Chapter 16
Constitutive Relations
Abstract In this chapter is devoted to constitutive laws describing equilibrium and
non-equilibrium properties in anisotropic media. Firstly, general principles, viz. the
Curie principle, energy requirements, positive entropy production and Onsager-
Casimir symmetry relations of irreversible thermodynamics are introduced. Sec-
ondly, phenomenological considerations and microscopic expressions are presented
for the elasticity coefficients describing linear elastic deformations of solids, with
emphasis on isotropic and cubic symmetries. Thirdly, the anisotropy of the viscous
behavior and non-equilibrium alignment phenomena are studied for various types of
fluids.TheinfluenceofmagneticandelectricfieldsareanalyzedforplaneCouetteand
plane Poiseuille flows. Results of the kinetic theory are presented for the Senftleben-
Beenakker effect of the viscosity. Consequences of angular momentum conservation
are pointed out for the antisymmetric part of the pressure tensor. The flow birefrin-
gence in liquids and in gases of rotating molecules is treated as well as heat-flow
birefringence in gases. The phenomenological description of visco-elasticity and of
non-linear viscous behavior is discussed. Vorticity-free flow geometries are consid-
ered. The fourth part of the chapter deals with the viscosity and alignment in nematic
liquid crystals. Viscosity coefficients are introduced which are needed to characterize
the anisotropy of the viscosity in an oriented liquid crystal as well as in a free flow.
Flow alignment and tumbling are considered. Model computations are presented for
the viscosity coefficients as well as the application of a generalized Fokker-Planck
equation for the non-equilibrium alignment. A unified theory for the isotropic and
nematic phases is introduced and limiting cases are discussed. Equations governing
the dynamics of the alignment in spatially inhomogeneous systems are formulated.
In addition to the general laws of physics, special relations are needed for the treat-
ment of physical phenomena in specific substances. Theseconstitutive relations
involvematerial coefficients. Examples already encountered are relations between
the electric polarization and the electric field, between the electric current density
and the electric field or between the friction pressure tensor and the velocity gradient.
The constitutive relations have to obey certain rules. These, as well as examples for
and applications of constitutive relations are presented here.
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_16
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