Distance of a stiffness tetrad to the symmetry classes of linear elasticity

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES

卷:156

Article

Weber, Martin¹  Gluege, Rainer¹  Bertram, Albrecht² 

[1] Otto Von Guericke Univ, Inst Mech, D-39106 Magdeburg, Germany

[2] Tech Univ Berlin, Inst Mech, D-10623 Berlin, Germany

关键词: Stiffness tetrad;    Elastic symmetry;    Anisotropic distance function;    Symmetry groups;   

DOI  :  10.1016/j.ijsolstr.2018.08.021

来源: Elsevier

【 摘 要 】

In the scope of linear anisotropic elasticity, the fourth-order elasticity tensor or tetrad has to be identified. This can be done either by measurements or by numerical simulations. An important task is then to identify a given tetrad, probably with some experimental or numerical scattering, with one of the symmetry classes. For this purpose one needs a distance function between a given tetrad and the class of all tetrads with a particular symmetry, which is zero if the tetrad obeys this symmetry, or non-zero otherwise. In this paper we present a fast method to solve these problems. We firstly introduce the 8th order projectors that map any stiffness tetrad into the part that is invariant under the action of a specific fixed symmetry group. For this purpose we consider the seven out of the eight symmetry classes that are distinguishable in linear elasticity. Secondly, since the symmetry axes of the specific stiffness tensor under consideration is generally not aligned with the used tensorial basis of the projector, we need to rotate the sample stiffness. The optimal orientation is obtained when the distance between the rotated stiffness and the rotated and projected stiffness is minimal. Thus, we need to apply only linear mappings and minimize over three Euler angles. The latter is quite simple, as the domain of the Euler angles is periodic, and the number of local minima is limited. This procedure has the advantage that it is applicable in an algorithmic manner, and does not require an a priori identification of symmetry planes, symmetry axes or component symmetries, which are only apparent under special choices for the tensorial basis. (C) 2018 Elsevier Ltd. All rights reserved.

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