The finder of the cannonbarrel fragments, GoDeep, asked me to comment. Unfortunately, as is often the case nowadays, I haven't been feeling well enough to write "detailed" replies to posts... which is why I haven't previously responded about this interesting Artillery find.
1- Contrary what some people in this discussion seem to believe, not even nearly all of the Historical cannons were Military. For example, there were many hundreds of "armed merchantmen" ships during the Colonial Era. For example, the famous East India Tea Company. Do you really think the pirates got all their cannons by capturing British/Spanish/French/Dutch NAVY WARSHIPS?
Also, many-many cannons were cast at privately-owned iron foundries, for use by the American Colonists. Most of those were "small-bore"... such as 1"-caliber Pivot Guns, etc.
All of that being said, the "privately made" larger-size cannons (1-Pounder/2.0"-bore and larger) tended to copy the the size of the Military calibers, so that the foundries wouldn't have to manufacture "unusual" sizes of cannonballs to supply both the Military and Civilian cannon owners.
GoDeep reported that his badly-burst cannon's caliber (bore diameter) is 2 inches. As another astute poster has already pointed out, that size is the Military's 1-Pounder caliber. Although 1-Pounders had become obsolete by the time of the civil war, the US Army still possessed enough of them to list 1-Pounders in the US Ordnance Manual of 1861.
Please pardon all that Historical Artillery trivia recitation. My point is, GoDeep's cannon corresponds to a widely used Military AND Civilian caliber... and that caliber was widely used from the Colonial Era into the early-1800s. It could be British, and it could be US Colonial Army or State Militia, OR a non-official military group, such as a town's local defense. Being made of LOW-QUALITY iron rather than brass, I lean toward it being early-American.
I emphasized the words LOW-QUALITY because of the "very-large-grained" look of the iron in the broken-open fragments. (As Smokeythecat called it, earlier in this discussion, "flawed iron" -- which was a major problem for the artillerymen of the Colonial American Army. The lower the quality of iron, the more brittle it is, and the more likely to burst into multiple pieces when subjected to excessive internal explosion.
On the subject of "multiple pieces"... I've never seen a cannon burst into so many pieces, fracturing to bits all the way from its muzzle (front end) to its breech (back end). As I indicated above, iron is more brittle than brass, which is why in the Colonial Era most Military cannons were made of brass. When a brass or high-quality iron cannon bursts, it usually ruptures only in the back half, where the firing propellant powder-charge was placed. This is clearly seen in the wonderful photos helpfully posted by AARC. (Thank you very much.)
Alan M said (on the 1st page of this 6-page discussion, "it is probable that it was overloaded intentionally in order to destroy it." Alan is correct. The fact that GoDeep's cannon burst into fragments literally from one end to the other strongly indicates deliberate destruction, to prevent it from being used after capture by enemy forces. To burst it so completely, it must have been packed with gunpowder topped with probably about 6 solid cannonballs. Joe Hunter was correct, it wasn't broken to pieces with a sledgehammer, "There’s no impact signs."
Relevent Chair said, "Resistance to pressure creates more pressure." Putting the weight of 6 solid balls in front of the powder charge would create enormous resistance. So, that was the typical method for destroying a cannon.
I should also mention, AARC made a valid point about "inspection" of cannons (also called "Proofing")... but:
National Armies and State Militia required that cannons be rigorously tested before being issued to troops. Private-foundry cannons manufactured for sale to civilians did not have that requirement.
The total lack of manufacturer's markings or other ID on a cannon can mean anything from modern-Reproduction to Wealthy-Man's Estate Lawn Cannon to Colonial manufacture for Militia OR "civilian merchant-ship use."
All of the above being said... the presence of "white paint" on at least one fragment of the cannon "could" suggest it wasn't destroyed during the Revolution or War Of 1812. That paint is the big mystery to me.
The white paint is "splotchy" and seems to be on only one fragment. (GoDeep posted a photo of it, and he did not say other fragments have the paint.) So perhaps this old 1-Pounder cannon was in somebody's barn for decades, and its owner decided to have some fun by destroying it. Or maybe that one fragment with paint was on the ground's surface and somehow somebody spilled paint on it. Why don't the rest of the fragments also have that white paint?
Those are my scattered observations and thoughts. I hope some of them they helpful to some folks. Please pardon if they are less than completely coherent. At my age, medical maladies (and medications) take a toll.