HDPE can only be thermally welded. However, the HDPE cages by Animal Plastics are held together with a combination of screws and rabet/dado joints. So to avoid purchasing a thermal plastic welder you'd have to use 1/2" HDPE which really would not save much weight over 1/2" plywood. It would still make a nice cage, however. A router would be helpful to cut the dados although not absolutely necessary. They can be cut with circular saws with a lot of patience.
A word of warning, the HDPE I've looked at tended to be warped. Sheets of PVC, ABS or styrene were always flat and true. Certainly there are sources for HDPE in better condition but I would not trust Lowes to find it.
High impact styrene can be theramally or chemically welded. I believe the thermal welds are tricky due to the how thin the material is, so it may not be something to learn on. Barrs and Precision both solvent weld their styrene cages if that tells you anything.
I really like expanded PVC (aka PVCX or EPVC) and believe this is the material of choice for the "do-it-yourselfer". But honestly, I would not hesitate to try to make a cage of any of those materials if it were available to me.
Lastly, have you checked plastic shops? I have found all of the above plastics and more at plastic shops in Lincoln, Nebraska. If I can find it there I think it would be easily found in any decent sized city.
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I was going to build a cage out of wood but have read some threads about using these plastics for material. I have looked at US Plastic Corp. online and to have the stuff shipped will be outragous but Lowes says they can order just about anything I want. Soooooo..... Now I need to decide which type of plastic I would like to use. Any advice would be GREAT!! Also, would I need to buy a plastic welder or is there another way to bond the plastic together? Thanks in advance.