably silicon carbide) which embedded into the copper tools and ground the granite. It was quite a labor intensive process.
Macroscopic: Driving a nail forward amounts to non-elastic deformation of the material. Generally, (almost) every material has some limit of force that causes only elastic deformation and you have to exert a greater force in order to make it deform non-elastically.
Then again, the above is an approximation. No deformation is ideally elastic even if the material is engineered to be highly elastic (that's why springs break after some use).
So, keep hitting.
One can think that there is a minimal energy needed to crack a single chemical bond. Even then, a thermal movement of the particles may supply the exact amount of energy to your bond minus the amount you apply by hammering so in general you make the "thermal decomposition" at the place of the tiny hammer hits more probable than elsewhere. That means, keep hitting - even with an energy less than a single chemical bond you can advance after enough hits. A great number of hits, actually.
While others have answered the questions here is a simple matter:
The strength of the nail must be far greater than the material into which it is being driven. The force applied to the nail must exceed the molecular cohesion and overcome the friction of material in contract with the nail.
The modulus of elasticity of the business end of the nail must be far greater than the plastic deformation limits of the material into which it is being driven that it continue to maintain its shape and strength.
There is an optimum shape to the nail point according to the material into which it is being driven. The sharpness angle of the point will determine the efficiency of the force required to penetrate the material.
So long as the force applied to the nail surface area exceeds molecular bond strength over the length of the nail, it should work as long as the nail used has higher elastic deformation than the plastic deformation limit of the material into which it is driven. The item used to drive the nail ought to be stronger than the nail or it will not complete the job.
The following is not totally what was asked about (unless you have an eg Iron or Aluminum surface) but is relevant enough to be useful:
Ferrous metals have "fatigue limits" - many* others do not.
This means that there is a limit below which you can deflect a ferrous"beam" repeatedly and never cause fatigue failure - it acts as a "spring" without destructively absorbing the deflection energy. Above this limit damage occurs which will lead to ultimate failure.
Aluminum, as probably the most commonly encountered example, has no lower fatigue limit. Any practical deflection** will result in some permanent damage and contribute to ultimate failure.
So eg a suitably corrosion resistant load bearing hook can be designed such that it will still be able to support a load below it's design limit after 2000 years of continuous use. An Aluminum hook can be designed to last 2000 years with N load applications at some maximum limit but it is progressively failing at each load application.
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*I said "many" rather than all as there may be others I am unaware of that behave similarly to iron. Iron and Aluminum are "good enough" examples in this context.
**"Any practical deflection" -> There may be an atomic scale limit. For real world applications there is no lower limit.
Assumptions:
In order for the nail to move forward, it needs to be pushed with a sufficient force. If you were using a heavy hammer, this is how that would work: You accelerate the hammer towards the nail, i.e. store kinetic energy in the hammer head. That kinetic energy is then transferred to the nail, thus pushing the nail into the material. The material pushes back, so the nail will compress a little.
The easiest way to visualise that is by imagining the nail as a (fairly stiff) spring: The hammer hits one end, compresses the spring from the "head" end, and on the "foot" end of the spring, we can measure a force that rapidly increases while the hammer is being slowed down, peaks when the hammer has been stopped, and then reduces again while it bounces back.
To decide whether the nail progresses at all, we need to look at the peak force during the bounce. If this is below the threshold for the concrete, we're not getting anywhere. If it is above, then as soon as the threshold is reached, the "foot" end of the nail will start to move into the concrete, thus doing useful work. This means some of the kinetic enrgy from the hammer will be used to enlarge the whole rather than just bounce the hammer back. This also means that the theoretical peak force won't actually be reached because the concrete gives way before it gets there.
So, what decides how high that peak force is, i.e. whether we reach the threshold required to drive the nail forward? Going back to the spring analogy: The peak force depends on how far the spring is compressed, and that in turn depends on two factors.
Think of hitting a trampoline with a hammer, to drive it into the ground: The hammer slows down very gently while the nail compresses very far. if you plotted force over time, it would take a long time to stop the hammer, and the forces would be very low. With a very stiff nail (think of a thick, straight metal rod), there's almost no compression, and the hammer is stopped abruptly, so the peak force is huge. This is why nails are usually straight metal rods, and why people prefer bouncing around on trampolines :)
Note: A critical look at the assumptions above shows that really we should be talking about the combined stiffness of the material/nail/hammer combination. Any elastic deformation makes reduces the peak load on the nail, as anyone knows who's tried to drive a nail into a thin, free-hanging plank of wood. -- I'll ignore this for now since the question was about nails in concrete.
Momentum is mass times speed. So if you double the mass of the hammer (while keeping the springiness of the nail constant), it will slow down the hammer at half the rate, meaning it will compress the nail twice as far, thus giving you twice the peak force. If you double the speed of a hammer, it will need twice as much slowing-down before it stops, so it would also compress the nail twice as much.
