What exactly happens when a material (particularly paper or even cloth or a metal) is folded to form a crease? Why is it that a creased material tends to retain form, while a lightly folded one, 'might' just happen to revert to an original configuration? Are bonds being broken here, then to what extent? Are they along a single molecular line or break on a bulk basis? Also, why does a rolled sheet retain shape through a variety of radii? Is there a critical radius to this sheet regaining its flatness? Thanks.
What happens during folding is that the material undergoes plastic deformation. When a sheet of material is bent slightly that deformation is usually elastic, meaning it will return to its original shape when the deforming stress is withdrawn.
But when the deformation is larger we enter the plastic zone: the material will no longer fully recover its original shape. A crease can be seen as a permanent deformation, for instance.
Plastic deformation causes some permanent (irreversible) changes at the atomic/molecular level but no significant bond breaking. The excessive deformation causes layers of the material at the atomic/molecular level to slide over each other in a non-reversible manner. Precisely what slides over what depends on the micro-crystalline structure of the material: mono-crystalline (silicon chip wafers, e.g.) or multi (micro) crystalline (steel, e.g.)
Re. rolled steel sheet, it's obvious that the degree of deformation depends on whether the sheet is closer to the edge or closer to the core of the roll. Undoubtedly there's a critical curvature (dependent on grade of steel) that would push the steel sheet into the plastic zone. I assume manufacturers of rolled steel sheet avoid that critical curvature by making the cores sufficiently large in radius.
In the case of inorganic matter, such as metal sheets, folding/creasing produces a substantial bulk stress in the material which can modify the molecular structure in a large number of complex (and not fully understood) ways. For example, it can break bonds, cause amorphisation, and propagate dislocations. These same mechanisms are at play when you cut a material with a knife, see: How does a knife cut things at the atomic level?
In the case of paper, however, which is an organic material, its structure consists of a complex network of microscopic fibres which are connected to each other via (weak) hydrogen bonds:
When you deform this structure, you cause the fibre-fibre bonds to break, and new ones to form, causing the paper to 'remember' its new shape.
I state this based on a simple duplicatable experiment, understanding thin sheet metal or paper also as a plastic material deformation.
The still remaining ( or remnant) of strained edges or ridges are tell-tale permanent deformations or folds after deformation. Plate & Shell theories in continuum mechanics should perhaps suffice for an explanation.
Going with mechanics of materials understanding... thin paper has some thickness. When unfolded/unravelled after creasing occurs small strain recovery is quite elastic but heavily creased edges must stay put with permanent set pinch marks.
To get a physical feel of this phenomenon take a new A4 paper sheet from printer and crumple it in the hand into as small a ball as possible and then attempt to flatten it out again.. to observe (tiny shallow origamis spread uniformly) the new folds better seen when light is incident laterally.
There are two types of folds discernable, schematically sketched here. Straight Ridge/crease and U-shaped semi arcs/ of local ring. When seen magnified their middle line has still retained the zero Gauss curvature $K$ geometrically.
By virtue of Gauss Egregium theorem $K$ is conserved at its initial zero value.
Gauss curvature $K= k_1\cdot k_2 = 0$ product of two curvatures.
The first type is visible/recognized as generator lines of cylinders or cones with $ k_1 = 0$
The second type is visible as the crown of a torus demarcating between $ K<0, K>0 $ regions with $k_2 = 0.$
Such folds also appear on Aluminum foil wraps.
Imho we are not into the molecular level here. Please comment.