Massive machine-type communications (mMTC) plays a crucial role in the fifth generation (5G) and beyond wireless networks. To tackle the massive connectivity challenge with limited radio resources, grant-free non-orthogonal multiple access (NOMA) has been proposed to support multiple devices access with the same spectrum. In grant-free NOMA, the devices can directly transmit data without going through the scheduling requests, so that the signaling overhead can be significantly reduced. Moreover, the reconfigurable intelligent surface (RIS) is deployed to assist grant-free NOMA transmission to further enable massive connectivity. In this paper, we investigate a RIS-assisted grant-free NOMA system with no direct link in which multiple devices transmit uplink signals to a base station via a RIS. In order to increase the number of the successfully accessed devices, we redesign the frame structure and optimize the RIS phase shifts. Hence, a novel frame structure is firstly proposed to enable RIS optimization based on channel state information. Then, we formulate a problem that optimizes the signal-to-interference-plus-noise-ratio of all devices while considering the fairness among devices, with optimizing the RIS phase shifts. To handle the non-convexity of the problem, we propose the semidefinite relaxation technique and bisection search method to solve the problem. Simulation results demonstrate that the proposed RIS-assisted grant-free NOMA scheme can substantially increase the minimum data rate by up to 44.3% and 623.1%, respectively, compared to: 1) the scheme where RIS is configured based on a priori positional information, and 2) the scheme where RIS is configured based on random phase.