As explained in Electric hammer and Hammering toolhead, a hammer is a simple machine, although Archimedes and Galileo were unaware of this because they lacked the modern concept of kinetic energy; its mechanical advantage is limited only by the fact that its impact is not instantaneous.
It turns out that it is possible to substantially improve this mechanical advantage.
As impact time approaches zero, the force and pressure of the impact approach infinity. Moreover, even during the impact, the deceleration of the hammer varies; as the deceleration approaches infinity, so does the force and pressure.
Thus you can beat an ice cube with a massive broomstick without damaging it, but a light whack with the bowl of a metal spoon can shatter it; and the tempered glass of a car window can withstand a baseball bat impact, but not a crackhead throwing a piece of sintered-aluminum-oxide spark plug at it.
Once the impact begins, the resulting deceleration does not affect the entire hammer at once, nor does the force affect the entire workpiece at once; instead, they travel through these bodies as sound waves. These waves reflect from other surfaces of the bodies in the same way that electrical waves reflect from the ends of transmission lines or sound waves reflect from the open ends of pipes, and may be focused or scattered in different parts of the solids. As a rule of thumb, the transmission line or hammer behaves like a rigid object or lumped-element node only over timescales about ten times the round-trip time of waves through it.
These sound waves are somewhat more complex than everyday sound waves. First, they involve both shear waves and compression waves, which travel at different speeds, each of which can produce the other under some circumstances. Second, their magnitude is often large enough to cause the usual linear approximations to break down; in particular, if there is a region within which the material stiffens with increased deformation, this can cause higher-amplitude waves to travel faster than, and overtake, lower-amplitude waves, and this can result in the propagation of self-sustaining discontinuous wavefronts through the material — in sound waves in air we call this a “shock wave” or a “detonation”, and a similar phenomenon in water channels is a “flash flood” or, on the beach, a “breaker”. Except when playing a percussion instrument, normally the point of using a hammer is to provoke some kind of nonlinear response from the thing you are hammering on, such as breaking or deforming plastically, so these phenomena are actually quite common in hammering.
Additionally, unless the impact is deep inside a hole in the workpiece, surface acoustic waves represent additional modes of vibration that may or may not carry significant energy dissipation.
One obvious approach to shortening the impact time and thus increasing the force is to shorten the distance from the impact side of the hammer head to the back side, shortening the timescale over which it behaves like a rigid object. Taken to the logical extreme, this suggests using a hammer that is just a thin, flat sheet of material, which is brought into contact with the workpiece at the same time over its whole surface.
One use of this approach is the “slapper foil” detonator, which does in fact work, but other examples include flat paddles, slappers, or leather spanking straps used to punish children or excite masochists. These enable the development of sufficiently high impact forces to cause localized tissue damage despite low energies (typically on the order of a joule) distributed over large areas (typically on the order of 100 J/m2). Even more prosaically, the same strategy is used for open-hand slapping, which the humans commonly use as a form of symbolic aggression to induce pain without any danger of more serious damage; and it is a factor in the effectiveness of spoon shattering of ice as well.
One difficulty of slapping things with paddles is that, in air, the large surface area of the paddle tends to accumulate an area of high-pressure air in front of it, soaking up most of the input energy. This can be desirable if the objective is merely to make a loud noise, but if maximal impact force is the objective, it is a drawback. Strategies to reduce this problem include putting air holes in the paddle, dividing the paddle into many strands (a “flogger”), or using a whip rather than a paddle.
During the impact, wavefronts of deformation are traveling through the workpiece as well as the hammer, and if the materials have reasonably similar acoustic impedances, they can even pass back and forth across the hammer-workpiece interface rather than being entirely reflected. Their particular behavior in the workpiece material depends on the properties of that material; for example, when a human is being slapped, I think much of the slap energy is dissipated by viscous and plastic behavior in the immediate neighborhood of the impact, so there is no tissue damage even millimeters away, while when using a bent wire to tap glass that has been scored on the other side, the stresses induced by the propagating wave are high enough to provoke the formation of a crack from the score.
(I don’t really know if that’s the reason slapping humans causes no damage except at submillimeter distances from the impact. Possibly dispersion also plays a role.)
The direction in which each wavefront travels through the workpiece depends on the speed of each sound in the workpiece material and on the relative phase of the impact at each part of the workpiece surface; since the speeds of sound in solids are typically around 1km/s and the speeds of movement of hammers are typically around 1m/s, even fairly small rotations between mated hammer and workpiece surfaces can result in fairly large differences in propagation directions. Consider a flat hammer face that is 10mm across striking a flat workpiece at 1 m/s with a compression-wave speed of sound of 1km/s with one milliradian of deviation from parallelism. So, when one side of the hammer face strikes, the other is still 10 microns away from striking; the contact area spreads across the hammer face gradually, as the hammer and workpiece deform, until reaching the opposite edge of the face 10 microseconds later. In those 10 microseconds, the waves from the initial impact have propagated 10 mm into the material, resulting in a planar wavefront propagating at a 60-degree angle from the surface normal, rather than in a nearly normal direction, as you might expect. The shear (or transverse or S) waves will propagate in a more nearly normal direction, since they propagate typically about 40% slower.
