Computing the optimal transversely isotropic approximation of a general elastic tensor

Abstract

Mathematically, 21 stiffnesses arranged in a 6 × 6 symmetric matrix completely describe the elastic properties of any homogeneous anisotropic medium, regardless of symmetry system and orientation. However, it can be difficult in practice to characterize an anisotropic medium's properties merely from casual inspection of its (often experimentally measured) stiffness matrix. For characterization purposes, it is better to decompose a measured stiffness matrix into a stiffness matrix for a canonically oriented transversely isotropic (TI) medium (whose properties can be readily understood) plus a generally anisotropic perturbation (representing the medium's deviation from perfect symmetry), followed by a rotation (giving the relationship between the medium's natural coordinate system and the measurement coordinate system). To accomplish this decomposition, we must find the rotated symmetric medium that best approximates a given stiffness matrix. An analytical formula exists for calculating the distance between the elastic properties of two anisotropic media. Starting from this formula, I show how to analytically calculate the TI medium with a [Formula: see text] symmetry axis that is nearest to a given set of 21 stiffness constants. There is no known analytical result if the symmetry axis is not fixed beforehand. I therefore present a simple search algorithm that scans all possible orientations of the nearest TI medium's symmetry axis. The grid is iteratively refined to optimize the solution. The algorithm is simple and robust and works well in practice, but it is not guaranteed to always find the optimal global answer if there are secondary minima that provide almost as good a fit as the optimal one.