November 14, 1998
Modified September 30, 1999
Pelcin identified two distinct types of flakes in his experimental work and we believe they are the result of the energy-rich and energy-poor propagation modes. Contterell and Kamminga (1987:692) referred to these modes as "stiffness-controlled" and "compression-controlled". I have been able to replicate these two types and this section is written to explain in detail how these two types occur.
Pelcin's controlled experiments consisted of dropping a steel ball on to a core made from plate glass (left image). The plate glass was 1/2 inch thick, 5.5 inches high on the left side and 3 inches across the bottom. The right edge varied in height depending on the external platform angle (EPA). The core was clamped to the table by the lower right corner.1 More details on the core's geometry and the FEA model can be found in the FEA model session.
The discussion on this page is based on replicated flakes from a core with an EPA of 55 degrees and an angle of blow (AOB) of 70 degrees. To date, this is the only geometry I have considered and my findings flow from this geometry. On the other hand, I feel confident my findings will hold for other geometries, yet the actual testing of them has not been done.
All materials are elastic which means they will deflect when loaded and returned to their original shape with unloaded. Steel and glass are no exceptions. When the steel ball impacts the glass core, it deflects similar to a tennis ball hitting a wall. At the point of impact, it is slightly flattened. At the same time, the glass core slightly deflects. Intuitively, we know this must happen because a steel ball will bounce when dropped on a concrete floor. It is the returning of the ball and the concrete to their original shapes that causes the ball to jump off the floor.
These minute elastic deflections resulting from the impact of the ball with the core represent stored potential energy in the ball and the core. This stored energy was converted from the kinetic energy of the falling ball prior to impact. An easy way to visualize stored potential energy is with a spring. When a spring is compressed, it is storing potential energy. The potential energy stored in a spring is equal to 0.5 times the deflection times the force that causes the deflection (potential energy = 0.5*distance*force).
In this image, I have replaced the blue ball with a mechanical analogy of two springs and a force. Technically, I should have showed the force as two resolved forces along the axes, but this would have cluttered the image. In all my FEA work this single force was resolved into a "Y" and "Z" component. (The "X" axis is coming out of the page.)
The two springs each have the value of 4.9E5 pounds per inch. This spring constant was determined during the replication work and its derivation will be presented below. The force was arbitrarily set to one (1) pound with an AOB of 70 degrees. The "Y" and "Z" components of this force then are -0.8192 and -0.5736, respectively.
Applying these forces and springs to the core at a location will cause a deflection of the core and the springs. For example, if the force and springs are applied at 9.5 millimeters2 from the left edge (TPT), the FEA model calculates the deflections of the core and springs to be -1.214E-6 inches in the "Y" direction and -5.130E-7 inches in the "Z". The total deflection of the core, which is also the deflection of the springs, can be calculated as:
delta = SQRT((dy*dy) + (dz*dz))
= SQRT((1.214E-6)*(1.214E-6) + (5.130E-7)*(5.130E-7)) = 1.318E-6 inches
If the forces and springs are applied at a different location the deflection in the core will be different. The reason it changes is because the core becomes stiffer (harder) as the point of application is moved toward the center of the core (increasing TPT). The following graph shows this lessening deflection as the application point is moved to the right (increasing TPT).
Deflection in the core is the addition of potential energy (strain energy) to the core. If the deflection varies with the point of application, so will the added potential energy. To calculate the potential energy in the core, consider the same example from above with the point of application at 9.5 millimeters from the left edge (TPT).
The following graph shows the potential energy in the core versus point of application of forces (TPT).
I hope the shape of this potential energy curve is unexpected to the reader. It was to me. There are two things that are surprising. First, the potential energy curve has an inflection point in it and the deflection curve did not. Second, for most of the image the potential energy increases while the deflection curve decreases. I would have expected the potential energy curve to decrease as the deflection curve did.
The reason there is an inflection point is because of the set of springs (Ky and Kz) that represent the elastic behavior of the ball. If the springs were not present, or had a value of zero, the potential energy curve would behave as the deflection curve. Both would decline as the point of application of the forces (TPT) was increased. However, as these springs become stiffer (spring constant increases) they begin to absorb some of the potential energy of the system (core, springs and forces) and the potential energy curve begins to have an inflection point at very small TPT values. As the spring constants are increased, the inflection point moves towards greater TPT values. At a spring constant of 4.9E5 pounds per inch the inflection point is at a TPT of 28 millimeters. This is the way I set (determined) the spring constant. I matched the transition point in Pelcin's real data.
