Verb suggestion: wounding by constant contact

Solution 1:

Abrading Dictionary.com
Lacerating Wiktionary
Both words are used in medicine, but, differently than in general writing. They both mean what the question requires, with lacerating being a stronger term.

A lacerating chain would cut whatever it was wrapped around by friction.

Serrating could be used, but is posibly more medical than the other two, as it isn't used much in general writingDictionary.com, but good for barbed wire wounds.

Gashing does not seem to have a medical meaning at all, so it could work:

With every move, the chain gashed deeper

My guess is gashing may be most appropriate.

Solution 2:

There is a word for this, especially in equine medicine. It is called galling.

Per Saunders Comprehensive Veterinary Dictionary, 2nd Edition (p. 483), a general definition of a gall is:

a sore caused by chafing; said commonly of horses

There are a couple of other related definitions in the same dictionary:

girth gall (p. 499): a skin abrasion on the chest just behind the elbow of a horse caused by pressure or movement of the girth while being ridden. Caused by sharp edges of the girth, girth adjusted too tightly or too loosely or a poor conformation of the horse.

saddle sore, saddle gall (p. 1005): a pressure sore caused by bad riding technique or more commonly a badly fitting or poorly stuffed saddle.

As for in human medicine, Dorland's Illustrated Medical Dictionary, 28th Edition, has one definition of gall as follows (p. 673):

a localized swelling or skin sore caused by friction.

As for the verb form of the word, gall is both a transitive and intransitive verb (see this entry on FreeDictionary.com)

v.t. to make sore by rubbing; chafe severely: The saddle galled the horse's back.

v.i. to be or become chafed.

Here are a couple of examples of gall being used in a sentence from the Wordnik entry for galled:

His flesh was galled by many days of contact with the haul-rope.

His shoulders and chest, galled by the pack-straps, made him think, and for the first time with understanding, of the horses he had seen on city streets.

Solution 3:

If looking for more esoteric terms I'd suggest looking for those which you can use in a figurative rather than literal sense.

A harrow is a spiked frame for breaking up soil. if this is analogous to the spiked barbed wire you could say that the wire harrowed his flesh.

To keep that agricultural theme you could say that the bonds 'like a scratch-plough, had opened a deep furrow' in his flesh.

Failing that you could consider Excoriate '. to strip off or remove the skin from, which

first appeared in English in the 15th century, comes from "excoriatus," the past participle of the Late Latin verb excoriare, meaning "to strip off the hide."