The crest you describe is a new one to me, and sounds very interesting. I have seen three others. Yours, with the gesture of cutting through oak would seem to correspond to the meaning of Eckhardt as "hard edge."
Many years ago I found an Eckhardt crest in a very old book. It has a shield decorated with an ax and a plow. This would correspond to another of the several meanings of Eckhardt, i.e. "edge of the forest: Those farming on the edge of the forest would need an ax and a plow.
However, I also have seen another version with three angles (they look like carpenters tools)and nothing else. This might also correspond to "hard edge."
There's a problem here, too, in that some of the "shake'n bake" companies who promise to tell you all about your family, quite literally make them up! One such, (my third example)which I have on a tee shirt, shows the usual helmet and mantling, but a boar's head against a backdrop of the modern German flag. It allegedly dates from 1661, but I don't think so.