We report on some unusual behavior of the measured current-voltage characteristics (CVC) in artificially prepared two-dimensional unshunted array of overdamped Nb- AlOx -Nb Josephson junctions. The obtained nonlinear CVC are found to exhibit a pronounced (and practically temperature independent) crossover at some current Icr = (1/2 ΒC-1) I C from a resistance R dominated state with VR =R I 2 - IC2 below Icr to a capacitance C dominated state with VC = /4eC I-IC above Icr. The origin of the observed behavior is discussed within a single-plaquette approximation assuming the conventional resistively shunted junction model with a finite capacitance and the Ambegaokar-Baratoff relation for the critical current of the single junction. © 2010 American Institute of Physics.