Stress is an important parameter in rock mechanics. As it is represented by a second-order tensor, the usual statistical methods for scalars and vectors strictly are not applicable. As a result, the correct method for calculation of a mean stress tensor is still not clear. Here, and using recent results from the field of diffusion tensor imaging (DTI), we examine two calculation approaches for the mean tensor – based on Euclidean and Riemannian geometries – and discuss their similarities and differences. We apply both methods to back-calculate the mean field stress around a circular opening, and conclude that the approaches give similar results. We go on to examine interpolation between stress tensors, and show that the Euclidean and Riemannian approaches can differ significantly. We conclude that it is currently not possible to identify which of these two approaches is most appropriate for engineering applications.
Stress is an important parameter in rock mechanics. As it is represented by a second-order tensor, the usual statistical methods for scalars (e.g. classical statistics) and vectors (e.g. directional statistics) strictly are not applicable. As a result, the correct method for calculation of a mean stress tensor – i.e., one that is faithful to its tensorial nature – is still not clear. Here, and using recent results from the field of diffusion tensor imaging (DTI), we examine two calculation approaches for the mean tensor – using Euclidean and Riemannian geometries – and discuss their similarities and differences.
Customarily in rock mechanics, stress magnitude and orientation are processed separately (Figure 1). Examples of this are found in the many publications concerning the relationship between stress magnitude/orientation and burial depth (e.g. Brown & Hoek 1978, Martin 1990). Other examples are where each principal stress has been considered separately as a vector (e.g. Markland 1974, Lisle 1989, Ercelebi 1997). All these methods violate the tensorial nature of stress and may yield unreasonable results (Gao & Harrison 2014). As a way of being faithful to the tensorial nature of stress, several researchers (e.g. Hyett et al. 1986, Hudson & Cooling 1988, Walker et al. 1990) have suggested use of a common Cartesian coordinate system when calculating the mean stress tensor. From a geometrical point of view, this approach calculates the Euclidean centroid in Euclidean space (Bhatia 2007).
