A discussion of blade sharpening for specific types of cutting requires an understanding of the slice cut vs the push cut. Cutting with a bladed tool can involve both motion perpendicular to the blade; pushing the blade through the material, and a slicing motion parallel to the blade. Occasionally the terms push and pull are used to describe slicing motion away from and towards the user; however, here we use the term push-cut to specifically refer to motion without slicing. Examples of push-cutting are shaving with a razor or planing wood. A common example of slice cutting is cutting bread with a serrated knife.
Mechanical cutting, or separating, is a very broad and complex subject, covering both the mechanical properties of the material being separated (toughness, fracture, deformation, etc) and the mechanics of the method of separation (scraping, shearing, grinding, piercing, etc). One of the most extensive overviews of this subject is the book “The Science and Engineering of Cutting” written by Tony Atkins (Professor Emeritus of Mechanical Engineering at the University of Reading).
In this article, I will use a paper written by Professor Atkins (Cutting, by ‘pressing and slicing,’ of thin floppy slices of materials illustrated by experiments on cheddar cheese and salami) as a jumping off point for the discussion. Unavoidably, this involves a bit of Mathematics, but I will summarize the key ideas as best I can.
The paper begins by defining the parameter “slice/push ratio” as the Greek letter xi. If we imagine cutting a steak with a knife held parallel to the plate, xi is the ratio of the speed of the knife in the horizontal (slicing) direction to the speed in the downward (pushing) direction. For a pure downward “push-cut” the ratio is zero. Since the pushing and slicing motions occur simultaneously, we can also define this by the displacement in each direction; for example, if the blade moves 10 cm in the horizontal for every 1 cm of cutting depth, the ratio, xi, is 10:1. It is common experience that it is easier to cut a steak with a slicing motion than with a straight downward push, even with a non-serrated blade.
To quantify the “effort” or ease of cutting requires calculation of the energy expended. In Physics, we define this effort or energy as the “work” done. Work is simply the force multiplied by the distance traveled. For example, lifting a 1.02kg requires a force of 10 Newtons (gravity exerts a force of 9.8 Newtons/kg and Newton is the unit of force). If that weight is lifted 1m off the ground, then the work done is 10 Newton-meters, or 10 Joules, where a Joule is the unit of energy, and a Joule is equivalent to a Newton-meter). It should be understood that work requires motion, so for example holding a heavy weight is not work, but lifting that weight from the floor to a table is. In the case of lifting the weight, that work-energy is stored as “potential energy” than can be released by lowering (or dropping) the weight.
In the simplified case, neglecting such things as friction and wedging of the blade, the force required (V) is proportional to the length of the cut (w):
V=Rw
where R is the fracture toughness. For example a 10cm wide cut requires twice the force of a 5cm wide cut, as expected. The “tougher” the material, the more force required to cut it.
There is a tremendous problem with this calculation, in that the sharpness/keenness of the blade is not considered. In the research of cutting, it is typically assumed that the blade causes a crack that propagates in front of the apex, the apex doesn’t contact the material, and keenness plays no role, provided the blade is not extraordinarily blunt. Or, possibly keenness only plays a role in initiating the cut. Almost anyone reading this article will know this is nonsense; that a keen knife cuts with minimal effort or that a fresh razor blade shaves better than a used but still very sharp one. However, this is the nature of Research and the Scientific Method, starting with a simplified idea/model/picture and iteratively determining what is missing, or where the model fails, and then add the missing or inadequate parameters in the next iteration. In this simple model, the sharpness is “contained” in the fracture toughness, so this calculation is OK for a given blade of given sharpness. The blade sharpness isn’t necessarily neglected, it’s just hidden in the fracture toughness parameter. A different, sharper blade would have a lower effective fracture toughness for the same material. An alternative way to understand this is that the “fracture mode” for a very keen cutting blade may be different from the fracture mode for a dull one, and each mode has a different fracture toughness. For example, paper can be separated by cleanly cutting individual fibers, or by pulling the fibers out as in tearing. In any case, simplified models and calculations like these can still be very informative, provided we identify their limitations.
A second problem is that the calculation doesn’t consider any advantage that may be realized due to texture along the blade, such as serrations or saw teeth. Nevertheless, this calculation will show that there is already a mechanical advantage to slice-cutting even in the absence of serrations.
The calculation essentially defines “fracture toughness”, R, as the ratio of the force required to cut (V) and the width of the cut (w), so V=Rw. The higher the fracture toughness, the more force required; the narrower the cut, the less force required. The calculation is only valid for describing how a blade of some particular geometry and sharpness will behave at varying slice-cut ratios.
The result of the calculation, is displayed by Prof. Atkins in his figure 3, reproduced below. The figure shows that the vertical (or downward) force V decreases rapidly as the slice-push ratio increases, while the horizontal force increases to a maximum at a slice-push ratio of one and then decreases. Obviously H (horizontal force) must be zero when there is no horizontal motion, as for a pure downward push cut. As an example, for slice-push ratio of 3 (say cutting a 1 inch thick steak with 3 inches of horizontal travel) the vertical force is reduced to 1/10 of that required for a push cut while the horizontal force is 3/10 of that nominal force required for a push cut. The important point here is that the total work (or effort) is the same the same as for the push cut; instead, we do 1/10 of the work (moving 1 inch at 1/10 the nominal) in the vertical direction and 9/10 of the work in the horizontal direction (moving 3 inches at 3/10 the nominal force).
In practice it is more likely that the user would apply a fixed downward force, insufficient to push cut, and then increases the slice ratio until the blade penetrates. Essentially finding the critical slice-push ratio for a given downward force V. There is an added complexity, not considered here, that initiating a cut may involve a different mechanism or effort than propagating the cut.
In this model, the fracture toughness, R, has units of energy per unit area (Joules/square meter), so the energy required to separate the material is determined entirely by the cross-sectional area of the cut. Neglecting friction, and energy that might be lost to distorting the off-cut, the total amount of work done by the blade is simply equal to the separation energy. What this means is that (within the model) slicing doesn’t reduce the amount work, but rather allow us to do the same amount of work by applying a lower force over a longer distance, just like a lever allows us to lift a heavy weight by applying a smaller force over a larger distance.
Now, where this gets interesting is when we consider the two components of cutting individually; with the slice cut potentially having a different microscopic mechanism of separation from the push cut. For example, when cutting paper, a keen razor-like blade will cut the individual fibers while a textured blade edge will tear or pull out the fibers. Both of these mechanisms may occur simultaneously.
As an extreme example, consider the Nogent micro-serrated knife shown below (designed never to need sharpening). This knife will slice paper, not by severing individual fibers within the paper, but instead by catching and ripping them apart.
Contrasted with with an image of paper cut by a keen knife or razor, where the individual fibers are cleanly cut, leaving a straight edge.
These can also be compared to paper cut by scissors, where a straight edge is produced by shearing and crushing the fibers. Each of these separations involves a different microscopic mechanisms, and likely different separation energy or fracture toughness.
Returning to the cutting of our steak, we know that the addition of slicing motion alone is insufficient when using a dull, non-serrated knife. No amount of sawing action with a butterknife will cut through through a tough piece of meat. However, even the notoriously dull restaurant steak knife is able to tear through the meat thanks to its serrations. The question is how do those serrations (or other non-uniformity) reduce the effort of cutting, beyond the simple mechanical advantage described earlier? Part of the answer is, as shown above, through a lower effort form of separation on the microscopic level.
In summary, there is an inherent mechanical advantage to the slicing action of a blade, and further advantage may be achieved through a non-uniformity of the the blade edge.
