Soft robotics has been receiving increasing attention due to its flexibility and adaptability offered by embodied intelligence.A soft robot may undergo complex motions including stretching,contraction,bending,twisting,and their intricate combinations.Among these basic motions,bending plays a central role when a robot accomplishes tasks such as locomotion,grasping and manipulation.Although a rich repertoire of bending mechanisms has been reported,a systematic and rational design framework is still lack.In this paper,we provide a novel design strategy for soft bending actuators which allows integral modeling and design optimization.Nowadays,metamaterials are emerging as a new tool for soft robots,by encoding the desired complex motions directly within the material architectures,leading to conformable monolithic systems.We combine pneumatic actuators and flexible metamaterials to provide an alternative solution to soft bending actuators,with advantages of compact design,large bending motion,and convenient fabrication.A regular pneumatic chamber is embedded inside auxetic and non-auxetic meta-materials,and bending is generated when inflated.We carry out dimensionless analysis to identify the key design variables.To provide insight into design optimization,we develop a computation framework by modeling metamaterial structures with beam elements and the inner chamber with shell elements as an integral part,allowing efficient simulation of the coupled system.We systematically investigate how the bending angle varies with the key design variables and find the optimal design parameters.The experimental results are well in line with the simulation,and a remarkable bending motion of 0.43°/mm is achieved.