Contributions to Tensor-based Stress Variability Characterisation in Rock Mechanics

In situ stress is an important parameter in rock mechanics, but localised measurements often display significant variability. To incorporate such stress variability into probabilistic related analyses in rock mechanics, robust approaches for stress variability characterisation are essential components and prerequisites. Currently, variability of stress is customarily characterised by processing principal stress magnitude and orientation separately using scalar and vector related approaches, respectively. This is erroneous, as stress is a second order tensor and should be processed using tensorial approaches. However, tensorial approaches are poorly developed. The result is that to date there seems to have been no mathematically rigorous proposal for and systematic analysis of stress variability characterisation in rock mechanics. This work presents a mathematically robust approach to characterise stress variability.\nThe work commences with an examination of the customary scalar/vector approach, and demonstrates why stress data should be processed in a tensorial manner. From this, fully tensorial approaches are proposed for calculating the mean stress (Euclidean mean), obtaining the statistical distribution model (multivariate distribution of distinct tensor components), assessing scalar valued stress dispersion (effective variance) and generating random stress tensors (generating multivariate random vector and forming the random stress tensor). The transformational consistency, or invariance, of these with respect to coordinate system change is derived in an analytical manner.\nA systematic examination of the applicability and efficacy of the proposed fully tensorial approaches for stress variability characterisation is presented using synthetic, actual and numerically simulated stress data. The calculations show that both the customary scalar/vector approach, which treats principal stress magnitudes and orientations as independent quantities, and existing quasi tensorial applications, which ignore the correlation between tensor components, may give incorrect results. The recommendation is thus made that stress tensors are referred to as comprising â six distinct componentsâ , rather than the customary â six independent componentsâ . This leads to the conclusion that stress variability should be characterised using multivariate statistics of the distinct tensor components referred to a common Cartesian coordinate system. Finally, transformational consistency, or invariance, of the proposed fully tensorial approaches is demonstrated, and this allows stress variability to be characterised in any convenient coordinate system.

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