Symmetry and Tensors [unknown]

2022 Elastic Waves in Solids 1   unpublished
This appendix deals with the symmetry of crystals and the representation of their physical properties by tensors. The anisotropy of crystals plays a very important role with respect to their macroscopic physical properties: crystals with the same point symmetry behave similarly under physical actions of the same orientation. Tensor analysis expresses this behavior well as it classifies physical quantities according to the laws for transforming their components when the reference axes are
more » ... . When this change of axes corresponds to a symmetry operation, it results from the identity of the macroscopic properties of the crystal in both reference frames, relations between the components of the tensors representing these properties. Consequently, there is a reduction in the number of independent components. The non-existence of properties represented by tensors of a given rank in various classes of crystals is verified. The reduction of the number of independent elastic constants is treated in the last section of this appendix. A2.1. Crystalline structure Among solids, we must distinguish between amorphous bodies and crystals. There are also intermediate cases such as polycrystalline materials, consisting of microscopic single crystals, randomly oriented, whose behavior at a macroscopic scale is homogeneous and isotropic. Amorphous bodies, such as resin or glass, have no characteristic geometrical forms and are, in reality, extremely viscous liquids. When heated, they undergo pasty fusion: their fluidity increases continuously with temperature. During cooling, the only visible change is a gradual increase in viscosity and the temperature decrease versus time presents no plateau. On the contrary, in the case of a crystal, during the cooling of the liquid, the temperature stabilizes as soon as solid seeds of polyhedral form appear: these are
doi:10.1002/9781119902942.app2 fatcat:a2tvrmxczzgqbmzh46gx2rdqmi