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Dunno about teddy bear fur, but I've mounted my 2x2 mdf boards to frames. 9mm thick mdf on 20mm square battens, with lots of filler, paint and pva on top. No warping.Though when I did the same process with bigger 2 x 4 ft boards, I did have warping problems.
Robh has it right.You need contact adhesive. wonderful stuff. wot warp your board. but you must realise you will never get the fur off...
Were the 2 x 3 ft boards fastened to batons too?
Yes, both the 2x2 and 2x4 boards are made the same way.The 2x4 ones warped somewhat, but I remedied it a bit by gluing newspaper on the underside with PVA. Still not perfectly level, but I put little wedges of cardboard under the corners when I set up a table to level things out nicely.The 2x2 boards have never warped - I guess they are sturdy little squares.When I do boards with recessed features such as rivers and ditches, the MDF becomes the bottom layer, and the battens become a 'border' to the 20mm blue foam top layer which the rivers etc are cut into.For flat boards, the MDF becomes the top layer and the battens are underneath, with no blue foam needed of course.
9mm.
I would paint the side of the board you are not gluing to, and as others have suggested contact glue like Copydex is good for fabrics.
Copydex is not a contact adhesive.