Performing tasks with a robot hand often requires a complete knowledge of the manipulated object, including its properties (shape, rigidity, surface texture) and its location in the environment, in order to ensure safe and efficient manipulation. While well-established procedures exist for the manipulation of rigid objects, as well as several approaches for the manipulation of linear or planar deformable objects such as ropes or fabric, research addressing the characterization of deformable objects occupying a volume remains relatively limited. The paper proposes an approach for tracking the deformation of non-rigid objects under robot hand manipulation using RGB-D data. The purpose is to automatically classify deformable objects as rigid, elastic, plastic, or elasto-plastic, based on the material they are made of, and to support recognition of the category of such objects through a robotic probing process in order to enhance manipulation capabilities. The proposed approach combines advantageously classical color and depth image processing techniques and proposes a novel combination of the fast level set method with a log-polar mapping of the visual data to robustly detect and track the contour of a deformable object in a RGB-D data stream. Dynamic time warping is employed to characterize the object properties independently from the varying length of the tracked contour as the object deforms. The proposed solution achieves a classification rate over all categories of material of up to 98.3%. When integrated in the control loop of a robot hand, it can contribute to ensure stable grasp, and safe manipulation capability that will preserve the physical integrity of the object.