Bend relief: A bend relief is a small notch cut into sheet metal where a bend ends next to an unbent edge, so the material can bend without tearing or bulging.
Where a bend line runs into an edge that stays flat, the metal at the boundary is pulled two ways at once. Without relief, it tears or bulges. A bend relief is a slot or rounded notch cut into the flat blank at that point. A common rule of thumb makes it at least as wide as the material is thick, and deeper than the bend radius plus the thickness. Corner reliefs do the same job where two flanges meet.
What it means for the price
Reliefs cost almost nothing to make, because they are cut as part of the flat pattern profile. Leaving them out is what costs money: the part can tear, be scrapped or need hand rework. A DFM check that catches a missing relief at quote time is far cheaper than a bad batch.
Example: on 2 mm aluminium with a 2 mm inside bend radius, a relief 2 mm wide and about 4 mm deep is a reasonable starting point. The figures are an example.
Common confusions
A bend relief is not a hinge cut or a living hinge, and it is not a kerf slot cut to make a bend easier. Shops may have their own relief shapes for their tooling, so follow their guidance where it exists. Read design for sheet metal.
Go deeper: Design for sheet metal: bend radii, reliefs, hole distances and flanges