Since the “propagation time” of the contact area across the material is on the order of the speed of sound in either material, we should not expect the initial impact to alter the conditions much for later parts of the impact.
By shaping the hammer face to time different parts of the impact relative to one another, we can change the propagation characteristics of the wave in the material. For example, a convex hammer face, like the bowl of a spoon, or the traditional spherical blacksmithing hammers of Dablo, Burkina Faso, shown in Christopher Roy’s documentary “From Iron Ore to Iron Hoe”, will tend to produce a spherical wave that propagates outwards in all directions in the workpiece from a common center, and inwards toward a focus inside the hammer, like the convergent ultrasound waves used in sonoluminescence. A concave hammer face will tend to produce a convergent wave within the workpiece, but as noted above, small misalignments in the hammer angle will produce large deviations in the direction in which the wave propagates.
You might think that a potentially more predictable way to focus waves within the workpiece might be to shape the workpiece surface to be concave instead, but that just makes matters worse; the same small angles of hammer misalignments will still produce the same large angles of deviation, and additionally now lateral translation of the hammer blow by similar amounts will also produce those same deviation angles.
A better approach is to use three or more small “outriggers” sticking out of the sides of the hammer face to rotate it into submicron alignment with the workpiece surface by elastically deforming over hundreds of microseconds before the main impact. This approach can also compensate for some degree of imperfection in the workpiece surface itself, but not for micron-scale concavity or convexity over the impact area.
Given a precise depth map of the workpiece surface, you could use a phased array of hammers in, for example, a hexagonal grid, each launched toward the surface to arrive at a time scheduled with submicrosecond jitter; this should enable the formation of a precisely customized wavefront within the workpiece material.
I had thought that perhaps you could use independent outriggers on a swarm of independent hammers (perhaps swung together using some kind of compliant coupling) to do this depth-mapping and timing mechanically rather than in software; the idea was that the outriggers would come into contact with the surface first, with the microsecond-scale jitter imposed by micron-scale surface roughness and milliradian-scale alignment imprecision, and would act to retard the subhammers that would otherwise make contact first and/or advance the subhammers that would otherwise make contact late, all by a few microseconds. But I don’t see how to make that work mechanically.
Given a known hammer geometry, it should be possible to arrange for focusing of deformation waves at or near the hammer surface as well. For example, a convex hammer surface can produce an expanding spherical pressure wave inside the hammer upon the initial impact, with minimal dependence on the precise angle and position of the impact; if the back surface of the hammer is an ellipsoid with one focus at the virtual source of this wave, it can then reflect this spherical wave into a convergent spherical wave focused on a particular point or points. (See Caustics for notes on designing light-reflective surfaces to form chosen focused patterns of intense light; this approach is also applicable to these sound waves.) The material at or near this focus point will experience stresses enormously greater than the stresses it would normally experience from an ordinary hammer blow, perhaps by two orders of magnitude or more. This point can be chosen to be within the hammer, at the interface with the workpiece, and probably most usefully, some distance inside the workpiece, thus limiting the damage to the hammer.
To be concrete, suppose we are using a steel hammer with a Young’s modulus of 200 GPa (which Plastic cutters and Wikipedia claim is typical of steels), weighing 1 kg, swung at 10 m/s, striking a flat piece of steel, and the hammer’s circular striking surface is shaped to be precisely 20 microns higher in the center than at its edges, which are 40 mm apart. (At 7.9 g/cc, the hammer head then must average 101 mm long to weigh 10 kg.) If the hammer strikes straight on, the center of its face will strike 2 microseconds earlier than its edges do. Supposing the speed of sound in steel is 6 km/s, that means the center of the wave will have a phase advance of 2 microseconds and 12 mm relative to the edges. The 12 mm of steel compressed by 20 microns amounts to 1.7 millistrains and thus is under some 300 MPa of pressure. (XXX this is wrong but I don’t know how to correct it yet; see below.)
A naive, untutored hammer of these dimensions used in this way will convert its 50 J of kinetic energy in a relatively straightforward fashion into 50 J of elastic deformation, with the pressure wave and other waves bouncing back and forth through the hammer and workpiece, augmenting the pressure by some 300 MPa on each bounce, over a timescale of a millisecond or two, during which time the hammer has come to a stop and begun to rebound violently — an average deceleration of about 5 km/s/s with a peak of about 10 km/s/s, producing about 10 kN of peak force. However, this only amounts to about 8 MPa of pressure, so clearly I have miscalculated in a very significant way above; ASTM A36 steel actually has a yield stress of only 250 MPa, so 8 MPa is a lot more plausible than 250 MPa.
