A general pneumatic soft gripper is proposed in this paper. Combined with the torque balance theory, the mathematical theoretical model of bending deformation of soft gripper is established based on Yeoh constitutive model and classical differential geometry. Assuming that the pressure in each inner cavity is evenly distributed, the input gas is in an ideal state, which is approximately treated as an isothermal condition, and all orifices experience blocked flow. In addition, compared with the mechanical work of gas, the energy related to gas flow and heat transfer is negligible. The nonlinear mechanical properties of silicone rubber are studied. It is regarded as isotropic and incompressible material, which is characterized by strain energy per unit volume. The material constant coefficients C10 and C20 are determined through the uniaxial tensile test, and the software gripper is simulated on the ABAQUS platform. The bending deformation models of grippers with three different force-bearing cavity structures are analyzed and compared, and the software clamping structure with the bending deformation most in line with the application conditions is selected. The limit input air pressure of the gripper and the situation of enveloping the clamping target object are analyzed. Through the bending deformation experiment, the maximum deformation angle is 72.4°. The relative error between the simulation analysis data and the prediction results of the mathematical model is no more than 3.5%, which verifies the effectiveness of the simulation and the correctness of the mathematical theoretical model of bending deformation. The soft manipulator proposed in this paper has good adaptability to grasping objects of different shapes and sizes. The minimum diameter of the target object that can be clamped is 0.1 mm. It can clamp the object weighing up to 1 kg. It has compact size, light weight, high ductility, and flexibility.