2-Norm Effective Isotropic Hookean Solids

In the physical realm, an elasticity tensor that is computed based on measured numerical quantities with resulting numerical errors does not belong to any symmetry class for two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry classes in question are properties of Hookean solids, which are mathematical objects, not measured physical materials. To consider a good symmetric model for the mechanical properties of such a material, it is useful to compute the distance between the measured tensor and the symmetry class in question. One must then of course decide on what norm to use to measure this distance. The simplest case is that of the isotropic symmetry class. Typically, in this case, it has been common to use the Frobenius norm, as there is then an analytic expression for the closest element and it is unique. However, for other symmetry classes this is no longer the case: there are no analytic formulas and the closest element is not known to be unique. Also, the Frobenius norm treats an n×n matrix as an n 2-vector and makes no use of the matrices, or tensors, as linear operators; hence, it loses potentially important geometric information. In this paper, we investigate the use of an operator norm of the tensor, which turns out to be the operator Euclidean norm of the 6×6 matrix representation of the tensor, in the expectation that it is more closely connected to the underlying geometry. We characterize the isotropic tensors that are closest to a given anisotropic tensor, and show that in certain circumstances they may not be unique. Although this may be a computational disadvantage in comparison to the use of the Frobenius norm—which has analytic expressions—we suggest that, since we work with only 6×6 matrices, there is no need to be extremely efficient and, hence, geometrical fidelity must trump computational considerations.