The abdominis rectis (abs) is just one muscle involved in bending forward. The 'core' is a much broader group of muscles in the torso. Straightening or arching the back, bending side to side, or twisting will use a whole range of core muscles other than the abs.
An analogy would be the difference between the quads (thigh muscles) and the entire lower body group - quads, hamstrings (back of thigh), glutes (butt), calves, etc.
Like all strength training, you should rest the core 48 hours between sessions. There are myths out there that you can allegedly work the core every day, but the truth is that core muscle fibers recover the same way as muscles in any other part of the body, and you should rest 48 hours between sessions.
Planks are one of the best core exercise available, as it works a whole range of core muscles, not just the abs. Add in something like bicycle crunches for the twisting motion, and you are pretty much done with the core in just those two moves. Core work does NOT burn belly fat, so a 30 or 40 minute core workout is ineffective and a waste of time.
Zorbs makes a good point - squats, deadlifts, pushups and pullups might classically be thought of as lower body and upper body moves, but also work the core. Interestingly, recent research suggests that of all the exercises you can possibly do, pull-ups are actually the exercise that work the abs the hardest.
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.
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