## 3 3D Kinematic Operators under Cylindrical Coordinates  
This paper focuses on linking the localization effects with kinematic effects as observed on the boundary of the specimen, such as compression, rotation, and/or straining, which may lead to specific mechanistic failure mechanisms. A cylindrical coordinate system was considered as the basis to maintain consistency with specimen geometry, and thought to be suitable for presenting the state of three-dimensional stresses. The observed displacement fields across the specimen boundary were decomposed in radial, tangential, and vertical directions. One challenge in calculating kinematics under current setting lies in the difficulty of computing directional derivatives along radial directions, since no contiguous measurements of internal deformations are retrieved by the DIC sampling method (i.e., DIC only measures boundary displacement fields). To overcome this issue, we introduced a series of auxiliary assumptions; namely, a set of reference origins along the axis of the specimen where the radial and tangential displacements were assumed to be zero; and the displacements along the axis were estimated to be equal to the average of all vertical displacements captured on the specimen boundary at each specimen height. This assumption was supported by experimental evidence revealed by [PERSON] et al. (2018)--a rigid cone delimited by a circular shear surface and multiple sets of planes extending outwards from the cone towards the specimen boundary imply that the radial and tangential displacements along the axis are negligible compared to vertical displacement. Figure 3 illustrates this assumption, in which the sampling area from a spatial surface is transformed into a series of 3D wedges by incorporating this set of auxiliary reference origins; thus, the localization effects can be properly evaluated from the kinematics of these 3D shapes. After discretizing the continuum media as described above, it follows to identify the kinematic operators under the current configuration. The conventional first-order kinematics include _gradient_, _divergence_, and _curl_ operators acting on the vector field \(U\)([PERSON], 1991). These quantities are defined through an operator called _del_, or _nubla_, and its tensor, dot, and cross product with the vector field \(U\). Equations (1)-(3) provide the expressions of _gradient_, _divergence_, and _curl_ under cylindrical coordinates. The _gradient_ of a 3D vector field \(U\) would generate a _deformation tensor_\(F\), composed of nine components representing translational or rotational changing rates of the neighboring local area affected by the vector field \(U\). The sum of diagonal terms presented in \(F\) is the _divergence_ indicating the total magnitude of sink (positive quantities) or source (negative quantities) of a local area. Moreover, the difference between every two off-diagonal terms in \(F\) gives each component of _curl_ representing  
Figure 3: Schematic illustration of 3D study objects: (**a**) the initial geometry of the specimen; (**b**) the assumed auxiliary origins along the axis of the specimen; (**c**) connecting boundary coordinates with added auxiliary origins to form the 3D study objects. local rotational behavior with respect to every axis. Full derivation of first-order kinematic operators under the cylindrical coordinates can be found in Appendix A. \[