2-Norm Effective Isotropic Hookean Solids

Abstract

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.

On Elastic Symmetry Identification for Polycrystalline Materials

“…One can distinguish a group of techniques, which are based on the separation of an additive part of the elasticity tensor (e.g., by way projecting), possessing one or another type of symmetry [34][35][36][37]. In this case, the identification problem is reduced to determination of the symmetric part, which is the closest in some metric to a given tensor, as e.g., in [38][39][40][41][42][43][44][45].…”

“…Nevertheless, optimization problems analogous to Problems 1-3, can be formulated in terms of quite arbitrary metric on E 4 3 . Various ways to define such metric are considered, e.g., in [34,36,37,40]. In [36], the symmetric approximations of elasticity tensors are constructed with the use of different approaches, which are based on the Frobenius, Log-Euclidean and Riemannian distance functions.…”

“…The last two, compared to the first one, possess some additional invariance properties: in particular, with regard to inversion of the tensor-arguments. In [34], the specific features of the operator norm, · , are investigated theoretically, when searching for the · -optimal elasticity tensor approximations. It was shown that the · -optimal approximation of a positive definite tensor, unlike the · E 4 Going back to the ECIP, suppose that Π…”

Effective Elasticity Tensors in Context of Random Errors

“…Underlying this question is the accuracy of the generally anisotropic tensor, and, hence, its effect on the reliability of information provided by the chosen model. Several researchers-among them, Voigt [25], Gazis et al [15], Moakher and Norris [20], Norris [21], Bucataru and Slawinski [6], Kochetov and Slawinski [18,19], Diner et al [11,12], Bos and Slawinski [5]-examined relations between a given generally anisotropic Hookean solid and its symmetric counterpart by invoking the concept of a norm in the space of elasticity tensors. In this approach, the term elasticity tensor is understood in a broader sense: the requirement of positive definiteness is omitted so the possible values of c form a vector space.…”

“…There are several natural norms on the space of elasticity tensors as discussed, for instance, by Norris [21] and by Bos and Slawinski [5]; the simplest among them is the Frobenius norm. Each of them has certain theoretical advantages and disadvantages; for instance, the Frobenius norm guarantees positive definiteness of the effective tensor.…”