In very general terms: The peak force is proportional to nail stiffness times hammer mass times hammer speed. So to find out if the nail can be driven, you could start with the weakest hammer blow that would still drive the nail just a tiny little bit, and in proportion to how much lighter it gets, you have to either move it accordinly faster or stiffen the nail proportionally. In practical terms, steel nails don't get much stiffer unless you make them thicker, but that would increase the driving force they require. Maybe a different material could help, but there are no huge gains to be made, even with very expensive materials. This leaves the "hammer" speed. For every halving of the mass, you'd have to double the speed. This means "dropping" small objects wouldn't work unless you drop them very far, and their speed is not limited by aerodynamic drag. Hitting the nail with fast-moving small objects could work, of course.
"hitting it with small, fast-moving objects" ... if you've ever used youtube, you'll have noticed slowmo videos of stuff being hit by bullets. And what they tell you is that some of the assumptions at the start of this post don't really hold up once you reach certain speeds:
You could mitigate the last problem by having a very lightweight stiff material (carbon fibre? ceramcis?) for the nail, with maybe some sort of impact protection at the head, but now we're getting a little bit crazy. On top of this, the stiffest nail in the world won't help you if it then turns out that the material you're driving through is suddenly less stiff than your new improved nail, and becomes able to absorb the momentum from the bullet by elastic deformation, while using the supernaturally robust nail to evenly distribute the impact across the contact area with the nail ... In that case, you should maybe just fire the nail directly at the material, but then you'd have to get it all the way in with a single blow, which needs extra momentum...
There is a lower threshold for how much momentum a blow needs, based on the elasticity of the system (nail, material, hammer/bullet), in order to make any progress. You can compensate for a lighter hammer by moving faster, but there's a limit to that where the impact can destroy the nail. You can also do a little bit by stiffening the nail/hammer/material but that's not going to get you very far without having to go to very impractical lengths.
I did not mention, of course, that this becomes a lot easier if the material you're driving the nail through is very soft, which lowers the momentum threshold, and reduces the load which the nail must be able to bear. So driving a steel needle into some balsa wood might well be possible by droping pennies onto it, although you'd need really good aim :)
Hammer is almost always able to transfer phonons, (this threshold is significantly lower than chemical bond energy).
To drill a chunk of concrete, put it in a thermally sealed container and use a non-conductive nail with a higher melting temperature.
Hit a nail, until heat dissipated in concrete melts it.
Disadvantages:
Ultimately, yes, you do need a certain minimum force. The best way to understand this is to think about what happens at the molecular scale.
"Driving in a nail" requires some of the material to be split apart and pushed aside. To do that, you have to break either intra- or inter-molecular bonds (which, depending on the kind of material you're trying to press the nail into may be any mixture of covalent, ionic, or dispersion ["van der Waals"] bonds). Hence, your question in the end amounts to "can an arbitrarily small force break a molecular bond?"
And the answer to that is no. A bond is nothing more than force exerted between atoms or molecules that attracts them together, and to break it, you have to counteract that force, which means applying at bare minimum an equal and opposite force. If you apply less force than this, the bond will stretch, but not break, because the attracting force will still win.
On the macro scale, stretching, but not breaking, the bonds is essentially elastic deformation. Breaking some bonds bot only a few, so as to allow internal parts of the material to slide around while not pulling away entirely, is plastic deformation (a permanent change of shape but still retaining continuity). Cleavage, or fracture, of material (where the material separates into two or more separate pieces) is when the bonds are broken completely.
Penetration of the material implies localized cleavage to create the hole, followed by plastic deformation of the surrounding material to widen it, and hence there must be enough force available at and around the tip to break all relevant bonds. If, instead, you tap the nail very, very lightly, there will only be elastic deformation: in effect, the material, no matter how hard it may seem, will simply flex up and down like a trampoline. You might not be able to see that with your eye for "hard" materials like wood or concrete, but that is still what is going on. You can super-light tap it as many times as you like, but the material will just keep flexing - even if you tapped it so many times that the total energy you expended in the tapping equalled that which could drive it in were it applied all at once.
In more general terms, both energy and power - that is, the rate of energy delivery - count when asking about damage. It's why that you can sit in the sunlight during daytime for, say, 100 seconds, and over that time maybe about 100 kilojoules will hit it (assuming you present $1\ \mathrm{m^2}$ to the Sun and you get near the full Solar constant so roughly $1\ \mathrm{kW/m^2}$ of radiant flux), but if you received that same 100 kilojoules in 1 second ($100\ \mathrm{kW/m^2}$, which is comparable to the open maw of a blast furnace or very close to a fire storm, I'd think), you'd suffer extreme - likely deadly - burns. The same principle applies with mechanical energy as well, as here.
