On the number of distinct elastic constants associated with certain anisotropic elastic symmetries

Abstract

Fedorov [1] observed, but did not demonstrate, that for monoclinic, tetragonal and trigonal linear elastic anisotropic symmetries the number of independent elastic constants (independent components of the elasticity tensor) may be reduced by one (13 to 12 for monoclinic and 7 to 6 for both tetragonal and trigonal) by the appropriate selection of the coordinate system. He also noted that for triclinic symmetry the number of independent elastic constants may be reduced by three (21 to 18) by the appropriate selection of the coordinate system. Thus, for these four symmetries the canonical, material symmetry determined minimum constant coordinate system, forms of the elasticity tensors given by Voigt [2], Love [3], Lekhnit-skii [4], Hearmon [5], Gurtin [6] and many others may be further simplified by selection of the reference coordinate system. The observation of Fedorov [1] for the monoclinic, tetragonal and trigonal symmetries is demonstrated here using a formulation of the anisotropic Hooke’s law in which the elasticity tensor is a second rank tensor in a space of six dimensions, rather than a fourth rank tensor in a space of three dimensions as is customarily the case.