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I'm looking around for adhesives that'll do PP. There are some, but it looks like they may be 'special'. That means expensive or you get 5 gallons of it when you need 5 ounces. It does look like it was welded together originally. If it was metal I'd say the penetration was insufficient.
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I'll tell ya there is nothing that sticks to polyPro, and that is simply the way that plastic works....it was designed to be slippery.... skis have that material (P-Tex) as the base material, and even though you could "bond" filler material to the base material to fill gouges and scrapes, welding the stuff is a nightmare.
I'll simply say, as rediculous as this sounds, adhesive backed foil tape, or ordinary duct tape, applied to the surface
after it is completely degreased using BrakeKleen, is the best solution. Yeah it sounds rediculous and crude, but it works. Unless you go along both flat surfaces and drill a bunch of small hole for the adhesive to bond like mushrooms, thru for retention, and then imbed screening into the adhesive (JB WELD works well, but will need help) nothing will last forever sticking to the outer surface only.
If you reallllly want to experiment with this, I'll give you a hint...
remove the box, clean it inside and out meticulously with BrakeKleen. ALLOW THIS TO COMPLETLY AIR OUT,....then take a "Hefty trash can liner" Bag and twist it up into a "rope". then, tie the liner into very tight knots, spaced closely together, to create a dense material to be used as "weld stock".
(
ACTUALLY DRY CLEANING BAGS WORK BEST FOR THIS....THEY BURN AND DRIP BETTER...)
place a piece of metalic foil tape inside surface of the split. leaving a bit of a gap for material introduced from the outside surface to fill into.
Working safely OUTDOORS. with a bernz-o-matic torch running nearby as an ignition source, light the "rope" you made from the trashcan liner on fire, and drip the flaming droplets into the split area to fill the crack, blowing out the flames as they hit and fill the crack...kinda like dripping drops of candle wax....but they are burning...it's dangerous, but it will work, just keep your concentration on the task.
Drip the flaming gooobers into the groove, and immediatly blow out the flames, repeating the process to fill the groove.This is very dangerous, and dripping molten burning plastic on your arm, or the floor, may result in bad, bad things.....think this over, and figure if you want to try this. If you consider what I'm saying, and can figure the process out, best of luck, because you will have conquered the problem and fixed the box....even though what I have written here is going to be controversial, and may result in major scar tissue.....