•A new soft robotic gripper with variable stiffness was designed. Its uniqueness lies in the structural decoupling of the deformation actuator for bending and the variable stiffness mechanism of the soft robotic gripper.•Based on the Cosserat theory, a grasping force model is built for the proposed soft robotic gripper with variable stiffness.•A series of experimental tests and simulations on the grasping force of the soft robotic gripper are investigated to discuss the modeling properties. The purpose of this research is to present a grasping force model for a soft robotic gripper with variable stiffness. The soft robotic gripper was made of shape memory alloys (SMAs) with contraction and variable stiffness properties. A variable stiffness mechanism with embedded sets of SMA fibers was developed; however, the response characteristics of its backbone did not comply with the constant-curvature model when it was subjected to complex forces/torques, such as gravity, grasping forces and driving torques. In this case, the Cosserat theory was used to implement real-time computations of the grasping force of the soft robotic gripper that was subjected to complex forces. Finally, a series of tests were conducted on the grasping force of the soft finger and the gripper. The elicited results showed that the grasping force is related to the stiffness and to the object's offset and friction coefficient. Moreover, experimental results showed that the grasping force of the soft robotic gripper increased by 48.7% when the Young's modulus of the SMA-2 wires increased from 25 GPa to 48 GPa.