Effective Elasticity Tensors in Context of Random Errors

Abstract

We introduce the effective elasticity tensor of a chosen material-symmetry class to represent a measured generally anisotropic elasticity tensor by minimizing the weighted Frobenius distance from the given tensor to its symmetric counterpart, where the weights are determined by the experimental errors. The resulting effective tensor is the highestlikelihood estimate within the specified symmetry class. Given two material-symmetry classes, with one included in the other, the weighted Frobenius distance from the given tensor to the two effective tensors can be used to decide between the two models-one with higher and one with lower symmetry-by means of the likelihood ratio test.

On Commutativity and Near Commutativity of Translational and Rotational Averages: Analytical Proofs and Numerical Examinations

“…, n , by 1 2 to find that, as expected, the commutator is multiplied by 1 4 . To quantify the strength of anisotropy, we invoke the concept of distance in the space of elasticity tensors (Danek et al [5,6], Kochetov and Slawinski [8,9]). In particular, we consider the closest isotropic tensoraccording to the Frobenius norm-as formulated by Voigt [10].…”

“…The Gazis et al [2] approach is reviewed and extended by Danek et al [5,6] in the context of random errors. Therein, elasticity tensors are not constrained to the same-or even different but known-orientation of the coordinate system.…”

On Backus Average for Generally Anisotropic Layers

“…If so, we might seek-using the method proposed by Gazis et al (1963) and elaborated by Danek et al (2015)-an elasticity tensor of a higher symmetry that is nearest to that medium. For such a study, Kelvin's notation-used in this paper-is preferable, even though one could accommodate rotations in Voigt's notation by using the Bond (1943) transformation (e.g., Slawinski (2015), section 5.2).…”

Distance to plane elasticity orthotropy by Euler-Lagrange method

“…Unfortunately, they are a priori useless in the common case of a triclinic/biclinic measured elasticity tensor. When experimental discrepancy has to be dealt with, the literature approaches are based on the concept of distance of a tensor to a considered symmetry class [17,14,16,15,21,19,11], starting from a given (usually measured) elasticity tensor C raw with no material symmetry, and sometimes from the additional quantification of the measurement errors [7,10,18].…”

“…The 3D case has, by far, been the most studied. It remains the most challenging case, and determining the distance to a 3D symmetry class is usually done numerically, with the risk of reaching a saddle point or a local minimum instead of the global minimum [15,19,10,21]. The 2D case has been shown to be more affordable [26].…”