I'm experiencing this myself. Twice now actually. After searching online this seems to be a somewhat common occurrence. A hairline fracture slowly develops in the area until it eventually snaps, there's absolutely a weak point in the headbands design here.
I too owned my LCD-1s for about half a year before this happened to me, on the right side. Sent them to Audeze and they replaced them but after a mere 3 months my new pair has developed small fractures on both the left and right side, in the same exact area, so it's only a matter of time before these too snap.
At this point I'd be willing to pay for a more robust headband replacement but sadly LCD1 owners are being neglected when it comes to options like this.