Modified September 30, 1999
This technical section discusses the details of constructing and using the FEA model. The model was created in ALGOR software and all the software terminology is ALGOR's. I have attempted to write this section in as much detail as necessary for the reader to reproduce my work. The first part is the creation of the FEA model. The second is how the model was used and the third is the differences in the application of the model for the energy-rich and the energy-poor flakes.
The creation of the FEA model, which is depicted in this image, began in ALGOR's Superdraw III. The "drawing", which is intended to simulate 1/2-inch plate glass, was made by creating a 5.5-inch high by 3-inch rectangle.1 Then the upper right corner was removed by drawing the diagonal from the upper left-hand corner toward the center of the right edge. This diagonal was drawn at a 55 degree angle from the vertical to replicate a core with an external platform angle (EPA) of 55 degrees. While the lower right portion of the rectangle was still present (not present in this drawing), a mesh of 88 cells vertically and 48 cells horizontally was automatically added. This mesh resulted in the largest cells located on the left edge being approximately 1.6 millimeters on a side. The lower right portion was then cut out and rigid boundary conditions were set on the cut out edges. This was done to simulate the clamping device that held the glass core.
The horizontal and vertical forces of -0.8192 and -0.5736 pounds were then added. These two forces yield a resultant of one (1) pound force applied at a 70 degree angle of blow (AOB). These were applied at the junction of the fifth and sixth cell for the replication of flake #2 in the Energy-Rich Flakes vs. Energy-Poor Flakes section. At this same junction two elastic boundary elements (springs) of 4.9E5 pounds/inch were added. These were necessary to make the energy-rich flakes and are believed to be associated with the hardness of the steel ball.
The Superdraw III drawing was then converted to a FEA model with the Element Data Control Tool and the following settings:
The FEA model was then "analyzed" (used) many times to produce one complete flake. The first time it was analyzed the model had no cracks in it and existed exactly as depicted in the image above. The only data obtained from this run was the horizontal and vertical displacements (strains) that resulted from the forces and springs. These displacements were measured at the point where the load and springs were attached to the core.
The Superdraw III drawing was then modified by placing a vertical crack one cell long to the right of the point of application of force and springs. (See the enlarged image depicted here.2) In drawing the crack, the mass was always removed from the core side, so as not to weaken the flake. The crack was drawn by dividing the line on the topside of the 7th cell into 10 equal parts and connecting to the first of the 10 lines. The actual mass removed from the 7th cell would be 0.5 times 1/10th or 1/20th of its mass.
After the crack was added a repetitive process began that can be described as:
When there is only one crack in the model then the only possible node from which a crack can propagate is at the tip. So the adjacent node to the tip (left, down, or the right) with the highest tensile stress is the direction for the next crack. In this example, that direction turned out to be to the right.
Repeating the above process four (4) more times (cracks), produced a crack geometry as depicted here. At this point the node within the crack with the highest tensile stress is located in the yellow. The crack can then propagate to the right, down or left. The color contours indicate that the crack should again go to the right, which is correct. The reader should not assume that the color contour map was used to decide the direction. The actual values at the three nodes were obtained and the direction for the next crack was based on these values.
There is one exception to the rules in the process above. This is that the crack is not allowed to return to itself which would create a hole in the model. This situation doesn't happen often, but it does happen. An example in this image would be to add the crack to the right and then add one more that is up. This returns the crack to itself and creates a hole. The hole weakens the model and doesn't effect the outcome of the flake because the next crack will be in the correct direction. Therefore, instead of adding the crack that returns to itself the direction of next highest tensile stress is chosen.
Energy-Rich flakes and energy-poor flakes are created slightly differently. The energy-rich flake proceeds as described above until the potential energy released is less than 9.0E-10 inch-pounds per millimeter.3 When the potential energy drops below this value then the springs (kx and ky) associated with the one pound force are removed.4 When this is done, the horizontal force is greatly increased and this breaks out the flake. Energy-poor flakes are created without the springs (kx and ky). Therefore, the potential energy does not have to be monitored to determine when to remove the springs.
The reader is probably thinking that including or not including the springs in the model requires prior knowledge of which type flake is to be produced. This is correct, and therefore it is best to begin with the springs applied. Then if the potential energy in the model is actually increasing, instead of decreasing, then the springs need to be removed because this is the condition for a energy-poor flake. See Energy-Rich Flakes, Energy-Poor Flakes and Potential Energy for more details.
#2 This was the location for the start of all cracks for the flakes produced during the research. It was chosen because it was closest to the force. In reality, the location of the initiation of the crack can not be predicted because of unknown flaws in each piece of glass. However, it is generally near the application of force.
#3 This value is based on the one (1) pound force that is applied to the model and it would be different if a different force was applied. See Energy-Rich Flakes, Energy-Poor Flakes and Potential Energy
#4 Instead of instantly removing the springs, it is best to taper the removal. I have found the best procedure is to reduce it to 25% for the first crack toward the edge, 10% for the next crack and then remove them altogether. This procedure overcomes some stress concentrations that can mislead the user.