Device-to-Device (D2D) communication is emerging as a viable solution for alleviating the severe capacity crunch in content-centric wireless networks. D2D encourages backhaul-free communication directly between devices with similar content requirements grouped into clusters. In this work, a self-sustaining D2D network is considered, where a set of commonly requested files are completely stored within the collective devices memories in a cluster and file requests from devices are serviced by local inter-device multicast transmissions. For such a network, new information theoretic converse results are developed, in the form of a lower bound on the minimum D2D multicast rate as a function of the storage per device. The proposed converse is then used to characterize the approximate tradeoff between the device storage and D2D multicast rate to within a constant multiplicative gap of 8.