Recovering the Normal Form and Symmetry Class of an Elasticity Tensor
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Recovering the Normal Form and Symmetry Class of an Elasticity Tensor S. Abramian1 · B. Desmorat2 · R. Desmorat1 · B. Kolev1 · M. Olive1
Received: 8 January 2020 © Springer Nature B.V. 2020
Abstract We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor. We produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants. An algorithm to detect the symmetry class of an Elasticity tensor is finally formulated. Keywords Elasticity tensor · Anisotropy · Natural basis · Symmetry classes · Covariants Mathematics Subject Classification 74E10 · 15A72 · 74B05
R. Desmorat, B. Kolev and M. Olive were partially supported by CNRS Projet 80|Prime GAMM (Géométrie algébrique complexe/réelle et mécanique des matériaux)
B M. Olive
[email protected] S. Abramian [email protected] B. Desmorat [email protected] R. Desmorat [email protected] B. Kolev [email protected]
1
ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et Technologie, Université Paris-Saclay, 91190, Gif-sur-Yvette, France
2
Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert, UMR 7190, 75005 Paris, France
S. Abramian et al.
1 Introduction The linear elastic properties of a given material are encoded into an Elasticity tensor E, a fourth-order tensor which relates linearly the stress tensor to the strain tensor. As it was clearly emphasized by Boehler and coworkers [7, 8], any rotated Elasticity tensor encodes the same material properties (in a different orientation). One shall say that the rotated tensor and initial one are in the same orbit. It should be emphasized here that this has not to be confused with a change of (orthonormal) basis once a basis has been fixed and the tensors expressed by their components in this basis. Here, the action of the rotation group is defined intrinsically and independently of any basis (no components are required to define this action). The elastic materials are classified by their eight symmetry classes [16] (isotropic, transversely-isotropic, cubic, trigonal, tetragonal, orthotropic, monoclinic, triclinic). Any non triclinic Elasticity tensor has a given symmetry class and a normal form. An orthonormal frame in which the matrix representation of this tensor belongs to such a normal form is called a proper or natural basis for E [15]. For instance, consider a cubic Elasticity tensor which is given in an arbitrary frame by its Voigt’s (matrix) representation [E] (not to be confused with the tensor E itself) as ⎞ ⎛ E1111 E1122 E1133 E1123 E1113 E1112 ⎜E2211 E2222 E2233 E2223 E2213 E2212 ⎟ ⎟ ⎜ ⎜E3311 E3322 E3333 E3323 E3313 E3312 ⎟ ⎟ (1.1) [E] = ⎜ ⎜E2311 E2322 E2333 E2323 E2313 E2312 ⎟ . ⎟ ⎜ ⎝E1311 E1322 E1333 E1323 E1313 E1312 ⎠ E1211 E1222 E1233 E1223 E121
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