The grasping dynamics between a soft gripper and a deformable object has not been investigated so far. To this end, a 3D printed soft robot gripper with modular design was proposed in this paper. The gripper consists of a rigid base and three modular soft fingers. A snap-lock mechanism was designed on each finger for easy attach-detach to the base. All components were 3D printed using the Objet260 Connex system. The soft finger is air-driven and the idea is based on the principle of fluidic elastomer actuator. A curvature sensor was integrated inside each finger to measure the curvature during grasping. The fingers integrated with sensors were calibrated under different pneumatic pressure inputs. Relationship between pressure loading and bending angle, and relationship between bending angle and sensor output (voltage) were derived. Experiments with the gripper grasping and lifting a paper container filled with peanuts were conducted and results were presented and discussed.