Energy efficiency has been known as the most significant problem in all facets of the wireless sensor network operations. For energy-aware routing scheme for the wireless sensor networks, it is necessary to use sub-optimal paths occasionally to increase the survivability of networks. In this paper, we present a new network lifetime definition and formulate the energy-aware routing problem with defined network lifetime. And then we propose an energy-aware routing scheme with the node relay willingness for wireless sensor networks. It considers not only to route packets through sensor nodes that have sufficient energy reserves, but also to route packets through lightloaded nodes. Simulation results show that the proposed routing algorithm can achieve longer network lifetime than that of the routing algorithm only considering the energy reserves.