Ceramics have the properties of high-stiffness, proof-corrosion, and light-weight superior to metal materials and so have been applied to a ball and a race in high performance ball bearing and etc. As a recent expansion of this application, a ceramics ball has been applied to the backflow valve, which is comprised in automotive fuel supply system. However, it must be considered sufficiently that ceramics balls can be kept a high confidence over long-term without being broken down by cyclic contact loadings, which generate between a ceramics ball and housing on switching of the backflow valve. In this investigation, the stress analysis by FEM was conducted for clarifying failure mechanism due to cyclic contact stress. As result of analysis, high stress place was seen in contact side neighborhood of a ball and housing. Ceramics is brittle material, predominant of the fracture is a microscopic crack in materials. Therefore, author clarified that an initial surface microscopic crack was a starting point of fracture.