Quantum spin Hall (QSH) insulators with gapless edge states have potential applications in designing low-dissipation devices.In spite of many predictions,to verify the QSH phase in graphene layered materials experimentally is still difficult due to the obstacle in achieving spin-orbit coupling strong enough.We propose a Rashba system of graphene bilayer gapped by dielectric layers and show it can host a valley-polarized QSH phase even when the Rashba interaction approaches zero.Such a system exhibits asymmetric topological quantum phase transitions under opposite interlayer biases,due to the dielectric-potential induced inversion asymmetry in the absence of interlayer bias.Specifically,the quantum valley Hall phase exists in zigzag-edged nanoribbons under the bias in one direction but is absent under the reverse bias.These topological phenomena can be well understood by the competition among the dielectric-induced potential,Rashba interaction,and interlayer bias in modulating the bulk band gap.Moreover,the phase diagram is given and the corresponding phase boundaries are derived analytically.Our findings provide a possible way to detect the QSH-related asymmetric topological quantum phenomena in graphene bilayer based on the current experimental technology.