Why would a dog lick and nibble at concrete?

Why do you think its something he needs in his diet. It could be another dogs scent or poo for that matter.

That is pica, which in humans can be psychological but in animals is pathological. From Wikipedia, "Unlike in humans, pica in dogs or cats may be a sign of immune-mediated hemolytic anemia, especially when it involves eating substances such as tile grout, concrete dust and sand. Dogs exhibiting this form of pica should be tested for anemia with a CBC or at least hematocrit levels."

Dogs sometimes just do weird things. It was probably an interesting smell or the dog was bored. Dogs are kind of like children...its like asking why a child would swallow a penny. They are curious.

Are you sure someone didn't spill something edible on that concrete? My dogs would be all over that especially if it was fatty/meaty.

he's after salt maybe.