This inflection point is the separation between the energy-rich and the energy-poor flakes. The energy-rich flakes are on the left of the point and the energy-poor flakes are on the right. Referring back to the deflection curve, which could also be thought of as a stiffness curve, the core deflects more as the TPT gets smaller. In different words, the stiffness of the core is proportional to the TPT.
Consider what happens to the core when it is loaded and a crack begins to form. The presence of the crack makes the core less stiff, which is equivalent to a smaller TPT or moving to the left on the potential energy curve. If the crack begins on the left of the inflection point, moving to the left is giving up potential energy. If the crack begins on the right of the inflection point, moving to the left requires more potential energy. The significance of this difference is the energy-rich flake cracks, from potential energy already stored in the core, while the ball is returning to its original shape. The energy-poor flake must wait for the ball to rebound and add more potential energy before its crack can proceed. To visualize this, consider the FEA created flakes and their associated energies as the crack propagates downward.
As is evident, the cores are releasing potential energy as the energy-rich flakes (#1, #2, and #3) are cracking downward. The energy-poor flake core (#4) requires more energy as the crack propagates downward. These two different mechanisms result in different vertical crack velocities which are 500 to 1000 meters per second for the energy-rich flakes (Cotterell and Kamminga 1987:680) and probably between 100 to 200 meters per second for the energy-poor flakes3.
When the energy-rich flake's crack stops its vertical travel, it stops all propagation. The crack does not propagate again until the ball rebounds and applies more potential energy. So the mechanism for the lateral propagation of the crack to the edge is the same for the energy-rich and the energy-poor flake. Both are caused by the ball adding more potential energy to the core.
By now the reader is probably wondering, "what makes the energy-rich flake's crack stop propagating?". It obviously is not the exhausting of the potential energy because the cores (Flakes #1, #2 and #3) in the above graph still had potential energy when vertical cracks stopped. It is not the same potential energy because the cracks stop at different values. This question plagued me for several months because its answer was essential for the replication of the flakes. I needed to know what was the "signal", so to speak, that I could use to know when to remove the restraining ball springs (Ky and Kz) and apply the full force of the ball to break out the flake.
The answer turned out to be logical just like every thing else in this research. All three energy-rich flakes stop at the same change in potential energy per millimeter of crack. As the crack progresses, the core is converting its potential energy to work to make the crack. This releasing process has a particular behavior of increasing and then decreasing. During the decreasing portion there appears to be a threshold of 9.0E-10 inch-pounds per millimeter below which the crack will not propagate. Consider this cluttered graph which is the slope of the potential energy curves on the previous graph.
This graph is cluttered because I have included regression lines (dotted lines) with the actual data. The actual data, which is the change in potential energy per millimeter of crack, is beginning to become unstable in the FEA model because the cell size is too large. So I used regressions lines to determine when to stop the cracks. The reader will note that all the cracks were stopped at a change in potential energy of 9.0E-10 inch-pounds per millimeter. There is nothing special about this value other than it is necessary to match Pelcin's real world flakes. Also, the reader should note that this value is dependent on the arbitrary one (1) pound force used in the model to represent the ball.
To conclude this section, I would like to briefly describe how I would create another flake with the FEA model. This also assumes the core will have an EPA of 55 degrees, a force of one (1) pound, and an AOB of 70 degrees. First, I would determine if the TPT is on the left or the right of the transition point (28 millimeters). This will tell me if I am making a energy-rich flake or a energy-poor flake. If it is to be a energy-poor flake, I just apply the one pound force, without springs, and start adding cracks to the core in the direction of maximum tension. If a energy-rich flake is to be created, I would apply the one pound force with the two restraining springs (K=4.9E5 pounds per inch). I would then start adding cracks to the core in the direction of maximum tension. However, I would be constantly checking the rate of change in potential energy in the core. When this value declined to 9.0E-10 inch-lbs per millimeter, I would remove the springs while leaving the one (1) pound force and continue adding cracks to the core. A more detailed discussion of this procedure can be found in the FEA model section.
#2 As an American engineer I created the FEA model in British units. Archaeologists use the metric system and Pelcin's work is in metric units. Instead of converting one system to the other, I chose to mix the two. This is then a mixing of the engineer and the archaeologist.
#3 The speed of the energy-poor flake is equal to the rebound speed of the steel ball and the glass core. I do not know what this value is. I do know Crabtree made Mesoamerican blades in the energy-poor mode and his propagation speed were 125 meters per second assuming a 10 centermeter blade (1968:474). Hutchings (1997:50) reported minimum velocities of 118 and 152 meter per second for metal and antler pressure flaking. It is possible steel balls could create energy-poor speeds as high as 200 meters per second, but I believe this would be the upper limit.