In this paper, a simple and accurate analytical framework for the duplex mode selection and the power allocation scheme for full duplex (FD) device-to-device (D2D) communications underlaying wireless cellular network is proposed. We first aim at maximizing the D2D links rate by properly selecting the duplex mode of the D2D pairs and allocating the power of the users while fulfilling the quality-of-service (QoS) of the cellular users (CUEs). Since the resulting optimization problem is non-convex, we derive a first-order optimal solution by leveraging an interesting optimization tool named as sequential convex optimization theory. We also discuss how to obtain an efficient sub-optimal solution in term of complexity and accuracy by deriving the optimal power ratio between the D2D devices assuming an interference limited system. The simulations have shown more than 95% accuracy of the proposed approaches and provided great insights on the solution design parameters taking into account the CU location and the self-interference threshold.