Graphical representation and analysis of second-order tensor anisotropy

Anisotropy arises in mechanics and other fields of science as the deviatoric-symmetric parts of various tensor quantities and is a frequently measurable physical proper ty of naturally deformed rocks. This presentation discusses homogeneous anisotropies of a real second-order tensor nature and asks whether any one graphical representation of form and intensity is fundamental and facilitates linear statistical analysis. The question has practical application to the problem of superposed anisotropies in geological materials and is of interest for representation of deviatoric second-order tensors generally. Geometric tensors such as homogeneous finite strain and second-order mineral-orientation density are members of what mathematicians call Lie groups that can be mapped into corresponding lineartensors by logarithmic operations. The deviatoric part that is of interest here lies at a unique position with respect to the projections of defined principal axes in an octahedral plane, sometimes called a three-axis plane. Thus vector addition and linear statistics can be applied to points in this plane, including points defined by the projections of logarithms of principal components of deviatoric geometric tensors. The logarithmic octahedral projection can assist not only in separating tectonic and initial components of anisotropy, such as in deformed sedimentary rocks, but also in providing information about the underlying lineartensor relating to average deformation styles and transport directions in tectonic zones. The method is discussed for monoclinic transpressive and transtensional non-coaxial progressive deformations of an orthogonal bedding anisotropy and compared with some of the more commonly used graphical plots.