Kvant Physics Problem 176
The system consists of a rigid tool, such as an awl, nail, or knife, interacting with a deformable material.
Verified: yes
Verdicts: PASS + PASS
Solve time: 11m22s
Source on kvant.digital
Problem
Erroneously published in the journal under number 166.
Why is it easier to pierce a hole with an awl if the awl is rotating? Why is it necessary to twist a nail in order to pull it out of a wall? Why, when you cut bread or meat, do you move the knife back and forth, whereas when cutting cheese you only press on the knife?
A. G. Kosourov
Setup and Assumptions
The system consists of a rigid tool, such as an awl, nail, or knife, interacting with a deformable material. Penetration or extraction is produced by an externally applied force. Resistance arises from pressure of the surrounding material on the tool surface, friction at the interface, and, in many materials, geometric interlocking of surface irregularities and fibers.
The normal pressure on a small surface element of the tool is denoted by $p$, the area of that element by $\mathrm dS$, and the local friction force by $\mathrm dF_f$. Coulomb friction gives
$$\mathrm dF_f \le \mu_s p,\mathrm dS$$
for sticking contact and
$$\mathrm dF_f = \mu_k p,\mathrm dS$$
for sliding contact.
For cutting, the material separates when the local stress near the cutting edge exceeds the material's fracture or yield strength. The force required depends not only on the magnitude of the applied force but also on how strongly that force is concentrated near the edge.
Physical Principles
Static contact permits the development of large tangential stresses over an extended region of contact. If relative motion is introduced, the locations of maximum stress continuously change, local sticking regions are destroyed, and interlocking asperities are successively broken rather than loaded simultaneously.
A sharp cutting edge acts by producing very large stresses in a small neighborhood of the edge. If the blade has both a normal velocity component and a tangential velocity component, the tangential motion generates additional shear stresses along the cutting line. The material is then separated progressively, point by point, instead of being compressed and fractured along the entire width at once.
Derivation
For a stationary awl pressed into a material, the applied force is directed mainly along the axis of the awl. The material exerts pressure on the conical surface of the awl, and this pressure has a resultant directed opposite to the motion. Friction on the lateral surface contributes to the resistance, but penetration is not governed by a balance of the form
$$F=\mu_s N.$$
The relevant point is that, when the awl does not rotate, each surface element of the awl remains in contact with the same portion of material. Tangential stresses and mechanical interlocking can build up simultaneously over the whole contact region.
When the awl rotates with angular velocity $\omega$, a surface element at radius $r$ acquires tangential speed
$$v_t=\omega r.$$
Each point of the material is then contacted only briefly before another point of the awl arrives. Local sticking zones are continually broken and reformed, and surface irregularities are sheared away successively. The contact resistance is distributed over time rather than accumulating over the entire contact area. Rotation also produces circumferential shear stresses that assist the material in yielding around the tip. The required axial force is consequently smaller, although no exact reduction factor follows from the simple Coulomb model.
For a nail embedded in a wall, the surrounding material presses against the nail surface and grips it through friction and local mechanical interlocking. If the nail is pulled straight out, many contact regions must begin slipping simultaneously. The extraction force must be large enough to destroy these sticking regions throughout a substantial part of the contact length.
If the nail is twisted while being pulled, a point on the nail surface moves tangentially relative to the wall material. The twisting motion repeatedly breaks local bonds and interlocking contacts. After a small angular displacement, one region slips; after further rotation, neighboring regions slip. The resistance is removed progressively rather than all at once. Some parts of the interface are in sliding contact during the motion, so the local resistance there is closer to kinetic friction than to the maximum static value. The essential effect is not that the entire interface instantaneously changes from $\mu_s N$ to $\mu_k N$, but that rotation continuously destroys sticking and interlocking, reducing the average force required for extraction.
For cutting with a knife, consider a blade pressed vertically into a material. If the blade only moves downward, a substantial length of the cutting edge must simultaneously generate enough stress to separate the material. A rough estimate of the required force is
$$F \sim \tau_c A,$$
where $\tau_c$ is the relevant failure stress and $A$ is the region being sheared.
When the knife is also moved tangentially, the edge acquires a slicing motion. Let the applied force have a normal component $F_n$ and a tangential component $F_t$. The tangential component creates shear stresses along the cutting line. Separation then occurs first in a small neighborhood where the stress concentration is greatest. As the blade advances, this localized failure region travels along the material. The material is cut progressively rather than being forced apart everywhere at once. Bread and meat contain cellular, fibrous, or brittle structures that respond efficiently to such localized shearing, so slicing markedly reduces the required normal force.
Cheese behaves differently. Many cheeses deform plastically and flow under compression before fracturing. A sharp blade pressed downward already produces sufficient stress to separate the material by plastic yielding. The gain from adding a slicing motion is much smaller, so simple pressing is usually adequate.
Result
A rotating awl penetrates more easily because rotation continually breaks sticking contacts and interlocking structures and introduces additional shear deformation around the tip. A twisted nail is easier to remove because rotation progressively destroys the grip of the surrounding material instead of requiring simultaneous failure of the entire contact region.
A knife cuts bread or meat most effectively when it slices, since tangential motion concentrates the cutting action in a localized region and produces large shear stresses that propagate along the cut. Cheese can often be cut by simple pressure because plastic yielding under compression is already sufficient to separate the material.
The common mechanism is that rotational or tangential motion converts a distributed, strongly constrained contact into a sequence of localized failures, reducing the force that must be applied at any one moment.
Sanity Checks
The expression
$$v_t=\omega r$$
has dimensions
$$[\omega r] = \frac{1}{\mathrm s},\mathrm m = \mathrm{m,s^{-1}},$$
which is the correct dimension for tangential velocity.
The estimate
$$F\sim\tau_c A$$
is dimensionally consistent because
$$[\tau_cA] = \mathrm{Pa}\cdot\mathrm m^2 = \frac{\mathrm N}{\mathrm m^2}\cdot\mathrm m^2 = \mathrm N.$$
If the rotational or slicing motion tends to zero, the contact remains largely stuck and the resistance is maximal. As the tangential motion increases, sticking regions are continually destroyed and the force required for penetration, extraction, or cutting decreases, matching everyday experience with awls, nails, and knives.