Distances of Stiffnesses to Symmetry Classes
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Distances of Stiffnesses to Symmetry Classes Oliver Stahn1 · Wolfgang H. Müller1 · Albrecht Bertram1
Received: 8 November 2019 © The Author(s) 2020
Abstract For a given elastic stiffness tetrad an algorithm is provided to determine the distance of this particular tetrad to all tetrads of a prescribed symmetry class. If the particular tetrad already belongs to this class then the distance is zero and the presentation of this tetrad with respect to the symmetry axes can be obtained. If the distance turns out to be positive, the algorithm provides a measure to see how close it is to this symmetry class. Moreover, the closest element of this class to it is also determined. This applies in cases where the tetrad is not ideal due to scattering of its measurement. The algorithm is entirely algebraic and applies to all symmetry classes, although the isotropic and the cubic class need a different treatment from all other classes. Keywords Symmetry class · Stiffness · Elasticity · Hooke’s law · Continuum Mechanics Mathematics Subject Classification 15A69 · 15A72 · 53A45 · 58D19 · 58J70 · 74A20 · 74A30 · 74B05 · 74E10 · 74E15 · 74E30
1 Introduction Many materials in engineering applications exhibit an inner structure, e.g., crystal lattices or fibers, which causes anisotropy. Such a structure induces certain material symmetries. Knowledge of these symmetries can be useful in order to apply representation theorems or to find the number of independent elastic constants that are needed to describe the elastic behavior fully. Based on material symmetries, materials are often categorized into symmetry classes. All material laws within the same symmetry class share the same symmetry group. However, it is a complicated task to determine the symmetry of a particular material. In the literature, see [14], the symmetry class of a stiffness is usually determined by the form of its VOIGT representation. Since the representation is non-unique, this method of identification is only applicable if the representation uses a particular base called symmetry base.
B O. Stahn 1
Technische Universität Berlin, Institut für Mechanik, Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Einsteinufer 5, 10587 Berlin, Germany
O. Stahn et al.
Furthermore, measured stiffnesses always show a certain amount of scattering. As a result, such stiffnesses appear to be triclinic (the “most anisotropic” case) even if represented in a special base. A practical method to determine the symmetry class of a material and the corresponding stiffness is needed. The method presented in this paper is able to identify the (exact) symmetry class of a material if it exists. Furthermore, in the presence of scattering, this method will determine the closest non-triclinic symmetry class of a stiffness. This is achieved by using the distance between the closest element in a non-triclinic symmetry class and the stiffness. The determination of the symmetry class of a stiffness as well as other closely related topics have been discussed for a long time. In [10] the closest
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