Is there a way to simulate a rubber band, say for a catapult. I'm not looking for detailed code, just some direction on how to approach this. Thanks.
Use springs. I did a rubber band with springs once, and I actually coded the springs myself. Then have a mesh with some bones that you then position to the simulated spring position. It will cause the mesh to smoothly bend as well, given you have skinned the model properly. It could probably also be done with configurable joints.
Take a look at this challenge, he teaches various video-lessons, and creates a challenge for you to try to get the skills you learned, I think it's similar to what you want:
http://www.unity3dstudent.com/2010/07/challenge-c02-beginner/
Open the link, and look at the tutorials necessary for a successful challenge, he uses hinge joint and other resources menu