Not all paper is made equally. For example, copy paper isn’t great for cutting with a cutting machine. Thin paper like that will most likely scrunch up and tear instead of cut. Not to mention when you attempt to lift it off the mat it might be too stuck to pull off cleanly.
There are just certain brands of paper that cut better. I know a lot of people swear by recollections brand from Michaels and American Crafts AC cardstock for use with their cutting machines.
Do a test cut on the bottom half of the paper. Use the arrows next to the test button to move the cutting machine. Test cut below the design, so you are able to check the test cut without having to unload the mat from the machine. If the rectangle or square does not appear to be completely cut, choose different cut settings, and try again. Continue this method until the test cut area can easily be lifted off the mat.
When cutting a more intricate design or using thicker cardstock, consider slowing down the speed. Silhouette automatically does this when you choose 110lb coverstock vs. 80lb cardstock. The speed is how fast the blade moves around cutting. It may help prevent tearing and squishing if slowing down the speed.
The force is how much downward pressure is being placed on the paper by the bar that holds the cutting mechanism. This can be adjusted, especially if the paper is moving around on the mat.
The blade depth is another option for changing. Paper will dull a blade faster than vinyl. When I know I’m working with a dull blade I will increase the depth of the blade to get a cleaner cut.
The amount of passes is also a consideration. I try to keep my cardstock to 1 pass, especially when cutting super-detailed designs. I’m afraid an extra pass might leave more room for tearing if things aren’t stuck down to the mat very well.