Foam core, or foam board, is a thin piece of foam sandwiched between two pieces of glossy paper. It is used in several types of art and crafting projects including 3-D models which often need to be glued together.
What Won't Work
Gluing foam core with a solvent based craft glue or tacky glue will work for a short period of time but the base of the glue will dampen the glossy paper and cause it to wrinkle. The security of the bond is also suspect with everyday craft glue.
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Hot Glue and PVA
A hot glue gun will adhere to the foam core without damaging either the board or the glossy covering. You will have to work quickly though, hot glue cools rapidly leaving relatively little time to make adjustments to your project. PVA glue (Elmer's Glue) works quite well as long as the board is held in place while it dries.
Spray adhesives are another good choice for gluing foam core. The bond is strong, there is no bubbling nor does the spray damage the surface. Use caution if the paper covering is compromised because the chemicals in the spray adhesive may eat away at the foam.