As a DM I used a simple rule for converting existing AC into DR. For D&D Next it would look like this:
-->AC: as per D&D Next
-->Dodge Defense (Dodge): Just 10 + whatever Dex (or whatever ability is used if changed from just Dex) bonus you get in that armor. So from the current playlets rules: 10 + Dex bonus if you use Light armor, 10 + half Dex bonus if you use Medium and 10 flat out if you use Heavy
-->Damage Reduction (DR): AC - Ref
You would have each PC and Mob record these 3 things where AC is currently recorded.
For example, say you have an 18 Dex fighter wearing Chain Shirt (AC 15+Dex) and using a Shield (+2 AC). His D&D Next AC is just 15+4+2=21.
His Dodge = 10+4 or 14.
Thus his DR = 21-14 or 7
So you record where AC us:
AC 21 (Dodge 14/DR 7)
Just add that extra bit in the parenthesis. Easy to add and easy to see. You just assume monsters with natural armor get their full Dex bonus to Dodge. Or D&D Next could have a module for this explaining how to do it or just have each monster already figured out with this extra bit.
How you use it in actual combat.
The only change is that each attack you make against AC requires you as the attacker to now decides to make this attack in 1 of 2 ways:
1) Default: you make a precision/power attack to try to hit a weak point in the armor or just hit so hard it gets through. This mean you attack AC as normal in D&D rules, and this changes nothing in the default rules.
2) You make an accurate attack. This means you just try to hit as the primary thing, and hope you cut through the armor. You then attack Dodge not AC. If you hit, you deduct the DR from your damage.
This approach is simple and it gives players the option to just use the old AC system if they like, by just using the original AC and ignoring the whole Dodge and DR in parenthesis. So it lets multiple players choose the method they like best. It lets one player play the normal D&D rules as they are, and another player choose the option 2 and play it that way. Each attacker would make a choice based on which method they prefer. It gives CharOp types fun in that they can try to figure out the best way to attack, while those who just want to roll and go use the default.
Using option 2, there is a risk you may do no damage on a Dodge hit if you roll less damage than the DR, that is the risk you take for using it. I would suggest that on a critical hit (since that only happens on a flat d20 roll, not like in older editions where you hit by 5+) you allow both choices to ignore DR, this way those who use option 2 do not get gimped on a critical hit by suffering the DR whereas the other guy who used option 1 gets no DR applied. It just makes critical hits, well, more critical! So a critical hit has no difference in effect because of your attack choice (option 1 or 2) and thus remains equally exciting for both.
The only note to add is that if the attack has any other effect than damage (like prone or push), that attack, while it may have hit Dodge, must have hit AC to have these other effects happen. This keeps the game balanced as intended for these other effects, only damage is altered in this rule. Not really that much more math, and if you like it, you do it, if it is too much work, you just use the default.
So this rule does not actually change the core, it just expands it, like a module you can use on the fly if you want to, but only if you declare it before you roll, if you do not specify before you roll, it is assumed default.
You can see it flavorwise as this:
1) If I attack Dodge I am just attacking normally for any opening you give me and your armor/shield reduce my damage if I hit your Dodge.
2) If I attack the normal AC (no different than normal D&D) then what I am really doing is either "power attacking" where I suffer a -1 to hit for each +1 damage I cause as I try to slam hard to get past your armor (which means I likely miss more often as my focus on power reduces my accuracy) or "precision attacking" you where I aim at joints or weak points in your armor, suffering the same -1 to hit but gaining a +1 damage to represent hitting vulnerable areas in your defenses. You can use the power attack flavor for big Str type weapons and the precision flavor for Dex type weapons.
I use this in my 2E games and 3.5E also, I find it works great.
The beauty of this system is how simple and easy to turn On/Off, even for each player, and even for each attack. All it takes is 2 extra stats next to AC to work, they are easy to calculate no matter how D&D Next decides to work armor and AC, you just use the concept as presented and you can add these 2 extra stats (Dodge and DR) easily. Then your player can either ignore them and just look at AC, or use them, and even do this on an attack by attack basis (deciding to try it out a few times and see how they like the feel, or using it only against BBEG types for flavor).
It almost does not need a module, it is so simple. The only benefit a module would give it is in the Monster Manual, where they record these 2 values for you so spare the DM that little bit of extra work (and I mean little) for each monster. It gives you the feel of armor as DR without any hassle or significant work/change of the existing rules.
As for the OPs specific request for typed DR (slash/pierce/blunt), this would work fine with the above approach. Instead of setting up a separate DR system for this, just incorporate it as a module that modifies the AC granted by each armor type, this way it can be used by any D&D player, those who use AC and those who like option 2 with Dodge and DR.
For example, you could make a simple system where Heavy armor grants +2/+1/+0 vs. (slash/pierce/blunt) on AC. Medium grants a +1/+0/+0, and Light grants nothing special here. You could make it more intense if you wanted by upping the bonuses, so Heavy could give +4/+2/+1, Medium: +2/+1/+0 and Light +1/+0/+0 this way Unarmored becomes the no adjustment case.
Another approach is to just say Slashing weapons suffer Disadvantage vs. Heavy and Medium. Piercing suffers Disadvantage vs. Heavy. Blunt is normal against all. Then have a new weapon property for certain piercing weapons, Armor Piercing, this grants Advantage vs. Heavy (canceling out the Disad and making the attack normal) and Medium. Certain Blunt weapons could have a new property, Penetrating, this gives the attack Advantage as well vs. Heavy and Medium. Make 2-handed weapons, like the great sword or axe, like a double weapon, you can choose to use the Slashing stat, say d12, or the Blunt one, say d8 depending on how you use it, as just a heavy piece of metal (like a metal staff) or a blade. Gives some options for the heavy slashing types to work better against armor. Or make those weapons Penetrating. Perhaps the Bastard Sword would get Penetrating when used 2-handed, etc.
Using a system like this would need to see the weapon damage modified so blunt weapons deal less damage and slashing more in general to keep balance.
Whatever module they use for this, the above AC/Dodge/DR system would work with them all.