I want to make a half-orc monk and I was wondering if the racial feature Savage Attacks from the half-orc race and unarmed strikes from the monk's Martial Arts feature work together. Or, is there some reason that unarmed strikes don't apply to this?
The Savage Attacks trait says:
Savage Attacks. When you score a critical hit with a melee weapon attack, you can roll one of the weapon’s damage dice one additional time and add it to the extra damage of the critical hit.
All it requires is that you get a critical hit with a melee weapon attack. An unarmed strike can be used to make a melee weapon attack, so yes, Savage Attacks will work if you get a critical hit with an unarmed strike.
No, the Savage Attacks trait doesn't benefit unarmed strikes.
The half-orc's Savage Attacks trait says (emphasis mine):
When you score a critical hit with a melee weapon attack, you can roll one of the weapon’s damage dice one additional time and add it to the extra damage of the critical hit.
Because it does say "the weapon's damage dice", it only applies on attacks made with weapons - not unarmed strikes.
Even though unarmed strikes are considered "melee weapon attacks", they do not involve a weapon, as mentioned in the "Melee Attacks" section of the rules (emphasis mine):
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).
As such, things that require a weapon don't work with unarmed strikes.
The same is true of the similarly worded (...and very similarly named) Savage Attacker feat. The description of the Savage Attacker feat says (PHB, p. 169; emphasis mine):
Once per turn when you roll damage for a melee weapon attack, you can reroll the weapon’s damage dice and use either total.
Page 8 of the Sage Advice Compendium includes the following official ruling on the Savage Attacker feat's interaction with unarmed strikes:
Does the Savage Attacker feat work with unarmed strikes?
No. Savage Attacker relies on a weapon’s damage dice, and an unarmed strike isn’t a weapon (a point that was clarified in the Player’s Handbook errata).
The half-orc's Savage Attacks trait includes the exact same phrase as the Savage Attacker trait: "the weapon's damage dice". Thus, by the same logic, Savage Attacks also requires an attack with a weapon - so unarmed strikes don't count. They're melee weapon attacks, but are not made with a weapon. (This is not changed by anything in the monk's Martial Arts feature either.)
As always, DMs can houserule otherwise.
(It likely does not break anything to allow this; even for monks, the highest damage die they can have on an unarmed strike is 1d10, whereas several weapons can also do 1d10 damage, and greataxes and lances do 1d12 damage. While monks can make more attacks per turn than some other classes, the benefit of the Savage Attacks trait in particular is limited to critical hits, so this does not seem particularly game-breaking to allow.)
That is dependent on the DM
A point of contention will always be whether or not unarmed strikes count as melee weapon attacks. A PHB errata and sage advice compendium muddy the answer to this question. This answer tries to assert that unarmed strikes count as a melee weapon attack using this SAC as its proof. However, in that very same compendium, that question is answered outright.
Does the Savage Attacker feat work with unarmed strikes? No. Savage Attacker relies on a weapon’s damage dice, and an unarmed strike isn’t a weapon (a point that was clarified in the Player’s Handbook errata)
Therefore, the operative question is: does Martial Arts override this, and allow unarmed strikes to be considered melee weapon strikes because of the nature of monks?
By definition, the answer is no.
But your DM may decide differently. Firstly, the way it's worded and described is VERY confusing, misleading, and generally uncool. Unarmed strikes are melee weapon attacks until they aren't, because for the above example unarmed strikes do not qualify even though the rule says melee weapon attack and unarmed strikes have been described as a melee weapon attack in that errata despite not involving a weapon and clearly being treated as an unarmed strike, which itself is being treated as separate from a melee weapon attack despite the...well, you get the loop from here. Your DM may choose fun over semantics. Secondly, damage die exist for unarmed strikes for monks, therefore bypassing the objection that damage dice are needed. But then there still comes the matter that unarmed strikes are simply not melee weapons -- and even that if having damage dice was enough, then half-orcs with the tavern brawler feat might also qualify their unarmed strikes to work with Savage Attacks. These complications may make for a stronger case that, even for monks, the RAW are also the RAI. Though the quote above addresses Savage Attacker and not Savage Attacks, both rely on damage die and trigger on melee weapon attacks for the added dice, ergo the answer seems to apply for both. So I'd say to be prepared for DMs to staunchly deny the extra crit damage die as there is nothing that supports monk unarmed strikes counting as melee weapon attacks, and everything supporting the RAW -- that monk unarmed strikes are still just unarmed strikes, and unarmed strikes are, beyond a shadow of a doubt, not eligible for Savage Attacks. But I know I would absolutely allow it because I wouldn't want to invalidate Half-Orcs as a race choice for my players' monks, nor would I want to restrict their RP/play style by forcing them to use weapons lest they lose out on their racial abilities. And if any exception to this rule should be allowed, it would be for monks and their martial arts.