Mean and dispersion of stress tensors using Euclidean and Riemannian approaches

• We introduce tensorial statistics, particularly mean and dispersion, for stress.
• We present both Euclidean and Riemannian approaches to tensorial statistics.
• Comparisons are conducted between Euclidean and Riemannian approaches.
• Recommendations are made for the use of these statistics in engineering.

Stress is central to many aspects of rock mechanics, and in the analysis of in situ stress measurement data the calculation of the mean value and an assessment of dispersion are important for statistical characterisation. Currently, stress magnitude and orientation are processed separately in such analyses. This effectively decomposes the second-order stress tensor into scalar (principal stress magnitudes) and vector (principal stress orientations) components, and calculation of mean and dispersion of stress data on the basis of these decomposed components, which violates the tensorial nature of stress, may either yield biased results or be difficult to conduct. Here, by introducing tensorial techniques, we examine two calculation approaches for the mean and dispersion for stress tensors – based on Euclidean and Riemannian geometries – and discuss their similarities, differences and potential applicability in engineering practice. We compare the two approaches using stress tensor superposition and interpolation, and the analysis of actual in situ stress data. The results indicate that Euclidean and Riemannian mean tensors are in general not equal, with the disparity increasing as stress tensor dispersion increases. Both Euclidean and Riemannian approaches are shown to be capable of characterising stress dispersion, although Euclidean dispersion is scale dependent and has units of stress whereas Riemannian dispersion is a scale independent unitless number. Finally, a paradox is revealed in that despite stress tensors being Riemannian entities, it is Euclidean mean stress that is the more meaningful for engineering applications.