(This 10 kN amounts to a mechanical advantage of about 1000, depending on how much space you have to swing the hammer in: a constant 10 N accelerates the 1 kg hammer up to 10 m/s in one second, requiring 5 meters of swing.)
Suppose our super-hammer, with its ellipsoidal back face, can focus the initial impact wave to two orders of magnitude higher strains within the workpiece, or some 800 MPa. This is enough to provoke cold steel into plasticity, which will prevent the 800 MPa figure from actually being reached. A significant fraction of the impact energy will then be converted into heat within about a gram of steel inside the workpiece, heating it by some tens of degrees.
If it is possible to provoke fracture in the workpiece by this approach, which should be easily feasible for brittle materials, the available concentration factors should increase dramatically: the bubble of vacuum thus formed inside the workpiece would experience very large forces and temperatures as its walls crashed together on the following oscillation, again similar to sonoluminescence or to the collapse of the void when a stone falls into water or fluidized sand.
A particularly interesting situation is when the focus is at or near the opposite face of a flat workpiece, because it seems plausible that this mechanism could be used to pit it, eject material from it, heat-treat it, or cold-weld it to something else. This is a particularly appealing prospect for heat-treating inaccessible places like the interior surfaces of structural tubing after welding, although it might turn out to be impractical to achieve high enough concentration factors in ductile metals to reach the necessary temperatures.
Better hammer materials might include things like copper and tungsten, with their lower speeds of sound, or even some kind of lead-filled steel foam. Graded-acoustic-impedance metamaterials should make it possible to reflectionlessly couple sonic energy into the workpiece even from a hammer (or other transducer) with a quite different acoustic impedance.
Above I mentioned that one use of paddles is for exciting masochists. A device popular in this connection is a thing known as a “slapper”: a double-layered paddle similar to the slapstick used in vaudeville or by Arlecchino. Shortly after the initial impact, the second layer of the paddle slaps into the first, producing an additional painful impact (as well as a much louder noise).
A potential advantage of this construction is that energy is not lost during the hammer swing to the air resistance of the second layer; the air between the layers moves along with the hammer, so it only absorbs any energy during the impact, when it is compressed between the layers to some degree, which is what gives the slapstick its loud slap.
This could be extended to many layers, but without further elaboration, the layers will produce many separate impacts as they pile up on a pile on the workpiece, starting with the ones closest to the workpiece. This may be useful for producing some sort of vibration or recorded-sound reproduction, but by itself it will not produce any kind of unusual impact; dead-blow hammers filled with shot already do something similar.
What is needed for giant pressure is for the layers to collide very nearly simultaneously, but in the reverse order, starting with the layers furthest from the workpiece, with the compression wave propagating through each layer just in time for the layer to hit the next layer. For 1-mm-thick layers of steel, the ideal timing should be about 170 ns apart; with 100 such layers the overall sequence would total about 17 microseconds.
However, even if the impacts are simultaneous, this would only broaden the peak of the shock wave traveling through the stack of layers by those 17 microseconds, which is already about a factor of 50 to 100 better than a standard hammer, exceeding the physical limits of steel, though perhaps not those of some technical ceramics.
A variety of well-known pantograph-like linkages could be pressed into service for getting the stack of layers to uniformly expand and contract; moreover, they could be activated by the contact of “outriggers” with the workpiece surface, thus completing the stackup as the frontmost layer slams into the surface. The difficulty will be to get them to perform with enough precision to significantly exceed the impact force of an ordinary hammer.
By slightly thinning the sheets in their centers, the impact shock wave can be produced as a convergent spherical shock wave, focused as before on a chosen position within the workpiece. In this case, however, the precision of the hammer’s positioning and angle are no longer so important, because the focusing of the shock wave does not depend on them, but only on the relative positioning of the layers in the stackup.
How much would you need to thin them? With the 40-mm-diameter, 101-mm-long geometry suggested above, and 6 km/s propagation velocity, the rearmost sheet would need a phase advance of about 2 mm and thus 333 ns at its outermost edge, if the focus is near the front of the hammer. If a 10 m/s impact velocity were split up evenly among 100 inter-sheet gaps, it would strike the sheet in front of it with a relative velocity of 0.1 m/s, so we’re talking about etching it 33 nanometers deep in the middle to get a 330-ns phase difference. Other sheets would be etched progressively deeper, but we’re still talking about a very difficult level of precision to reach any meaningful kind of focus.
By perforating the sheets it should be possible to provide low-resistance air paths from the center of the stack to outside of it, thus reducing the loss of potential impact energy to air compression. Simple aligned round perforations would provide lengthwise channels, but elongated and partly-aligned perforations could provide diagonal and radial air channels, which would be shorter and thus lower-resistance for many hammer geometries.