"The ship cracked, leaking oil into the sea."

"The ship cracked up, leaking oil into the sea."

Could one of them be incorrect, or could they have slightly different meanings?

They are both OK but the first is probably best

To a London ear, the second one can be slightly odd because "cracked up" is slang for "..I fell about laughing.."..

- When she described her husband's driving, I just cracked up!

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The first one suggests a hull failure (a failure of the material the hull is made of, because of extreme temperature conditions, or repeated stresses from waves, etc.), with no collision or accident. The second suggests a collision leading to hull damage.
So, "crack up" is a technical term?
So, "crack up" is not used correctly in my example?
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No! absolutely not. To my knowledge, ships don't "crack up" as you described, unless the ship ran a ground, collided with another ship ,or hit by a missile. However, the area which may likely develop cracks is the weldings between steel plate that made of the ship body. They can crack under sharp impact or great stress, like running into an iceburg.

A crack- up is a funny person.