The problem is this: I have an abstract class that does some work in its constructor, and a set of child classes that implement the abstract class:
class Abs
This sounds to me like a mixture of concerns - something Java is not well equipped to handle.
While it is not the answer you were hoping for or one that I am proud to type, you could simply create ConcreteClass2 that mimics ConcreteClass1 and uses the AbstractClass's constructor.
As @TofuBeer said, this is not something that Java supports. This is why some modern languages (i.e. Scala w/ Traits) are gaining passionate developers.