Cone Flakes, Bending Flakes and Potential Energy

Tony Baker and Andrew Pelcin

September 20 1998

Pelcin identified two distinct types of flakes in his experimental work and believes they are the cone (Hertzian) and bending flakes discussed by Contterell and Kamminga. I have been able to replicate these two types and this section is written to explain in more detail how these two types occur.

Pelcin's controlled experiments consisted of dropping a steel ball onto a core made from plate glass. The plate glass was 1/2 inch thick, 5.5 inch 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

The reader should know that the discussion presented on this page will be base on a core with an EPA of 55 degrees and an angle of blow ( AOB) of 70 degrees. This geometry makes the right edge 2.9 inch high. I am writing about only this geometry because it is the only geometry I have explored up to this time. I feel confident that the findings presented here will hold for other geometries based on their internal consistenancy. Yet, the actual testing of them against other geometries has not been done.

All materials are elastic which means they will deform when loaded and returned to their original shape with unloaded. Steel and glass are no exceptions althought they are not normally considered elastic except to engineers. When the steel ball impacts the glass core, it deforms similar to a tennis ball hitting a wall. At the point of impact, it is slightly flattened. Additionally, the glass core slightly deforms. 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 glass to their original shapes that causes the ball to jump off the floor.

These elastic defomations resulting from the impact of the ball with the core are extremely minute, yet they represent stored potential energy in the ball and the core. This stored energy was converted from the kenetic energy of the falling ball prior to impact. An easy way to visulize stored potential energy is with a spring. When a spring is compressed or stretch, it is storing potential energy. The potential energy stored in a spring is equal to the distance of its deformation times the force that causes the deformation (potetial energy = force*distance).

In this image, I have replaced the blue ball with a mechanical analogy of two blue springs and a blue force. Technically, I should have first presented the analogy with only one spring oriented in the same direction as the force. However, further in the paper it is important that the single spring be resolved into the two located along the principal axis, so I have shown them this way.

In my FEA replication work, the blue force has a value of one (1) pound. The detemination of the cracks that defined the flakes was base on the location of maximum tensile stress and not the actual tensile forces required to cause the glass to fail. The one (1) pound force are arbitrary chosen for all the replication work and this value then determined all the other values that will be present here.

The two blue springs each have the value of 4.9E5 pounds per inch and they are the resolutions, along the axes, of the single spring that parallels the force. The spring constant was detemine during the replication work and its derivation with be presented below.

In this image, I now have replace the brown glass core with two brown springs. Unlike the blue ball springs, these springs have different values determined by the location of the application of the force. For example, if the force is applied near the edge the core making the TPT small, the Kmy spring was significantly weaker than the the Kmz spring. However, as the application of the force is moved away from the edge, creating a larger TPT, then the Fmy springs becomes much stronger.


Notes

#1 In the image the core has been rotated to the right for discription purposes. In really, the ball fell vertically downward onto the core which was rotated to the left.