Snake models are extensively used from its debut in image processing and motion tracking, but its poor convergence on concave boundary is a handicap for object location. Although, the GVF Snake model shows high performance for this problem, but it suffers from costly computation by virtual of PDE's and another so-called critical point problem for the initial contour selection. In order to improve the performance of the traditional Snake model for concavity segmentation, a new external force based on the local curvature of the discrete contour and a two-stage Snake-based algorithm are proposed. The local curvature of the discrete contour, which characterizes the bending of a contour associated with a direction, is defined using the center of the inscribed circle of the triangle derived from three consecutive contour nodes. The first stage of the new method is a traditional Snake, and in the second stage the new force would drive the contour into the concave region. This new force can also be generalized to enlarge the capture range of the Snake model. In this case, it can be considered as a generalization of the balloon force. In order to overcome the difficulty of determining the magnitude, the magnitude is set to be small and the gradient-based force is first used as resistance; when the contour is converged, the gradient-based force swerves to attract the contour. Generalized in this way, the capture range is enlarged and there is no critical point problem. The experimental results validate the performance of this method.