"Re-bowing" the seat piece might prove an impossible task. It was likely either originally made by bending green wood that has straightened as it dried, or else by steam bending sir-dried wood that is now returning to it's original form. You could try steam bending it over a form that's got a tighter radius than what you want the finished product to be (compensates for springback).
In another post, you mentioned a failed re-gluing of the back splat. From that picture (which I could briefly see, but which later disappeared ) it looks like the best repair would be to rip the piece in two, cutting right down the split line, then planing the cuts to match, and regluing. Use a good wood glue, like Titebond®. You'll loose maybe 1/8~3/16" of width, but that shouldn't be much of a problem.
Also, from your pictures, the wood does look like oak.
Hope this helps.
Now we see the pictures. It is unlikely the cracks can be closed. Those look to be separate boards and over they years they have shrunk. Soaking may get some water in the outer cells but not for long and likely never to the inner cells. That is the nature of wood.
As for the top. Likely that was originally steam bend. Could it be again? Perhaps, but I'd take a hard look at the whole project. I'd think this one takes considerable skill, money and time to make new again. You didn't tell us your skill level or experience. So it is hard to know where to begin.
Dave Hawksford is our resident refinishing guy. Hopefully he will pop in with some thoughts.
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Looks like a lost cause to me. More effort than it's worth.
If it were mine I'd give it to my daughter. She would paint it a funky color and put it in the garden, put plants on it and use it as a decoration piece .
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Jim - This may indeed end up in the flower garden. I got it out of a garden. Maybe it was put there for a reason. Your idea of splitting and re-gluing is a doable option but then I still have the bend to figure out.
Carol - The back piece looks to be one piece. It is grooved (where the crack is) to accept the four pieces of the seat. On a scale of 1 - 10, my skill level is -3. I mostly like unfinished wood. My house is full of old barn siding and wood shingles on the walls. Fun stuff to work with. I make an occasional pen on one of my three lathes. My workshop is on my side of the two car garage (minus room for the two motorcycles). I've done a couple of refinish things this summer and I have wood working 'bug' again. And I'm just learning about refinishing.
Bob - Much as I hate to admit defeat from a piece of wood, it just may be a garden piece. That was my wife's original idea. DEFEATED by a piece of wood. Just the thought of it.... Don't know if I can take that lightly. It's oak - for crying out loud. How can you throw that in the garden?
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I had given that a thought but I already have so many oak blanks I'll probably never use them all. My thinking cap is on. I have a couple of ideas. We'll see what happens in the next day or so.
The back top piece looks like it could have been just band sawed from solid stock. The nail can be pulled out of the cracked board, and if the crack can be closed up with minimum clamp pressure, to reglue and clamp up. The holes can be drilled out and wood plugs can be glued back in to drill a clean hole.