Embedding stress difference in parameter space for stress tensor inversion

This paper proposes a rearrangement of Fryʹs sigma-space which has translated stress tensor inversion into a concise geometric problem. The kernel of our modification is in the normalisations of tensor invariants and in the adoption of weighting factors used in the studies of crystal plasticity. After describing a fault-slip datum as a strain tensor, we mapped both stress and strain tensors onto the modified parameter space. There are two main benefits. First, the geometry is simplified. The points representing normalised tensors are located on the five-dimensional unit sphere and their relative arrangement is independent from the coordinate selection in physical space. Second, the Euclidean metric of the space was equated to the so-called stress difference, a useful measure of difference between normalised stress tensors. This metric led us to a straightforward method for quantifying the confidence region of stress tensor deduced through inversion.